[    {        "title": "2005.07730v1.Slow_magnetosonic_wave_absorption_by_pressure_induced_ionization_recombination_dissipation.pdf",        "content": "Slow magnetosonic wave absorption by pressure induced\nionization-recombination dissipation\nTodor M. Mishonov1,a)and Albert M. Varonov1,b)\nInstitute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee Blvd., BG-1784 So\fa,\nBulgaria\n(Dated: 15 May 2020, 21:05)\nA new mechanisms for damping of slow magnetosonic waves (SMW) by pressure induced oscillations of the\nionization degree is proposed. An explicit formula for the damping rate is quantitatively derived. Physical\nconditions where the new mechanism will dominate are brie\ry discussed. The ionization-recombination\ndamping is frequency independent and has no hydrodynamic interpretation. Roughly speaking large area of\npartially ionized plasma are damper for basses of SMW while usual MHD mechanisms operate as a low pass\n\flter. The derived damping rate is proportional to the square of the sine between the constant magnetic \feld\nand the wave-vector. Angular distribution of the spectral density of SMW and Alfv\u0013 en waves (AW) created\nby turbulent regions and passing through large regions of partially ionized plasma is qualitatively considered.\nThe calculated damping rate is expressed by the electron impact cross section of the Hydrogen atom and in\nshort all details of the proposed damping mechanisms are well studied.\nI. SHORT INTRODUCTION\nBehind purely fundamental interest for plasma physics\npropagation of hydromagnetic (nowadays known as mag-\nnetohydrodynamic ) waves attracted signi\fcant attention\nand was strongly simulated by the development of the\nphysics of solar atmosphere and the eternal problems\nrelated to its heating.1{3It has already been con\frmed\nthat the magnetohydrodynamic (MHD) waves (both in-\ncompressible and compressible) are present in the solar\natmosphere and they have already been considered for\nheating of the solar chromosphere and corona.4{7Models\nadapted to study heating problems of solar atmosphere\ninclude two \ruids coupled through collisions and chemi-\ncal reactions, such as impact ionization and radiative re-\ncombination with imposed initial thermal and chemical\nequilibrium. Within this approach the plasma heating is\ndominantly wave-based and the main energy source for\nheating are the excited fast magnetosonic \ructuations,8\nwhile older studies of FMW heating can be found in\nRef. 9 for instance.\nIt is worthwhile to mention also works on overre\rection\nor swing ampli\fcation in shear \row of slow magnetosonic\nwaves (SMW); see for example Refs. 10 and 11.\nIn a review on partially ionized astrophysical plasmas12\nit is shown that viscosity plays no important role in the\ndamping of chromospheric Alfv\u0013 en waves and recently it is\nconcluded that the solar corona electrical resistivity has\nonly very small impact, while and thermal conduction\nand viscosity contribute equally.13Therefore, the ques-\ntion of chromospheric heating due to the ion-neutral in-\nteraction will require further studies in the future. A\ncomplete review on the problem requires many hundred\ncitations, but here we mention the importance of two\na)E-mail: mishonov@bgphysics.eu\nb)E-mail: varonov@issp.bas.bg\ruid approach for consideration of MHD waves in par-\ntially ionized plasmas.14\nA. Scenario\nWhen a slow magnetosonic wave (SMW) propagates\nthrough partially ionized plasma, the oscillations of the\npressure creates oscillations of the temperature and gen-\nerates small oscillations of the degree of ionization \u000b.\nThose pressure induced deviations of the chemical equi-\nlibrium gives an extra entropy production and energy\ndissipation of the SMW.\nThis additional mechanism does not work for the\nAlfv\u0013 en waves (AW) and in spite of common dispersion\nand damping of AW and SMW in (MHD) approach at\nsmall magnetics \feld their ionization-recombination ab-\nsorption can be completely di\u000berent.\nThe purpose of the present work is to present an ex-\nplicit formula for the chemical damping and to consider\nin short when the predicted new damping mechanism is\nimportant and dominates and how it can be observed.\nThe article is organized as follows. In order to create\nthe necessary system of notions and notations following\nLandau and Lifshitz3in the next Sec. II we will recall\nthe physics of SMW. Then we derive in Sec. III our new\nresult for ionization-recombination absorption. Finally\nwe will discus in Sec. IV\nII. RECALLING SMW\nIn this section we will repeat these details which are\ncommon for Alfv\u0013 en waves (AW) and SMW. The di\u000ber-\nences between AW and SMW which are our new result\nwe derive after that.arXiv:2005.07730v1  [physics.plasm-ph]  15 May 20202\nA. Dispersion of MHD Waves\nLow density hydrogen plasma we approximate as a\ncocktail of ideal gases of electrons, protons and neutral\natoms with pressure pand mass density \u001a\np=nT; n =ne+np+n0; (1)\n\u001a=n\u001aM; n \u001a=np+n0; (2)\nwhere temperature Tis written in energy units and M\nis the proton mass. The sound velocity is de\fned by the\nadiabatic compressibility\ncs=s\u0012@p\n@\u001a\u0013\ns; (3)\nfor which the standard expression from the averaged\natomic mass of the cocktail hMi\ncs=s\n\rpT\nhMi;hMi=npM+n0M+nem\nnp+n0+ne; (4)\ncv= 3=2; cp=cv+ 1 = 5=2; \r =cp=cv= 5=3 (5)\nwherecvandcpare the heat capacities per atom and m\nis the electron mass.\nSmall amplitude MHD waves we treat as small varia-\ntions of the magnetic \feld b, density\u001a0, pressurep0and\ntemperature T0\nB=B0+b; \u001a =\u001a0+\u001a0; (6)\np=p0+p0; T =T0+T0(7)\nfrom their constant values. The index 0 we omit where\nit is obvious. The variations of the pressure are related\nwith variations of the density\np0\u0019c2\ns\u001a0; (8)\naccording the de\fnition of the sound speed. Here we\nrecall also the equations of state pV\r= const and\nTV1=cv= const for S= const (constant entropy S) and\ngive the relations between variations of the temperature,\npressure and density15\nT0\nT=1\ncv\u001a0\n\u001a=1\ncpp0\np;\u001a0\n\u001a=1\n\rp0\np; \r\u0011cp\ncv=5\n3:(9)\nFor a weak amplitude plane wave the variations of all\nvariables are proportional to the imaginary exponent\n/ei(k\u0001r\u0000!t)and phase velocity u\u0011!=k the ratio be-\ntween the frequency !and the modulus k=jkjof the\nwave-vector. For a plane wave the time tand space r\nderivatives are reduced to multiplication\n@t=\u0000i!;r= ik (10)\nand the MHD equations, omitting index 0 and imaginary\nunit i, reads3\n\u0000!\u001av=\u0000c2\nsk\u001a0+B\u0002(k\u0002b)=\u00160; \u0016 0= 4\u0019;(11)\n\u0000!b=k\u0002(v\u0002B); !\u001a0=\u001ak\u0001v; \"0= 1=4\u0019;(12)where k\u0001b= 0. We use Gaussian units but in SI&C:\n\u00160\u0003= 10\u00007and\"0==c210\u00007, i.e. all formulae in the\npresent work are written in system invariant form. In\nHeaviside{Lorentz units \u00160= 1 and\"0= 1. Thex-axis\nis chosen along the wave-vector k=kex, andy-axis is in\nthe plane of the wave-vector and the constant magnetic\n\feld\nB=Bxex+Byey; Bx=Bcos\u0012; By=Bsin\u0012;\nsee Fig. 1. The unit vector along the external magnetic\nx,k B\nyθ\nvgr=VAcosθkB=|k·B|/|B|\nω=VAkB\nFIG. 1: Geometry of propagating of SMW in a constant\nexternal magnetic \feld B. The group velocity of the\npropagating wave packet vgris along the external\nmagnetic \feld. The wave vector kis along the normal\nof wave fronts (equiphase planes) shown here with lines\nparallel to the yaxis.\n\feld is\neB=B=B= cos\u0012ex+ sin\u0012ey (13)\nDividing by k, the nonzero components of the MHD equa-\ntions Eqs. 11 and 12 read\n\u001au\u0012\n1\u0000c2\ns\nu2\u0013\nvx=Byby=\u00160; (14)\n\u001auvy=\u0000Bxby=\u00160; (15)\nuby=Byvx\u0000Bxvy: (16)\nExpressing velocity components vxandvyfrom the \frst\ntwo equations and substituting it in the third one gives a\nquadratic equation for the phase velocity u=!=kwhich\nhave the solutions describing fast (f) and slow (s) mag-\nnetosonic waves3\nu2\nf;s=1\n2n\nV2\nA+c2\ns\u0006\u0002\n(V2\nA+c2\ns)2\u00004c2\n\u0012V2\nAc2\ns\u00031=2o\n:(17)\nFor small magnetic \felds for SMW wave we have\nu=us\u0019VAc\u0012\u001ccs; c\u0012\u0011jcos\u0012j; (18)3\nwhere\nVA\u0011Bp\u00160\u001a; \u001aV2\nA=B2=\u00160 (19)\nis the speed of AW and uA\u0011VAc\u0012is the modulus of\nits projection along the x-axis. In such a way for the\ndispersion of SMW we have\n!=jVA\u0001kj;vgr\u0011@!\n@k=VAsgn(B\u0001k): (20)\nThe frequency of SMW can be expressed by the projec-\ntion of Alfv\u0012 en speed along the wave vector uA\n!=uAk=VAkB; (21)\nu\u0019uA\u0011VAc\u0012\u001ccs; kB=kc\u0012 (22)\nor by projection of the wave-vector along the magnetic\n\feldkB. For SMW the last inequality substituted in\nEqs. (14) and (15) gives\n\u0012c2\ns\nu2\u00001\u0013\n\u0019c2\ns=u2\u001d1; (23)\nand we have approximate expressions for the components\nof the velocity\nvy=\u0000Bx\n\u00160\u001auby;k\u0001v=\u0000Bxk\n\u00160\u001auby (24)\nvx\u0019\u0000u\nc2sBy\n\u00160\u001aby; bx= 0: (25)\nThen from Eq. (12) we obtain the variation of the density\n\u001a0\n\u001a=k\u0001v\n!=Byby\n\u00160\u001ac2s; (26)\nand from Eq. (8)\np0=c2\ns\u001a0=\u0000Byby\n\u00160=\u0000B\n\u00160s\u0012by; s\u0012\u0011sin(\u0012);(27)\nwe express the variations of the pressure proportional to\nthe small wave component of the magnetic \feld\nby=b0cos(kx\u0000!t);b=b0cos(kx\u0000!t): (28)\nThe unit vector eb\u0011b0=b0=eyalong the oscillating\ncomponent of the magnetic \feld has angle \u0019=2\u0000\u0012with\nthe constant one\n(eb\u0001eB)2=s2\n\u0012: (29)\nNow we can express the averaged density of the wave\nenergy which according to the virial theorem is twice the\naveraged density of the magnetic energy\nE= 2\u001cb2\n2\u00160\u001d\n=b2\n0\n2\u00160;q=Evgr (30)and the averaged density of the pressure oscillations\nwhich is one important ingredient of the forthcoming\nanalysis\n\n(p0)2\u000b\n=B2\n\u00160s2\n\u0012b2\n0\n2\u00160=B2\n\u00160Es2\n\u0012: (31)\nWe mention that the pressure oscillations disappear for\nwave-vector parallel to the magnetic \feld ( kkBor\nsin\u0012= 0) and v\u0001k=b\u0001k= 0, the waves are purely\ntransverse.\nAfter the consideration of dissipationless wave prop-\nagation in the next subsection we recall the results for\nSMW damping.\nB. MHD Absorption\nIn the WKB approximation we suppose that wave am-\nplitudes have small exponential decay e\u0000\rttas a function\nof time or space extinction e\u0000\rxif we trace a traveling\nwave packet.\nFor the energy \rux and density we have quadratic de-\npendence e\u00002\rxxand the extinction\n\rx=QMHD\n2qx(32)\nis given by the ratio of the time averaged power of MHD\ndissipation QMHD and the energy \rux qx.3In dissipation-\nless approximation the substitution of\nby=b0cos(kx\u0000!t); bx=bz= 0 (33)\nand the derived velocity vyEq. (24) in the formula for\nthe Pointing vector in MHD\nq\u0019S\u0019B\u0002(v\u0002B)=\u00160 (34)\ngives\nS=VAE; (35)\nE=\nb2=2\u00160+\u001av2=2\u000b\n=b2\n0=2\u00160; (36)\nin agreement with Eq. (30). For the x-component we\nhave\nqx=uAE: (37)\nThe small dissipation is proportional to the dissipative\ncoe\u000ecients\nQMHD =\u0017k\u001a\u0012@v\n@x\u00132\n+\u0017m\u0012@b\n@x\u00132\n=\u00160 (38)\nparamererized by kinematic \u0017k=\u0011=\u001aand magnetic dif-\nfusivity\u0017m=\"0c2%, where\u0011is viscosity coe\u000ecient and %\nis the Ohmic resistivity. Expressing vyfrom Eq. (15) and\nassumingvx\u00190 from Eq. (14) meaning that div v\u00190\nafter some algebra we obtain\n\rx=\rt\nua; \rt=1\n2(\u0017k+\u0017m)k2: (39)4\nIf we trace a wave packet of SMW propagating along\nmagnetic force lines Bat distance l=x=c\u0012for the energy\ndamping/e\u00002\rllwe have the extinction\n\rl=QMHD\n2VAE=\rt\nVA=(\u0017k+\u0017m)k2\n2VA: (40)\nThis space damping rate does not depend on the angle\n\u0012. For AW we have velocity vzand magnetic \feld bz\noscillations only normal to the ( k-B) plane direction but\nfor small magnetic \feld VA\u001ccsthe dispersion and wave\ndamping are the same. The di\u000berence appears when we\nanalyze the chemical damping of SMW.\nAfter this recall of the well-known result we analyze in\nthe next section the chemical damping.\nIII. IONIZATION-RECOMBINATION ABSORPTION\nThe degree of the ionization\n\u000b=np\nn\u001a(41)\nis a result of the continuous balance of ionization and\nrecombination processes\ndnp\ndt=\fn0ne\u0000\rrecnpn2\ne (42)\nwith temperature dependent rates of electron impact ion-\nization\f(T) and two electron recombination \r(T):For\ndense enough plasma the radiative processes have neg-\nligible contribution, especially for optically thin plasma\nregions.16,17\nIn thermal equilibrium the degree of ionization is given\nby Saha equation15,18\n\u0016np\u0016ne\n\u0016n0=nS\u0011\u0012mT\n2\u0019~2\u00133=2\ne\u0000I=T; (43)\nwhereIis the ionization energy. The rates of ionization\nand recombination processes in equilibrium are equal\n\u0017=\f\u0016ne\u0016n0=\rrec\u0016n2\ne\u0016np: (44)\nThe variable \u0017(T) gives the number of reactions\n\f: H + e\u0000!p + e + e; (45)\n\rrec: p + e + e\u0000!H + e (46)\nper unit volume and unit time. From this rate one can\ncreate a temperature dependent variable\nQ\u0013\u0011T\u0017 (47)\nwith dimension of power density; energy per unit volume\nand unit time.\nWhen MHD waves propagate through the plasma os-\ncillations of the pressure p0, density\u001a0and the tempera-\ntureT0perturbate the chemical equilibrium and inducevariations of the chemical composition and ionization de-\ngree\u000b. This extra chemical chaos creates an additional\nmechanism of increasing of entropy and wave energy dis-\nsipation\nQion=Q\u0013\n\u001f2\u000b\n; (48)\nwhere brackets denotes wave period averaging.\nThe main detail of the chemical energy dissipation is\nthe deviation from the chemical equilibrium\n\u001f\u0011nenp\nn0nS\u00001 =nenp\nn0\u0016n0\n\u0016ne\u0016np\u00001 (49)\ndescribed in detail in a recent Ref. 19. We suppose that\nthe variations of the chemical chomposition are relatively\nsmall, and the frequency of SMW is high enough\n!\u001d\u000b(1\u0000\u000b)\fn\u001a; (50)\n\u0016ne= \u0016np=\u000b\u0016n\u001a;\u0016n0= (1\u0000\u000b)n\u001a: (51)\nIn equilibrium \u0016 \u001f= 0 and we have to calculate the small\nsmall change of the variable \u001fdescribing the deviation\nfrom the chemical equilibrium substituting in Eq. (49) all\nnecessary details\nne= \u0016ne+n0\ne; np= \u0016np+n0\np; n 0= \u0016n0+n0\n0;(52)\nnS(T+T0) =nS+n0\nS=nS(T) +dnS\ndTT0: (53)\nFor linearized waves and small j\u001fj\u001c1 we have\n\u001f\u0019n0\ne\nne+n0\np\nnp\u0000n0\n0\nn0\u0000n0\nS\nnS: (54)\nAll relative changes of the variables can be expressed by\nthe relative change of the pressure\nn0\ne\nne=n0\np\nnp=n0\n0\nn0=\u001a0\n\u001a=1\n\rp0\np(55)\nand the Saha density\nn0\nS\nnS=dnS\nnSdTT0=\u0012I\nT+cv\u0013T0\nT=\u0012I\nT+cv\u0013p0\ncpp:(56)\nDue to detailed text-book recalling of the SMW dynamics\nwe easily arrive at a simple result\n\u001f=\u0012I\ncpT+2\n\r\u0013p0\np(57)\nand its square can be easily averaged using Eq. (31)\n\n\u001f2\u000b\n=\u0012I\ncpT+2\n\r\u00132\n(p0)2\u000b\np2(58)\n=\u0012I\ncpT+2\n\r\u00132B2\n\u00160pE\nps2\n\u0012: (59)5\nMultiplying with the power density rate we \fnally derive\nthe main result of the present work: the mean energy\ndissipation of a SMW propagating in magnetized plasma\nQion=Q\u0013\u0012I\ncpT+2\n\r\u00132B2\n\u00160pE\nps2\n\u0012: (60)\nNow for the time damping we obtain\n~\rt=Qion\n2E=Q\u0013\np\u0012I\ncpT+2\n\r\u00132B2=2\u00160\nps2\n\u0012; (61)\n(62)\nand for the extinction at low temperatures T\u001cIwe\nhave an additional chemical term\n~\rl\u0019Qion\n2EVA=Q\u0013\npVA\u0012I\ncpT\u00132B2=2\u00160\nps2\n\u0012; (63)\nwhich disappears at small angles \u0012\u001c1:In the next \fnal\nsection we will discuss the di\u000berence between two damp-\ning mechanisms giving total SMW extinction\n\rtot=\rl+ ~\rl: (64)\nIV. DISCUSSION AND CONCLUSIONS\nThe angular dependence of the chemical damping ob-\ntained in Eq. (63) ~ \rl/sin2\u0012is the main di\u000berence be-\ntween the chemical damping and the MHD one. Here we\nwish to emphasize also that the derived new ionization-\nrecombination damping is frequency independent and has\nno hydrodynamic sense as second viscosity, for example.\nThe MHD damping according to Eq. (40) \rl/k2/!2is\nproportional to the square of the wave-vector and square\nfrequency.\nRoughly speaking MHD damping is a low pass \fl-\nter while ionization-recombination mechanism is a bass\ndamper.\nImagine that turbulence generates broad distribution\nof MHD waves and the angular distribution of the spec-\ntral density is almost constant at small angles between\nthe wave-vector and constant magnetic \feld\ncos\u0012=k\u0001B\nkB: (65)\nIf then SMW pass through a partially ionized region with\nlengthlthe chemical damping gives transmission coe\u000e-\ncient\n~TSMW = e\u00002~\rll= exp\u0012\n\u0000\u00122\n2\u00122\n0\u0013\n; (66)\n1\n2\u00122\n0=2\u0017Tl\npVA\u0012I\ncpT\u00132B2=2\u00160\nps2\n\u0012; (67)\n\u00120=cpT\n2Ir\npVA\fpl\n\u0017Tl\u001c1; \f pl\u0011p\npB; (68)\npB=B2=2\u00160; p = (\u0016ne+ \u0016np+ \u0016n0)T: (69)In other words, strong ionization-recombination absorp-\ntion gives a cumulative small angle distribution of SMW.\nWaves with signi\fcant angles \u0012are absorbed and domi-\nnantly AW will pass through large area of partially ion-\nized plasma.\nHow this can be checked by observations. Imagine that\nin an observation point we have a good record of the time\ndependence of the magnetic \feld B(t). Time averaging\ncan give mean value B0and orientation eBof the con-\nstant component of the magnetic \feld\nB0=hB(t)i;eB=B0=jB0j:\nThen we can make Fourier analysis and calculate the\nwave components of the magnetic \feld for all frequen-\ncies!\nb0\n!=h(B(t)\u0000B0) cos(!t)i;e0\n!=b0\n!=jb0\n!j;\nb00\n!=h(B(t)\u0000B0) sin(!t)i;e00\n!=b00\n!=jb00\n!j:\nFor the considered in Sec. II example we have\neB= cos\u0012ex+ sin\u0012ey;e0\n!=ey;eB\u0001e0\n!= sin\u0012:\nIn the general case we have di\u000berent angles for all Fourier\nfrequencies\nsin(\u00120\n!) =eB\u0001e0\n!;sin(\u001200\n!) =eB\u0001e00\n!;\nand it is worthwhile the study probability distribution\nfunction of angles \u0012. Our simple consideration predicts\nGaussian distribution\nP(\u0012)/exp(\u0000\u00122=2\u00122\n0) (70)\ncreated by long regions of partially ionized plasma.\nRoughly speaking SMW \fltered by large regions of par-\ntially ionized plasmas will be almost transverse.\nEvery similarity with phenomena in the magnetized at-\nmosphere even in the nearest star is random. We present\na purely academic study.\nLast but not least the ionization rate \f=hv\u001biis given\nby the Maxwell velocity vaveraging of the electron im-\npact ionization cross-section \u001band all details of the pro-\nposed new damping mechanisms are well studied.\nThe considered in the present work the chemical damp-\ning of the pressure oscillations in some sense belongs to\nthe notions of plasma multi-\ruid approach. Not only\nrelative velocity between di\u000berent \ruids12,14creates dis-\nsipation as a friction forces. Periodic oscillations around\nthe Saha equilibium for some MHD modes of partially\nionized plasmas can give even bigger dissipation and in-\ndispensably have to be taken into account in the arsenal\nof the plasma physics notions.\nACKNOWLEDGMENTS\nThe authors appreciate stimulating discussions cor-\nrespondence with Dantchi Koulova, Kamen Kozarev,\nHassan Chamati, Yavor Boradjiev, Nedko Ivanov, and\nStanislav Varbev.6\nDATA AVAILABILITY STATEMENT\nData sharing is not applicable to this article as no new\ndata were created or analyzed in this study, which is a\npurely theoretical one.\n1H. Alfv\u0013 en, \\Existence of Electromagnetic-Hydrodynamic\nWaves\", Nat. 150, 405 (1942).\n2H. Alfv\u0013 en, \\Granulation, magnetohydrodynamic waves, and the\nheating of the solar corona\", Mon. Not. Roy. Astr. Soc. 107, 211\n(1947).\n3L. D. Landau and E. M. Lifshitz, Electrodynamics in Continuous\nMedia in L. D. Landau and E. M. 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In\ruence of electronic momentum\", Eq. (46.1a),\nSec. 104, \\Ionization equilibrium\".\n16L. P. Pitaevskii, \\Electron Recombination in a Monatomic gas\",\nJETP 15(5), 919 (1962).\n17E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics in\nL. D. Landau and E. M. Lifshitz, Landau-Lifshitz Course on The-\noretical Physics, Vol. X (Pergamon, New York, 2002), Sec. 24,\n\\Recombination and ionization\".\n18M. N. Saha, \\On a physical theory of stellar spectra\", Proc. R.\nSoc. Lond. A, 99(697), 135-153 (1921).\n19T. M. Mishonov, A. M. Varonov, \\Sound Absorption in Par-\ntially Ionized Hydrogen Plasma and Heating Mechanism of Solar\nChromosphere\", arXiv:2005.05056 [physics.plasm-ph]."    },    {        "title": "1111.7060v1.Radiation_Damping_for_Speeding_up_NMR_Applications.pdf",        "content": " 1LA-UR-11-12094 \n \n \n \n  \nRadiation Damping for Speeding-up NMR Applications \n \nGennady P. Berman1, Michelle A. Espy2, Vyacheslav N. Gorshkov1,3,    \nVladimir I. Tsifrinovich4,  and Petr L. Volegov2 \n \n1 T-4, Theoretical Division, MSB213, LANL, Los Alamos, NM 87544 \n2 P-21, Applied Modern Physics, D454, LANL, Los Alamos, NM 87544 \n3 National Technical University of Ukraine “KPI,” 37 Peremogy Av., Bldg 7, Kiev-56, \n03056 Ukraine \n4 Department of Applied Physics, Polytechnic Institute of NYU, 6 MetroTech Center, \nBrooklyn, NY, 11201  \n \nAbstract \n \nWe demonstrate theoretically and numerically how to control the NMR relaxation \nrate after application of the standard  spin echo technique. Using radiation \ndamping, we return the nuclear magnetization to its equilibrium state during a time interval that is negligible compared to the relaxation time. We obtain an estimate for optimal radiation damping which is consistent with our numerical simulations.   \n \n1. Introduction  \n   The problem of low repetition rate is one of the most important in many NMR applications [1]. \nThere are two main approaches for controlling the nuclear magnetic relaxation [1-6]. The first is \nthe application of an additional rf pulse in the process of the echo formation. This pulse can return \nthe nuclear magnetization to its equilibrium pos ition. However, in order to implement this \nmethod, the phase of the rf phase must be accurately adjusted relative to the phase of the \ntransverse magnetization. The second method relies on the radiation field created by the resonant \nLCR coil interacting with the nuclear magne tization. In this method, the phase of the rf field \nproduced by the coil is automatically adjusted re lative to the transverse magnetization. However, \nthe effect of the radiation field is, normally, small because the radiation field is much smaller than \nthe rf field.  \n   In this paper we demonstrate that, by incr easing the duration of the spin echo and optimizing \nthe coil inductance, one can restore the equilib rium position of the nuclear magnetization during \nthe time of the echo formation, which is neglig ible compared with the relaxation time. Our \nconsideration was done in the “dynamical” regime, when the characteristic time of the relaxation \nin the coil that produces the relaxation damping is much larger than the spin echo time. Our \napproach is valid for both, weak and strong magnetic fields. We obtained an analytical estimate for the optimal conditions for radiation damping, consistent with our numerical simulations.    22. Fast radiation damping with low-field setup \n \nConsider a sample containing an ensemble of protons. This sample can interact with four \ncoils. (See Fig. 1.) Coil “0” produces a permanent non-uniform magnetic field, B 0, in the \npositive z-direction. Coil “1 x” produces an oscillating field B 1x (rf-pulses) along the x-axis. It \nis also used for NMR detection (with weak radiation damping). Coil “1 y” produces an \noscillating field B 1y (rf-pulses) along the y-axis. It also can be us ed for NMR detection (with \nweak radiation damping).  Coil 2 produces the oscillating field, B 2, along the x-axis, which  \ncauses radiation damping. Note that we use two coils, 1\nx and 1 y, in order to produce circularly \npolarized rf-pulses. This is especially important \nif one is going to use an ultra-low permanent magnetic field, B\n0. In the opposite case of high \npermanent magnetic field, instead of the circular polarized rf-pulses one can safely apply linearly \npolarized rf-pulses. In this case one of the coils \n(1\nx or 1 y) can be removed.  \n  \nWe propose the following scheme:  \n1. At t = 0 a rectangular rf  π/2-pulse of \nduration, say, 10 μs is applied to the \nsample. At time t = 1010 μs, a \nrectangular rf  π-pulse of duration 10 μs \nis applied to the sample. At time t = 3020 \nμs, a rectangular rf  π-pulse of duration \n10 μs is again applied to the sample. \n2. After the first pulse one observes a \ndecaying signal of magnetic induction. \nBetween the second and the third pulses \none observes a spin echo. After the third \npulse one again observes a spin echo. \n(See Fig. 2.)      \n3. Coil 3 is a part of an LCRS-circuit (S st ands for a switch) with capacitance, C, and \nresistance, R. The switch, S, is  closed (turned on) at t = 3030 μs, i.e. after the third \npulse. Before the third pulse, coil 3 does not  influence the spin system. During the \nduration of the second spin echo, the radi ation damping due to coil 3 drives the \nnuclear magnetization toward its equilibrium state.  \nBelow we write the equations  of motion for 0 < t < 10 μs together with definitions of \nnotations: \n  \n \n \nFig. 1. Suggested setup. \n  3\n\n\n01 2\n10 2\n11 1\n00 0\n11 0\n11 0\n00 1 1 1 1\n0() / ,\n() / ,\n() () 1 / ,\n() ( ,) ( ) ,\n() c o s ,\n() s i n ,\n(0) (0) 0,\n(0) 1,\n,,\n() 1 /2 e x pxy z y x\nyz x x y\nzx y y x z\nx\ny\nxy\nz\nxx yymm b m b tm T\nmm b t m b m T\nmm b tm b t mT\nmt d bmb tgb\nbt a b t\nbt a b t\nmm\nm\nbB b B b B\ngb b \n\n \n \n \n\n\n\n\n\n \n \n\n\n\n22\n00 /2 . b \n                            (1) \n \nHere and below the equations of motion are written for a normalized dimensionless magnetic \nmoment; b 0, b 1x, and b 1y denote the corresponding magnetic fields, B 0, B 1x, and B 1y in \nfrequency units of s-1; g(b 0) is the Gaussian distribution for a permanent magnetic field, b 0,  \nwith the average value 0b; γ is, for definiteness, the pr oton gyromagnetic ratio. In our \nnumerical experiments described below, we used  an averaging in Eqs (1) over an ensemble \nthat included 25 thousand elementary magnetic moments, ()im. \n \nIn our simulations, we use the following parameters: 313.5 10 s  and 4110s , 0B= \n1,mT T1 = T 2 = 1 s, the amplitude, a 1, of a π/2-pulse is found from the condition: \n1(10 ) / 2.as                                 (2) 0 1000 2000 3000 4000 50000.00.20.40.60.81.0\n-pulse\nt, smxy-pulses\nFig. 2. Suggested spin echo sequence.  4110 .s    40B = 1 mT. \n \nThe equations of motion for 10 < t < 1010 μs are: \n \n02\n02\n1/,\n/,\n1/ .xy x\nyxy\nzzmm b m T\nmm b m T\nmm T\n \n \n\n (3) \nThe equations of motion for 1010 < t < 1020 μs are: \n \n01 2\n10 2\n11 1() / ,\n() / ,\n() () 1 / .xy z y x\nyz x x y\nzx y y x zmm b m b tm T\nmm b t m b m T\nmm b tm b t mT\n \n \n\n (4) \nHere the amplitude, a 1, of the π-pulse is found from the condition: \n1(10 ) .as             (5) \n \nThe equations of motion for 1020 < t < 3020 μs are equations (3). \nThe equations of motion for 3020 < t < 3030 μs are equations (4). \nThe equations of motion for 3030 μs < t are the following: \n 02\n20 2\n21\n20\n0/,\n() / ,\n() 1 / ,\n() ,\n/0 .xy x\nyz x y\nzy z\ncxmm b m T\nmm b t m b m T\nmm b t m T\nbt n I\nIC R I L I n V M m\n\n \n  \n\n \n\n\n   (6) \nAdditional initial conditions at t = 3030 μs for a current in the coil 2 are: \n                        0. II                                              \n(7)  \nHere (for coil 2) n is the number of turns per unit length, and Vc is the volume of the coil; η is \nthe filling factor (the ratio of the sample volume to the coil volume); and M is the nuclear \nmagnetization of the sample. \nOur objective is to find the optimal conditions fo r a rapid return of the nuclear magnetization \nto  the  state, 1zm. We varied the parameter of the Gaussian distribution Γ, the resistance R,  5and the inductance, 2\n0 c L nV , keeping the resonant condition 1/ 2\n0 () .LC b  Also, we use \nthe following values of the parameters: \n 0 cnV M = 10-6 Wb, 0n = 3.36x106 rad/C.                                   (8)      \nThese parameters are computed using the values  \n 7\n0\n41\n0\n7\n0\n3/ 2 42.6 / ,\n41 0 / ,\n10 ,\n,\n1.33 10 ,\n750 .c\ncMHz T\nHm\nnm\nnV M\nM T\nVc m\n\n \n\n\n\n\n\n\n\n (9)  \n3. Results of numerical simulations \n \nThe sequence of echo signals described above was simulated numerically.  The dynamics of \nthe average transverse magnetic moment,  1/22 2() () ()xy x ymt m t m t , with no LCRS \ncircuit is presented in Fig. 3.  \n \n \n  \nThen, the numerical experiment for time, \n33030 tt s   , was done taking into account the \nLCRS circuit assuming that the switch is closed.  \n 0 1000 2000 3000 4000 50000.00.20.40.60.81.0\n - pulse\nt, smxy-pulses\n                                         Fig. 3. Dynamics of ()xymt for Γ = 3.5x103 s-1.   6 The physical process of fast radiation da mping (FRD) can be described as follows.  \nThe induced transverse magnetic field, 2()xBt, oscillating with the frequency, 0b,  initially \ndrives the nuclear magnetization toward the z-axis. The variation of the envelope, ()xymt , at \nthe initial stage of this process is caused by two factors: \n \n(i) the spin echo effect which increases the value of ()xymt ;  \n(ii) the rotation of the magneti zation vector  in the field, 2()xBt,  which decreases the \nvalue of  ()xymt . \n \nThe second factor is defined at the final stage of the FRD process.    \n One could assume that at \n() 1zmt , () 0xmt the state of the system would \napproach the  equilibrium as 2() 0xBt. In fact, the maximal values of ()zmt  (and, \ncorrespondingly, the smallest values of ()xymt ) correspond to the maximal value of the current \nin the coil 2 (see Fig. 4). As a result, the vector of magnetization continues to rotate in the field, \n2()xBt, and the value of  ()zmt  decreases.  At first, this leads to an increase of the transverse \nmagnetization, ()xymt , which then disappears due to the inhomogeneity of the field, 0()br. \n(See Fig. 4.)  \nThe inductance of the LCRS circuit chosen in Fig. 4 is not optimal. Our numerical simulations \nshow that the optimal condition for the FRD can be described by the following formula:  \n \n \nFig. 4.  The dynamics of th e system after the second -pulse for Γ = 3.5x103 s-1,\n3000Q and 0.15LH .  The time is counted here and below from 33030ts . The\namplitude of current, 0()It, is presented in mA. ,0()xymt is the transversal magnetizatio n\nwhen switch in LCRS circuit is open. \n  7                                                              0 2\n4opt echobL .                                                  (10) \n \nThis relation can be obtained from simple considerations. (See Appendix.) For the field, \n01B mT , 610Wb , 63.36 10 / rad C  , and 1echo ms  (see Fig. 3) the relation (10) \nleads to 0.072optL H . The corresponding optimal capacitance can be found from the \nresonance condition 2\n0 LC b . Substituting,  2\n0 c L nV , we can rewrite the condition (10) \nfor optimal radiation damping in the form: \n \n \n   2\n00 /4 1echo Mb   .                    (11) \n \n The values of our parameters (9) are chosen to match this condition. In applications, for a \ngiven values of \n and , one can vary 0, ,  and echo Mb by changing the pre-polarization field, \nthe mean field 0B, and the non-uniformity of 0B, correspondingly.  \n \nOur estimate (10) appears to be very good, as the results of numerical experiment demonstrate. \nIn Fig. 5, the results of the numerical simulations  are presented when the current in the circuit \nincreases with advancing rate relative to the optimal regime. The maximal effect of FRD is \nrealized at 0.075LH (see Fig. 6). In this case, the increase of ()xymt  in the process of spin \necho formation is maximally used. As the results  of numerical simulations demonstrate, the \noptimal choice of L leads to the symmetric dependence of ()xymt  relative to the moment of \ntime when 0()It is maximal. Note that the FRD is not very sensitive to the value of L. As one \n    \n  \nFig. 5                                                                         Fig. 6 \n \nThe dynamics of the system after the second -pulse for 3000Q and 313.5 10xs .\nThe amplitude of the current, 0()It in mA. 0.025LH (Fig. 5) and 0.075LH (Fig. 6).  8can see from Figs. 4-6, changing of the inductan ce by a factor 6  does not  significantly affect \nthe FRD.   \n \nNow we verify the correctness of the relation \n(10) which states that \noptL does not depend on \nthe resistance of the circuit, R.  For this, we \nchanged the quality factor, Q, by a factor в 3  \n(decreased the resistance, R, by a factor three) \nat 0.075LH . (The quality factor of the \nLCRS circuit is given by 0/ Qb L R r as \n0bis equal to the resonant frequency of the \ncircuit.)  The dynamics of the system, which is \nshown in Fig. 7, remained practically unchanged in the most important time-interval. If, in addition to the above described modification, we decrease by a factor of three \nparameter, \n, and simultaneously decrease by \na factor of three the inductance, then we obtain \nthe same results (the dependences shown by dotted lines in Fig. 7). \nFinally, we will check relation (10) by changing the duration of the spin echo. For this we \nincrease the parameter of the Gaussian distribution from \n313.5 10 s   to 4110s . The \nspin echo signal of the reduced duration is shown in Fig. 8.  \n   \n \nFig. 7.  The dynamics of the system for a \ngiven parameter, , and the inductance, L. \nContinuous curves: 3000Q , dotted lines: \n1000Q . 1500 2000 25000.00.20.40.60.81.0\nt, smxy(t)\n \nFig. 8. Spin echo of the reduced duration. \n(This is a fragment of Fig. 2.) 0.325echo ms  is\nhalf of the time-interval between the re d\narrows.  \n \n600 800 1000 1200 1400 160 00.00.20.40.60.81.0 mz\nmxy\nt, s\nFig. 9. Dynamics of the system for 0. 3echo\nand 0.008L . The value of ,maxzm is ,maxzm\n  9For 0.325echo ms  from formula (10) we obtain 0.008optL . Fig. 9 demonstrates the \ndynamics of the system for 0.325echo ms  and 0.008L . One can see again that formula (10) \nprovides a reliable estimate for the optimal inductance. \n Certainly, the current in the coil 2 should be  terminated when the longitudinal nuclear \nmagnetization reaches its maximal value. As an  example, in Fig. 4 the current should be \nterminated at \n1160 safter the end of the second -pulse.  \n \n \n4. Conclusion \n \nWe considered here the “dynamical” regime of the fast radiation damping (FRD). We have \nshown that FRD during the process of the spin  echo formation can effectively restore the \nlongitudinal nuclear magnetization. We have obtai ned an estimate for the optimal choice of the \ncircuit parameters for the FRD, and confirmed it in our numerical experiments. \n \nAcknowledgement   \nThis work was carried out under the auspices of the NNSA of the U. S. DOE at LANL under  \nContract No. DEAC52-06NA25396. We thank the LDRD program at LANL for funding this \nresearch.  \nAppendix  \n \nWe will consider the time interval fr om the beginning of the echo formation, \n't,  until the \ninstant, ''t, when the longitudinal magnetization reach es its maximum value. (See the left \ngreen peak in Fig. 6.) For our set of parameters, the time constant of the LCRS circuit \n(relaxation time) 2/LR  is much greater than the time of the echo formation, echo, so we can \nsafely put 0R in Eqs (6). We will express xm and the current, I, in terms of their \namplitudes, () ,  a n d   () : fft J Jt  \n \n \n0\n0exp ,\nexp .xmf i b t\nIJ i b t\n                    (A1) \n \nSubstituting these expressions into the last e quation in (6), we obtain the equation for the \namplitudes: \n 2\n00 02( / ) 2 .Ji b J L f ib fb f                                  (A2) \n \nTaking into consideration that the characteristic time of the amplitude variation is much longer \nthan the period of the Larmor precession, 0 2/ b , we can reduce this equation to \n \n2\n002( / )ib J L b f  .                                    (A3) \n  10This equation gives the relation between the amp litude of the transverse magnetization and the \nderivative of the amplitude of current, \n 0/2 Jib L f  .                                      (A4) \n \nIntegrating (A4) over the time interval (from ' t o  ' 'tt ) we obtain, \n''\n0\n'( '') / 2 ( )t\ntJt i b L ftd t  .                                       (A5) \n \nThe integral in (A5) is, obviously, the area below the green peak in Fig. 6. We approximate this \narea with a triangle of height 1, midline , and base '' ' 2tt . Then, the integral in Eq. (5) \nequals .  It follows from Eq. (A5) and Eqs (6) th at during the considered time-interval the \nnuclear magnetization experiences a linearly polarized rf pulse of maximum amplitude, \n  \n0/2 . bL                                   (A6) \n \nApproximating this pulse as a tria ngle (see Fig. 6) with the base, 2, and height (A6), we \nestimate its dimensionless area as 2\n0 /2 bL  . The area of the corresponding pulse of the \ncircular polarized field is 2\n0 /4 bL  . Next, we approximate 2as a half of the \ncorresponding value 2\necho of the unperturbed echo (with no radiation damping): 22/2echo  \n(see Fig. 6). For optimal radiation damping, the ar ea of the circular polarized pulse, which is \nequal to the angle of rotation of the nuclear magnetization, should be equal to /2 .  From \nequation, \n  2\n0 /8 /2echobL    ,                                     (A7) \n \nwe obtain condition (10).                                                                                          \n \nReferences  \n1. S.Y. Huang, T. Witzel, and L.L. Wald, A ccelerated Radiation-Damping for Increased \nSpin Equilibrium: A new method for controlling the recovery of longitudinal magnetization, Magn. Reson. Med., 60, 1112 (2008). \n2. N. Bloembergen and R.V. Pound, Radia tion Damping in Magnetic Resonance  \nExperiments, Phys. Rev., 95, 8 (1954). \n3. C.R. Bruce, R.E. Norberg, and Q.E. Pake , Radiation Damping and Resonance Shapes \nin High Resolution Nuclear Magnetic Resonance, Phys. Rev., 104, 419  (1956).  \n4. S. Bloom, Effects of Radiation Damp ing on Spin Dynamics, J. Appl. Phys., 28, 800 \n(1957).  \n5. A. Szöke and S. Meiboom, Radiation Damp ing in Nuclear Magnetic Resonance, Phys. \nRev., 113, 585 (1958). \n6. M.P. Augustine, Transient Properties of radiation Damping, Prog. in Nuc. Magn. Res., \n40, 111 (2002).\n "    },    {        "title": "1601.06213v1.Nonlinear_magnetization_dynamics_of_antiferromagnetic_spin_resonance_induced_by_intense_terahertz_magnetic_field.pdf",        "content": " 1Nonlinear magnetization dyna mics of antiferromagnetic \nspin resonance induced by inte nse terahertz magnetic field \n \nY Mukai1,2,4,6, H Hirori2,3,4,7, T Yamamoto5, H Kageyama2,5, and K Tanaka1,2,4,8 \u001f \n1 Department of Physics, Graduate School of  Science, Kyoto University, Sakyo-ku, Kyoto \n606-8502, Japan  \n2 Institute for Integrated Cell-Material Scien ces (WPI-iCeMS), Kyoto University, Sakyo-ku, \nKyoto 606-8501, Japan \n3 PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan \n4 CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan \n5 Department of Energy and Hydrocarbon Chemistr y, Graduate School of Engineering, Kyoto \nUniversity, Nishikyo-ku, Kyoto 615-8510, Japan \n \nE-mail: \n6 mukai@scphys.kyoto-u.ac.jp \n7 hirori@icems.kyoto-u.ac.jp \n8 kochan@scphys.kyoto-u.ac.jp \n We report on the nonlinear magnetization dynamics of a HoFeO\n3 crystal induced by a strong \nterahertz magnetic field resonantly enhanced with a split ring resonator and measured with \nmagneto-optical Kerr effect microscopy. The terahertz magnetic field induces a large change (~40%) in the spontaneous magnetization. The frequency of the antife rromagnetic resonance \ndecreases in proportion to the square of the magnetization change. A modified \nLandau-Lifshitz-Gilbert equation with a phenomenological nonlinear damping term \nquantitatively reproduced the nonlinear dynamics. \nPACS: 75.78.Jp, 76.50.+g, 78.47.-p, 78.67.Pt \n    21. Introduction \nUltrafast control of magnetization dynamics by a femtosecond optical laser pulse has \nattracted considerable attention from the persp ective of fundamental physics and technological \napplications of magnetic recording and inform ation processing [1]. The first observation of \nsubpicosecond demagnetization of a fe rromagnetic nickel film demonstrated that a femtosecond \nlaser pulse is a powerful stimulus of ultrafast magnetization dynamics [2], and it has led to numerous theoretical and experimental inves tigations on metallic and semiconducting magnets \n[3-8]. The electronic state created by the laser pulse has a strongly nonequilibrium distribution \nof free electrons, which consequently leads to  demagnetization or even magnetic reversal \n[1,2,9-11]. However, the speed of the magnetizat ion change is limited by the slow thermal \nrelaxation and diffusion, and an alternative t echnique without the limits of such a thermal \ncontrol and without excessive thermal energy would be desirable.  \nIn dielectric magnetic media, carrier heating hardly occurs, since no free electrons are present \n[12]. Consequently, great effort has been devoted  to clarifying the spin dynamics in magnetic \ndielectrics by means of femtosecond laser pulses. A typical method for nonthermal optical \ncontrol of magnetism is the inverse Faraday effect, where circularly polarized intense laser \nirradiation induces an effective magnetic field in  the medium. Recently, new optical excitation \nmethods avoiding the thermal effect such as the ma gneto-acoustic effect is also reported [13,14]. \nIn particular, these techniques have been used in  many studies on antiferromagnetic dielectrics \nbecause compared with ferromagne ts, antiferromagnets have inhe rently higher spin precessional \nfrequencies that extend into the terahertz (THz) regime [12,15]. Additionally, ultrafast \nmanipulation of the antiferromagnetic order parame ter may be exploited in order to control the \nmagnetization of an adjacent ferromagnet through the exchange interaction [16]. The THz wave \ngeneration technique is possibly a new way of optical spin control through direct magnetic \nexcitation without undesirable thermal effects [17-19]. As yet however, no technique has been \nsuccessful in driving magnetic motion excited directly by a magnetic field into a nonlinear \ndynamics regime that would presumably be fo llowed by a magnetization reversal [20-22]. \n \nIn our previous work [23], we demonstrated that the THz magnetic field can be resonantly \nenhanced with a split ring resonator (SRR) and may become a tool for the efficient excitation of \na magnetic resonance mode of antiferromagnetic dielectric HoFeO\n3. We applied a Faraday \nrotation technique to detect the magnetization ch ange but the observed Faraday signal averaged  3the information about inhomogeneous magnetiza tion induced by localized THz magnetic field \nof the SRR over the sample thickness [23]. In th is Letter, we have developed a time-resolved \nmagneto-optical Kerr effect (MOKE) micr oscopy in order to access the extremely \nfield-enhanced region, sample surface near th e SRR structure. As a result, the magnetic \nresponse deviates from the linear response in the strong THz magnetic field regime, remarkably \nshowing a redshift of the antiferromagnetic r esonance frequency that is proportional to the \nsquare of the magnetization change. The observe d nonlinear dynamics could be reproduced with \na modified Landau-Lifshitz-Gilbert (LLG) e quation having an additional phenomenological \nnonlinear damping term.  \n2. Experimental setup \nFigure 1 shows the experimental setup of MOKE microscopy with a THz pump pulse \nexcitation. Intense single-cycle THz pulses were  generated by optical rectification of \nnear-infrared (NIR) pulses in a LiNbO\n3 crystal [24-26]; the maximum peak electric field was \n610 kV/cm at focus. The sample was a HoFeO 3 single crystal polished to a thickness of 145 µm, \nwith a c-cut surface in the Pbnm  setting [27]. (The x-, y-, and z-axes are parallel to the \ncrystallographic a-, b-, and c-axes, respectively. ) Before the THz pump excitation, we applied a \nDC magnetic field to the sample to saturate its magnetization along the crystallographic c-axis. We fabricated an array of SRRs on the crystal surface by using gold with a thickness of 250 nm. \nThe incident THz electric field, parallel to the metallic arm with the SRR gap (the x-axis), drove \na circulating current that resulted in a strong  magnetic near-field normal to the crystal surface \n[23,28,29]. The SRR is essentially subwavelength LC circuit, and the current induces magnetic \nfield B\nnr oscillating with the LC resonance frequency (the Q-factor is around 4). The right side \nof the inset in figure 1 shows the spatial distribut ion of the magnetic field of the SRR at the LC \nresonance frequency as calculated by the fin ite-difference time-domain (FDTD) method. \nAround the corner the current density in the metal is very high, inducing the extremely \nenhanced magnetic field in the HoFeO 3 [29]. \n \nAt room temperature, the two magnetizations mi (i=1,2) of the different iron sublattices in \nHoFeO 3 are almost antiferromagnetically aligned along the x-axis with a slight canting angle \n0(=0.63°) owing to the Dzyaloshinskii fiel d and form a spontaneous magnetization MS along \nthe z-axis [30]. In the THz region, ther e are two antiferromagnetic resonance modes \n(quasiantiferromagnetic (AF) and quasiferroma gnetic (F) mode [31]). The magnetic field Bnr  4generated along the z-axis in our setup causes AF-mode motion; as illustrated in figure 2(a), the \nZeeman torque pulls the spins along the y-ax is, thereby triggering precessional motions of mi \nabout the equilibrium directions. The precessional motions cause the macroscopic magnetization M=m\n1+m2 to oscillate in the z-direction [32,33]. The resultant magnetization \nchange Mz(t) modulates the anti-symmetric off-diagonal element of the dielectric tensor \nεxyaሺൌ െεyxaሻ and induces a MOKE signal (Kerr ellipticity change \u001f [34,35] (see Appendix A \nfor the detection scheme of the MOKE measuremen t). The F-mode oscillation is also excited by \nTHz magnetic field along the x or y-axis. Howe ver, the magnetization deviations associated \nwith the F-mode, Mx and My, do not contribute to the MOKE in our experimental geometry, \nwhere the probe light was incident no rmal to the c-cut surface of HoFeO 3 (the xy-plane) [34,35]. \nIn addition, the amplitude of the F-mode is much smaller than AF-mode because the F-mode \nresonance frequency ( F~0.37 THz) differs from the LC resonance frequency ( LC~0.56 THz). \n-10010x position (µm)\n-10 0 10\ny position (µm)\nTHz pump HoFeO3 \nz y \nx \nSRR  \nObjective lens \nNonpolarized \nbeam splitter Quarter \nwave plateWollaston  \nprism \nLens  Balanced photodiodes \nVisible probe Bin Bnr \nEin \n-10 0 10\ny position (µm)\n120\n80\n40\n0\nFigure 1. Schematic setup of THz pump-visible MOKE measurement. The left side\nof the inset shows the photograph of SRR fabricated on the c-cut surface of the\nHoFeO 3 crystal and the white solid line indicat es the edge of the SRR. The red soli d\nand blue dashed circles indicate the probe spots for the MOKE measurement. The\nright side of the inset shows the spatia l distribution of the enhancement facto r\ncalculated by the FDTD method, i.e., the ratio between the Fourier amplitude at LC\nof the z-component of Bnr (at z=0) and the incident THz pulse Bin.  5To detect the magnetization change induced onl y by the enhanced magnetic field, the MOKE \nsignal just around the corner of the SRR (indicated by the red circle in figure 1’s inset), where \nthe magnetic field is enhanced 50-fold at the LC  resonance frequency, was measured with a 400 \nnm probe pulse focused by an objective lens (spot diameter of ~1.5 µm). Furthermore, although \nthe magnetic field reaches a maximum at the surface and decreases along the z-axis with a \ndecay length of lTHz~5 µm, the MOKE measurement in refl ection geometry, in contrast to the \nFaraday measurement in transmission [23], can evaluate the magnetization change induced only \nby the enhanced magnetic field around the sample  surface since the penetration depth of 400 nm \nprobe light for typical orthoferrites is on the orde r of tens of nm [35]. (The optical refractive \nindices of rare-earth orthoferrites in th e near ultraviolet region including HoFeO 3 are similar to \neach other, regardless of the rare-earth ion speci es, because it is mostly determined by the strong \noptical absorption due to charge transfer and orbital promotion transitions inside the FeO 6 \ntetragonal cluster [35].) All experiments in this  study were performed at room temperature. \n \n3. Results and discussions   \nFigure 2(a) (upper panel) shows the calculated  temporal magnetic waveform together with \nthe incident magnetic field. The maximum peak am plitude is four times that of the incident THz \npulse in the time domain and reaches 0.91 T. Th e magnetic field continues to ring until around \n25 ps after the incident pulse has decayed away. The spectrum of the pulse shown in figure 2(c) \nhas a peak at the LC resonance frequency ( LC=0.56 THz) of the SRR, which is designed to \ncoincide with the resonance frequency of the AF-mode ( νAF0=0.575 THz). Figure 2(a) (lower \npanel) shows the time development of the MOKE signal  for the highest THz excitation \nintensity (pump fluence I of 292 µJ/cm2 and maximum peak magnetic field Bmax of 0.91 T). The \ntemporal evolution of  is similar to that of the Faraday rotation measured in the previous \nstudy and the magnetization oscillates harmonically with a period of ~2 ps [23], implying that \nthe THz magnetic field coherently drives the AF-mode motion.  \n As shown in figure 2(b), as th e incident pump pulse intensity increases, the oscillation period \nbecomes longer. The Fourier transform spectra  of the MOKE signals for different pump \nintensities are plotted in figure 2(c). As the ex citation intensity increases, the spectrum becomes \nasymmetrically broadened on the lower freque ncy side and its peak frequency becomes \nredshifted. Figure 2(d) plots the center-of-mass fre quency (open circles) and the integral (closed \ncircles) of the power spectrum P(\n) as a function of incident pulse fluence. The center  6frequency monotonically redshifts and P() begins to saturate. As shown in figure 2(c), the \nMOKE spectra obtained at the center of the SRR (indicated in the inset of figure 1) does not \nshow a redshift even for the highest intensity excitation, suggesting that the observed redshift \noriginates from the nonlinearity of the precessional spin motion rather than that of the SRR \nresponse. We took the analytic signal approach  (ASA) to obtain the time development of the \ninstantaneous frequency (t) (figure 3(c)) and the envelope amplitude 0(t) (figure 3(d)) from \nthe measured magnetization change (t)=Mz(t)/|MS| (figure 3(b)) (see Appendix B for the \ndetails of the analytic signal approach). As is  described in the Appendix C, the MOKE signal \n6\n4\n2\n0\nIntegral of P( ) \n(arb. units)\n300 200 100 0\nFluence (µJ/cm2)0.575\n0.570\n0.565Frequency (THz)1.0\n0.8\n0.6\n0.4\n0.2\n0.0Intensity P( ) (arb. units)\n0.60 0.58 0.56 0.54\nFrequency (THz)  50%\n   100%\n 100% (x 3.7)\n (center)\n \n  10%\n |Bnr|2\n \nz \nm2 m1 \nM \nx Bnr \ny (a) (c) \n(d) (b) -0.020.000.02∆(degrees)\n40 30 20 10 0\nTime (ps)1.0\n0.5\n0.0\n-0.5B (T)  Bnr\n  Bin (x 3)\n0.08\n0.06\n0.04\n0.02\n0.00∆(degrees)\n24 20 16 12 8\nTime (ps)10%100%\nFigure 2. (a) Upper panel: Incident magnetic field of the THz pump pulse Bin estimated by\nelectro-optic sampling (dashed line) and the THz magnetic near-field Bnr calculated by the\nFDTD method (solid line). The illustration s hows the magnetization motion for the AF-mode.\nLower panel: The MOKE signal for a pump fluence of 292 µJ/cm2 (100%). (b) Comparison o f\ntwo MOKE signals for different pump fluences, vertically offset for clarity. (c) The FFT powe r\nspectrum of the magnetic near-field Bnr (black solid line). The spectra P() of the MOKE\nsignals for a series of pump fluences obtained at th e corner (solid lines) and at the center (blue\ndashed circle in the inset of figure 1) for a pump fluence of 100% (dashed line). Each spectru m\nof the MOKE signal is normalized by the peak amplitude at the corner for a pump fluence o f\n100%. (d) Intensity dependence of the center-of-m ass frequency (open circles) and the integral\n(closed circles) of the P().  7(t) is calibrated to the magnetization change (t) by using a linear relation, i.e., (t)=g(t), \nwhere g (=17.8 degrees−1) is a conversion coefficient. The tim e resolved experiment enables us \nto separate the contributions of the applied magnetic field and magnetization change to the \nfrequency shift in the time domain. A comparison of the temporal profiles between the driving \nmagnetic field (figure 3(a)) and the frequency e volution (figure 3(c)) shows that for the low \npump fluence (10%, closed blue circles), the frequency is redshifted only when the magnetic \nfield persists ( t < 25 ps), and after that, it recovers  to the constant AF mode frequency \n(νAF0=0.575 THz). This result indicates that the signals below t = 25 ps are affected by the \npersisting driving field and the redshift may orig inate from the forced oscillation. As long as the \n0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz)\n50 40 30 20 10 0\nTime (ps)-0.4-0.20.00.20.4Magnetization \nchange 0.5\n0.0\n-0.5Bnr (T)\n0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0  Experiment\n100%\n  10%\n  Experiment\n100%\nSimulation\n100% (1=0)\n100% \n  10% Simulation(a) \n(b) \n(c) \n(d) \nFigure 3. (a) FDTD calculated magnetic field Bnr for pump fluence of 100%. (b) Temporal\nevolution of the magnetization change obtained from the experimental data (gray circles) an d\nthe LLG model (red line). (c) Instantaneous frequencies and (d) envelope amplitudes fo r\npump fluences of 100% and 10% obtained by the analytic signals calculated from the\nexperimental data (circles) and the LLG simulation with nonlinear damping paramete r\n(1=1×10−3, solid lines) and without one ( 1=0, dashed line).  8magnetic response is under the linear regime, the instantaneous frequency is independent on the \npump fluence. However, for the high pump fluenc e (100%) a redshift (a maximum redshift of \n~15 GHz relative to the constant frequency νAF0) appears in the delay time ( t < 25 ps) and even \nafter the driving field decays away ( t > 25 ps) the frequency continues to be redshifted as long \nas the amplitude of the magnetization change is large. These results suggest that the frequency \nredshift in the high intensity case depends on the magnitude of the magnetization change, \nimplying that its origin is a nonlinear precessional spin motion with a large amplitude. \n \nThe temperature increase due to  the THz absorption (for HoFeO 3 T=1.7×10−3 K, for gold \nSRR T=1 K) is very small (see Appendix D). In ad dition, the thermal relaxation of the spin \nsystem, which takes more than a nanosecond [36], is much longer than the frequency \nmodulation decay (~50 ps) in figure 3(c). Therefore, laser heating can be ignored as the origin of the redshift.  \n \nFigure 4 shows a parametric plot of the instantaneous frequency \n(t) and envelope amplitude \n0(t) for the high pump fluence (100%). The instantaneous frequency shift for t > 25 ps has a \nsquare dependence on the amplitude, i.e., νAF=νAF0(1െCζ02). To quantify the relationship \nbetween the redshift and magnetization change, it  would be helpful to have an analytical \nexpression of the AF mode frequency AF as a function of the magnetization change, which is \nderived from the LLG equation based on the two- lattice model [32,33]. The dynamics of the \nsublattice magnetizations mi (i=1,2), as shown in the inset of figure 2(a), are described by \n \n dRi\ndt=െγ\n(1+α2)ቀRi×[B(t)+Beff,i]െαRi×൫Ri×[B(t)+Beff,i]൯ቁ,  (1) \n \nwhere Ri=mi/m0 (m0=|mi|) is the unit directional vector of the sublattice magnetizations, \n=1.76×1011 s−1T−1 is the gyromagnetic ratio, V(Ri) is the free energy of the iron spin system \nnormalized with m0, and Beff,i is the effective magnetic field given by െ∂V/∂Ri (i=1,2) (see \nAppendix E). The second term represents the ma gnetization damping with the Gilbert damping \nconstant \u001f \n \nSince Beff,i depends on the sublattice magnetizations mi and the product of these quantities \nappears on the right side of Eq. (1), the LLG e quation is intrinsically nonlinear. If the angle of  9the sublattice magnetization precession is sufficien tly small, Eq. (1) can be linearized and the \ntwo fixed AF- and F-modes for the weak excitation can be derived. However, as shown in figure \n3(b), the deduced maximum magnetization change  reaches ~0.4, corresponding to precession \nangles of 0.25° in the xz-plane and 15° in th e xy-plane. Thus, the magnetization change might \nbe too large to use the linear approximation. For such a large magnetization motion, assuming \nthe amplitude of the F-mode is zero and =0 in Eq. (1), the AF mode frequency AF in the \nnonlinear regime can be deduced as  \n \n νAF =νAF0ට1ିζ02tan2β0\nK(D),       ( 2 )  \n D =ඨ\t\t\t\t\t\tζ02(rAF2ି1) tan2β0\n1ିζ02tan2β0,      ( 3 )  \n \n0.575\n0.570\n0.565Frequency (THz)\n0.4 0.3 0.2 0.1 0.0\nAmplitude 0Experiment\n t > 25 ps\n t < 25 ps\n \n Analytic Solution\n 2nd order expansion\n \nFigure 4.  Relation between instantaneous frequency  and envelope amplitude 0 obtaine d\nfrom the magnetization change; for t < 25 ps (open circles) and for t > 25 ps (closed circles),\nthe analytic solution (blue line) and second orde r expansion of the analytic solution (gree n\ndashed line). Errors are estim ated from the spatial inhomogeneity of the driving magnetic\nfield (see Appendix H).  10where K(D) is the complete elliptic integral of the first kind, rAF(≈60) is the ellipticity of the \nsublattice magnetization precession trajectory of the AF-mode (see Appendix F), and 0 is the \namplitude of the (t). As shown in figure 4, the analytic solution can be approximated by the \nsecond order expansion νAF≈νAF0(1െtan2β0(rAF2െ1)ζ024⁄) and matches the observed redshift \nfor t > 25 ps, showing that the frequency appr oximately decreases with the square of (t). The \ndiscrepancy of the experimental data from the theoretical curve ( t < 25 ps) may be due to the \nforced oscillation of the AF-mode caused by the driving field.  \n \nTo elaborate the nonlinear damping effects, we compared the measured (t) with that \ncalculated from the LLG equation with the damping term. As shown in figures 3(c) and 3(d), the \nexperiment for the high intensity excitation devi ates from the simulation with a constant Gilbert \ndamping (dashed lines) even in the t > 25 ps time region, suggesting nonlinear damping \nbecomes significant in the large amplitude region. To describe the nonlinear damping \nphenomenologically, we modified the LLG equa tion so as to make the Gilbert damping \nparameter depend on the displacement of th e sublattice magnetization from its equilibrium \nposition, (Ri)=0+1Ri. As shown in figures 3(b)-(d), the magnetization change (t) derived \nwith Eq. (1) (solid line) with the damping parameters ( 0=2.27×10−4 and 1=1×10−3) nicely \nreproduces the experiments for both the high (100%) and low (10%) excitations.1 These results \nsuggest that the nonlinear damping plays a signifi cant role in the large amplitude magnetization \ndynamics. Most plausible mechanism for the nonlinear damping is four magnon scattering \nprocess, which has been introdu ced to quantitatively evaluate the magnon mode instability of \nferromagnet in the nonlinear response regime [37]. \n \n4. Conclusions  \nIn conclusion, we studied the nonlinear magnetization dynamics of a HoFeO 3 crystal excited \nby a THz magnetic field and measured by MOKE microscopy. The intense THz field can induce \nthe large magnetization change (~40%), and the ma gnetization change can be kept large enough \n                                            \n1 The damping parameter 0 (=2.27×10−4) and conversion coefficient  g (=17.8 degrees−1) are \ndetermined from the least-squares fit of the calculated result without the nonlinear damping \nparameter 1 to the experimental MOKE signal for the low pump fluence of 29.2 µJ/cm2. The \nnonlinear damping parameter 1 (=1×10−3) is obtained by fitting the experimental result for the \nhigh intensity case ( I=292 µJ/cm2) with the values of 0 and g obtained for the low excitation \nexperiment. The estimated value of g is consistent with the stat ic MOKE measurement; the Kerr \nellipticity induced by the spontaneous magnetization MS is ~0.05 degrees ( g~20 degrees−1). See \nAppendix G for details on the static Kerr measurement.  11to induce the redshift even after the field has gone , enabling us to separate the contributions of \nthe applied magnetic field and ma gnetization change to the frequency shift in the time domain. \nThe resonance frequency decreases in proportion to  the square of the magnetization change. A \nmodified LLG equation with a phenomenologi cal nonlinear damping term quantitatively \nreproduced the nonlinear dynamics. This suggest s that a nonlinear spin relaxation process \nshould take place in a strongly driven regime. Th is study opens the way to the study of the \npractical limits of the speed and efficiency of magnetization reversal, which is of vital \nimportance for magnetic recording and information processing technologies. \n   12Acknowledgments \nWe are grateful to Shintaro Takayoshi, Masah iro Sato, and Takashi Oka for their discussions \nwith us. This study was supported by a J SPS grants (KAKENHI 26286061 and 26247052) and \nIndustry-Academia Collaborative R&D grant fro m the Japan Science and Technology Agency \n(JST). \n   13Appendix A. Detection sche me of MOKE measurement \nWe show the details of the detection scheme  of the MOKE measurement. A probe pulse for \nthe MOKE measurement propagates along the z direction. By using the Jones vector [38], an electric field E\n0 of the probe pulse polarized linearly along the x-axis is described as  \n \n E0 =ቀ1\n0ቁ.       ( A . 1 )  \n \nThe probe pulse E1 reflected from the HoFeO 3 surface becomes elliptically polarized with a \npolarization rotation angle and a ellipticity angle . It can be written as \n \n E1 =R(െ߶)MR(θ)E0ൌ൬cos θ cos ߶െ\t݅ sin θ sin ߶\ncos θ sin ߶\t݅ sin θ cos ߶൰,  (A.2) \n  \nwhere  M is the Jones matrix describing \u001f\u001f phase retardation of the y component with \nrespect to the x component \n \nM=ቀ10\n0െiቁ,       ( A . 3 )  \n \nand R(ψ) is the rotation matrix \n \nR(ψ)=൬cosψ sinψ\nെsinψcosψ൰.      (A.4)  \n \nThe reflected light passes through the quarter wave plate, which is arranged such that its fast \naxis is tilted by an angle of 45° to the x-axis. The Jones matrix of the wave plate is given by \n \nRቀെπ\n4ቁMRቀπ\n4ቁ.        ( A . 5 )  \n Thus, the probe light E\n2 after the quarter wave plate is described as follows, \n  14E2 = ൬E2,x\nE2,y൰=Rቀെπ\n4ቁMRቀπ\n4ቁE1  \n=1\n2൬cosሺθ߶ሻെsin (θെ߶+)i(െcosሺθെ߶ሻsin (θ߶))\ncosሺθെ߶ሻsin (θϕ)+i(c o sሺθ߶ሻsin (θെ߶))൰.  (A.6)  \n \nThe Wollaston prism after the quarter wave plat e splits the x and y-polarization components of \nthe probe light E2. The spatially separated two pulses are incident to the balanced detector and \nthe detected probe pulse intensity ratio of the di fferential signal to the total corresponds to the \nKerr ellipticity angle as follows,    \n〈หாమ,ೣหమ〉ି〈หாమ,หమ〉\n〈หாమ,ೣหమ〉ା〈หாమ,หమ〉ൌെsin2θ.      ( A . 7 )  \n \nIn the main text, we show the Kerr ellipticity change =w−wo, where the ellipticity angles \n(w and wo) are respectively obtained with and without the THz pump excitation. \n \nAppendix B. Analytic signal approach and short time Fourier transform \nThe Analytic signal approach (ASA) allows the extraction of the time evolution of the \nfrequency and amplitude by a simple procedure and assumes that the signal contains a single \noscillator component. In our study, we measure only the MOKE signal originating from the \nAF-mode and it can be expected that the single oscillator assumption is valid. In the ASA, the \ntime profile of the magnetization change (t) is converted into an analytic signal (t), which is a \ncomplex function defined by using the Hilbert transform [39]; \n \nψ(t)=ζ0(t)exp( i߶(t))=ζ(t)+i ζ෨(t),     (B.1) \nζ෨(t)ൌ1\nπ pζ(t)\ntିτ∞\n-∞ dτ.       ( B . 2 )  \n \nwhere the p is the Cauthy principal value. The real part of (t) corresponds to (t). The real \nfunction 0(t) and (t) represent the envelope amplitude  and instantaneous phase of the \nmagnetization change. The instantaneous frequency (t)(=2(t)) is given by (t)=d(t)/dt. In \nthe analysis, we averaged 0(t) and (t) over a ten picosecond time range. \n \nTo confirm whether the ASA gives appropriate results, as shown in figure B.1 we compare  15them with those obtained by the short time Fourie r transform (STFT). As shown in figure B.1(a), \nthe time-frequency plot shows only one oscillato ry component of the AF-mode. As shown in \nfigures B.1(b) and (c), the instantaneous freque ncies and amplitudes obtained by the ASA and \nthe STFT are very similar. Because the ASA provides us the instantaneous amplitude with a \nsimple procedure, we showed the time evolu tions of frequency and amplitude derived by the \nASA in the main text. \n \nAppendix C. Determination of conversion coefficient g and linear damping parameter 0 \nThe conversion coefficient  g and the linear damping parameter 0(=) in Eq. (1) are \ndetermined by fitting the experimental MOKE signal (t) for the low pump fluence of 29.2 \nµJ/cm2 with the LLG calculation of the magnetization change (t). Figure C.1 shows the MOKE \nsignal (t) (circle) and the calculated magnetization change (t) (solid line). From the \nleast-squares fit of the calculated result to th e experiment by using a linear relation, i.e., \n(t)=g(t), we obtained the parameters g(=17.8 degrees−1) and 0(=2.27×10−4). 0.575\n0.570\n0.565\n0.560\n0.555\n0.550Frequency (THz)\n50 40 30 20 100\nTime (ps)ASA\n 100%\n   10%\nSTFT\n 100%\n   10%\n 1.0\n0.8\n0.6\n0.4\n0.2\n0.0\nFourier am plitude (arb. units)\n50 40 30 20 100\nTime (ps)0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0ASA\n   100%\n     10% \nSTFT\n   100%\n     10%\n (a) (b) (c) \n1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Frequency (THz)\n5040302010\nTime (ps)(arb. units)\n1.0 0.0\nFigure B.1.  (a) Time-dependence of the power spectrum of the magnetization \noscillation for the highest THz excitation ( I=292 µJ/cm2) obtained by the STFT. \nComparison of (b) instantaneous frequencies and (c) amplitudes obtained by the ASA \nand STFT with a time window with FWHM of 10 ps.   16 \n \nAppendix D. Laser heating effect \nThe details of the calculation of the temperature change are as follows: \n \nFor HoFeO 3:  \nThe absorption coefficient abs of HoFeO 3 at 0.5 THz is ~4.4 cm−1 [40]; the fluence IHFO \nabsorbed by HoFeO 3 can be calculated as IHFO=I(1−exp(−absd)), where d (=145 µm) is the \nsample thickness and I is the THz pump fluence. For the highest pump fluence, I=292 µJ/cm2, \nIHFO is 18.1 µJ/cm2. Since the sample thickness is much smaller than the penetration depth, \nd≪abs−1, we assume that the heating of the sample due to the THz absorption is homogeneous. \nBy using the heat capacity Cp of 100 J mol−1 K−1 [27], and the molar volume v of ~1.4×102 \ncm3/mol [27], the temperature change T can be estimated as\u001f T=IHFOv/Cpd ~1.7×10−3 K. \n \nFor gold resonator (SRR): \nThe split ring resonator has an absorption band (center frequency ~0.56 THz, band width ~50 \nGHz) originated from the LC resonance (figure 2( c)). Assuming the SRR absorbs all incident \nTHz light in this frequency band, the absorbed energy accounts for 3 % of the total pulse energy. \nHence, for the highest THz pump fluence, I=292 µJ/cm2, the fluence absorbed by the SRR is \nIgold=8.76 µJ/cm2. By using the heat capacity Cp of 0.13 J g−1 K−1 [41], the number of the SRRs \nper unit area N of 4×104 cm−2, and the mass of the SRR m of 1.6×10−9 g, the temperature change -10x10-3-50510degrees)\n50 40 30 20 10 0\nTime (ps)-0.10.00.1\nMagnetization \nchange  Experiment\n Simulation\nFigure C.1.  Experimentally observed MOKE signal \u001f(circle) and LLG simulatio n\nresult of the magnetization change \u001f(solid line) for the pump fluence of 29.2\nµJ/cm2.  17T can be estimated as\u001f T=Igold/CpNm ~ 1 K \n \nAppendix E. Free energy of HoFeO 3 \nThe free energy F of the iron spin (Fe3+) system based on the two-lattice model is a function \nof two different iron sublattice magnetizations mi, and composed of the exchange energy and \none-site anisotropy energy [32,33]. The free en ergy normalized by the sublattice magnetization \nmagnitude, V=F/m0 (m0=|mi|), can be expanded as a power series in the unit directional vector of \nthe sublattice magnetizations, Ri=mi/m0=(Xi,Yi,Zi). In the magnetic phase 4 (T > 58K), the \nnormalized free energy is given as follows [32,33]: \n \nV=ER1·R2+D(X1Z2െX2Z1)െAxx(X12+X22)െAzz(Z12+Z22),  (E.1) \n \nwhere E(=6.4×102 T) and D(=1.5×10 T) for HoFeO 3 are respectively the symmetric and \nantisymmetric exchange field [42]. Axx and Azz are the anisotropy constants. As mentioned in \nAppendix F, the temperature dependent values of the anisotropy constants can be determined \nfrom the antiferromagnetic resonance frequencies. The canting angle of Ri to the x-axis β0 \nunder no magnetic field is given by \n \ntan 2β0=D\nE+AxxିAzz.        ( E . 2 )  \n \nAppendix F. Linearized resonance modes and anisotropy constants ( Axx and Azz) \nThe nonlinear LLG equation of Eq. (1) can be  linearized and the two derived eigenmodes \ncorrespond to the AF and F-mode. The sublatti ce magnetization motion for each mode is given \nby the harmonic oscillation of mode coordinates; for the AF-mode ( QAF, \nPAF)=((X1−X2)s i nβ0+(Z1+Z2)c o sβ0, Y1−Y2), and for the F-mode ( QF, \nPF)=((X1+X2)sinβ0−(Z1−Z2)cosβ0, Y1+Y2), \n \nQAF=AAFcosωAFt,       ( F . 1 )  \nPAF=AAFrAFsinωAFt,      ( F . 2 )  \n \nQF=AFcosωFt,       ( F . 3 )  \nPF=AFrAFsinωFt,       ( F . 4 )   18 \nwhere AAF,F represents the amplitude of each mode. AF,F, and rAF,F are the resonance frequencies \nand ellipticities, which are given by    \n \nωAF=γට(b+a)(d-c),        ( F . 5 )  \nωF=γට(b-a)(d+c),        ( F . 6 )  \n rAF=γටሺௗିሻ\n(b+a),        ( F . 7 )  \n rF=γටሺௗାሻ\n(b-a),        ( F . 8 )  \n \nwhere =1.76×1011 s−1T−1 is the gyromagnetic ratio, and \n \n a=െ2Axxcos2β0െ2Azzsin2β0െEcos 2β0െDsin 2β0,  (F.9) \n b=E,         ( F . 1 0 )  \n c=2Axxcos2β0െ2Azzcos2β0+Ecos 2β0+Dsin 2β0,    (F.11) \n d=െEcos 2β0െDsin 2β0.       ( F . 1 2 )  \n \nSubstituting the literature values of the exchange fields ( E=6.4×102 T and D=1.5×10 T [42]) and \nthe resonance frequencies at room temperature ( AF/2=0.575 THz and F/2=0.37 THz) to \nEqs. (F.5) and (F.6), Axx and Azz can be determined to 8.8×10−2 T and 1.9×10−2 T. \n \nAppendix G. MOKE measurement for the spontaneous magnetization \nFigure G.1 shows time-development of the MOKE signals for the different initial condition \nwith oppositely directed magnetization. We applied the static magnetic field (~0.3 T) to saturate \nthe magnetization along the z-axis before the TH z excitation. The spontaneous magnetization of \nsingle crystal HoFeO a can be reversed by the much smaller magnetic field (~0.01 T) because of \nthe domain wall motion [27]. Then, we separately measured the static Kerr ellipticity angle \n\u001f\u001f\u001f\u001f\u001f\u001f  and THz induced ellipticity change  for different initial magnetization Mz=±Ms \nwithout the static magnetic field \u001f In figure G.1 we plot the summation of the time resolved \nMOKE signal \u001fand the static Kerr ellipticity \u001f\u001f\u001f\u001f\u001f\u001f  The sings of the ellipticity offset angle  19\u001f\u001f\u001f\u001f\u001f\u001f  for the different spontaneous magnetization (±M S) are different and their magnitudes \nare ~0.05 degrees. The conversion coefficient g(=1/~\u001f/0.05 degrees) is estimated to be ~20 \ndegrees−1, which is similar to the value dete rmined by the LLG fitting (~17.8 degrees−1). In the \ncase of the AF-mode excitation, the phases of the magnetization oscillations are in-phase \nregardless of the direction of the spontaneous magnetization M=±Ms, whereas they are \nout-of-phase in the case of the F-mode excitation. We can explain this claim as follows: In the \ncase of AF-mode excitation, the external THz magne tic field is directed along the z-direction as \nshown in the inset of figure 2(a), the signs of the torques acting on the sublattice magnetization \nmi (i=1,2) depends on the direction of mi, however, the resultant oscillation of the macroscopic \nmagnetization M= m1+m2 along the z-direction has same phase for the different initial condition \nM=±Ms. In the case of the F-mode excitation with the external THz magnetic field along the x \nor y-direction, the direction of the torques acting on the magnetization M depends on the initial \ndirection and the phase of the F-mode osc illation changes depending on the sign of the \nspontaneous magnetization ±Ms. \n \nAppendix H. Influence of the spatial distri bution of magnetic field on magnetization \nchange \nAs shown in the inset of figure 1, the pump magnetic field  strongly localizes near the metallic \narm of the SRR and the magnetic field strength significantly depends on the spatial position r \nwithin the probe pulse spot area. The intensity distribution of the probe pulse Iprobe(r) has an 0.05\n0.00\n-0.05Kerr ellipticity  (degrees)\n25 20 15 10 5\nTime (ps) +MS\n -MS\n \nFigure G.1. The MOKE signals, the temporal change of the Kerr ellipticity , measured \nfor different initial conditions with oppositely directed magnetizations.       20elongated Gaussian distribution with spatial widt hs of 1.1 µm along the x-axis and 1.4 µm along \nthe y-axis [full width at half maximum (FWHM)  intensity]. The maximum magnetic field is 1.2 \ntimes larger than the minimum one in the spot  diameter, causing the different magnetization \nchange dynamics at different positions. To take into account this spatial inhomogeneity to the \nsimulation, the spatially weighted average of magnetization change ζ̅(t) has to be calculated as \nfollows:  \n \n ζ̅(t)=ζ(r,t)Iprobe(r)dr\nIprobe(r)ௗr ,       ( H . 1 )  \n \nwhere (r,t) is a magnetization change at a position r and time t.  \n \nFigure H.1(a) shows the simulation result of the spatially averaged magnetization change ζ̅(t) \nand the non-averaged (r0,t) without the nonlinear damping term ( 1=0), where r0 denotes the \npeak position of Iprobe(r). For the low excitation intensity (10%), ζ̅(t) is almost the same as \n(r0,t) as shown in figure H.1(a). On the other hand, for the high excitation intensity, the spatial \ninhomogeneity of magnetization change dyna mics induces a discrepancy between the ζ̅(t) and (a) (b) (c) \n-0.100.000.10Magnetization change\n50 40 30 20 100\nTime (ps)-0.6-0.4-0.20.00.20.4 Averaged\n Non-averaged\n Averaged\n Non-averaged100%  10%\n0.575\n0.570\n0.565\n0.560\n0.55550 40 30 20 100\nTime (ps)0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz) Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\nExperiment\n100%  10%\n0.4\n0.3\n0.2\n0.1\n0.0\n50 40 3020 100\nTime (ps)0.12\n0.08\n0.04\n0.00Amplitude   Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\n Experiment100%  10%\nFigure H.1.  Comparison of the spatially averag ed and non-averaged magnetization \nchange for the different pump fluences of 10% and 100%. (a) Temporal evolutions of \nthe magnetization change, (b) instantaneous  frequencies and (c) normalized envelope \namplitudes. Open circles show the experimental results.      21(r0,t). This discrepancy is caused by the quasi-i nterference effect between the magnetization \ndynamics with different frequencies and amplit udes at different positions. Figures H.1(b) and \n(c) show the instantaneous freque ncy and envelope amplitude obt ained from the data shown in \nfigure H.1(a) by using analytic signal approach with the experimental result. For the averaged \nmagnetization change, the frequency redshift is more emphasized (figure H.1(b)) and the decay \ntime becomes shorter (figure H.1(c)). Nonetheless,  neither spatially averaged nor non-averaged \nsimulation reproduces the experimental result of  the instantaneous frequency (figure H.1(b)) \nwithout nonlinear damping term.  \n \n   22References \n[1] Kirilyujk A, Kimel A V and Rasing T 2010 Rev. Mod. Phys.  82 2731 \n[2] Beaurepaire E, Merle J-C, Daunois A and Bigot J-Y 1996 Phys. Rev. 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Phys. Soc. Jpn.  57 4418 "    },    {        "title": "1802.10494v3.Global_in_time_Stability_of_2D_MHD_boundary_Layer_in_the_Prandtl_Hartmann_Regime.pdf",        "content": "arXiv:1802.10494v3  [math.AP]  7 Jul 2018Global-in-time stability of 2D MHD boundary layer in the\nPrandtl-Hartmann regime\nFeng Xie\nSchool of Mathematical Sciences, and LSC-MOE,\nShanghai Jiao Tong University, Shanghai 200240, P.R.China\nTong Yang\nDepartment of Mathematics, City University of Hong Kong,\nTat Chee Avenue, Kowloon, Hong Kong;\nand School of Mathematical Sciences\nShanghai Jiao Tong University, Shanghai 200240, P.R.China\nAbstract: In this paper, we prove global existence of solutions with an alytic regularity to\nthe 2D MHD boundary layer equations in the mixed Prandtl and H artmann regime derived\nby formal multi-scale expansion in [10]. The analysis shows that the combined effect of the\nmagnetic diffusivity and transveral magnetic field on the boun dary leads to a linear damping\non the tangential velocity field near the boundary. And this d amping effect yields the global in\ntime analytic norm estimate in the tangential space variabl e on the perturbation of the classical\nsteady Hartmann profile.\n2000 Mathematical Subject Classification : 76N20, 35Q35, 76W05, 35M33.\nKeywords : MHD boundary layer, Prandtl-Hartmann regime, global stab ility, analytic regular-\nity.\n1 Introduction\nThe following mixed Prandtl and Hartmann boundary layer equ ations from the classical in-\ncompressible MHD system were derived in [10] for flat boundar y in two space dimensions (2D)\nwhen the physical parameters such as Reynolds number, magne tic Reynolds number and the\nHartmann number satisfy some constraints in the high Reynol ds numbers limit:\n\n\n∂tu1+u1∂xu1+u2∂yu1=∂yb1+∂2\nyu1,\n∂yu1+∂2\nyb1= 0,\n∂xu1+∂yu2= 0, x∈R, y∈R+.(1.1)\nHere, (u1,u2) denotes the velocity field of the boundary layer and b1is the corresponding tan-\ngential magnetic component.\n1E-mail address: tzxief@sjtu.edu.cn (F. Xie), matyang@cityu.edu.hk (T. Ya ng)\n1For the classic Hartmann bounday layer system, there is a fam ily of steady solutions called\nHartmann layer. It turns out that Hartmann layer is also a sol ution to the above system. In this\npaper, we will study the global-in-time stability of the Har tmann layer in the analytic function\nspace. For this, consider the system (1 .1) with initial data\nu1(t= 0,x,y) =u10(x,y), (1.2)\nand the no-slip boundary conditions\nu1|y=0= 0, u2|y=0= 0. (1.3)\nAnd the far field is taken as a uniform constant state. Consequ ently, the pressure term vanishes\nin equation of (1 .1)1. Denote\nlim\ny→+∞u1= ¯u,lim\ny→+∞b1=¯b. (1.4)\nIntegrating the equation of (1 .1)2overyyields\n−u1(t,x,y)+ ¯u=∂yb1. (1.5)\nThus, the equations (1.1) can be written as\n/braceleftbigg∂tu1+u1∂xu1+u2∂yu1=−u1(t,x,y)+ ¯u+∂2\nyu1,\n∂xu1+∂yu2= 0.(1.6)\nRecall that the classical Hartmann boundary layer is given b y\nu1= (1−e−y)¯u, u 2= 0, (1.7)\nwhich is a steady solution to (1.6). Without loss of generali ty, set ¯u= 1 and denote the\nperturbation by ( u,v):\nu1= (1−e−y)+u, u 2=v. (1.8)\nObviously, ( u,v) satisfies\n/braceleftbigg∂tu+(1−e−y+u)∂xu+v∂y(−e−y+u) =−u+∂2\nyu,\n∂xu+∂yv= 0,(1.9)\nwith initial data\nu0(x,y) =u10(x,y)−(1−e−y), (1.10)\nand boundary condition\nu|y=0= 0, v|y=0= 0. (1.11)\n2Before we state the main result, let us first introduce the fol lowing weighted analytic regu-\nlarity function spaces. For some r >1, denote an analytic weight Mmby\nMm=(m+1)r\nm!.\nWith a parameter τ >0, set\nXm=/bardbleαy∂m\nxg/bardblL2(R2\n+)τmMm, Zm=/bardbleαy∂y∂m\nxg/bardblL2(R2\n+)τmMm, (1.12)\nYm=/bardbleαy∂m\nxg/bardblL2(R2\n+)τm−1/2m1/2Mm, Dm=/bardbleαy∂m\nxg/bardblL∞yL2xτmMm,(1.13)\nand define\n/bardblg/bardbl2\nXrτ,α=/summationdisplay\nm≥0X2\nm,/bardblg/bardbl2\nZrτ,α=/summationdisplay\nm≥0Z2\nm, (1.14)\n/bardblg/bardbl2\nYrτ,α=/summationdisplay\nm≥0Y2\nm,/bardblg/bardbl2\nDrτ,α=/summationdisplay\nm≥0D2\nm. (1.15)\nHere,τdenotes the analytic radius.\nTheorem 1.1 Let the initial data u10(x,y)be a small perturbation of the Hartmann profile\n(¯u(1−e−y),0)satisfying the compatibility conditions and\n/bardbl∂yu10+u10−¯u/bardblXrτ0,α≤δ0, (1.16)\nwithr >1,0< α <√\n2/2for some small constant δ0>0. Then there exists a unique global-in-\ntime solution (u1,u2)to the problem (1.1)-(1.4) satisfying\n/bardblg/bardblXr\nτ(t),α≤e−2(1−2α2)tδ0,withτ(t)> τ0/2,\nfor all time t≥0, whereg=∂yu1+u1−¯u.\nRemark 1.1 The initial analytic radius τ0and the size of initial perturbation δ0should satisfy\nthe constraint of (3.19).\nNow let us briefly review the background and some related work s. First of all, the Prandtl\nequations derivedby Prandtl [27] in1904 describethefluidp henomenanear aboundarywithno-\nslip boundary condition through the high Reynolds number li mit of the incompressible Navier-\nStokes equations. For this system, so far the mathematical t heories are basically limited to\ntwo space dimensions except in the framework of analytic or G evrey function spaces or under\nsome structure constraints. And the justification of the Pra ndtl ansatz has been extensively\ninvestigated with only a few results, cf. [14, 22, 28, 29] and references therein. In fact, under the\nmonotonicity condition on the tangential velocity compone nt in the normal direction, Oleinik\nfirstly obtained the local existence of classical solutions in the two space dimensions by using\nthe Crocco transformation, cf. [25]. This result together w ith some other works in this direction\nare presented in Oleinik-Samokhin’s classical book [26]. R ecently, this well-posedness result\n3was re-proved by using energy method in the framework of weig hted Sobolev spaces in [1] and\n[23] independently, by observing the cancellation mechani sm in the system. By imposing an\nadditional favorable pressure condition, a global in time w eak solution was obtained by Xin and\nZhang in [30].\nWhen this monotonicity condition is violated, seperation o f the boundary layer is expected.\nFor this, E-Engquist constructed a finite time blowup soluti on to the Prandtl equations in [5].\nAnd this kind of blowup result is extended to the van Dommelen -Shen type singularity in [16]\nwhen the outer Euler flow is spatially periodic. In addition, some interesting ill-posedness (or\ninstability) phenomena of solutions to both the linear and n onlinear Prandtl equations around\na shear flow have been studied, cf. [7, 8, 12, 13] and the survey paper [6].\nIn the framework of analytic functions, Sammartino and Cafli sch [28, 29, 3] proved the local\nwell-posedness result of the Prandtl system and justified th e Prandtl ansatz. The analyticity\nrequirement in thenormal variable ywas later removed by Lombardo, CannoneandSammartino\nin [21]. The main argument used in [21, 28] is to apply the abst ract Cauchy-Kowalewskaya\ntheorem. For recent development of mathematical theories i n Gevrey function space to Prandtl\nequations, one can refer to [9, 17].\nA natural question then is whether global existence of smoot h (or strong) solution can\nbe achieved for the Prandtl equations in either analytic or G evrey regularity function spaces.\nHowever, the answer to this is still not known. In this direct ion, a lower bound of the life-span\nfor small analytic solution to the Prandtl equations with sm all perturbation analytic initial data\nwas given by Zhang ang Zhang in [31]. Precisely, when the outfl ow velocity is of the order of\nε5/3, and the initial perturbation is of the order of ε, then the Prandtl system admits a unique\nanalytic solution with life-span greater than ε−4/3. On the other hand, when the initial data\nis a small perturbation of a Guassian error function, almost global existence for the Prandtl\nequations is obtained by Ignatova and Vicol in [15], where th e cancellation observed in [23]\nand the monotonicity of background solution are essentiall y used to have a linear time decay\ndamping effect.\nBack to the MHD system, it is believed that suitable magnetic field can stabilize the bound-\nary layer in some physical regime [24, 2, 4, 11]. One can also r efer to some very recent results\nin [10, 18, 19, 20] for the derivation of MHD boundary layer eq uations, stability analysis of\nmagnetic field on the boundary layer from the mathematical po int of view.\nThe purpose of this paper to show a global-in-time existence of solution to a mixed Prandtl\nand Hartmann MHD boundary layer equations derived in [10]. T he key observation is that the\ncombined effect of the magentic diffusivity and transveral magn etic field on the boundary leads\nto a linear damping on the tangential velocity field near the b oundary. This damping effect\nyields a time exponential decay in analytic regularity norm of the solution when we consider\nperturbation near the classical Hartmann profile.\nFinally, the rest of the paper is organized as follows. We wil l reformulate the problem\nby using the cancellation mechanism observed in [1, 23] for t he study of Prandtl equations in\nSobolev space in Section 2. The uniform estimates on the solu tion in analytic norm will be given\nin Section 3. Based on the uniform estimates, the global exis tence and uniqueness of solution\nto (1.1) will be proved in the last section. Throughout the pa per,C,¯C,C0andC1are used to\ndenote some generic constants.\n42 Preliminaries\nAs for the classical Prandtl equation, one needs to use the ca ncellation mechanism in the system\nto overcome the loss of derivative in order to close the a prio ri estimate. For this, note that the\nvorticity in the boundary layer ω=∂yusatisfies\n∂tω+(1−e−y+u)∂xω−ve−y+v∂yω=−ω+∂2\nyω. (2.1)\nAsin[1,23], onecanusethevorticity tocancel sometermwit hessential difficultyintheequation\nforu. Precisely, by noticing the Hartmann layer ( u1,0) has the propertyu1yy\nu1y=−1, set\ng=ω+u, (2.2)\nthen the new unknown function gsatisfies\n∂tg+(1−e−y+u)∂xg+v∂yg=−g+∂2\nyg. (2.3)\nThe relation between the new unknown function ganduis\nu=e−y/integraldisplayy\n0ezg(t,x,z)dz, (2.4)\nand the initial data of gis\ng(0,x,y) =∂yu10(x,y)+u10(x,y)−1. (2.5)\nAs for the boundary condition on g, note that from (1.9) and (1.11), we have\n(∂yω−u)|y=0= 0,\nwhich implies\n(∂yg−g)|y=0= 0. (2.6)\nIn the rest of the paper, we consider the reformulated proble m (2.3)-(2.6).\nRemark 2.1 If/bardblg/bardblXrτ,α<∞, then it follows from (2.4) that /bardblu/bardblXr\nτ,α′<∞and/bardblu/bardblZr\nτ,α′<∞\nwith0≤α′< α. In addition, /bardblu/bardblDrτ,α<∞.\n3 Uniform Estimates\nIn this section, we will derive the uniform estimates on the s olution to (2.3)-(2.6) in analytic\nregularity norm through energy method.\nApplying the operator ∂m\nxon (2.3), multiplying the resulting equation by e2αy∂m\nxgand\nintegrating it over R2\n+yield\n/integraldisplay\nR2\n+∂m\nx(∂tg+(1−e−y+u)∂xg+v∂yg+g−∂2\nyg)e2αy∂m\nxgdxdy= 0. (3.1)\n5We estimate the above equation term by term. Firstly,\n/integraldisplay\nR2\n+∂m\nx∂tge2αy∂m\nxgdxdy=1\n2d\ndt/bardbleαy∂m\nxg/bardbl2\nL2, (3.2)\n/integraldisplay\nR2\n+∂m\nxge2αy∂m\nxgdxdy=/bardbleαy∂m\nxg/bardbl2\nL2, (3.3)\nand\n−/integraldisplay\nR2\n+∂2\ny∂m\nxge2αy∂m\nxgdxdy\n=/integraldisplay\nR∂y∂m\nxg(t,x,0)∂m\nxg(t,x,0)dx+/bardbleαy∂y∂m\nxg/bardbl2\nL2+2α/integraldisplay\nR2\n+∂y∂m\nxge2αy∂m\nxgdxdy\n=/bardbl∂m\nxg(t,x,0)/bardbl2\nL2x+/bardbleαy∂y∂m\nxg/bardbl2\nL2−α/bardbl∂m\nxg(t,x,0)/bardbl2\nL2x−2α2/bardbleαy∂m\nxg/bardbl2\nL2\n=(1−α)/bardbl∂m\nxg(t,x,0)/bardbl2\nL2x+/bardbleαy∂y∂m\nxg/bardbl2\nL2−2α2/bardbleαy∂m\nxg/bardbl2\nL2, (3.4)\nwhere in the second equality, we have used the boundary condi tion (2.6).\nFor the two mixed nonlinear terms in (3.1), we firstly have\n/integraldisplay\nR2\n+∂m\nx((1−e−y+u)∂xg)e2αy∂m\nxgdxdy\n=m/summationdisplay\nj=0(m\nj)/integraldisplay\nR2\n+∂m−j\nxu∂j+1\nxge2αy∂m\nxgdxdy/definesR1,\nand\n|R1| ≤[m/2]/summationdisplay\nj=0(m\nj)/bardbl∂m−j\nxu/bardblL2xL∞y/bardbleαy∂j+1\nxg/bardblL∞xL2y/bardbleαy∂m\nxg/bardblL2\n+m/summationdisplay\nj=[m/2]+1(m\nj)/bardbl∂m−j\nxu/bardblL∞xy/bardbleαy∂j+1\nxg/bardblL2/bardbleαy∂m\nxg/bardblL2.\nFor 0≤j≤[m/2], by (2.4), one has\n/bardbl∂m−j\nxu/bardblL2xL∞y=/bardbl∂m−j\nx/integraldisplayy\n0e−(y−z)g(t,x,z)dz/bardblL2xL∞y\n≤/bardbleαy∂m−j\nx/integraldisplayy\n0e−(y−z)g(t,x,z)eαze−αzdz/bardblL2xL∞y\n=/bardbl/integraldisplayy\n0e−(1−α)(y−z)∂m−j\nxg(t,x,z)eαzdz/bardblL2xL∞y≤C/bardbleαy∂m−j\nxg/bardblL2,\nprovided that α <1. Using the Agmon inequality gives that\n/bardbleαy∂j+1\nxg/bardblL∞xL2y≤C/bardbleαy∂j+1\nxg/bardbl1/2\nL2/bardbleαy∂j+2\nxg/bardbl1/2\nL2.\n6For [m/2]+1≤j≤m,\n/bardbl∂m−j\nxu/bardblL∞xy=/bardbl∂m−j\nx/integraldisplayy\n0e−(y−z)g(t,x,z)dz/bardblL∞xy\n≤/bardbleαy∂m−j\nx/integraldisplayy\n0e−(y−z)g(t,x,z)eαze−αzdz/bardblL∞xy\n=/bardbl/integraldisplayy\n0e−(1−α)(y−z)∂m−j\nxg(t,x,z)eαzdz/bardblL∞xy\n≤C/bardbleαy∂m−j\nxg/bardblL2yL∞x≤C/bardbleαy∂m−j\nxg/bardbl1/2\nL2/bardbleαy∂m−j+1\nxg/bardbl1/2\nL2.\nConsequently,\n/summationdisplay\nm≥0|R1|τ2mM2\nm\n≤C\n(τ(t))1/2\n\n/summationdisplay\nm≥0[m/2]/summationdisplay\nj=0Xm−jY1/2\nj+1Y1/2\nj+2Ym(m\nj)Mm\nMm−jM1/2\nj+1M1/2\nj+2(j+1)1/4(j+2)1/4m1/2(3.5)\n+/summationdisplay\nm≥0m/summationdisplay\nj=[m/2]+1X1/2\nm−jX1/2\nm−j+1Yj+1Ym(m\nj)Mm\nM1/2\nm−jM1/2\nm−j+1Mj+1(j+1)1/2m1/2\n\n.\nThe second nonlinear term can be estimated as follows. Note t hat\n/integraldisplay\nR2\n+∂m\nx(v∂yg)e2αy∂m\nxgdxdy\n=m/summationdisplay\nj=0(m\nj)/integraldisplay\nR2\n+∂m−j\nxv∂j\nx∂yge2αy∂m\nxgdxdy/definesR2,\nand\n|R2| ≤[m/2]/summationdisplay\nj=0(m\nj)/bardbl∂m−j\nxv/bardblL2xL∞y/bardbleαy∂j\nx∂yg/bardblL∞xL2y/bardbleαy∂m\nxg/bardblL2\n+m/summationdisplay\nj=[m/2]+1(m\nj)/bardbl∂m−j\nxv/bardblL∞xy/bardbleαy∂j\nx∂yg/bardblL2/bardbleαy∂m\nxg/bardblL2.\nFor 0≤j≤[m/2],\n/bardbl∂m−j\nxv/bardblL2xL∞y=/bardbl/integraldisplayy\n0(e−z/integraldisplayz\n0es∂m−j+1\nxg(t,x,s)ds)dz/bardblL2xL∞y\n=/bardbl/integraldisplayy\n0e−αz(/integraldisplayz\n0e(1−α)(s−z)∂m−j+1\nxg(t,x,s)eαsds)dz/bardblL2xL∞y\n7≤C/bardbleαy∂m−j+1\nxg/bardblL2,\nwhere we have used the fact that 0 < α <1. Note that\n/bardbleαy∂j\nx∂yg/bardblL∞xL2y≤C/bardbleαy∂j\nx∂yg/bardbl1/2\nL2/bardbleαy∂j+1\nx∂yg/bardbl1/2\nL2.\nFor [m/2]+1≤j≤m,\n/bardbl∂m−j\nxv/bardblL∞=/bardbl/integraldisplayy\n0(e−z/integraldisplayz\n0es∂m−j+1\nxg(t,x,s)ds)dz/bardblL∞\n=/bardbl/integraldisplayy\n0e−αz(/integraldisplayz\n0e(1−α)(s−z)∂m−j+1\nxg(t,x,s)eαsds)dz/bardblL∞\n≤C/bardbleαy∂m−j+1\nxg/bardbl1/2\nL2/bardbleαy∂m−j+2\nxg/bardbl1/2\nL2.\nConsequently,\n/summationdisplay\nm≥0|R2|τ2mM2\nm\n≤C\n(τ(t))1/2\n\n/summationdisplay\nm≥0[m/2]/summationdisplay\nj=0Ym−j+1Z1/2\njZ1/2\nj+1Ym(m\nj)Mm\nMm−j+1M1/2\njM1/2\nj+1(m−j+1)1/2m1/2(3.6)\n+/summationdisplay\nm≥0m/summationdisplay\nj=[m/2]+1Y1/2\nm−j+1Y1/2\nm−j+2ZjYm(m\nj)Mm\nM1/2\nm−j+1M1/2\nm−j+2Mj(m−j+1)1/4(m−j+2)1/4m1/2\n\n.\nCombining the above estimates and using the definitions of Xr\nτ,α,Zr\nτ,αandYr\nτ,αgive\n1\n2d\ndt/bardblg/bardbl2\nXrτ,α−˙τ/bardblg/bardbl2\nYrτ,α+/bardblg/bardbl2\nZrτ,α+(1−2α2)/bardblg/bardbl2\nXrτ,α+(1−α)/bardblg(t,x,0)/bardbl2\nXrτ,α\n≤/summationdisplay\nm≥0|R1|τ2mM2\nm+/summationdisplay\nm≥0|R2|τ2mM2\nm. (3.7)\nSince\n(m\nj)Mm\nMm−jM1/2\nj+1M1/2\nj+2(j+1)1/4(j+2)1/4m1/2\n=(m+1)r(j+1)(j+2)1/2\n(m−j+1)r(j+2)r/2(j+3)r/2(j+1)1/4(j+2)1/4m1/2\n≤C(1+j)1/2−r(3.8)\nfor all 0≤j≤[m/2], and\n(m\nj)Mm\nM1/2\nm−jM1/2\nm−j+1Mj+1(j+1)1/2m1/2\n8=(m+1)r(m−j+1)1/2(j+1)1/2\n(m−j+1)r/2(m−j+2)r/2(j+2)rm1/2\n≤C(m−j+1)1/2−r, (3.9)\nfor all [m/2]+1≤j≤m. Consequently,\n/summationdisplay\nm≥0|R1|τ2mM2\nm≤C\n(τ(t))1/2\n\n/summationdisplay\nm≥0[m/2]/summationdisplay\nj=0Xm−jY1/2\nj+1Y1/2\nj+2Ym(1+j)1/2−r\n+/summationdisplay\nm≥0m/summationdisplay\nj=[m/2]+1X1/2\nm−jX1/2\nm−j+1Yj+1Ym(m−j+1)1/2−r\n\n\n≤C\n(τ(t))1/2/bardblg/bardblXrτ,α/bardblg/bardbl2\nYrτ,α. (3.10)\nNote that in the above second inequality, we have used the fol lowing discrete Young’s inequality\n/bardblζ·(η∗ξ)/bardbll1≤C/bardblζ/bardbll2/bardblη/bardbll1/bardblξ/bardbll2\nwithζk=Yk,ηk=Y1/2\nkY1/2\nk+1k1/2−r,ξk=Xkfor the first term on the right hand side in (3.10),\nand then H¨ older inequality for /bardblη/bardbll1≤C/bardblg/bardblYrτ,α, provided r >1. And for the second term on\nthe right hand side of (3.10), we can choose of ζk=Yk,ηk=X1/2\nkX1/2\nk+1(k+1)1/2−r,ξk=Yk+1\nwith/bardblη/bardbll1≤C/bardblg/bardblXrτ,α, provided r >1. In conclusion, (3.10) holds if r >1.\nSimilarly,\n/summationdisplay\nm≥0|R2|τ2mM2\nm≤C\n(τ(t))1/2\n\n/summationdisplay\nm≥0[m/2]/summationdisplay\nj=0Ym−j+1Z1/2\njZ1/2\nj+1Ym(1+j)1/2−r\n+/summationdisplay\nm≥0m/summationdisplay\nj=[m/2]+1Y1/2\nm−j+1Y1/2\nm−j+2ZjYm(m−j+1)1/2−r\n\n\n≤C\n(τ(t))1/2/bardblg/bardblZrτ,α/bardblg/bardbl2\nYrτ,α, (3.11)\nprovided that r >1.\nIt follows, from (3.7), (3.10) and (3.11), that\n1\n2d\ndt/bardblg/bardbl2\nXrτ,α+/bardblg/bardbl2\nZrτ,α+(1−2α2)/bardblg/bardbl2\nXrτ,α+(1−α)/bardblg(·,0)/bardbl2\nXrτ,α\n≤(˙τ+C\n(τ(t))1/2(/bardblg/bardblXrτ,α+/bardblg/bardblZrτ,α))/bardblg/bardbl2\nYrτ,α. (3.12)\nSet\n˙τ+C\n(τ(t))1/2(/bardblg/bardblXrτ,α+/bardblg/bardblZrτ,α) = 0. (3.13)\n9Then we have\n1\n2d\ndt/bardblg/bardbl2\nXrτ,α+/bardblg/bardbl2\nZrτ,α+(1−2α2)/bardblg/bardbl2\nXrτ,α+(1−α)/bardblg(·,0)/bardbl2\nXrτ,α≤0.(3.14)\nWhen 0< α <√\n2/2, we have\n1\n2d\ndt/bardblg/bardbl2\nXrτ,α+/bardblg/bardbl2\nZrτ,α+(1−2α2)/bardblg/bardbl2\nXrτ,α≤0. (3.15)\nIt follows that\ne2(1−2α2)td\ndt/bardblg/bardbl2\nXrτ,α+2e2(1−2α2)t/bardblg/bardbl2\nZrτ,α+2(1−2α2)e2(1−2α2)t/bardblg/bardbl2\nXrτ,α,≤0,(3.16)\nthat implies\ne2(1−2α2)t/bardblg/bardbl2\nXrτ,α+/integraldisplayt\n02e2(1−2α2)s/bardblg(s)/bardbl2\nZrτ,αds≤ /bardblg(0)/bardbl2\nXrτ0,α. (3.17)\nFrom (3.13), one has\nτ(t)3/2−τ3/2\n0=−C/integraldisplayt\n0(/bardblg(s)/bardblXrτ,α+/bardblg(s)/bardblZrτ,α))ds\n≥−C/integraldisplayt\n0/bardblg(0)/bardblXrτ0,αe−(1−2α2)sds−C/integraldisplayt\n0/bardblg(s)/bardblZrτ,αe(1−2α2)se−(1−2α2)sds\n≥−C1/bardblg(0)/bardblXrτ0,α. (3.18)\nHence, if the initial perturbation data is suitably small su ch that\nτ0\nK> C2/3\n1/bardblg(0)/bardbl2/3\nXrτ0,α, (3.19)\nwithK= (2√\n2)2/3/(2√\n2−1)2/3. Then (3.18) implies that τ(t)> τ0/2 for all t≥0. Conse-\nquently, we have\nProposition 3.1 Under the same conditions of Theorem 1.1, suppose that gis a solution to\n(2.3)-(2.6) with analytic regularity in the norm Xr\nτ,α, then\ne2(1−2α2)t/bardblg/bardbl2\nXrτ,α+/integraldisplayt\n02e2(1−2α2)s/bardblg(s)/bardbl2\nZrτ,αds≤ /bardblg(0)/bardbl2\nXrτ0,α(3.20)\nwithτ(t)> τ0/2for allt≥0.\n4 The Proof of Theorem 1.1\nBy the uniform estimates on the solution to (2.3)-(2.6) give n in Proposition 3.1, and the local\nexistence of solutions in analytic function space that can b eobtained by usingthe argument used\nin [15, 31], the global existence of solution to (2.3)-(2.6) follows. Then by the relation (2.4), the\n10global existence of uto the initial-boundary value problem (1.9)-(1.11) is prov ed. In addition,\naccording to Remark 2.1 and Proposition 3.1, it follows that /bardblu/bardblXr\nτ,α′+/bardbl∂yu/bardblXr\nτ,α′<∞with\n0≤α′< αandτ > τ0/2. As consequence, the proof of global existence part in Theo rem 1.1 is\ncompleted.\nWe now turn to prove the uniqueness. For this, it suffices to sho w the uniqueness of solution\nto (2.3)-(2.6). Assume that there are two solutions g1andg2to (2.3)-(2.6) with the same initial\ndatag0satisfying /bardblg0/bardblXrτ0,α≤δ0. Denote the radii of analytic regularity for g1andg2byτ1(t)\nandτ2(t) respectively. Define τ(t) by\n˙τ+C\n(τ(t))1/2(/bardblg1/bardblXrτ1,α+/bardblg1/bardblZrτ1,α) = 0 (4.1)\nwith initial data τ(0) =τ0\n4.By the estimates given in Section 3, the analyticity radius τ(t)\nsatisfies\nτ0\n8≤τ(t)≤τ0\n4≤min{τ1(t),τ2(t)}\n2,fort≥0. (4.2)\nNote that ¯ g=g1−g2satisfies\n∂t¯g+(1−e−y+u1)∂x¯g+(v1−v2)∂yg1=−¯g+∂2\ny¯g+R, (4.3)\nwith\nR=−(u1−u2)∂xg2−v2∂y¯g. (4.4)\nThe initial data and the boundary condition for ¯ gare\n¯g(t= 0,x,y) = 0,(∂y¯g−¯g)|y=0= 0. (4.5)\nFollowing the arguments used in Section 3, we have\n1\n2d\ndt/bardbl¯g/bardbl2\nXrτ,α+/bardbl¯g/bardbl2\nZrτ,α+(1−2α2)/bardbl¯g/bardbl2\nXrτ,α+(1−α)/bardbl¯g(·,0)/bardbl2\nXrτ,α\n≤(˙τ+C\n(τ(t))1/2(/bardblg1/bardblXrτ,α+/bardblg1/bardblZrτ,α))/bardbl¯g/bardbl2\nYrτ,α+C0\n(τ(t))1/2/bardblg2/bardblYrτ,α(/bardbl¯g/bardbl2\nXrτ,α+/bardbl¯g/bardbl2\nZrτ,α).(4.6)\nFrom (4.1), we have\n˙τ+C\n(τ(t))1/2(/bardblg1/bardblXrτ,α+/bardblg1/bardblZrτ,α)≤0, (4.7)\nwhere we have used the facts that τ(t)≤τ1(t) and the norms Xr\nτ,αandZr\nτ,αare increasing with\nrespect to τ. Moreover,\nC0\n(τ(t))1/2/bardblg2/bardblYrτ,α≤C\nτ/bardblg2/bardblXr\n2τ,α≤C\nτ/bardblg2/bardblXrτ2,α\n≤C1\nτ0/bardblg(0)/bardblXrτ0,αe−2(1−2α2)t≤¯Cδ1/3\n0, (4.8)\n11where in the second inequality 2 τ≤τ2is used, and in the third inequality (3.19) and (3.20) are\nused. 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Twamley1\n1)Quantum Machines Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan\n2)School of Mathematical and Physical Science, Macquarie University, 2109 NSW, Australia\n(*Electronic mail: jason.twamley@oist.jp)\n(Dated: 17 November 2022)\nResearchers seek methods to levitate matter for a wide variety of purposes, ranging from exploring fundamental prob-\nlems in science, through to developing new sensors and mechanical actuators. Many levitation techniques require\nactive driving and most can only be applied to objects smaller than a few micrometers. Diamagnetic levitation has the\nstrong advantage of being the only form of levitation which is passive, requiring no energy input, while also support-\ning massive objects. Known diamagnetic materials which are electrical insulators are only weakly diamagnetic, and\nrequire large magnetic field gradients to levitate. Strong diamagnetic materials which are electrical conductors, such\nas graphite, exhibit eddy damping, restricting motional freedom and reducing their potential for sensing applications.\nIn this work we describe a method to engineer the eddy damping while retaining the force characteristics provided by\nthe diamagnetic material. We study, both experimentally and theoretically, the motional damping of a magnetically\nlevitated graphite plate in high vacuum and demonstrate that one can control the eddy damping by patterning the plate\nwith through-slots which interrupt the eddy currents. We find we can control the motional quality factor over a wide\nrange with excellent agreement between the experiment and numerical simulations.\nKeywords: magnetic levitation, diamagnet, graphite, damping, eddy currents\nThe levitation of matter has a variety of applications rang-\ning from quantum science and technology through to indus-\ntrial development of levitated actuators, motors, and robots.\nTypically to levitate or trap an object requires a source of\npower, e.g. optical trapping uses strong laser fields, while\ndynamic electric fields can hold charged objects in Paul\nor Penning traps. These so-called “active” systems suf-\nfer from various types of noise, and thereby motional heat-\ning. In contrast “passive” diamagnetic levitation, which re-\nquires no active driving or energy input, has the potential\nfor very low noise, low heating, levitation and motional con-\ntrol. Moreover, macroscopic objects can easily be supported\nand manipulated1. For this reason diamagnetic levitation\nhas recently attracted much attention and since its original\nexposition2–4, it is now firmly regarded as one of the main\ntechniques in levitodynamics5,6,33.\nLevitation can be used to explore many fundamental ques-\ntions, including testing ideas in thermodynamics7and alter-\nnative theories of quantum mechanics8, and searching for\nnew types of forces in nature9–11. Tabletop experiments have\nbeen proposed which explore the relationship between quan-\ntum theory and gravity12–17. Another major application of\nlevitodynamics is the development of novel sensors, includ-\ning inertial sensors18–23, detecting gravitational waves24and\ndark matter/energy25, magnetometry26, measuring mass27,\nand light pressure sensing28. It is only in levitated systems\n(using optical trapping) that a macroscopic quantum superpo-\nsition has ever been achieved29–32. Within levitodynamics,\ngraphite plates are gaining increasing attention as a highly\npromising platform. These have already been used to exper-\na)These three authors contributed equallyimentally test theories of dark matter25, and may outperform\nother types of levitated systems53.\nBesides quantum science, another important area of levi-\ntodynamics is the development of levitated actuators, motors,\nand robots. This mostly uses magnetic levitation, due to its\nability to support large mass loads against gravity. An ex-\nample includes the development of miniature robots which\nmove on a planar surface and can be controlled via localised\ncurrents34–38. Researchers have recently developed photo-\nthermal methods to actuate the motion of a diamagnetically\nlevitated graphite plate, which has led to a large interest in de-\nveloping photo-activated 2D positioners39–46. We particularly\nnote recent works using diamagnetic levitated graphite for op-\ntical energy harvesting47, and a multiple-degree-of-freedom\nnanopositioner48.\nDiamagnetic materials which are electrical insulators, in-\ncluding diamond, polymers, some glasses, and many organic\nmaterials49–51, tend to be only very weakly diamagnetic. For\ndiamagnetic levitation of macroscopic masses, for example\nthose used in actuators, the ideal material is Highly Oriented\nPyrolytic Graphite (HOPG). The magnetic susceptibility of\nHOPG is highly aniostropic, with the direction of strongest\ndiamagnetic susceptibility normal to the slab face1,52. How-\never, HOPG is an excellent conductor, and thus eddy currents\nare induced as it moves through magnetic fields. These damp\nthe motion, reducing the quality factor of the motional mode,\nand producing heat leading to thermal noise.\nFor applications such as robots and actuators it may not be\nnecessary to completely eliminate eddy damping, which may\neven be useful in some schemes. For other applications how-\never, such as motional sensors, it is desirable to completely re-\nmove the eddy damping. In electrical transformers, engineers\nreduce eddy losses either by interrupting the currents with lay-arXiv:2211.08764v1  [physics.app-ph]  16 Nov 2022Controlling the motional quality factor of a diamagnetically levitated graphite plate 2\ners of laminated material, or by making the core of the trans-\nformer out of a highly resistive ferrite magnetic material. The\nlatter route was recently adapted to the diamagnetic levitation\nof graphite53, where micrometer-scale graphite particles were\nencapsulated in an electrically insulating resin. Eddy currents\nand the associated damping were significantly reduced, lead-\ning to quality factors of 5 ×105. At the same time however,\nthis results in a lowering of the diamagnetic lift. The filling\nfraction of graphite within the resin cannot be too high with-\nout compromising structural integrity, and random orientation\nof the particles within the resin lowers the effective magnetic\nsusceptibility.\nIn our work we seek to control eddy damping using a\nmethod more similar to the lamination technique in electri-\ncal transformers. We consider the diamagnetic levitation of\na solid slab of HOPG, which we then pattern with very nar-\nrow through-cut slots, the purpose of which is to interrupt the\npath of eddy currents. We machine several samples with in-\ncreasing slot densities. We hypothesise that as we modify the\ndensity of slotted interruptions to the eddy currents, then these\ncurrents will be modified, and so too will the associated eddy\ndamping. The advantage of this method to control the eddy\ndamping is that one retains virtually all the diamagnetic lift of\nthe original HOPG slab, as the through-slots are very narrow.\nIn the following we outline the experiment, analysis, and\nsimulation of the levitodynamics of the slotted graphite plates.\nWe find that the through-cut slots allow us to systematically\ncontrol the motional quality factor in a highly predictable\nmanner, with excellent agreement between theory and experi-\nment.\nA sketch of the experimental setup is presented in Fig. 1. A\nplate of pyrolytic graphite is levitated by four NdFeB mag-\nnets placed in an alternating polarity checkerboard pattern.\nThe magnets are rigidly held within a holder which is fixed\nto a five axis vacuum-compatible motorised stage. The mag-\nnet/motor platform is mounted on a small optical breadboard\nwhich in turn sits on four vibration isolation supports. The po-\nsition of the graphite sample is monitored by an interferomet-\nric displacement sensor. This displacement sensor is based on\na compact Michelson interferometer, and enables high preci-\nsion measurements in real time with a resolution of picometers\nat a high bandwidth.\nThe whole structure is positioned in a vacuum chamber,\nwhich is evacuated by a system consisting of a turbopump,\nan ion pump, and associated roughing pump. During the mea-\nsurement periods the turbopump is switched off to avoid un-\nwanted mechanical vibrations, while the ion-pump operates\ncontinuously to maintain high vacuum (10−7hPa). The vac-\nuum chamber and ion pump are supported by a damped and\nvibration isolated optical table, while the turbopump is sup-\nported by a separate vibration damped and isolated platform.\nThe experiment aims to increase the motional quality factor\nof a diamagnetically levitated slab of pyrolytic graphite. Four\n10mm ×10mm samples were machined from a single piece\nof graphite, to ensure they all possessed similar electric and\nmagnetic properties. The thickness of each sample was ap-\nproximately 0 .77mm, however this was not uniform since the\ngraphite surface was very coarse. Each plate had a pattern of\nFIG. 1. Experimental setup. (a) A plate of pyrolitic graphite is\nlevitated by four NdFeB N52 magnets with polarities arranged in a\ncheckerboard pattern. The magnets are fixed on a five-axis motorized\nstage, which is supported by a breadboard resting upon four vibration\nisolation mounts within the vacuum chamber. The vacuum chamber\nand ion pump lie on a vibration isolation optical table, while the tur-\nbopump is on a separate vibration isolation platform. A small mirror\nis fixed to the graphite sample, which is used by an interferometer\nto measure the vertical displacement. The interferometer is aligned\nusing the five-axis stage. (b) Photograph of the platform which sits\ninside the vacuum chamber.\nring-like slits machined into it whose purpose was to interrupt\nthe eddy currents, and hence lower the resulting eddy damp-\ning forces. These slits were created by femtosecond laser ma-\nchining. The slit designs, and photographs of the machined\nsamples, are shown in Fig. 2. A small mirror was then glued\nonto the centre of each piece, to allow an interferometer to\nmeasure vertical displacement.\nEach graphite sample was levitated for a period of twenty\nminutes, and its vertical position recorded using the interfer-\nometer. The resulting power spectral density was then anal-\nysed as described below. Further details of the setup are givenControlling the motional quality factor of a diamagnetically levitated graphite plate 3\nFIG. 2. Slit designs and photographs of the machined graphite samples. (a)-(d) CAD design patterns of ring-like slits. These are named\nN=0, 3, 6, and 8, with Nreflecting a parameter used to generate the ring patterns. As Ngrows larger slit density increases, leading to\nstronger suppression of eddy currents and an increase in motional quality factor. (e)-(h) Photos of the machined graphite plates, measuring\n10mm ×10mm. The thickness is approximately 0 .77mm, however this is not uniform due the visible surface roughness. After machining, a\nmirror (not shown, made from aluminum coating on glass) was glued to the center of each plate for optical measurement of the motion.\nin the Supplementary Material §1.\nDue to its positioning, the interferometer is most sensitive\nto vertical motion of the plate. Using the normal mode simu-\nlation methods outlined above we find, and in agreement with\nChen et al.1, motion corresponding to three motional modes:\nvertical oscillation along the z-axis, and torsion about the x-\nandy-axes. These are all expected to have frequencies of\naround 17Hz.\nAs the graphite plate moves, the magnetic field it experi-\nences, and hence the magnitude and direction of the force\non the plate, vary in a complex manner. This will cause a\nnontrivial coupling between all of the motional modes. For\nsimplicity, we will approximate the three vertical modes as a\nsingle effective mode of a one-dimensional oscillator. This\noscillator is primarily damped by eddy currents, induced by\nmotion of the plate through the magnetic field. As we discuss\nin the Supplementary Material §3, damping due to air should\nbe negligible at the pressures we consider.\nThe power spectral density for a harmonic oscillator is\ngiven in §10 of Ref. 54, and discussed further in our Sup-\nplementary Material §3. The power spectral densities for the\nexperimental data traces were fitted to the theoretical values,\nallowing us to extract the effective damping rates γand natu-\nral frequencies f0, from which we could calculate the quality\nfactor Qof each oscillator. These are shown in Fig. 3.\nAs expected, the oscillation frequencies were all approx-\nimately 17Hz. As the number of slots increases, there is aslight upwards shift in f0. This is most likely due to the slots\nremoving material from the graphite plate, leading to a de-\ncrease in levitation height, and hence change in the magnetic\ntrapping force. In total, the rings are able to increase the os-\ncillator quality factor by a factor of approximately forty.\nTo understand the frequencies observed in the power spec-\ntral density, we simulated the motional modes for the n=0\nplate. These modes are determined by the forces and torques\nexperienced as it moves through the inhomogenous magnetic\nfield above the magnet array. As discussed in the Supplemen-\ntary Material §2, there are six modes in total. Three of them,\noscillations in the horizontal plane or rotation about the ver-\ntical axis, have frequencies bunched around 4Hz. The other\nthree involve vertical motion, namely vertical oscillation and\ntilting about the horizontal axes, and are predicted to have fre-\nquencies around 17Hz. It is these vertical modes that our in-\nterferometric setup will be most sensitive to.\nTo estimate the increase in quality factor due to the slits,\nwe simulated the eddy currents in each of the graphite plates\nin Fig. 2. The currents are induced by an effective electric\npotential which depends on the geometry of the plate and its\nmotion through the external magnetic field1,55–57. This cur-\nrent then exerts a force on the plate due to the magnetic field,\nwhich can be integrated to find the induced eddy damping. We\ndeveloped both a two-dimensional model in Mathematica, and\nthree-dimensional COMSOL simulation, the details of which\nare described in Supplementary Material §4.Controlling the motional quality factor of a diamagnetically levitated graphite plate 4\nFIG. 3. Power spectral density of the vertical motion. In (a)-(d) the\nsolid, coloured traces show the experimental data. The dashed black\nline denotes the theoretical fit, with insets showing the fitted quality\nfactors ( Q) and resonance frequencies ( f0). Errors come from the\ncovariance matrix of the fit. With increased slit density, the quality\nfactor increases from QN=0∼66 to QN=8∼2476. In (e) we overlap\nthe power spectral densities. As the number of slits increases the res-\nonance peaks get sharper, and there is a slight increase in frequency.\nThe two-dimensional model used the finite element method\nin Mathematica 13.0 to simulate the currents. Moving from\nn=0 ton=8, the total current was found to have decreased\nby an order of magnitude. Around the edges of a slit very large\ncurrents could occur, however as these occur in infinitesimal\nareas they do not contribute significantly to eddy damping.\nAt low pressures, eddy currents are solely responsible for\nmotional damping γeddy of the plate. The quality factor of\na mode is inversely proportional to the damping rate: qn∝\nmn/γeddyn , for sample nwith mass mn. Hence we can predict\nthe ratio of quality factors of the different plates, by calculat-\ning the ratios of their corresponding γeddyvalues and masses.\nThese are shown in Fig. 4. We can see that the simulated val-\nues agree well with experiment. The n=8 sample does appear\nto have a slightly larger quality factor than predicted, which as\nwe will discuss later is most likely due to the machined slots\nbeing wider than the design.\nWe also built a three-dimensional model using the commer-\ncial FEM package COMSOL. The eddy currents are plotted in\nFig. 5 (these do not significantly differ from those generated\nin Mathematica). Using this we were able to predict the ab-\nFIG. 4. Plate quality factors relative to n=0. The solid blue trace\nis obtained from fitting the experimental power spectral density. The\ndashed lines show the simulated quality factors due to eddy damp-\ning. The orange trace shows the two-dimensional model in Mathe-\nmatica, while the green is the three-dimensional COMSOL model.\nThe simulations are all within experimental error, indicating that the\nobserved increase in quality factor is due to suppression of eddy cur-\nrents.\nsolute values of the quality factors , rather than simply their\nratios. The results of the COMSOL simulation are also plot-\nted in Fig. 4, agreeing with the results from the experiment\nand Mathematica.\nThe motional quality factor of the graphite plates was mea-\nsured to increase as more slots were cut into the surface. This\nincrease was consistent with both the two-dimensional Math-\nematica and three-dimensional COMSOL simulations of the\neddy current damping. This indicates that the increase in qual-\nity factor is indeed due to suppression of eddy currents. Over-\nall currents were suppressed by an order of magnitude, corre-\nsponding to an increase in quality factor of forty.\nIn Fig. 4 we can see that for N=8, the simulated quality\nfactors are slightly lower than the experimental value. We at-\ntribute this primarily to a discrepancy between the designed\nslot patterns, and what is created by the femtosecond laser\ncutting. In Fig. 2, the slot patterns of the laser-cut samples are\nclearly wider than the CAD designs. Moreover, the machin-\ning process carves V-shaped slits which remove more graphite\nthan expected, an effect which is is more pronounced at high\nslit density. Wider slots yield less eddy currents and thus re-\nduced eddy damping, leading to higher quality factors in the\nactual samples than what is predicted by our models.\nGraphite is one of the strongest diamagnetic materials\nknown, and has great potential for use in levitated technolo-\ngies. However, due to its high electrical conductivity it ex-\nhibits strong eddy damping. The ability to engineer this damp-\ning while retaining a strong diamagnetic susceptibility will\npermit researchers in a wide range of disciplines the ability\nto apply such conducting diamagnetic materials to situations\nwhere fast motional control is required. We show that by pat-\nterning the graphite plate with through-slots we can interrupt\nthe eddy currents in a controlled manner, and gain detailed\ncontrol over the eddy damping while retaining the strong dia-\nmagnetic lift. In this study we have not optimized the slotted\npattern, and it is an interesting question whether one can pro-Controlling the motional quality factor of a diamagnetically levitated graphite plate 5\nj (A/m2)\n(b) (a)\n(c) (d)\nx[mm] x[mm]y[mm] y[mm]\nFIG. 5. Eddy currents simulated by COMSOL on the bottom surface of each plate. The plate is assumed to move vertically downwards away\nfrom its equilibrium position. The currents simulated by the two-dimensional Mathematica model are similar. We choose v=−6×10−6m/s\nforn=0 and n=3, and v=−12×10−6m/s for n=6 and n=8 (the different velocities are to ensure the currents are still visible at higher\nNvalues). Colour denotes the current magnitude, while arrows show the current direction. Around the edges of the slits the current can attain\nvery large values not shown on our colour scale, but these occur in vanishingly small regions. As the number of slits increases, the eddy\ncurrents are significantly suppressed.\nduce designs which remove the least material, maintain the\nstructural integrity of the plate, and control the eddy damping\nto the maximal extent.\nACKNOWLEDGEMENTS\nThis work was supported by the Okinawa Institute of Sci-\nence and Technology Graduate University, Japan and Mac-quarie University, Sydney, Australia. The authors acknowl-\nedge technical assistance from E. Elerabi and P. Kennedy from\nthe OIST Engineering Section. The graphite slab machining\nwork was performed in part at the OptoFab node of the Aus-\ntralian National Fabrication Facility (ANFF) utilising Com-\nmonwealth and NSW State Government funding.Controlling the motional quality factor of a diamagnetically levitated graphite plate 6\nI. DATA AVAILABILITY STATEMENT\nThe data and simulation codes that support the findings of\nthis study are available from the corresponding author upon\nreasonable request.\n1X. Chen, A. 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Jayanth, Diamagnetically levitated nanopositioners\nwith large-range and multiple degrees of freedom, Nature Communications\n13, 3334 (2022).\n49C. W. Lewandowski, T. D. Knowles, Z. B. Etienne, and B. D’Urso, High-\nSensitivity Accelerometry with a Feedback-Cooled Magnetically Levi-\ntated Microsphere, Physical Review Applied 15, 10.1103/PhysRevAp-\nplied.15.014050 (2021).\n50R. Nakashima, Diamagnetic levitation of a milligram-scale silica using\npermanent magnets for the use in a macroscopic quantum measurement,Physics Letters A 384, 126592 (2020).\n51Y . Leng, R. Li, X. Kong, H. Xie, D. Zheng, P. Yin, F. Xiong, T. Wu, C.-K.\nDuan, Y . Du, Z.-q. Yin, P. Huang, and J. Du, Mechanical Dissipation Below\n1µHz with a Cryogenic Diamagnetic Levitated Micro-Oscillator, Physical\nReview Applied 15, 024061 (2021).\n52C. Niu, F. Lin, Z. M. Wang, J. Bao, and J. Hu, Graphene levitation and\norientation control using a magnetic field, Journal of Applied Physics 123,\n044302 (2018).\n53X. Chen, S. K. Ammu, K. Masania, P. G. Steeneken, and F. Alijani, Dia-\nmagnetic Composites for High-Q Levitating Resonators, Advanced Science\n2203619 , 2203619 (2022).\n54M. C. Wang and G. E. Uhlenbeck, On the theory of the Brownian motion\nII, Reviews of modern physics 17, 323 (1945).\n55M. Kirpo, T. Boeck, and A. Thess, Eddy-current braking of a translating\nsolid bar by a magnetic dipole, Pamm 10, 513 (2010).\n56E. V . V otyakov and A. Thess, Interaction of a magnetic dipole with a slowly\nmoving electrically conducting plate, Journal of Engineering Mathematics\n77, 147 (2012).\n57M. Carlstedt, K. Porzig, M. Ziolkowski, R. P. Uhlig, H. Brauer, and\nH. Toepfer, Comparison of Lorentz force eddy current testing and com-\nmon eddy current testing-measurements and simulations, Studies in Ap-\nplied Electromagnetics and Mechanics 39, 218 (2014).Supplementary Material\nP. Romagnoli,1,a)R. Lecamwasam,1,a)S. Tian,1,a)J.E. Downes,2and J. Twamley1\n1)Quantum Machines Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan\n2)School of Mathematical and Physical Science, Macquarie University, 2109 NSW, Australia\n(*Electronic mail: jason.twamley@oist.jp)\nI. EXPERIMENTAL SETUP\nThe NdFeB magnets are grade N52 (B888-N52) from K&J\nMagnetics. The magnets are rigidly held within a holder\nwhich is fixed to a five axis vacuum-compatible motorised\nstage. The magnet/motor platform is mounted on a small op-\ntical breadboard which in turn sits on four wire rope vibration\nisolation supports.\nThe graphite samples have a size of 10 mm ×10 mm×∼\n0.77 mm. These were machined from a single piece of\ngraphite (PG3, K&J Magnetics), to ensure that all samples\npossessed similar electric and magnetic properties. Note that\nthe thickness of the graphite is not uniform due to the very\ncoarse surface. The ring-like slit patterns were machined by a\nfemtosecond laser machining through the graphite samples.\nA small mirror is fixed to the centre of each machined\ngraphite sample. These mirrors are fabricated by sputtering\nan aluminium coating onto a microscope coverglass (thick-\nness∼0.17 mm), and then cut in to pieces of dimension\n3 mm×3 mm. The mirrors then are attached to the graphite\nplate by optical glue. Following machining and gluing of\nthe mirror, the masses of the plates N= (0,3,6,8)were\n(147,137,93,79)mg respectively.\nA PICOSCALE laser interferometic displacement sensor is\nused to measure the vertical displacement of the sample rel-\native to an optomechanical cage system arranged to hold the\nsensor head above the graphite sample. The cage structure\nis fixed to the optical breadboard. The laser is aligned using\nthe 5-axis stage to reflect from the mirror which is affixed to\nthe centre of the graphite sample. The displacement sensor\nis based on a compact Michelson interferometer, and enables\nhigh precision measurements in real time with a resolution of\npicometers at a high bandwidth. A graphical user interface\nof the interferometer is used to acquire position and velocity\ndata, and data is streamed to a PC for storage and later analy-\nsis. The position is also output from the interferometer to an\noscilloscope for monitoring.\nThe whole structure (magnets/motor/optical sen-\nsor/breadboard/vibration isolators) is positioned inside a\nvacuum chamber. The optical signal from the in-vacuum\ninterferometer sensor head is fed via an optical fibre through\na feedthrough to the external interferometer. The cham-\nber is evacuated by a system consisting of a turbopump\n(HiCube300H, Pfeiffer), an ion pump (VacIon Plus 300,\nAgilent), and associated roughing pump. The vacuum\nchamber and ion pump are supported by a vibration iso-\nlated/damped optical table, while the turbopump is supported\na)These three authors contributed equallyby a separate vibration isolation platform. The ion pump has\nno moving parts and is motionally extremely quiet, while\nthe turbopump rests on its own vibration isolated support\nplatform. The turbopump is used initially to reach a base\npressure (10−6hPa), then we switch to the ion pump to reach\nhigh vacuum (10−7hPa). During measurement periods a\ngate-valve connected to the turbopump is closed and the\nturboppump is switched off to avoid unwanted mechanical\nvibrations. The ion-pump then operates continuously to\nmaintain high vacuum.\nII. ESTIMATING MOTIONAL FREQUENCIES FOR THE\nN=0PLATE\nTo understand the measured power spectral density, we\nsimulated the oscillation frequencies of the N=0 plate. These\nmodes are determined by the forces and torques the plate\nexperiences in the inhomogenous magnetic field above the\ncheckerboard magnet array. Due to the small size of the slits,\nthe modes are not expected to be significantly different for\nother values of N. In the following, we will use \"laboratory\nframe\" to denote the reference frame of the magnets.\nA diamagnetic material sitting in a magnetic field Bexpe-\nriences an effective induced magnetization M= (1/µ0)χ·B,\nwhere χis the magnetic susceptibility tensor for the mate-\nrial. In the body frame of the plate this tensor is diagonal:\nχ=diag(χx,χy,χz), with values χjgiven in Table I. The force\ndensity on the graphite plate is then1f= (M·∇)B. If the\nplate’s normal is vertical in the laboratory frame, then we have\nf=1\n2µ0∇(χxB2\nx+χyB2\ny+χzB2\nz), (1)\nbut if the plate is tilted then the χ−tensor, which is diagonal\nin the body frame of the plate, has to be transformed into the\nlaboratory frame.\nAnalytic forms for the magnetic fields arising from the\ncheckerboard array are given in §2.5 of Ref. 2, though the ex-\npressions are too complex to be reproduced here. We then nu-\nmerically (in Matlab) evaluate the force density over a dense\n3D grid throughout the graphite plate, which we integrate nu-\nmerically to obtain the net force on the centre of mass of the\nplate as a function of the position and angular tilt. Setting the\nnet force Fnet=Fmag+Fgrav=0, we obtain the equilibrium\nposition of the centre of mass of the plate as /vectorr= (0,0,zequil)\n(see Table I).\nTo evaluate the equilibrium orientation of the plate one can\nuse the torque density exerted by the interaction of the induced\nmagnetization of the plate with the magnetic field of the mag-arXiv:2211.08764v1  [physics.app-ph]  16 Nov 20222\n<latexit sha1_base64=\"KU7er+7/BVrnYYJ7lw7pVxG2qX0=\">AAAB7HicdVBNS8NAEJ3Ur1q/qh69LBbBU0hKaeqt6MVjBdMW2lA22227dLMJuxuxlP4GLx4U8eoP8ua/cdNGUNEHA4/3ZpiZFyacKe04H1ZhbX1jc6u4XdrZ3ds/KB8etVWcSkJ9EvNYdkOsKGeC+pppTruJpDgKOe2E06vM79xRqVgsbvUsoUGEx4KNGMHaSH5fpIP7Qbni2BcN1/Nc5NjOEhmpek69htxcqUCO1qD83h/GJI2o0IRjpXquk+hgjqVmhNNFqZ8qmmAyxWPaM1TgiKpgvjx2gc6MMkSjWJoSGi3V7xNzHCk1i0LTGWE9Ub+9TPzL66V61AjmTCSppoKsFo1SjnSMss/RkElKNJ8Zgolk5lZEJlhiok0+JRPC16fof9Ku2m7drt3UKs3LPI4inMApnIMLHjThGlrgAwEGD/AEz5awHq0X63XVWrDymWP4AevtE0BAjv8=</latexit>⌫x\n<latexit sha1_base64=\"pncMpJgywnzUz+3rU/xZz9MArtM=\">AAAB7HicdVBNS8NAEJ3Ur1q/qh69LBbBU0lKMXorevFYwbSFNpTNdtMu3WzC7kYIob/BiwdFvPqDvPlv3LYRquiDgcd7M8zMCxLOlLbtT6u0tr6xuVXeruzs7u0fVA+POipOJaEeiXksewFWlDNBPc00p71EUhwFnHaD6c3c7z5QqVgs7nWWUD/CY8FCRrA2kjcQ6TAbVmt23V4ArRC3edVwHeQUSg0KtIfVj8EoJmlEhSYcK9V37ET7OZaaEU5nlUGqaILJFI9p31CBI6r8fHHsDJ0ZZYTCWJoSGi3U1YkcR0plUWA6I6wn6rc3F//y+qkOL/2ciSTVVJDlojDlSMdo/jkaMUmJ5pkhmEhmbkVkgiUm2uRTMSF8f4r+J51G3bmoN++atdZ1EUcZTuAUzsEBF1pwC23wgACDR3iGF0tYT9ar9bZsLVnFzDH8gPX+BSBpjuo=</latexit>⌫y\n<latexit sha1_base64=\"mTd2MQr/zNEBsIXbl233wAKDXh8=\">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkqMeiF48VTFtoQ9lsN+3SzSbsToRa+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MJXCoOt+O4W19Y3NreJ2aWd3b/+gfHjUNEmmGfdZIhPdDqnhUijuo0DJ26nmNA4lb4Wj25nfeuTaiEQ94DjlQUwHSkSCUbSS31VZ76lXrrhVdw6ySrycVCBHo1f+6vYTlsVcIZPUmI7nphhMqEbBJJ+WupnhKWUjOuAdSxWNuQkm82On5MwqfRIl2pZCMld/T0xobMw4Dm1nTHFolr2Z+J/XyTC6DiZCpRlyxRaLokwSTMjsc9IXmjOUY0so08LeStiQasrQ5lOyIXjLL6+S5kXVu6zW7muV+k0eRxFO4BTOwYMrqMMdNMAHBgKe4RXeHOW8OO/Ox6K14OQzx/AHzucP93OOzQ==</latexit>⌫z<latexit sha1_base64=\"BwmGTDP2tm0lLd2ZwHBItOmx1aI=\">AAAB73icdVDLSgNBEOyNrxhfUY9eBoPgaZlNgom3oBePEcwDkiXMTibJkNnZdWZWDEt+wosHRbz6O978GycPQUULGoqqbrq7glhwbTD+cDIrq2vrG9nN3Nb2zu5efv+gqaNEUdagkYhUOyCaCS5Zw3AjWDtWjISBYK1gfDnzW3dMaR7JGzOJmR+SoeQDTomxUrsrk156P5n28gXsesViBVcQds9LpTIuWYKLuFr1kOfiOQqwRL2Xf+/2I5qETBoqiNYdD8fGT4kynAo2zXUTzWJCx2TIOpZKEjLtp/N7p+jEKn00iJQtadBc/T6RklDrSRjYzpCYkf7tzcS/vE5iBlU/5TJODJN0sWiQCGQiNHse9bli1IiJJYQqbm9FdEQUocZGlLMhfH2K/ifNouudueXrcqF2sYwjC0dwDKfgQQVqcAV1aAAFAQ/wBM/OrfPovDivi9aMs5w5hB9w3j4B8SWQlg==</latexit>⌫xy<latexit sha1_base64=\"+Kv7271/OzMFj/+tGqzYuNI+c0M=\">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuwiQttu6KblxWsA9oQplMJ+3QySTMQyyhv+HGhSJu/Rl3/o3Th6CiBy4czrmXe++JMs6URujDWVldW9/YLGwVt3d29/ZLB4dtlRpJaIukPJXdCCvKmaAtzTSn3UxSnEScdqLx1czv3FGpWCpu9SSjYYKHgsWMYG2lIBAm0Nj08/vJtF8qI9fz/RqqQeReVCpVVLEE+ahe96DnojnKYIlmv/QeDFJiEio04VipnocyHeZYakY4nRYDo2iGyRgPac9SgROqwnx+8xSeWmUA41TaEhrO1e8TOU6UmiSR7UywHqnf3kz8y+sZHdfDnInMaCrIYlFsONQpnAUAB0xSovnEEkwks7dCMsISE21jKtoQvj6F/5O273rnbvWmWm5cLuMogGNwAs6AB2qgAa5BE7QAARl4AE/g2THOo/PivC5aV5zlzBH4AeftExRkkmQ=</latexit>⌫⌧xy\n<latexit sha1_base64=\"8uXVpCFbZu2WNl5cT7YBYejD/0I=\">AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKqMegF48RzEOyS5idzCZDZmaXeQgx5Cu8eFDEq5/jzb9xkuxBEwsaiqpuurvijDNtfP/bK6ysrq1vFDdLW9s7u3vl/YOmTq0itEFSnqp2jDXlTNKGYYbTdqYoFjGnrXh4M/Vbj1Rplsp7M8poJHBfsoQRbJz0EEobGmy7T91yxa/6M6BlEuSkAjnq3fJX2EuJFVQawrHWncDPTDTGyjDC6aQUWk0zTIa4TzuOSiyojsazgyfoxCk9lKTKlTRopv6eGGOh9UjErlNgM9CL3lT8z+tYk1xFYyYza6gk80WJ5cikaPo96jFFieEjRzBRzN2KyAArTIzLqORCCBZfXibNs2pwUT2/O6/UrvM4inAEx3AKAVxCDW6hDg0gIOAZXuHNU96L9+59zFsLXj5zCH/gff4AFaiQmw==</latexit>⌫⌧z<latexit sha1_base64=\"jtoM/XmIFDGPakTRd208Hi7IS3A=\">AAAB8HicdVBNSwMxEM36WetX1aOXYBE8ld1SrN6KXjxWsB/SXUo2zbah2eySTISl9Fd48aCIV3+ON/+NabtCFX0w8Hhvhpl5YSq4Btf9dFZW19Y3Ngtbxe2d3b390sFhWydGUdaiiUhUNySaCS5ZCzgI1k0VI3EoWCccX8/8zgNTmifyDrKUBTEZSh5xSsBK9740PhDTz/qlsltx58BLpF67rNY97OVKGeVo9ksf/iChJmYSqCBa9zw3hWBCFHAq2LToG81SQsdkyHqWShIzHUzmB0/xqVUGOEqULQl4ri5PTEisdRaHtjMmMNK/vZn4l9czEF0EEy5TA0zSxaLICAwJnn2PB1wxCiKzhFDF7a2YjogiFGxGRRvC96f4f9KuVrzzSu22Vm5c5XEU0DE6QWfIQ3XUQDeoiVqIohg9omf04ijnyXl13hatK04+c4R+wHn/Aj6PkLg=</latexit>⌫⌧y<latexit sha1_base64=\"KFvUtJgaPW29MWBwLSK9oP/MS2Y=\">AAAB8HicdVDLSsNAFJ34rPVVdelmsAiuQlJKU3dFNy4r2Ic0oUymk3boZBLmIZbQr3DjQhG3fo47/8ZJG0FFD1w4nHMv994TpoxK5Tgf1srq2vrGZmmrvL2zu7dfOTjsykQLTDo4YYnoh0gSRjnpKKoY6aeCoDhkpBdOL3O/d0eEpAm/UbOUBDEacxpRjJSRbn2ufYX08H5YqTr2edP1PBc6trNATmqe06hDt1CqoEB7WHn3RwnWMeEKMyTlwHVSFWRIKIoZmZd9LUmK8BSNycBQjmIig2xx8ByeGmUEo0SY4gou1O8TGYqlnMWh6YyRmsjfXi7+5Q20ippBRnmqFeF4uSjSDKoE5t/DERUEKzYzBGFBza0QT5BAWJmMyiaEr0/h/6Rbs92GXb+uV1sXRRwlcAxOwBlwgQda4Aq0QQdgEIMH8ASeLWE9Wi/W67J1xSpmjsAPWG+fXmaQzQ==</latexit>⌫⌧x\nFIG. 1. Schematic of normal mode analysis: we illustrate the\ndisplacement (left) and torsional (right), motions of the levitated\ngraphite plate (orange), above the 2x2 checkerboard magnet array\n(purple and green): (a) we show the displacement restoring frequen-\ncies along the { x,y,z,x+y} directions (red, blue, black, green); (b)\nwe show the torsional restoring frequencies for oscillations about the\nsame axes as in (a). The normal mode motional frequencies for the\nsolid graphite plate are reported in Table I.\nnet array:\nτ=M×B+r×f, (2)\nwhere one again must take care to transform the χ-tensor into\nthe laboratory frame. The total torque on the plate is obtained\nby integrating this torque density over the volume of the plate.\nWe find that this vanishes and yields a stable equilibrium value\nfor the configuration where the plate is horizontal but rotated\nbyπ/4, around the z-axis relative to the checkerboard frame,\nas shown in Fig 1.\nTo determine the oscillation frequencies of the plate when\nsubject to small displacements or tilts from its equilibrium\nconfiguration, we evaluated the net force or torque as a func-\ntion of these small perturbations, then evaluated the spring\nstiffness via:\nδFNET=−kδr,δτNET=−kδθ. (3)\nThe oscillation frequencies where then given by\n2πνj=/radicalbigg\nk\nm,2πντ j=/radicalBigg\nk\nIj, (4)\nwhere mis the total mass of the plate and Ijis the moment of\ninertia of the plate about axis j.\nThe various directions of displacements and axes for tor-\nsional oscillations studied are shown in Fig 1, which can be\ncompared with the numerical in Table I. We performed an ex-\nperiment to confirm these numerical predictions, and observed\nvisually the center of mass of the N=8 graphite oscillating\nin the vertical oscillation mode of motion with a frequency\nνz∼17Hz (see Supplementary video).The simulations also\nshowed evidence for a small coupling between the displace-\nment and torsional modes.\nIII. MODELLING THE POWER SPECTRAL DENSITY\nWe approximate the plate’s dynamics as a one-dimensional\nharmonic oscillator with some effective damping rate γ,whose natural frequency is approximately f0=17Hz. This\nhas equation of motion (see §10 of Ref. 3)\nz/prime/prime(t)+γ\nmz/prime(t)+(2πf0)2z(t) =/radicalbigg\n2γkBT\nmξ(t), (5)\nwhere mis the effective mass, kBBoltzmann’s constant, and\nTthe effective temperature of the mode. The term ξ(t)repre-\nsents white noise with correlation /angbracketleftξ(t)ξ(t/prime)/angbracketright=δ(t−t/prime), and\ndescribes a stochastic force due to thermal fluctuations in the\nelectrons in the graphite plate, which are heated by the eddy\ncurrents.\nThe power spectral density Szzis given by3\nSzz(f) =8kBTγ/m\n((2πf0)2−(2πf)2)2+(2πfγ/m)2. (6)\nThis forms a shape similar to that of a Lorentzian4, whose\nwidth increases with γ/m. The distribution is peaked at fre-\nquency\nf=/radicalbigg\nf2\n0−γ2\n8m2π2. (7)\nWe took position data for 1200 seconds with time inter-\nval∆t=0.41 ms, and performed a Welch Peridogram analy-\nsis over blocks of size 60 s, which averages out noise in the\nPSD. This data was fitted to Eq. (6), to obtain estimates of\nthe parameters f0andγ. The resulting PSDs and fittings are\nshown in the main text in Fig. 3 We note that as the slot den-\nsity increases the mechanical damping decreases, and there is\na small shift in f0towards larger values. This shift is most\nlikely due to the small change in trapping forces and sample\nmass due to the removal of the slot material and various vari-\nations in the sample thickness.\nAnother possible source of damping of the resonator is col-\nlisions with background gas molecules. To estimate the qual-\nity factor Qgassociated with the gas damping, we will sim-\nplify the geometry of our sample and assume it is a graphite\nsphere of radius of r=1 mm surrounded by nitrogen gas\nmolecules ( mg=4.65×10−26Kg), at a pressure of pg=\n1×10−7hPa. For a graphite sphere of density ρundergoing\nBrownian motion in a harmonic trap of frequency ωz=2πνz,\nthe quality factor is given by5\nQg=πωz\n6ρr\npg/radicalBigg\n3kBT\nmg. (8)\nUsing values for ρandωzfrom Table I, we obtain Qg∼6×\n109. This is much larger than the quality factor expected from\neddy damping. Thus we may treat air damping as negligible.\nIV. SIMULATING THE EDDY CURRENTS\nA. Two-dimensional Mathematica model\nThe Mathematica simulation modelled the eddy currents by\nsolving Poisson’s equation for an effective electric potential,3\nfollowing the method in Ref. 6. As a conductor moves with\nvelocity vthrough a magnetic field B, the delocalised elec-\ntrons feel a Lorentz force proportional to v×B. This moves\nthe electrons and causes charge to collect at the boundaries,\nresulting in an electric potential Ve. The eddy current density\njis formed by the sum of these electric and magnetic forces:\nj=−σ∇Ve+σ(v×B), (9)\nwhere σis the 3×3 electrical conductivity matrix. Only the\ndiagonal entries σx,σy,σzare non-zero, describing how easily\nelectrons in the material may flow in each Cartesian direction.\nOn the right hand side, B(the external magnetic field) and v\n(the plate velocity) are known, so to find jwe must solve for\nthe electric potential Ve.\nMaxwell’s equations imply conservation of charge: ∇·j=\n0. Applying this to Eq. (9), we find that the potential satisfies\nPoisson’s equation:\n∇·[−σ∇Ve+σ(v×B)] = 0. (10)\nThe conductivity σdoes not cancel, since it is a diagonal\nmatrix whose entries are not all equal. The boundary con-\ndition comes from the fact that current must flow parallel to\nthe edges of the conductor. Thus if ˆnis the unit vector for the\nboundary, we have j·ˆn=0, which expressed in terms of the\npotential gives:\nˆn·(σ∇Ve) =ˆn·σ(v×B). (11)\nThe partial differential equation (10), with the Neumann\nboundary condition (11), has a solution Vewhich is unique\nup to addition of a constant. This constant does not affect the\neddy currents, since jis proportional to the gradient ∇Ve.\nFor graphite, the conductivity in the xy-plane is much larger\nthan that along the z-direction. We can thus define σxy=σx=\nσy, and approximate σz≈0. Then σcancels from (10), which\nbecomes\n(∂2\nx+∂2\ny)Ve(x,y) =∇·(v×B)xy, (12)\nwith boundary condition\nˆn·∇Ve(x,y) =ˆn·(v×B)xy, (13)\nwhere (v×B)xydenotes the projection of the Lorentz force\ninto the xy-plane. The resulting eddy current density is then\nj(x,y) =−σxy[∇Ve(x,y)+(v×B)xy]. (14)\nBoth quantities on the right-hand side of Eq. (12) are\nknown. The magnetic field is generated by the checkerboard\nof permanent magnets, evaluated at the z-value corresponding\nto the equilibrium position of the levitated plate. An analytical\nexpression for the magnetic field of a rectangular prism can be\nfound in §10 of Ref. 2, from which we can calculate Bfor the\narray. Since we are considering vertical motion, the velocity\ncan be taken to be v= (0,0,1). Note that the magnitude of v\ndoesn’t matter, since from Eq. (12) and Eq. (13) we can see\nthatVeis linearly proportional to v.Eq. (12) and Eq. (13) were solved using the Mathematica’s\nfinite element method functionality. The results were found to\nbe stable for a grid size going from 1mm to 0 .05mm, indicat-\ning convergence of the numerical method.\nThe current density jcan then be used to calculate the eddy\ndamping6–9, and hence the estimated change in quality factor.\nThe electrons in the current will feel a Lorentz force j×B\ndue to the permanent magnets below. Integrating this over the\nentire plate gives us the net force due to the eddy currents:\nFEddy=/integraldisplay\n(j×B)dS, (15)\nwhere/integraltextdSrepresents an integral over the two-dimensional\nplate. We find the net eddy force to be directed downwards,\nopposing the upwards motion v. Since Veis linearly propor-\ntional to v, so is j, and hence Feddy. Thus we may write\nFeddy=−γeddyv, (16)\nwhere γeddyis a positive constant associated with each plate\ngeometry, which we recognise as the damping on the plate\ndue to eddy currents.\nB. Three-dimensional COMSOL model\nA three-dimensional COMSOL model was constructed us-\ning the parameters in Table I. The static magnetic force on\nthe graphite plate was calculated within COMSOL using the\nforce equations described in Section II. The equilibrium levi-\ntation height determined by balancing the magnetic force with\ngravity agrees well with both the calculations performed in\nSection II, and the experiment. Then, using a time-dependent\nstudy in COMSOL, we simulated the eddy current damping\nby calculating the Lorentz force at the equilibrium position as\na function of velocity. We show the ratio of these COMSOL\nderived Q−factors in the main text Fig. 4. while examples of\nthe current distributions in a 2D horizontal slice are shown in\nthe main text Fig. 5.\n1C. Niu, F. Lin, Z. M. Wang, J. Bao, and J. Hu, “Graphene levitation and\norientation control using a magnetic field,” Journal of Applied Physics 123,\n044302 (2018).\n2J. M. Camacho and V . Sosa, “Alternative method to calculate the magnetic\nfield of permanent magnets with azimuthal symmetry,” Revista Mexicana de\nFisica E 59, 8–17 (2013).\n3M. C. Wang and G. E. Uhlenbeck, “On the theory of the Brownian motion\nII,” Reviews of modern physics 17, 323 (1945).\n4This distribution is often referred to as Lorentzian , however it is slightly\ndifferent since γon the denominator is multiplied by f.\n5T. Wang, S. Lourette, S. R. Oâ C™Kelley, M. Kayci, Y . Band, D. F. J. Kim-\nball, A. O. Sushkov, and D. Budker, “Dynamics of a Ferromagnetic Parti-\ncle Levitated over a Superconductor,” Physical Review Applied 11, 044041\n(2019).\n6X. Chen, A. Ke¸ skekler, F. Alijani, and P. G. Steeneken, “Rigid body dy-\nnamics of diamagnetically levitating graphite resonators,” Applied Physics\nLetters 116, 243505 (2020).\n7M. Kirpo, T. Boeck, and A. Thess, “Eddy-current braking of a translating\nsolid bar by a magnetic dipole,” Pamm 10, 513–514 (2010).\n8E. V . V otyakov and A. Thess, “Interaction of a magnetic dipole with a slowly\nmoving electrically conducting plate,” Journal of Engineering Mathematics\n77, 147–161 (2012).4\nTABLE I. Quantities and parameter values appearing in this manuscript - bold items are the results of simulation while non-bold symbols are\nmeasured or taken from the literature.\nQuantity Symbol Value Unit\nDisplacement frequency along x: νx 3.9 Hz\nDisplacement frequency along y: νy 3.9 Hz\nDisplacement frequency along z: νz 17.2 Hz\nDisplacement frequency along x+y: νxy 4.1 Hz\nTorsional frequency about x: ντx 17.5 Hz\nTorsional frequency about y: ντy 17.5 Hz\nTorsional frequency about z: ντz 5.9 Hz\nTorsional frequency about x+y: ντxy 16.9 Hz\nMagnetic Susceptibility along xin body frame of plate: χx −85×10−6NA\nMagnetic Susceptibility along yin body frame of plate: χy −85×10−6NA\nMagnetic Susceptibility along zin body frame of plate: χz −450×10−6NA\nElectrical Conductivity along x in body frame of plate: σx 200 S /m\nElectrical Conductivity along y in body frame of plate: σy 200 S /m\nElectrical Conductivity along z in body frame of plate: σz 200 000 S /m\nCheckerboard Cube Magnet Sidelength: magnetlength 12.7 mm\nCheckerboard Magnet Magnetization: M0 1.18×106A/m\nSidelength of square graphite solid slab: platedims xy 10 mm\nThickness of graphite solid slab:aplatedims z 0.77 mm\nEquilibrium height of solid plate:bzequil 1.46 mm\nDensity of Graphite: ρgraphite 2070 kg /m3\nMass of Graphite resonator: m 79−147 mg\naThe thickness change from area to area due the coarse surface of graphite plates.\nbFrom top surface of the magnets to the center of the plate. It also changes with different resonators.\n9M. Carlstedt, K. Porzig, M. Ziolkowski, R. P. Uhlig, H. Brauer, and\nH. Toepfer, “Comparison of Lorentz force eddy current testing and com-\nmon eddy current testing-measurements and simulations,” Studies in Ap-plied Electromagnetics and Mechanics 39, 218–225 (2014)."    },    {        "title": "0708.3323v1.Enhancement_of_the_Gilbert_damping_constant_due_to_spin_pumping_in_noncollinear_ferromagnet_nonmagnet_ferromagnet_trilayer_systems.pdf",        "content": "arXiv:0708.3323v1  [cond-mat.mes-hall]  24 Aug 2007Enhancement of the Gilbert damping constant due to spin pump ing in non-collinear\nferromagnet / non-magnet / ferromagnet trilayer systems\nTomohiro Taniguchi1,2, Hiroshi Imamura2\n1Institute for Materials Research, Tohoku University, Send ai 980-8577,\n2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan\n(Dated: October 29, 2018)\nWe analyzed the enhancement of the Gilbert damping constant due to spin pumping in non-\ncollinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert\ndamping constant depends both on the precession angle of the magnetization of the free layer and\non the direction of the magntization of the fixed layer. We find the condition to be satisfied to\nrealize strong enhancement of the Gilbert damping constant .\nPACS numbers: 72.25.Mk, 75.70.Cn, 76.50.+g, 76.60.Es\nThere is currently great interest in the dynamics of\nmagnetic multilayers because of their potential applica-\ntions in non-volatile magnetic random access memory\n(MRAM) and microwave devices. In the field of MRAM,\nmuch effort has been devoted to decreasing power con-\nsumption through the use of current-induced magnetiza-\ntion reversal (CIMR) [1, 2, 3, 4, 5, 6, 7]. Experimentally,\nCIMR is observed as the current perpendicular to plane-\ntype giant magnetoresistivity (CPP-GMR) of a nano pil-\nlar, in which the spin-polarized current injected from the\nfixed layer exerts a torque on the magnetization of the\nfree layer. The torque induced by the spin current is\nutilized to generate microwaves.\nThe dynamics of the magnetization Min a ferromag-\nnet under an effective magnetic field Beffis described by\nthe Landau-Lifshitz-Gilbert (LLG) equation\ndM\ndt=−γM×Beff+α0M\n|M|×dM\ndt,(1)\nwhereγandα0are the gyromagnetic ratio and the\nGilbert damping constant intrinsic to the ferromagnet,\nrespectively. The Gilbert damping constant is an im-\nportant parameter for spin electronics since the critical\ncurrent density of CIMR is proportional to the Gilbert\ndamping constant [8, 9] and fast-switching time magne-\ntization reversal is achieved for a large Gilbert damp-\ning constant [10]. Several mechanisms intrinsic to ferro-\nmagnetic materials, such as phonon drag [11] and spin-\norbit coupling [12], have been proposed to account for\nthe origin of the Gilbert damping constant. In addition\nto these intrinsic mechanisms, Mizukami et al.[13, 14]\nand Tserkovnyak et al.[15, 16] showed that the Gilbert\ndampingconstantinanon-magnet(N) /ferromagnet(F)\n/ non-magnet(N) trilayersystem is enhanced due to spin\npumping. Tserkovnyak et al.[17] also studied spin pump-\ning in a collinear F/N/F trilayer system and showed that\nenhancement of the Gilbert damping constant depends\non the precession angle of the magnetization of the free\nlayer.\nOn the other hand, several groups who studied CIMR\nin a non-collinear F/N/F trilayer system in which theFIG. 1: (Color online) The F/N/F trilayer system is schemat-\nically shown. The magnetization of the F 1layer (m1) pre-\ncesses around the z-axis with angle θand angular velocity ω.\nThe magnetization of the F 2layer (m2) is fixed with tilted\nangleρ. The precession of the magnetization in the F 1layer\npumpsspin current Ipump\nsintotheNandF 2layer, andcreates\nthe spin accumulation µNin the N layer. The spin accumu-\nlation induces the backflow spin current Iback(i)\ns(i= 1,2).\nmagnetization of the free layer is aligned to be perpen-\ndicular to that of the fixed layer have reported the reduc-\ntion of the critical current density [5, 6, 7]. Therefore, it\nis intriguing to ask how the Gilbert damping constant is\naffected by spin pumping in non-collinear F/N/F trilayer\nsystems.\nIn this paper, we analyze the enhancement of the\nGilbert damping constant due to spin pumping in non-\ncollinear F/N/F trilayer systems such as that shown in\nFig. 1. Following Refs. [15, 16, 17, 18], we calculate the\nspin current induced by the precession of the magnetiza-\ntion of the free layer and the enhancement of the Gilbert\ndamping constant. We show that the Gilbert damping\nconstant depends not only on the precession angle θof\nthe magnetization of a free layer but also on the angle ρ\nbetweenthemagnetizationsofthefixedlayerandthepre-\ncession axis. The Gilbert damping constant is strongly\nenhanced if angles θandρsatisfy the condition θ=ρor\nθ=π−ρ.\nThe system we consider is schematically shown in Fig.\n1. A non-magnetic layer is sandwiched between two fer-\nromagnetic layers, F 1and F 2. We introduce the unit2\nvectormito represent the direction of the magnetiza-\ntion of the i-th ferromagnetic layer. The equilibrium\ndirection of the magnetization m1of the left free fer-\nromagnetic layer F 1is taken to exist along the z-axis.\nWhen an oscillatingmagnetic field is applied, the magne-\ntization of the F 1layer precesses around the z-axis with\nangleθ. The precession of the vector m1is expressed\nasm1= (sinθcosωt,sinθsinωt,cosθ), whereωis the\nangular velocity of the magnetization. The direction of\nthe magnetization of the F 2layer,m2, is assumed to be\nfixed and the angle between m2and thez-axis is repre-\nsented byρ. The collinear alignment discussed in Ref.\n[17] corresponds to the case of ρ= 0,π.\nBefore studying spin pumping in non-collinear sys-\ntems, we shall give a brief review of the theory of\nspin pumping in a collinear F/N/F trilayer system [17].\nSpin pumping is the inverse process of CIMR where the\nspin current induces the precession of the magnetization.\nContrary to CIMR, spin pumping is the generation of\nthe spin current induced by the precession of the mag-\nnetization. The spin current due to the precession of the\nmagnetization in the F 1layer is given by\nIpump\ns=/planckover2pi1\n4πg↑↓m1×dm1\ndt, (2)\nwhereg↑↓is a mixing conductance [18, 19] and /planckover2pi1is the\nDirac constant. Spins are pumped from the F 1layer\ninto the N layer and the spin accumulation µNis cre-\nated in the N layer. Spins also accumulate in the F 1\nand F 2layers. In the ferromagnetic layers the trans-\nverse component of the spin accumulation is assumed to\nbe absorbed within the spin coherence length defined as\nλtra=π/|k↑\nFi−k↓\nFi|, wherek↑,↓\nFiis the spin-dependent\nFermi wave number of the i-th ferromagnet. For fer-\nromagnetic metals such as Fe, Co and Ni, the spin co-\nherence length is a few angstroms [20]. Hence, the spin\naccumulation in the i-th ferromagnetic layer is aligned to\nbe parallel to the magnetization, i.e., µFi=µFimi. The\nlongitudinal component of the spin accumulation decays\non the scale of spin diffusion length, λFi\nsd, which is of the\norder of 10 nm for typical ferromagnetic metals [21].\nThe difference in the spin accumulation of ferromag-\nnetic and non-magnetic layers, ∆ µi=µN−µFimi(i=\n1,2), induces a backflow spin current, Iback(i)\ns, flowing\ninto both the F 1and F 2layers. The backflow spin cur-\nrentIback(i)\nsis obtained using circuit theory [18] as\nIback(i)\ns=1\n4π/braceleftbigg2g↑↑g↓↓\ng↑↑+g↓↓(mi·∆µi)mi\n+g↑↓mi×(∆µi×mi)/bracerightbig\n,(3)\nwhereg↑↑andg↓↓are the spin-up and spin-down con-\nductances, respectively. The total spin current flowing\nout of the F 1layer is given by Iexch\ns=Ipump\ns−Iback(1)\ns\n[17]. The spin accumulation µFiin the F ilayer is ob-\ntained by solving the diffusion equation. We assume\nthat spin-flip scattering in the N layer is so weak thatwe can neglect the spatial variation of the spin current\nwithin the N layer, Iexch\ns=Iback(2)\ns. The torque τ1\nacting on the magnetization of the F 1layer is given by\nτ1=Iexch\ns−(m1·Iexch\ns)m1=m1×(Iexch\ns×m1). For\nthe collinear system, we have\nτ1=g↑↓\n8π/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\nm1×dm1\ndt,(4)\nwhereν= (g↑↓−g∗)/(g↑↓+g∗) is the dimensionless\nparameter introduced in Ref. [17]. The Gilbert damping\nconstant in the LLG equation is enhanced due to the\ntorqueτ1asα0→α0+α′with\nα′=gLµBg↑↓\n8πM1dF1S/parenleftbigg\n1−νsin2θ\n1−ν2cos2θ/parenrightbigg\n,(5)\nwheregLis the Land´ e g-factor,µBis the Bohr magneton,\ndF1is the thickness of the F 1layer andSis the cross-\nsection of the F 1layer.\nNext, we move on to the non-collinear F/N/F trilayer\nsystem with ρ=π/2, in which the magnetization of the\nF2layer is aligned to be perpendicular to the z-axis. Fol-\nlowing a similar procedure, the LLG equation for the\nmagnetization M1in the F 1layer is expressed as\ndM1\ndt=−γeffM1×Beff+γeff\nγ(α0+α′)M1\n|M1|×dM1\ndt,(6)\nwhereγeffandα′are the effective gyromagnetic ratio\nand the enhancement of the Gilbert damping constant,\nrespectively. The effective gyromagnetic ratio is given by\nγeff=γ/parenleftbigg\n1−gLµBg↑↓νcotθcosψsinωt\n8πMdF1Sǫ/parenrightbigg−1\n,(7)\nwhere cosψ= sinθcosωt=m1·m2and\nǫ= 1−ν2cos2ψ−ν(cot2θcos2ψ−sin2ψ+sin2ωt).(8)\nThe enhancement of the Gilbert damping constant is ex-\npressed as\nα′=gLµBg↑↓\n8πMdF1S/parenleftbigg\n1−νcot2θcos2ψ\nǫ/parenrightbigg\n.(9)\nItshouldbenotedthat, fornon-collinearsystems, both\nthegyromagneticratioandtheGilbert dampingconstant\nare modified by spin pumping, contrary to what occurs\nin collinear systems. The modification of the gyromag-\nnetic ratio and the Gilbert damping constant due to spin\npumping can be explained by considering the pumping\nspincurrentandthe backflowspincurrent[SeeFigs. 2(a)\nand 2(b)]. The direction of the magnetic moment car-\nried by the pumping spin current Ipump\nsis parallel to\nthe torque of the Gilbert damping for both collinear and\nnon-collinear systems. The Gilbert damping constant is\nenhanced by the pumping spin current Ipump\ns. On the\notherhand, the directionofthe magneticmoment carried3\nFIG. 2: (Color online) (a) Top view of Fig. 1. The dotted\ncircle in F 1represents the precession of magnetization M1\nand the arrow pointing to the center of this circle represent s\nthe torque of the Gilbert damping. The arrows in Ipump\nsand\nIback(1)\nsrepresent the magnetic moment of spin currents. (b)\nThe back flow Iback(1)\nshas components aligned with the di-\nrection of the precession and the Gilbert damping.\nby the backflow spin current Iback(1)\nsdepends on the di-\nrection of the magnetization of the F 2layer. As shown in\nEq. (3), the backflowspin current in the F 2layerIback(2)\ns\nhas a projection on m2. Since we assume that the spin\ncurrent is constant within the N layer, the backflow spin\ncurrent in the F 1layerIback(1)\nsalso has a projection on\nm2. For the collinear system, both Ipump\nsandIback(1)\ns\nare perpendicular to the precession torque because m2\nis parallel to the precession axis. However, for the non-\ncollinear system, the vector Iback(1)\nshas a projection on\nthe precession torque, as shown in Fig. 2(b). Therefore,\nthe angular momentum injected by Iback(1)\nsmodifies the\ngyromagnetic ratio as well as the Gilbert damping in the\nnon-collinear system.\nLet us estimate the effective gyromagnetic ratio using\nrealistic parameters. According to Ref. [17], the con-\nductancesg↑↓andg∗for a Py/Cu interface are given\nbyg↑↓/S= 15[nm−2] andν≃0.33, respectively. The\nLand´ eg-factor is taken to be gL= 2.1, magnetization is\n4πM= 8000[Oe] and thickness dF1= 5[nm]. Substitut-\ning these parameters into Eqs. (7) and (8), one can see\nthat|γeff/γ−1| ≃0.001. Therefore, the LLG equation\ncan be rewritten as\ndM1\ndt≃ −γM1×Beff+(α0+α′)M1\n|M1|×dM1\ndt.(10)\nThe estimated value of α′is of the order of 0.001. How-\never, we cannot neglect α′since it is of the same order\nas the intrinsic Gilbert damping constant α0[22, 23].\nExperimentally, the Gilbert damping constant is mea-\nsuredasthe width ofthe ferromagneticresonance(FMR)\nabsorptionspectrum. LetusassumethattheF 1layerhas\nno anisotropy and that an external field Bext=B0ˆzisapplied along the z-axis. We also assume that the small-\nangle precession of the magnetization around the z-axis\nis excited by the oscillating magnetic field B1applied in\nthexy-plane. The FMR absorption spectrum is obtained\nas follows [24]:\nP=1\nT/integraldisplayT\n0dtαγMΩ2B2\n1\n(γB0−Ω)2+(αγB0)2,(11)\nwhere Ω is the angular velocity of the oscillating mag-\nnetic field, T= 2π/Ω andα=α0+α′. Sinceαis very\nsmall, the absorption spectrum can be approximately ex-\npressedasP∝α0+∝an}bracketle{tα′∝an}bracketri}htandthehighestpointofthepeak\nproportional to ∝an}bracketle{t1/(α0+α′)∝an}bracketri}ht, where∝an}bracketle{tα′∝an}bracketri}htrepresents the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant. In Fig. 3(a), the time-averaged value\n∝an}bracketle{tα′∝an}bracketri}htfor a non-collinear system in which ρ=π/2 is plot-\nted by the solid line as a function of the precession an-\ngleθ. The dotted line represents the enhancement of\nthe Gilbert damping constant α′for the collinear system\ngiven by Eq. (5). The time-averaged value of the en-\nhancement of the Gilbert damping constant ∝an}bracketle{tα′∝an}bracketri}httakes\nits maximum value at θ= 0,πfor the collinear system\n(ρ= 0,π). Contrary to the collinear system, ∝an}bracketle{tα′∝an}bracketri}htof the\nnon-collinear system in which ρ=π/2 takes its maxi-\nmum value at θ=π/2.\nAs shown in Fig. 2(b), the backflow spin current\ngives a negative contribution to the enhancement of the\nGilbert damping constant. This contribution is given by\nthe projection of the vector Iback(1)\nsonto the direction\nof the torque of the Gilbert damping, which is repre-\nsented by the vector m1×˙m1. Therefore, the condition\nto realize the maximum value of the enhancement of the\nGilbert damping is satisfied if the projection of Iback(1)\ns\nontom1×˙m1takes the minimum value; i.e., θ=ρor\nθ=π−ρ.\nWe can extend the above analysis to the non-collinear\nsystemwith anarbitraryvalue of ρ. After performingthe\nappropriate algebra, one can easily show that the LLG\nequation for the magnetization of the F 1layer is given\nby Eq. (6) with\nγeff=γ/bracketleftBigg\n1−gLµBg↑↓νsinρsinωt(cotθcos˜ψ−cscθcosρ)\n8πMdS˜ǫ/bracketrightBigg−1\n(12)\nα′=gLµBg↑↓\n8πMdS/braceleftBigg\n1−ν(cotθcos˜ψ−cscθcosρ)2\n˜ǫ/bracerightBigg\n,\n(13)\nwhere cos ˜ψ= sinθsinρcosωt+ cosθcosρ=m1·m2\nand\n˜ǫ=1−ν2cos2˜ψ\n−ν{(cotθcos˜ψ−cscθcosρ)2−sin2˜ψ+sin2ρsin2ωt}.\n(14)\nSubstituting the realistic parameters into Eqs. (12) and\n(14), we can show that the effective gyromagnetic ratio4\n\u0013\n\u0003\u0013\u0011\u0013\u0013\u0015\u001a\u0003\u0013\u0011\u0013\u0013\u0016\u0015\u0003\u0013\u0011\u0013\u0013\u0016\u001a\n\u000bD\f\n\u000bE\f\u0013\u0011\u0013\u0013\u0016\u001a\n\u0013\u0011\u0013\u0013\u0016\u0015\n\u0013\u0011\u0013\u0013\u0015\u001a/c50/c0f/c12 /c50\n/c51/c1c/c41\n/c1e\n\u0013 /c50/c0f/c12 /c50\n/c52\u0013/c50/c0f/c12/c50\n/c51/c1c/c41\n/c1e\nFIG. 3: (Color online) (a) The time-averaged value of the en-\nhancement of the Gilbert damping constant α′is plotted as a\nfunction of the precession angle θ. The solid line corresponds\nto the collinear system derived from Eq. (9). The dashed\nline corresponds to the non-collinear system derived from E q.\n(5). (b) The time-averaged value of the enhancement of the\nGilbert damping constant α′of the non-collinear system is\nplotted as a function of the precession angle θand the an-\ngleρbetween the magnetizations of the fixed layer and the\nprecession axis.γeffcan be replaced by γin Eq. (6) and that the LLG\nequation reduces to Eq. (10). Figure 3(b) shows the\ntime-averaged value of the enhancement of the Gilbert\ndamping constant ∝an}bracketle{tα′∝an}bracketri}htof Eq. (13). Again, the Gilbert\ndamping constant is strongly enhanced if angles θandρ\nsatisfy the condition that θ=ρorθ=π−ρ.\nIn summary, we have examined the effect of spin\npumping on the dynamics of the magnetization of mag-\nnetic multilayers and calculated the enhancement of the\nGilbertdampingconstantofnon-collinearF/N/Ftrilayer\nsystems due to spin pumping. The enhancement of the\nGilbert damping constant depends not only on the pre-\ncession angle θof the magnetization of a free layer but\nalso on the angle ρbetween the magnetizations of the\nfixed layerand the precession axis, as shown in Fig. 3(b).\nWe have shown that the θ- andρ-dependence of the en-\nhancement of the Gilbert damping constant can be ex-\nplained by analyzing the backflow spin current. The con-\ndition to be satisfied to realizestrongenhancement of the\nGilbert damping constant is θ=ρorθ=π−ρ.\nThe authors would like to acknowledge the valuable\ndiscussions we had with Y. Tserkovnyak, S. Yakata, Y.\nAndo, S. Maekawa, S. Takahashi and J. Ieda. This work\nwas supported by CREST and by a NEDO Grant.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[4] A. Deac, K. J. Lee, Y. Liu, O. Redon, M. Li, P. Wang,\nJ. P. Nozi´ eres, and B. Dieny, J. Magn. Magn. Mater.\n290-291 , 42 (2005).\n[5] A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys.\nLett.84, 3897 (2004).\n[6] K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86,\n022505 (2005).\n[7] T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl.\nPhys. Lett. 89, 172504 (2005).\n[8] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[9] J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J. M. George,\nG. Faini, J. B. Youssef, H. L. LeGall, and A. Fert, Phys.\nRev. B67, 174402 (2003).\n[10] R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett.\n92, 088302 (2004).\n[11] H. Suhl, IEEE Trans. Magn. 34, 1834 (1998).\n[12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[13] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn.\nMater.239, 42 (2002).[14] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[15] Y. Tserkovnyak and A. Brataas and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[16] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n[18] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur.\nPhys. J. B 22, 99 (2001).\n[19] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[20] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[21] J. Bass and W. P. Jr., J. Phys.: Condens. Matter 19,\n183201 (2007).\n[22] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[23] F. Schreiber, J. Pflaum, Th. M¨ uhge, and J. Pelzl, Solid\nState Commun. 93, 965 (1995).\n[24] S. V. Vonsovskii, ed., FERROMAGNETIC RESO-\nNANCE (Israel Program for Scientific Translations Ltd.,\nJersalem, 1964)."    },    {        "title": "1809.09429v1.Theory_of_damping_in_magnetization_dynamics__dispelling_a_myth_and_pointing_a_way_forward.pdf",        "content": "arXiv:1809.09429v1  [cond-mat.mtrl-sci]  25 Sep 2018Theory of damping in magnetization dynamics, dispelling a m yth and pointing a way\nforward\nD M Edwards\nDepartment of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom\nThere is a widely-held belief amongst theoreticians that th e Gilbert damping parameter αin\nmagnetization dynamics is infinite for a pure metal at T=0. Th e basic error leading to this belief\nis pointed out explicitly and the various methods of calcula tion used are viewed in a unified way\nbased on the Lorentzian lineshape of ferromagnetic resonan ce spectra. A general torque formula\nforαis proposed as a good starting-point for treating inhomogen eous materials such as alloys,\ncompounds and layered structures. Local spin density funct ional theory provides a simple physical\npicture, in terms of a non-uniform precessional cone angle i n ferromagnetic resonance, of how such\ninhomogeneity contributes to the damping. In acomplementa ry many-bodytheory this contribution\nis given by a vertex correction to the torque-torque respons e function.\nThe damping of magnetization dynamics in ferromagnetic metals and a lloys is of critical importance in spintronic\ndevices. Damping largely controls the speed at which a device can ope rate and its energy requirement. In device\nphysics damping is usually treated phenomenologically by means of a Gilb ert term in the Landau-Lifshitz-Gilbert\nequation [1, 2] and many quantum-mechanical calculations of the Gilb ert parameter have been made for specific\nmaterials [3–10]. A reliable treatment of damping in transition metals an d alloys would be an invaluable guide in\nthe search for materials with very low damping [11, 12], as required fo r the future development of devices such\nas magnetic random access memory(MRAM). Most recent work in th is direction is concerned with the important\nintrinsic contribution arising from spin-orbit coupling(SOC) and it is th is which concerns us here. A satisfactory\ntheory should work in the limit of a pure metal but almost all existing ca lculations predict that the Gilbert damping\nparameter αdiverges to infinity for a pure metal at T=0. This would mean that in th e pure metals Fe, Co and Ni\nat low temperature the linewidth in a ferromagnetic resonance (FMR ) experiment would be much too large for the\nresonance to be observed. The prediction or acceptance of infinit e damping has been made by so many authors [3–\n10, 13, 14] over the last forty years that it has acquired the stat us of a myth. A very recent paper [15] repeats it\nonce again. It is noteworthy that no experimentalist seems to have troubled to investigate the problem by work\non high purity metals and dilute alloys at low temperature. The aim of th is article is not only to dispel the myth\nbut to formulate a firm starting-point for future calculations of αin technically important materials such as alloys,\ncompounds and layered structures.\nThe most direct method to investigate damping,both experimentally a nd theoretically, is to study the ferromagnetic\nresonance (FMR) linewidth. In FMR a uniform static magnetic field His applied and the absorption of a transverse\nmicrowave field of angular frequency ωpeaks around the frequency bex//planckover2pi1wherebex= 2µBHis the Zeeman energy.\nForHin the z direction the absorption is determined by the imaginary part o f the dynamical transverse susceptibility\nχ−+(ω). This susceptibility, which must include the effect of SOC, can be calc ulated by standard many-body theory\nusing the Kubo formula or by time-dependent spin-density function al theory (SDFT). In practice the many-body\nmethod is usually based on a tight-binding approximation and employs t he random phase approximation (RPA)\nwith a short-range screened Coulomb interaction. This is then equiv alent to a time-dependent Hartee-Fock mean-\nfield theory. The long-range interaction can also be included if care is taken that it does not enter the exchange\nterms [16, 17]. SDFT is approximated similarly as a time-dependent mea n-field theory in the local spin density\napproximation (LSDA) and the long-range Coulomb interaction pres ents no problem since it is effectively screened\nin the exchange-correlation functional. It is useful to consider bo th these methods in parallel. In a system with\nvarying direction of magnetization SDFT is based on a density matrix o f order 2 [18] rather than just spin and\nparticle densities. χ−+(ω) is then coupled to fifteen other response functions which determ ine the longitudinal spin\nsusceptibility as well as the charge response and mixed charge-spin responses [19, 20]. These last relate to phenomena\nlike the spin-Hall effect. Some of these response functions, includin g the longitudinal spin susceptibility, involve the\nlong-range Coulomb interaction importantly even in the absence of S OC [16, 17, 20]. Costa and Muniz [22], following\nan earlier paper [23], show how SOC produces mode coupling in the RPA m any-body approach. However the long-\nrange Coulomb interaction is left untreated. Their paper is particula rly important for being the first to challenge the\nmyth of infinite damping.\nWe firstdiscussthe caseofaBravaislattice whichisappropriatefor puremetalswith acubic structurelikeFe andNi\nat T=0. In both the approaches described above the dynamical su sceptibility is related to mean-field susceptibilities\nof the general form\nχ0(ω) =N−1/summationdisplay\nkmnMmn(k)fkn−fkm\nEkm−Ekn−/planckover2pi1ω+iη. (1)2\nHereEkmis the energy of the one-electron state with wave-vector kin bandm, calculated in the presence of SOC,\nfkmis the corresponding occupation number, Mmn(k) is a product of matrix elements and ηis a small positive\nconstant which ultimately tends to zero. As in usual time-dependen t perturbation theory equation (1) represents the\nresponse to a perturbing field of angular frequency ωin which transitions occur between occupied and unoccupied\nstates. ”Intraband transitions” with m=nclearly do not occur for ω/negationslash= 0 owing to the cancellation of the Fermi\nfunctions. These transitions between identical states, which are not really transitions at all, can play no role in a\ndynamical process. Hankiewicz et al have made a similar point [24]. How ever, in nearly all calculations of the Gilbert\ndamping parameter α, intraband transitions appear and lead to the infinite damping discus sed above.\nTo dispel a myth effectively it is necessary to see how it has arisen. It is instructive to review, in a unified way, some\nmethods which have been used to calculate α. We start from the Lorentzian form of the FMR lineshape which is\nwell-established experimentally [21] and theoretically [22]. Near the re sonance the dynamical transverse susceptibility\nis dominated by a pole at /planckover2pi1ω=bex+/planckover2pi1∆ωwhere ∆ω∼ξ2,ξbeing the SOC parameter, so that\nχ−+(ω) =−2/angbracketleftSz/angbracketright/N\n/planckover2pi1(ω−∆ω)−bex. (2)\nHereSzis thezcomponent of total spin and Nis the number of atoms in the crystal. Near the resonance the FMR\nabsorption is determined by\nℑ(χ−+(ω)) =−2(/angbracketleftSz/angbracketright/N)ℑ(/planckover2pi1∆ω)\n(/planckover2pi1ω−ℜ(/planckover2pi1∆ω)−bex)2+(ℑ(/planckover2pi1∆ω))2. (3)\nℜ(/planckover2pi1∆ω) corresponds to a shift in the resonance frequency and ℑ(/planckover2pi1∆ω) determines the linewidth, both due to SOC.\nThe Gilbert damping factor αis given by ℑ(/planckover2pi1∆ω)/bex(e.g. [25]). The most direct way to calculate αis a brute-force\nnumerical RPA calculation of ℑ(χ−+(ω)), with SOC included, as a function of ωaround the resonance. Costa and\nMuniz [22] performed such calculations using the tight-binding appro ximation and found perfect Lorentzians from\nwhich they deduced α. Taking a monnolayer of Co as an example they found no tendency fo rαto diverge in the pure\nlimit of sharp electronic states. This method of calculating αis very computer intensive and more economic methods\nexist if one assumes a Lorentzian curve from the outset.\nIt follows immediately from (2) that\nα=ℑ(/planckover2pi1∆ω)/bex=2/angbracketleftSz/angbracketright\nNbexℑ(1\nχ−+(bex//planckover2pi1)). (4)\nThis new formula for αmay be regarded as exact. A full treatment of the transverse su sceptibility includes coupling\nto other modes and leads to a rather complex expression in terms of sixteen mean-field susceptibilities of the form (1)\nwith different sets of matrix elements [20]. There is an enormous simplifi cation in the case of a Bravais lattice if we\ncalculate αonly to second order in the SOC parameter ξ. Following the arguments of [20] it is readily found that\ncoupling of the transverse susceptibility to other modes is then elimin ated and that χ−+in (4) may be replaced by\nthe mean-field susceptibility χ0\n−+. This elimination depends on inversion symmetry, which is a property o f a Bravais\nlattice. Without this symmetry, coupling of the transverse suscep tibility to other modes occurs in general even to\norderξ2, as discussed later. It follows further that to order ξ2\nα= (N∆2/2/angbracketleftSz/angbracketrightbex)ℑ(χ0\n−+(bex//planckover2pi1)) (5)\nwhere ∆ is the exchange splitting in the band structure. It is usually s ufficient to calculate the last factor to first\norder in bexso we may take the unphysical limit bex→0, but with due care as discussed below. Then\nα= (N∆2/2/angbracketleftSz/angbracketright)[∂ωℑ(χ0\n−+(ω)]ω=0 (6)\nwhere the electronic state energiesand matrixelements in ℑ(χ0\n−+(ω) are calculatedwith bex= 0. Beforeproceedingto\nthe static ω→0 limit it is essential not to include contributions from ”intraband tran sitions”, as pointed out after (1).\nThis precaution was not taken in [14], where a similar formula was obtain ed, so the spurious infinite damping for a\npure metal appeared. Sometimes it is preferable to keep the physic al non-zero Zeeman field to remove all danger of\nincluding intraband transitions. This also gives the option of calculatin g the frequency-swept FMR linewidth as a\nfunction of Zeeman field. This has been measured [21] and can be con verted to a frequency dependence of α. Such\na dependence has been discussed by Costa and Muniz [22]. However the low-field limit is usually sufficient and here\nwe take the limit bex→0, with the precaution mentioned above, to compare with other the oretical work. Following\n[14], but excluding intraband terms, we find the following two express ions forαat T=0:\nα= (π∆2/2/angbracketleftSz/angbracketright)/summationdisplay\nk/summationdisplay′\nmn|/angbracketleftkm|S−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF)\n= (πξ2/2/angbracketleftSz/angbracketright)/summationdisplay\nk/summationdisplay′\nmn|/angbracketleftkm|T−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF).(7)3\nHereSSS= (Sx,Sy,Sz) is the total spin operator, S−=Sx−iSy,ξhsois the total spin-orbit interaction, T−= [S−,hso]\nis a torque operator and EFis the Fermi energy. The prime on the sum over bands means m/negationslash=nand the sum over k\nis to be carried out as an integral over the Brillouin zone as usual. As p ointed out these expressions are only correct\nto orderξ2so that in the second expression we must evaluate the electronic st ates and energies with ξ= 0. The prime\non the summation sign may then be omitted since the m=nterms are zero owing to inversion symmetry [10]. The\nresulting expression, which can now be written in terms of one-part icle Green functions if desired, is just the version\nof Kambersky’s torque formula [13] for a Bravais lattice derived in tw o ways by Edwards [20]. It is the mean-field\napproximation to a much more general formula [20], valid for an orde red or disordered system,\nα=−(ξ2/2bex/angbracketleftSz/angbracketright)ℑ[χξ=0\nT(bex//planckover2pi1)]. (8)\nWe shall refer to this as the general torque formula. It is exact to orderξ2and we have left open the option of taking\nthe limit bex→0. Here the torque-torque response function is given by the Four ier transform of a retarded Green\nfunction using the Kubo formula\nχT(ω) =/integraldisplay\n/angbracketleft/angbracketleftT−(t),T+/angbracketright/angbracketrighte−iωtdt. (9)\nThe wide application of (8) is discussed later and we recall that the se cond expression in (7), corresponding to the\nmean-field approximation to χT, is only valid for a Bravais lattice. To evaluate the integral over kin the formula (7)\nnumerically it is usual to replace the delta-functions by Lorentzians of width proportional to an inverse relaxation\ntime parameter τ−1. This broadening of the electron states may be regarded as a crud e representation of the effect\nof impurity and/or phonon scattering. The limit τ−1→0 of a perfect crystal at T=0 leads to a finite value of αbut\nis quite tricky to perform numerically [26]. If we wrongly retain SOC in ca lculating the electron states in the second\nexpression of (7) the diagonal matrix elements are non-zero and le ad to the notorious infinite damping parameter\nα. The only work which deals correctly with αin pure metals is reported in fig.1 of [10] and in [22, 26].(In [26] the\ncaption of fig.1 should read ”with and without SOC included in calculating electronic states”).\nWenowturntothetaskofestablishingafirmbasisforcalculatingthe dampingparameter αintechnicallyimportant\nmaterials, which are typically random alloys or layered structures. T his task is greatly simplified if we are satisfied\nwith calculating αto second order in the SOC parameter ξ. This should be sufficient in nearly all systems of interest.\nAt room temperature the ξ2dependence of αis well-established experimentally in several alloy systems, including\nsome containing Pt with its large SOC [27, 28]. The general torque for mula (8) is a very convenient starting-point. Its\nderivation in Appendix A of [20] is for a completely general ferromag netic material, either ordered or disordered, and\nagain relies only on the universal FMR Lorentzian lineshape. The deriv ation proceeds by comparing an exact relation\nbetween χ−+(ω) andχT(ω) with an expansion of (2) in the limit /planckover2pi1∆ω/(/planckover2pi1ω−bex)→0 followed by /planckover2pi1ω→bex. This\norder of limits is essential and results in the form (8) where χTis evaluated in the absence of SOC. A similar formula\nwas derived by Kambersky [13] in another way where crucially the pre scription ξ= 0 did not become apparent. The\nformula is remarkable for describing the essence of a phenomenon a rising solely from SOC without the need to include\nSOC in the calculation.\nThe calculation of χTin a disordered system is still a very demanding problem. It may be app roached using the\nRPA of standard many-body theory or, less obviously, using time-d ependent LSDA. A diagrammatic RPA treatment\nofχTinvolves a sum of ladder diagrams and the first term, without an inter action line, corresponds to the mean-field\napproximation χ0\nT. The remaining terms constitute a vertex correction and we have s hown above that in a monatomic\nBravais lattice this vanishes. In a disordered system like an alloy, or a metal at finite temperature in a frozen phonon\npicture, this is not the case. However this great simplification persis ts if, in a very crude approximation, the system is\nreplaced, at the outset, by an effective medium with the full transla tional symmetry of the lattice but finite electron\nlifetime. We are then led to the Kambersky-like formula (7) for αwith a Lorentzian broadening of the delta-functions\ndetermined by relaxation times which may be dependent on spin and te mperature. This is the background to a\nrecent calculation of αin bulk Ni at room temperature [26] which is in reasonable agreement w ith experiment. A\nproper treatment of χTin a disordered material must deal simultaneously with the RPA verte x correction and any\nvertex corrections which arise in connection with methods of taking a configurational average, such as the coherent\npotential approximation (CPA). There is a small literature on this pr oblem as applied to χ−+, notχT, in a one-band\nmodel [29, 30]. Santos and Costa [30] find that for dilute non-magne tic impurities the RPA vertex correction is\nparticularly important. However as yet the many-body approach is far from being able to provide reliable results for\nαin real disordered materials. The time-dependent LSDA method see ms more promising. As shown below, it gives\na clear physical picture of the RPA vertex correction and separat es it from the configurational averaging problem.\nIn a FMR experiment the local magnetization vector sweeps out a co ne as it precesses around the Zeeman field\ndirection and in the presence of SOC the cone angle θ(rrr) is a function of position. In the time-dependent LSDA θ(rrr)\nsatisfies an integral equation whose solution is avoided in [14] by tak ing a spatially-independent averaged cone angle4\nθ(rrr). This approximation enforces a uniform precession, as occurs in t he absence of SOC, and removes the possibility\nof coupling between transverse and longitudinal susceptibilities. It is very reasonable for a monatomic Bravais lattice\nwhere the variation of θ(rrr) within a unit cell is largely an artificial consequence of the local appr oximation. In the\ntight-binding framework of [20] it would not be an approximation at all for a monatomic Bravais lattice. However, in\ncompounds, alloys and layered structures, variation of the cone a ngle between different types of atom and different\nlayers may be very important. In the many-body approach the ver tex correction in χTis the difference between\nthe fullχTand the mean-field approximation χ0\nT. Since we have seen that the mean-field approximation works well\nfor a homogeneous system, like a monatomic Bravais lattice, we conc lude that the vertex correction corresponds to\nthe effect of the spatial variation of the cone angle, which can be st udied with the LSDA approach. This will be\ndemonstrated explicitly in a forthcoming publication. This productive interplay between standard many-body theory\nand density-functional theory is quite unusual.\nI would like to acknowledge a useful exchange of e-mails with Filipe Guima res on the subject-matter of this paper.\n[1] L.D. Landau, E.M. Lifshitz and L.P. Pitaevski, Statistical Physics, Part 2 (Oxford: Pergamon 1980).\n[2] T.L. Gilbert, Phys. 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Vignale and Y. Tserkovnyak, Phys. R ev. B75174434 (2007).\n[25] D.M. Edwards and O. Wessely, J. Phys. Condens. Matter 21146002 (2009).\n[26] A. Umerski and D.M. Edwards, IOP Conf. Series: Journal o f Physics: Conf. Series 903012056 (2017).\n[27] C. Scheck, L. Cheng, I. Barsukov, Z. Frait and W.E. Baile y, Phys. Rev. Lett. 98117601 (2007).\n[28] P. He, X. Ma, J.W. Zhang, H.B. Zhao, G. L¨ upke, Z. Shi and S .M. Zhou, Phys. Rev. Lett. 110077203 (2013).\n[29] H. Yamada and M. Shimizu, J. Phys. Soc. Japan 311344 (1971).\n[30] E.B. Santos and A.T. Costa, J. Mag. Magn. Mat. 46969 (2019)."    },    {        "title": "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": "2006.07944v2.A_general_formulation_for_the_magnetic_oscillations_in_two_dimensional_systems.pdf",        "content": "arXiv:2006.07944v2  [cond-mat.mtrl-sci]  12 Jul 2021A general formulation for the magnetic oscillations in two d imensional systems\nF. Escudero, J. S. Ardenghi, and P. Jasen\nDepartamento de F´ ısica, Universidad Nacional del Sur,\nAv. Alem 1253, B8000CPB Bah´ ıa Blanca, Argentina\nInstituto de F´ ısica del Sur, Conicet,\nAv. Alem 1253, B8000CPB Bah´ ıa Blanca, Argentina\nWe develop a general formalism for the magnetic oscillation s (MO) in two dimensional (2D)\nsystems. We consider general 2D Landau levels, which may dep end on other variable or indices,\nbesides the perpendicular magnetic field. In the ground stat e, we obtain expressions for the MO\nphase and amplitude. From this we use a Fourier expansion to w rite the MO, with the first term\nbeing a sawtooth oscillation. We also consider the effects of finite temperature, impurities or lattice\nimperfections, assuming a general broadening of the Landau levels. We develop two methods for\ndescribing these damping effects in the MO. One in terms of the occupancy of the Landau levels, the\nother in terms of reduction factors, which results in a gener alization of the Lifshits-Kosevich (LK)\nformula. We show that the first approach is particularly usef ul at very low damping, when only the\nstates close to the Fermi energy are excited. In contrast, th e LK formula may be more convenient at\nhigher damping, when only few terms are needed in its harmoni c expansion. We compare different\ndamping situations, showing how the MO are broadened in each case. The general formulation\npresented allows to relate the properties of the MO with thos e of the 2D systems.\nI. INTRODUCTION\nOne of the most important consequences of the Lan-\ndau quantization are the oscillations of the thermody-\nnamic potentials. For instance, the de Haas-van Alphen\neffect is the oscillation of the magnetization as a func-\ntion of the magnetic field [ 1]. This phenomenon was first\nobservedexperimentally in bismuth in 1930, and theoret-\nically explained by Landau [ 2], but for many years it was\nconsidered to be just a curiosity. It was later shown that,\nby analysing the magnetic oscillations (MO) frequencies,\none could obtain information about the Fermi surface of\nthe material [ 3]. In fact, the MO profile depends on the\nLandau levels (LL) of the system. In ideal conditions,\nany distinctive feature of the LL should be observed in\nthe properties of the corresponding MO [ 4,5]. This is\nparticularly revealing in two dimensional (2D) materi-\nals, which have many unique properties not seen in the\nconventional 2D electron gas [ 6–8].\nFor example, in graphene-like systems [ 9], at low en-\nergy the LL are relativistic and thus not equidistant\n[10,11]. Moreover, whereas the hexagonal lattice of\ngraphene is planar [ 12], in other materials such as sil-\nicene [13–16] or germanene [ 17–19], the hexagonal lattice\nis buckled. Hence the energy bands can be split by ap-\nplying a potential difference between them [ 20–22]. The\nenergy of these systems can also depend strongly on the\nspin-orbit interaction (SOI) [ 23], which in turn can make\nthem a topological insulator [ 24–26]. When the SOI is\nstrong enough, is possible to observe the quantum spin\nHall effect [ 27,28]. The LL are also modified when other\nexternal fields are applied, such as parallel electric field,\nor when the lattice itself is deformed [ 11,29]. In addi-\ntion, the LL may depend on the spin or valley properties\nofthe 2Dsystem [ 25,30,31], which can alsoinfluence the\nelectronic and magnetic properties [ 32,33]. Thus, there\ncan be a rich manipulation of the LL in 2D materials,all of which should be reflected in the corresponding MO\n[34].\nDamping effects, due to temperature or impurity scat-\ntering, tend to broaden and reduce the MO amplitude\n[35–37]. Besides the need for relatively strong magnetic\nfields, this damping of the MO is the reason why they\nare difficult to observe [ 38]. The damping effects are\nusually considered as reduction factors in the MO ampli-\ntude, which results in the known Lifshits-Kosevich (LK)\nformula [ 39]. This formula is particularly useful in the\ncase of relatively high damping, where the infinite series\nthat gives the LK formula can be exactly solved [ 40],\nor well approximated with the first few harmonic terms.\nThat is commonly the case in 3D metals, where the os-\ncillations can be treated as simple harmonic. Neverthe-\nless, the higher the damping, the more details are lost\nin the MO. In 2D materials, this loss can be significant,\nas many properties of the system, such as the interplay\nbetween valley and spin, would no longer be observed\nin the MO. Another approach to treat the damping of\nthe MO, recently developed in pristine graphene and 2D\nmaterials [ 37,38], considers the temperature effect by its\nmodification to the MO in the ground state. At zero\nimpurities, this modification is given by Fermi-Dirac dis-\ntribution functions, and thus they represent the damping\nin the MO due to the change in the occupancy of the LL.\nThis proved to be useful at very low temperatures, when\nonly the states nearby the Fermi energy are excited and\nthedampingislowenoughtoobservetheMOfinedetails.\nDespite the recent advances in the study of the MO in\n2D materials [ 41–48], the analysis usually considers spe-\ncial cases, such as assuming only a perpendicular mag-\nnetic or electric field [ 49–54]. There is no general for-\nmulation of the MO, in case the LL depend on other\nvariables or indices. When damping effects are consid-\nered, they are usually described using the LK formula,\nassuming a Lorentz distribution for the LL broadening2\n[55]. However, there is no general consensus on how is\nthe disorder in 2D materials, in the presence of a mag-\nnetic field [ 56,57]. Depending on the situation, one im-\npurity distribution could be more suitable than another\n[58–62]. Thus, it is important to have a detailed analy-\nsis of how different broadening distributions modify the\nMO. In fact, careful measurements of the MO can be one\nof the most reliable methods to obtain information about\nthe LL broadening [ 63].\nMotivatedbythis, in thispaperwedevelopedageneral\nformulation for the magnetic oscillations in 2D systems,\nconsidering a general expression for the LL, and taking\ninto account damping effects such as finite temperature,\nimpurities or other lattice imperfections. We have or-\nganized this work as follow: in section IIwe define the\ngeneral LL considered, assuming that they may depend\ninanyvariablesandindices, andwedevelopageneralfor-\nmulation to obtain the corresponding ground state MO.\nWe obtain expressions for the sawtooth oscillation am-\nplitudeAand the oscillating phase ψ. From these we use\na Fourier expansion which allow us to write the MO in\nterms ofAandψ. In section IIIwe consider the damp-\ning effects in the MO. We model the effect of impurities\nor lattice imperfections by considering a general density\nof states that broadens each LL. From this we obtain an\nexpression for the damping effects in the MO, which de-\npendsontheoccupancyofeachLL.Wethenshowthat, if\nthe broadeningis the sameforallLL, the dampingeffects\ncan also be described by reduction factors, which leads\nto a generalization of the LK formula. As an example\nwe consider the MO in graphene-like systems, comparing\nboth approachesfor different damping cases. Finally, our\nconclusions follow in section IV.\nII. GROUND STATE MAGNETIC\nOSCILLATIONS\nWe consider a 2D system under a perpendicular mag-\nnetic fieldB, with discrete LL εn;γ(B;X), wherenis the\nLL index. We denote with γall the other indices with\nwhich the energy levels may depend, such as the spin\ns, the valley η, or the wave vector k. WithXwe de-\nnote all the other variables (besides B) with which εn;γ\nmay also depend, such as the perpendicular (parallel)\nelectric field E⊥/parenleftbig\nE/bardbl/parenrightbig\nand/or the parallel magnetic field\nB/bardbl. It is worth emphasizing that in all cases there is\na perpendicular magnetic field Bwhich causes the dis-\ncretenLL, which is why we separate them. The other\nindicesγand variables Xmay or not be present, depend-\ning on the specific case studied. In general, each energy\nlevel has a degeneracy ̺, which could be the degeneracy\nD=ABe/h(whereAis the 2D system area), in case the\nenergy levels do not depend on the wave vector, and/or\nany additional degeneracyof the system, such as the spin\nor valley degeneracy; the specific value of ̺depends on\nthe particular system. We shall set the zero energy to be\nbetween the valence band (VB) and the conduction band(CB).\nIntroducing the decreasing sorting index mfor the CB\nenergy levels εm, for a constant chemical potential µ>0,\nthe ground state grand potential Ω0is\nΩ0= Ω0\nVB+f/summationdisplay\nm=1̺(εm−µ), (1)\nwhere Ω0\nVBis the valence band contribution and fis the\nlast position such that εf≤µ<εf+1. The oscillation in\nΩ0is produced whenever fchanges, and hence only the\nCB contributes to the quantum oscillations. Therefore,\nsince we are interested in the MO, we will omit the VB\nand work only with the CB grand potential. The magne-\ntization isM0=−A−1/parenleftbig\n∂Ω0/∂B/parenrightbig\nµ, whereAis the 2D\nsystem area. For the CB we get\nM0\nCB=−∂̺\n∂BΩ0\nCB\nA̺−f/summationdisplay\nm=1̺\nA∂εm\n∂B, (2)\nwhere Ω0\nCB=/summationtextf\nm=1̺(εm−µ). Given that Ω0is always\ncontinuous, when fchanges to f+1 we have\n∆M0=−̺\nA∂εf+1\n∂B(Bf). (3)\nwhereBfis the magnetic field when the change occur\n(determined by εf+1(Bf) =µ). This last equation gives\nthe discontinuityin the groundstatemagnetizationwhen\nthe energy levels crossthe chemical potential. The minus\nsign just tell us that the magnetization decreases when\nthe number of filled states increases, given our initial as-\nsumptionf→f+1.\nWe shall now separate the magnetization peaks by the\nγindices, as previously defined. For each γindex, we\nconsiderthemagnetizationpeaksduetothechangeofthe\nnLL index. To see how this works, consider in general\nthe peak that occurs when εn;γ=µ, for eachγindex.\nThen we can solve εn;γ=µforn, obtaining in general\nεn;γ=µ−→n=ψγ(µ,B,X). (4)\nThe function ψγwill depend on µand on the energy\nlevels, but in all cases ψγdetermines when the peaks\noccurs, namely when ψγ=nwithna positive integer.\nFor each index γ, in the small-field limit the oscillat-\ning part of Ω γandMγcan be written in a Fourier series\n(see appendix A). ForMγ, the first and dominant term\nin the series is a sawtooth oscillation (SO). The SO am-\nplitudeAγis the discontinuity in Mγgiven by equation\n(3). Thus, considering the function ψγ, we have\nAγ=−̺\nA∂εn;γ\n∂B/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nn=ψγ. (5)\nThen, the SO associated with each γcan be written as\nM0\nSO,γ=Aγ∞/summationdisplay\np=11\nπpsin(2πpψγ). (6)3\nThe next order (non-sawtooth) term in the MO can be\neasily obtained considering that Ω0is a continuous func-\ntion andM0\nosc=−A−1/parenleftbig\n∂Ω0\nosc/∂B/parenrightbig\nµ. This implies that\nΩ0\noscis of the form Ω0\nosc=/summationtext∞\np=1C2cos(2πpψγ)/(πp)2,\nwhereC2is apindependent factor that satisfies\n2C2(∂ψγ/∂B)/A=Aγ. Therefore we obtain\nΩ0\nosc=/summationdisplay\nγAAγ\n2ψ(1)\nγ∞/summationdisplay\np=1cos(2πpψγ)\n(πp)2(7)\nM0\nosc=/summationdisplay\nγ/braceleftigg\nM0\nSO,γ+1\n2ψ(1)\nγ/bracketleftigg\nAγ\nψ(1)\nγψ(2)\nγ−/parenleftbigg∂Aγ\n∂B/parenrightbigg/bracketrightigg\n×∞/summationdisplay\np=1cos(2πpψγ)\n(πp)2/bracerightigg\n, (8)\nwhere we noted ψ(n)\nγ= (∂nψγ/∂Bn), thenth derivative\nofψγwith respect to B. These harmonic expansions\nare in agreement with the small-field limit expressions\nobtained in classical [ 40] and graphene-like [ 51,57] 2D\nsystems. The expression ( 8) implies that, for any 2D en-\nergy levels εn;γ, the corresponding MO are determined\nby just two parameters: the dimensionless phase func-\ntionψγand the SO amplitude Aγ, which can be easily\nobtained from equations ( 4) and (5). The second term\ninM0\noscaffects the MO by changing the curvature of the\noscillations, but usually it can neglected if ψγ≫1. To\nsee this, consider that in general ( ∂Aγ/∂B) is very small\nifAγdepends on B, whereasψγdecreases with Bifεn;γ\nincreases with B, as usual. Then the amplitude ratio\nbetween the cosine and sine series in equation ( 8) is of\norder|Acos/Asin| ∼1/ψγ. Therefore at high occupancy\nthe cosine series can be neglected and the MO can be\napproximated as a pure sawtooth oscillation. Hence, if\nψγ≫1 we simply have\nM0\nosc≃/summationdisplay\nγAγ\nπarctan[cot( πψγ)]. (9)\nIt is worth noting that equation ( 9) is valid at high oc-\ncupancy, which generally requires relatively low Band\nhighµ.\nIII. MAGNETIC OSCILLATIONS WITH\nDAMPING\nWe shall consider the damping effects due to impu-\nrity scattering, or any other lattice imperfection, by the\nbroadening of the energy levels, so each εn;γwill have a\ndensity of states (DOS) ρn;γ(ε,εn;γ,Γn;γ), where Γ n;γis\nthe parameter that defines the broadening. In general,\ndepending on how is the DOS and the damping mecha-\nnism, the parameter Γ n;γmay depend on the same vari-\nables and indices as the the energy levels εn;γ. We will\nneglect the damping effects due to any many-body inter-\naction, such as the scattering of electrons by phonons.This is a reasonable assumption, given the high Debye\ntemperature of common 2D systems [ 64–69], which are\nwell above the very low temperature required to observe\nthe MO. Then the grand potential is\nΩ =−̺\nβ/integraldisplay∞\n−∞ρln/bracketleftig\n1+eβ(µ−ε)/bracketrightig\ndε, (10)\nwhereβ= 1/kBT,̺is the degeneracy of each energy\nlevel (as in the ground state) and ρ=/summationtext\nn;γρn;γis the\nDOS. Notice that we separate the degeneracy ̺and the\nbroadening ρof the LL; the total DOS would be ̺ρ. In-\ntroducing again the energy sorting index m, as done in\nthe previous section, we can write Ω =/summationtext\nmΩm, with\nρm= (ε,εm,Γm). Then, for each mwe have\n/parenleftbigg∂Ωm\n∂B/parenrightbigg\nµ=∂̺\n∂BΩm\n̺−̺\nβ/integraldisplay∞\n−∞∂ρm\n∂Bln/bracketleftig\n1+eβ(µ−ε)/bracketrightig\ndε.\n(11)\nWeshallassumetheDOS ρmtobesymmetricaround εm,\nsuch that∂ρm/∂εm=−(∂ρm/∂ε). Considering that Γ m\nmay also depend on B, integrating by parts we obtain\n/parenleftbigg∂Ωm\n∂B/parenrightbigg\nµ=I1,m+I2,m+I3,m, (12)\nwhere\nI1,m=−1\nβ∂̺\n∂B/integraldisplay∞\n−∞ρmln/bracketleftig\n1+eβ(µ−ε)/bracketrightig\ndε (13)\nI2,m=̺∂εm\n∂B/integraldisplay∞\n−∞ρm1\n1+eβ(ε−µ)dε (14)\nI3,m=−̺\nβ∂Γm\n∂B/integraldisplay∞\n−∞∂ρm\n∂Γmln/bracketleftig\n1+eβ(µ−ε)/bracketrightig\ndε.(15)\nFrom this, we will first consider the damping effects as a\nmodification of the ground state MO, which depends on\nthe occupancy of the LL. Then we will prove that this is\nequivalent to describe the damping effects with reduction\nfactors, as in the LK formula.\nA. Damping effects in terms of the occupancy of\nthe Landau levels\nThe idea is to describe the MO with damping in terms\nof its modification to the ground state MO. From equa-\ntions (1) and (12), summing over mwe can separate\n/parenleftbigg∂Ω\n∂B/parenrightbigg\nµ=/parenleftbigg∂ΩVB\n∂B/parenrightbigg\nµ+/parenleftbigg∂Ω0\nCB\n∂B/parenrightbigg\nµ+∞/summationdisplay\nm=1I3,m\n+f/summationdisplay\nm=1/bracketleftbigg\nI1,m−∂̺\n∂B(εm−µ)/bracketrightbigg\n+∞/summationdisplay\nm=f+1I1,m\n+f/summationdisplay\nm=1/bracketleftbigg\nI2,m−̺∂εm\n∂B/bracketrightbigg\n+∞/summationdisplay\nm=f+1I2,m,(16)4\nwherefis the last sorted position such that εf≤µ <\nεf+1, as considered in the previous section. Since we\nare interested only in the MO, we shall neglect the non-\noscillatory contribution of the VB. In the appendix Bwe\nshow that if βµ≫1 andµ/Γm≫1, we can neglect the\nterms with I1,mandI3,m.\nAs done previouslyin the groundstate, it is convenient\nto separate the MO in the indices γ. For eachγ, the final\npositionfγ, determined by εfγ;γ≤µ < εfγ+1;γ, can be\nobtained from the ψγfunction as fγ= floor(ψγ) = [ψγ]\n(see equation ( 4)). We define the occupancy factor Fn;γ,\nfor each energy level εn;γ, as\nFn;γ=/integraldisplay∞\n−∞ρn;γ1\n1+eβ(ε−µ)dε. (17)\nAt low damping the occupancy factor Fn;γonly changes\n(relative to its ground state value) when εn;γis close to\nµ. Thus it is good approximation to expand each εn;γ\naroundn=ψγand take (̺/A)∂εn;γ/∂B=−Aγ, where\nAγis the SO amplitude in the ground state, given by\nequation ( 5). In this way, from equations ( 14) and (16)\nwe obtain, for each γ, the MO with damping\nMosc,γ=M0\nosc,γ+Aγ/summationdisplay\nn≤fγ(Fn;γ−1)+Aγ/summationdisplay\nn>fγFn;γ.\n(18)\nThe total magnetization is obtained by summing over all\nγ. The MO, as expressed by equation ( 18), allows a sim-\nple interpretation. First of all, we should notice that in\nthe groundstatewe have Fn;γ= 1ifn≤fγandFn;γ= 0\nifn > fγ, soMosc,γ→M0\nosc,γas expected. At non zero\ndamping, equation ( 18) tell us that the MO are modified\nby the inclusion of the last two terms. Each one of these\ncan be viewed as representing the change in the MO due\nto the modification of the available states. This is deter-\nmined by the occupancy factor Fn;γof each state εn;γ,\nand the corresponding modification to the MO, given by\nthe oscillation amplitude Aγ. In other words, the states\nn≤fγ(fully occupied in the ground state) are broad-\nened and thermally excited, which tends to vacate them\n(Fn;γ<1), changing the MO by a factor Aγ(Fn;γ−1).\nThe states n > fγ(empty in the ground state) are also\nbroadened and thermally excited, which tends to occupy\nthem (Fn;γ>0), changing the MO by a factor AγFn;γ.\nEquation ( 18) can be conveniently rewritten in differ-\nent ways, depending on the situation. For instance, if\nψγ≫1 such that M0\nosc,γis given by equation ( 9), then\none can use the properties of the arc tangent function to\nrewrite [37]\nMosc,γ=Aγ\nπarctan/bracketleftigg\ncot/parenleftigg\nπψγ−π/summationdisplay\nnFn;γ/parenrightigg/bracketrightigg\n.(19)\nThis expression would be particularly useful if one is in-\nterested in the overall profile of the MO, when many\npeaks are considered. On the other hand, when one anal-\nyses the damping effects over each MO peak, it is usefulto rearrange Mosc,γconsidering only the nearby states,\nbecausethesearetheonesthatprimarilymodifytheMO.\nFor instance, around the magnetization peak at ψγ=n0,\nit isgoodapproximationtoconsideronlythe factors Fn;γ\nclose ton=n0. In this way we can approximate\nMosc,γ≃M0\nosc,γ+AγFn0;γ\n+Aγ(Fn0−1;γ+Fn0+1;γ−1)+...(20)\nThe first order approximation is Mosc,γ≃M0\nosc,γ+\nAγFn0;γ, which is just the damping effects due only to\nthen0state. Around the n0magnetization peak, this is\nalways the dominant term modifying the MO. Thus, at\nvery low damping, the modification of the MO around\nthen0peak can be studied only by analysing the occu-\npancy factor Fn0;γ.\nB. Damping effects as reduction factors\nWeshallshowthat, ifthe broadeningisthesameforall\nLL (so that Γ does not depend on n), then the damping\ncan be described with reduction factors. We start by\nrewriting equation ( 18) as\nMosc,γ(µ) =/integraldisplay∞\n−∞δ(ε−µ)M0\nosc,γ(ε)dε−∞/summationdisplay\nn=1̺\nA∂εn;γ\n∂B\n×/integraldisplay∞\n−∞[Fγ−H(ε−µ)]δ(ε−εn;γ)dε,\n(21)\nwhereH(ε−µ) is the Heaviside function and\nFγ=/integraldisplay∞\n−∞ργ(ε′−ε)nF(ε′−µ)dε′,(22)\nwherenF=/bracketleftig\n1+eβ(ε′−µ)/bracketrightig−1\nis the Fermi-Diracdistribu-\ntion. Integrating by parts the second integral in equation\n(21) result\nMosc,γ(µ) =/integraldisplay∞\n−∞δ(ε−µ)M0\nosc,γ(ε)dε+∞/summationdisplay\nn=1̺\nA∂εn;γ\n∂B\n×/integraldisplay∞\n−∞/bracketleftbig\nF′\nγ+δ(ε−µ)/bracketrightbig\nH(ε−εn;γ)dε,\n(23)\nwhere\nF′\nγ=/integraldisplay∞\n−∞ργ(ε′−ε)∂nF\n∂ε′(ε′−µ)dε′.(24)\nThus, from equation ( 2) we obtain\nMosc,γ(µ) =−/integraldisplay∞\n−∞ργ(ε′−ε)/bracketleftbigg\n−∂nF\n∂ε′/bracketrightbigg\nM0\nosc,γ(ε)dεdε′\n(25)5\nHence the damping effects are expressed as convolutions\nof the ground state magnetization, which is the basis of\nthe LK formula. From this, the MO with the reduction\nfactors can be obtained by replacing in equation ( 24) the\nSO ofM0\nosc,γ(ε), given by equation ( 6). This is done in\nthe appendix C, where it is shown that\nMosc,γ=Aγ∞/summationdisplay\np=1RT,γ(p)RΓ,γ(p)sin(2πpψγ)\nπp,(26)\nwhereRT,γ(p) andRΓ,γ(p) are the temperature and\nbroadening reduction factors, respectively, given by\nRT,γ(p) =/parenleftbigg2π2p\nβ∂ψγ\n∂µ/parenrightbigg\ncsch/parenleftbigg2π2p\nβ∂ψγ\n∂µ/parenrightbigg\n(27)\nRΓ,γ(p) =/integraldisplay∞\n−∞ργ(x)cos/bracketleftbigg\n2πpx∂ψγ\n∂µ/bracketrightbigg\ndx. (28)\nThe MO, as given by equation ( 26), is a generalization of\nthe LK formula in 2D, for a general system with energy\nlevelsεn;γ. ThetotalMOisobtainedbysummingoverall\nγ. The expression for RT,γ(p) is the known temperature\nreduction factor of the LK formula, generalized to be\ndependent on ψγ. The expression for RΓ,γ(p) depends\non how is the Landau level broadening. In the appendix\nDwe obtainRΓfor some common DOS distributions.\nEquation ( 26) is particularly useful when the infi-\nnite sum can be solved, as it is the case in some spe-\ncial situations. For instance, for a Lorentz distribu-\ntion, the broadening factor is the known Dingle factor\nRΓ,γ(p) = exp( −2πpΓγ∂ψγ/∂µ) (see appendix D), for\nwhichatzerotemperaturetheinfinitesumgivenbyequa-\ntion (26) can be solved. Likewise when the temperature\nis relatively high and RT,γ(p) can be approximated as a\nsimple exponential. This is a very good approximation\nforthe MOin metals[ 40], but it doesnot holdat the very\nlow temperatures required to observe the fine details of\nthe MO in 2D materials [ 37,38].\nFrom equations ( 27) and (28) we can already see that\nthereductionfactorsdepend ontheenergylevelsthrough\nthe term (∂ψγ/∂µ). For classical 2D LL we have ψγ∼µ,\nwhereas for relativistic LL (like in graphene) we have\nψγ∼µ2(see equation ( 4)). Thus in the classical 2D\nelectron gas, the reduction factors do not depend on the\nchemical potential, whereas in graphene they are propor-\ntional toµ[57].\nC. Analysis of the two damping descriptions\nWe shall compare and discuss the two methods of de-\nscribing the damping effect in the MO, as developed in\nsections IIIAandIIIB, considering broadening distri-\nbutions with constant Γ. As an example, we consider\nrelativistic 2D Landau levels, which occur in graphene-\nlike systems. For the sake of simplicity we ignore any\neffect of spin splitting or valley mixing. The energy lev-\nels areεn=a√\nBn, whereais material dependent con-\nstant. The corresponding phase of the ground state MOisψ= (µ/a)2/B(see equation ( 4)). We consider Band\nµsuch thatψis relatively high, in which case M0\noscis\ngiven by equation ( 9).\nWe will consider the simple cases of (a) finite tem-\nperature and zero broadening, (b) zero temperature with\nGaussianbroadeningand(c)zerotemperaturewithsemi-\nelliptic broadening. In each case, the occupancy factor\nFnand the reduction factors Rare obtained in the ap-\npendixD. It should be noted that we are considering\ncases in which the infinite series of equation ( 26) cannot\nbe solved. Thuswe do not consider, for instance, the case\nofbroadeningwith aLorentzdistributionat zerotemper-\nature (see above discussion). For each case, the MO are\ngiven by equations ( 18) and (26). In general, each ex-\npression can be approximated. We consider a magnetic\nfieldBsuch thatn0−1<ψ<n 0so [ψ] =n0−1, where\nn0≫1. Then for equation ( 18) we shall use the first\norder in equation ( 20), whereas in equation ( 26) we take\nthe upper limit pMin the summation.\nIn Figure 1it is shown the MO as a function of ψ,\nwith fixedµ= 0.25 eV. For the three cases, we show the\ndamping effects around the peak n0= 50, considering\nn0−1/2< ψ < n 0. In each case we plot the SO M0\nosc\n(black solid line) and the MO given by the LL occupancy\napproach (top figures) and the reduction factor approach\n(bottom figures), considering its approximations. The\ntemperature Tand broadening parameter Γ were specif-\nically chosen to show a similar MO amplitude reduction,\nconsidering the two cases of lowandhighdamping. As\nwe can observe, the two ways to describe the MO with\ndamping give the same result, as expected. Neverthe-\nless, they differ substantially in how effective they are,\nparticularly depending if the damping is high or low.\nAt low damping, we see that the occupancy factor de-\nscription (top figures) can be done by just considering\nthe occupancy factor of the n0LL. This can be very con-\nvenientin orderto study howthe increaseoftemperature\nor impurities affects the observation of the fine details in\nthe MO of 2D materials, such as those due to the spin\nsplitting or valley mixing. In other words, at low damp-\ning the broadening of any SO is entirely dictated by how\nis the occupancy factor of the LL that causes the SO\npeak. In contrast, at low damping the reduction factor\ndescription of the MO requires more terms. As can be\nseen in the bottom figures, at low damping one needs\nto consider many terms in the harmonic summation of\nequation ( 26), in order to converge the series. This im-\nplies that one cannot describe the MO with just the first\nterms in the series, as is usually done when using the LK\nformula in metals.\nAt higher damping we observe a different behaviour.\nInthetopfiguresweseethattheMOcannotbedescribed\nby just the occupancy of the n0LL. This can be seen in\nthe (a) and (b) cases, although it should be mentioned\nthat the MO can still be perfectly described by the next\nterm in the expansion given by equation ( 20). That is,\nforT= 4 K and Γ /kB= 9.5 K (Gauss broadening), we\nhaveMosc/A=M0\nosc/A+Fn0+ (Fn0−1+Fn0+1−1).6\nFIG. 1. MO of a system with 2D relativistic Landau levels εn=a√\nBnwithµ= 0.25 eV, as a function of the MO phase\nψ= (µ/a)2/B, around the peak at ψ=n0= 50. In the top figures we plot the MO with the damping effects in terms of\nthe Landau level occupancy. In the bottom figures we plot the M O with the damping effects as reduction factors. In both\napproaches we show the approximations given by the occupanc y factor of the n0Landau level (top figures) and the sum of\nthe firstpMharmonic terms (bottom figures). We consider the damping cas es of: (a) finite temperature and zero broadening;\nzero temperature and broadening Γ with (b) gaussian and (c) s emi-elliptic distribution. In each case we consider lowandhigh\ndamping situations, with the damping parameters chosen to r epresent similar MO amplitude reduction. In all cases it is a lso\nplotted, in black solid line, the ground state magnetic sawt ooth oscillation.\nThus the difference between the first order approxima-\ntion (blue dot-dashed line) and the total MO (red solid\nline) is caused by the change in the occupancy of the\nnearbyn0−1 andn0+1 LL. For the (c) case we observe\nthat even at relatively high damping, the MO depend\non just the occupancy of the LL n0. This is expected\nconsidering the nature of the semi-elliptic broadening, in\nwhich if Γ <∆εnthen the broadening of each level is\nindependent. On the other hand, in the bottom figures\nwe observe that at highdamping the reduction factor\ndescription requires fewer harmonic terms. This can be-\ncome convenient, especially when the reduction factors\ncan be approximated so that the summation in equation\n(26) can be solved.\nD. Comparison between different broadening\ndistributions\nThe damping of the MO can depend significantly on\nthe broadening of the LL. This not only depends on the\ntype of broadening distribution, but also on the broad-\nening parameter Γ itself. In Figure 2we show the MO of\na system with 2D relativistic Landau levels εn=a√\nBn\nwithµ= 0.25 eV (as in Figure 1), at zero temperature\nand for broadening of type Gauss (red solid line), semi-\nelliptic (blue dashed line) and Lorentz (green dot-dashed\nline). Specifically, in Figure 2(a) we show the MO as a\nfunction of the phase ψ= (µ/a)2/B, around the peak at\nψ=n0= 50, considering different broadening parame-ters Γ. In Figure 2(b) we show the MO as a function of\nthe Γ, at different fixed phases ψ(which implies constant\nmagnetic field).\nAs we can see in Figure 2(a), the shape of the MO\nwith damping parameter Γ depends strongly on the type\nof broadening distribution. The Lorentz distribution de-\ncays always faster, whereas the semi-elliptic distribution\ndecays always slower. At low damping, for both the\nGauss and the semi-elliptic distributions, the MO are\nnot damped if the phase is not close to n0= 50. In\nother words, further from ψ= 50, the MO reduce to the\nground state magnetization (sawtooth oscillation in this\ncase). The same behaviour was also seen in Figure 1,\nwhere at low damping the MO eventually reduces to the\nground state MO (black solid line). The reason is that\nat low damping, both the Gauss and semi-elliptic distri-\nbutions decay fast and thus Fn0≪1 unlessψis close\nton0. This is not the case for the Lorentz distribution,\ndue to its long rangewhich implies that Fn0decaysmuch\nslower.\nThe dependence of the MO decay with the type of\nbroadening can be also seen in Fig 2(b). We observe that\nthe decay of the MO depends on how far is the phase ψ\nfromn0, at whichthe energylevel crossthe Fermi energy.\nIn the three considered ψ, the top value in the figure cor-\nresponds to the value of the ground state magnetization\nM/A= arctan[cot( πψ)]/π, which for 49 .5< ψ <50\nimpliesM/A=ψ−49.5. Hence, for the Gauss and semi-\nelliptic distributions, we see that the further ψis from\nn0(for 49.5< ψ <50), the larger is the broadening Γ7\nFIG. 2. MO of a system with 2D relativistic Landau levels\nεn=a√\nBnwithµ= 0.25 eV, for different broadening distri-\nbutions at zero temperature. In (a) we plot the MO in terms\nof the MO phase ψ= (µ/a)2/B, around the peak at ψ= 50,\nconsidering different broadening parameters Γ. In (b) we plo t\nthe MO as function of the Γ, at different phases ψ.\nrequired to damp the MO. For instance, at ψ= 49.75 we\nobserve the MO are damped only if Γ /kB>3 K for the\nGauss distribution and Γ /kB>7 K for the semi-elliptic\ndistribution. In contrast, for the Lorentz distribution the\nMO are practically always damped, regardless of ψand\nΓ.\nE. Chemical potential oscillations\nThe presented formulation assumes a constant chemi-\ncal potential µ, which necessarily implies that the num-\nber of electrons is not constant as the magnetic field is\nvaried. The satisfaction of this condition depends on the\nconfiguration of the system, such as the application of a\ngatevoltage. Nevertheless, inmostcasesit istheelectron\ndensityne=N/Aof the system that is kept constant. In\nthat situation one has to obtain how should µoscillate as\na function of Bin order to maintain neconstant [ 70,71].\nIn three dimensional (3D) metals, the oscillation in µ\nis very small and usually can be neglected [ 40]. In 2D\nsystems, these oscillations can be appreciable and thus\none needs to be careful about which situation (constant\nchemical potential or electron density) best represents\nthe system [ 35]. The oscillations of µare obtained from\nN=−(∂Ω/∂µ), which using equations ( 10) and (17),\ncan be written as\nN=̺/summationdisplay\nγ/summationdisplay\nnFn;γ. (29)\nFrom this equation one should solve µ=µ(ne,B,β,Γ),\nwhich in general can only be done numerically, although\nat low damping it is possible to make some simplifica-\ntions. For instance, it is known that in the ground state,µreduces to the last partially occupied energy level [ 35].\nThus at low damping it is good approximation to con-\nsider only the µdependence in the Fn;γwithεn;γclose\nto the last occupied energy level, considering for the rest\nFn;γtheir ground state value (1 or 0).\nIV. CONCLUSIONS\nWe developed a general formalism to obtain the mag-\nnetic oscillations (MO) in two dimensional (2D) systems.\nWe considered a general expression for the 2D Landau\nlevels (LL), which may depend not only on the perpen-\ndicularmagneticfieldthat causesthem, but alsoonother\nvariables and indices. This may included perpendicular\nand parallel electric fields, or the spin, valley and wave\nvector indices. In the ground state of the system, we\nobtained expressions for the MO amplitude and phase.\nUsing this and a Fourier expansion, we wrote the MO as\na sum of elemental oscillations. The first term is a saw-\ntooth oscillation, which becomes dominant when many\nenergy levels are occupied.\nThe effects of temperature, impurities or lattice im-\nperfections were also considered. We assumed a general\ndensity of states that broaden each energy level, taking\ninto account that the broadening parameter may depend\non the magnetic field and the LL index. From this we\nobtained an expression for the damping effects in the\nMO, which depends on the occupancy of the LL. We\nthen showed that, if the broadening parameter does not\ndepend on the LL, the damping effects can also be de-\nscribed using reduction factors, which resulted in a gen-\neralization of the known Lifshits-Kosevich (LK) formula.\nWe compared both approaches, for graphene-like LL and\ndifferent types of damping effects. At low damping, we\nshowedthattheMOdependonlyontheoccupancyfactor\nof the LL closest to the Fermi energy, whereas at higher\ndamping one needs to consider also the occupancy of the\nnearby levels. In contrast, the lower the damping the\nmore harmonic terms one needs to consider in the LK\nformula in order to converge the series; at higher damp-\ning, only the first few harmonic are needed.\nThegeneralformulationpresentedgivesarelativelydi-\nrect way to obtain the MO in 2D systems, by calculating\nits more important properties: the amplitude and phase\nof the oscillations. The expressions obtained show that\nthe MO is mostly dependent on the properties of the en-\nergy levels. Hence, by measuring the MO profile, one\ncould map the corresponding LL properties. The general\ndescription of the damping effects in the MO can be use-\nful to study how is the scattering in 2D materials under\na perpendicular magnetic field. In particular, how are\nthe LL broadened. Careful magnetization measurements\ncouldbe one ofthe most reliablemethods, asthere areno\nperturbations other than the magnetic field that causes\nthe LL.8\nACKNOWLEDGMENTS\nThis paper was partially supported by grants of CON-\nICET (Argentina National Research Council) and Uni-\nversidad Nacional del Sur (UNS) and by ANPCyT\nthrough PICT 2014-1351. Res. N 270/15. N: 2014-\n1351, and PIP 2014-2016. Res. N 5013/14. C´ odigo:\n11220130100436COresearchgrant, aswell asby SGCyT-\nUNS., J. S. A. and P. J. are members of CONICET, F.\nE. acknowledge research fellowship from this institution.\nAppendix A: Fourier expansion of the ground state\nMO\nWritten in terms of the phase ψγdefined in equation\n(4), the ground state magnetization M0\nγoscillates as a\nfunction of ψγin integer intervals. The oscillation is de-\ntermined by the change in the last LL occupied when\nψγis an integer and thus floor( ψγ) changes. Consider-\ning that in general εn;γincreases with nandB, thenψγ\ndecreases with increasing B. This implies that in the\nsmall-field limit, when many energy levels are occupied\nandψγis large, the magnetization varies little over each\ninterval (n0,n0+1), where n0= floor(ψγ). Thus for any\nx0∈(n0,n0+1) we can take the quadratic approxima-\ntion\nM0\nγ≃M0\nγ(x0)+∂M0\nγ\n∂ψγ(x0)(ψ−x0)\n+∂2M0\nγ\n∂ψ2γ(x0)(ψ−x0)2\n2. (A1)\nUsing this approximation, the oscillating part of M0\nγ\ncan be expanded in a Fourier series. Because M0\nγos-\ncillates discontinuously, the linear term gives a saw-\ntooth oscillation, whereas the quadratic term gives a\ncos(2πpψ)/(πp)2term. That is\nM0\nosc,γ∼C1∞/summationdisplay\np=1sin(2πpψ)\nπp+C2∞/summationdisplay\np=1cos(2πpψ)\n(πp)2,(A2)\nwhereC1andC2arepindependent factors. In this way,\nthe oscillation part of the magnetization is written as\nthe sum of elemental oscillating functions. The expan-\nsion given by equation ( A2) is in general valid for any\noscillating interval ( n0,n0+1), as long as the small-field\nlimit condition is satisfied. Hence the equation ( A2) can\nbe generalized to all ψby considering the general de-\npendence of the coefficients C1andC2withψ. ForC1\nwe already have C1→Aγ, whereAγis the amplitude\nof the discontinuities in Mγ, given by equation ( 5). It\nis worth noting that, if necessary, the next terms in the\nexpansion ( A2) can be found by considering higher or-\nders in the approximation given by equation ( A1) and\nexpanding each one in a Fourier series. This results in el-\nementaloscillationsoftheformsin(2 πpψ)/(πp)2n+1andcos(2πpψ)/(πp)2n, wherenis a positive integer. How-\never, in generalthe need oftheseterms wouldbe required\nonly in the extreme situation of high magnetic fields and\nvery low occupancy. Finally, it should be mentioned that\na similar expansion can be made for the grand potential\nΩosc,γ, only that then there is no sawtooth oscillation\nbecause Ω osc,γvaries continuously and thus C1= 0.\nAppendix B: Approximation of the MO for βµ≫1\nandµ/Γ≫1\nWe will show that if βµ≫1 andµ/Γ≫1, then we\ncanneglectthetermswith I1,mandI3,mintheoscillating\npartofequation( 16). We startwith the separationofthe\nterms form≤fandm≥f+1, as done in equation ( 16).\nWe defineIi,>=/summationtext∞\nm=f+1Ii,mfori={1,2,3}, and\nI1,<=f/summationdisplay\nm=1/bracketleftbigg\nI1,m−∂̺\n∂B(εm−µ)/bracketrightbigg\n(B1)\nI2,<=f/summationdisplay\nm=1/bracketleftbigg\nI2,m−̺∂εm\n∂B/bracketrightbigg\n(B2)\nI3,<=f/summationdisplay\nm=1I3,m. (B3)\nThe idea is to show that, if βµ≫1 andµ/Γ≫1, the os-\ncillating amplitudes of I1andI3are much smaller than\nthe oscillating amplitude of I2. In each case, this am-\nplitude is determined when fchanges, which happens\nwhen the energy levels εmcross the chemical potential\nµ. Suppose fchanges to f+ 1 atB=Bf, such that\nεf+1(Bf) =µ. Then the corresponding amplitudes are\n∆I1,<=I1,f+1=−∆I1,> (B4)\n∆I2,<=I2,f+1−̺∂εf+1\n∂B,∆I2,>=−I2,f+1(B5)\n∆I3,<=I3,f+1=−∆I3,>, (B6)\nwhere, defining y= (ε−εf+1) = (ε−µ), from equations\n(13), (14) and (15) we can write\nI1,f+1=−1\nβ∂̺\n∂B/integraldisplay∞\n−∞ρf+1ln/bracketleftbig\n1+eβy/bracketrightbig\ndy (B7)\nI2,f+1=̺∂εf+1\n∂B/integraldisplay∞\n−∞ρf+11\n1+eβydy (B8)\nI3,f+1=−̺\nβ∂Γf+1\n∂B/integraldisplay∞\n−∞∂ρf+1\n∂Γln/bracketleftbig\n1+eβx/bracketrightbig\ndy.(B9)\nNow, given that/integraltext∞\n−∞ρf+1dy= 1, we have\n/integraldisplay∞\n−∞ρf+1dy=/integraldisplay∞\n−∞ρf+1/bracketleftbigg1\n1+eβy+1\n1+e−βy/bracketrightbigg\ndy\n=2/integraldisplay∞\n−∞ρf+11\n1+eβydx= 1.(B10)9\nThen we get I2,f+1=̺(∂εf+1/∂B)/2, and in order\nof magnitude ( ∂εf+1/∂B)∼µ/B. Thus from equation\n(B8) we obtain\n|∆I2,<|=|∆I2,>| ≡ |∆I2| ∼̺µ\nB,(B11)\nindependently of the damping parameters Tand Γ. This\nis the same order as the sawtooth amplitude in the\nground state (see equation ( 5)), which explains why the\ndamped MO falls to zero when the energy levels cross\nµ. On the other hand, for I1andI3, the order of\nmagnitude of the oscillating amplitude depends on the\ndamping parameters Tand Γ. This can be seen by in-\nspection of equations ( B7) and (B9), whose value de-\npends onTand Γ. For instance, without broadening\nsuch thatρf+1(y)→δ(y), one has |I1,f+1| ∼̺/Bβ\n(∂̺/∂B=̺/Bif̺depends on B) andI3,f+1= 0. Like-\nwise, at zero temperature, one has |I1,f+1| ∼̺Γ/Band\nalso|I3,f+1| ∼̺Γ/B. In the general case, for any tem-\nperatureandbroadening,wecansay |I1,f+1| ∼̺{Γ}1/B\nand|I1,m| ∼̺{Γ}2/B, where{Γ}1and{Γ}3are terms\nthat depend on the damping parameters kBTand Γ. We\nare not interested exactly in how are these terms, but\nrather in the fact that {Γ}1and{Γ}3will always in-\ncrease if Γ and Tincrease. Then, from equations ( B4)\nand (B6) we can say\n|∆I1,<|=|∆I2,>| ≡ |∆I1| ∼̺{Γ}1\nB(B12)\n|∆I3,<|=|∆I3,>| ≡ |∆I3| ∼̺{Γ}3\nB.(B13)\nFrom equations ( B11), (B12) and (B13) we obtain the\nratios\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆I1\n∆I2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼{Γ}1\nµ(B14)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆I3\n∆I2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼{Γ}3\nµ. (B15)\nTherefore, under the condition βµ≫1 andµ/Γ≫1, we\nhave|∆I1/∆I2| ≪1 and|∆I3/∆I2| ≪1, so the terms\nI1,mandI3,mcan be neglected in the oscillating part\nof equation ( 16). It is worth noting this low damping\ncondition is usually required in order to observe the MO.\nAppendix C: MO with damping effects as reduction\nfactors\nWe will obtain, from equation ( 24), the MO with the\ndamping effects as reduction factors. We start by taking\nthe SO ofM0\nγ(ε), given by equation ( 6). We have\nM0\nosc,γ(ε) =Aγ(ε)∞/summationdisplay\np=11\nπpsin[2πpψγ(ε)],(C1)where we explicitly indicated that Aγandψγare ob-\ntained considering a chemical potential equal to ε. Re-\nplacing in equation ( 24) we get\nMosc,γ(µ) =/integraldisplay∞\n−∞/bracketleftbigg\n−∂nF\n∂ε′(ε′−µ)/bracketrightbigg\nYγ(ε′)dε′,(C2)\nwhere\nYγ(ε′) =/integraldisplay∞\n−∞ρ(ε′−ε)Aγ(ε)∞/summationdisplay\np=1sin[2πpψγ(ε)]\nπpdε\n=∞/summationdisplay\np=1/integraldisplay∞\n−∞ρ(x)Aγ(ε′−x)sin[2πpψγ(ε′−x)]\nπpdx.\n(C3)\nAt relatively low damping, the DOS ρ(x) is non zero\nonly for |x| ≪1, so is good approximation to take\nAγ(ε′−x)≃Aγ(ε′) outside the integral. We can also\ntakeψγ(ε′−x)−ψγ(ε′)≃(∂ψγ/∂ε′)x, so the sine func-\ntion can be decomposed as\nsin[2πpψγ(ε′−x)] =sin[2πpψγ(ε′)]cos/bracketleftbigg\n2πp∂ψγ\n∂ε′x/bracketrightbigg\n+cos[2πpψγ(ε′)]sin/bracketleftbigg\n2πp∂ψγ\n∂ε′x/bracketrightbigg\n.\n(C4)\nThen, given that ργ(x) is a symmetric function (as as-\nsumed), we have\nYγ(ε′) =Aγ(ε′)∞/summationdisplay\np=11\nπpsin[2πpψγ(ε′)]\n×/integraldisplay∞\n−∞ρ(x)cos/bracketleftbigg\n2πp∂ψγ\n∂ε′x/bracketrightbigg\ndx. (C5)\nFromthiswecanidentifythebroadeningreductionfactor\nRΓ, given by the integral in the last equation. That is,\nRΓ,γ(p,ε′) =/integraldisplay∞\n−∞ρ(x)cos/bracketleftbigg\n2πp∂ψγ\n∂ε′x/bracketrightbigg\ndx.(C6)\nReplacing in equation ( C2) and changing variables we\nobtain\nMosc,γ(µ) =/integraldisplay∞\n−∞/bracketleftbigg\n−∂nF\n∂x(x)/bracketrightbigg\nAγ(x+µ)\n×∞/summationdisplay\np=1RΓ,γ(p,x+µ)\nπpsin[2πpψγ(x+µ)]dx.\n(C7)\nAt low temperature, the function ( ∂nF/∂x) is non zero\nonly for|x| ≪1, so we can take Aγ(x+µ)≃Aγ(µ) and\nRΓ,γ(p,x+µ)≃RΓ,γ(p,µ) outside the integral. Then,\ndecomposing again the sine function as in equation ( C4),10\nand considering that ( ∂nF/∂x) is a symmetric function,\nequation ( C7) becomes\nMosc,γ(µ) =Aγ(µ)∞/summationdisplay\np=1RΓ,γ(p,µ)\nπpsin[2πpψγ(µ)]\n×/integraldisplay∞\n−∞/bracketleftbigg\n−∂nF\n∂x/bracketrightbigg\ncos/bracketleftbigg\n2πp∂ψγ\n∂µx/bracketrightbigg\ndx.(C8)\nHence we can identify the temperature reduction factor\nRT, given by\nRT,γ(p,µ) =/integraldisplay∞\n−∞/bracketleftbigg\n−∂nF\n∂x(x)/bracketrightbigg\ncos/bracketleftbigg\n2πp∂ψγ\n∂µx/bracketrightbigg\ndx.\n(C9)\nIn this way, we finally obtain\nMosc,γ(µ) =Aγ∞/summationdisplay\np=1RΓ,γ(p,µ)RT,γ(p,µ)sin[2πpψγ(µ)]\nπp.\n(C10)\nThus the damping effects are described as reduction fac-\ntors in the MO amplitude.\nAppendix D: Examples of damping effects in the MO\nHere we obtain the damping effects in the MO, as de-\nveloped in sections IIIAandIIIB, for some particular\ncases. We do not particularize to any system, so the\nresult correspond to any general 2D energy levels εn;γ.\nIn each situation we obtain the occupancy factor Fn;γ\nandthe reductionfactor RΓ,γ(p), givenbyequations( 17)\nand (28), respectively. The temperature reduction factor\nRT,γ(p) has always the same general expression, given\nby equation ( 27).\n1. The pristine case\nIn the pristine case, the DOS is just the Dirac delta\nρn;γ=δ(ε−εn;γ). Then the occupancy factor simply\nbecomes\nFn;γ=1\n1+eβ(εn;γ−µ), (D1)\nwhich is just the Fermi-Dirac distribution for the energy\nlevelεn;γ[37,38]. For the reduction factors, at Γ = 0\nwe haveRΓ,γ(p) = 1, while the temperature reduction\nfactor is given by equation ( 27).2. Non zero broadening at zero temperature\nAt zero temperature, β→ ∞so equation ( 17) becomes\nFn;γ=/integraldisplayµ\n−∞ρn;γdε. (D2)\nTo proceed one needs specify how is the broadening of\nthe levels, that is, how is ρn;γ. Some common cases are\nthe Lorentz (L), Gauss (G) and semi-elliptic (E) distri-\nbutions, which have the form\n(L)ρn;γ=Γn;γ\nπ/bracketleftig\n(ε−εn;γ)2+Γ2n;γ/bracketrightig (D3)\n(G)ρn;γ=1\nΓn;γ√πe−(ε−εn;γ)2/Γ2\nn;γ (D4)\n(E)ρn;γ=2\nπΓ2n;γRe/bracketleftbigg/radicalig\nΓ2n;γ−(ε−εn;γ)2/bracketrightbigg\n,(D5)\nwhere Γn;γis the broadening parameter which in general\ncan depend on the Landau level and other variables such\nas the magnetic field B. Then equation ( D2) gives\n(L)Fn;γ=1\nπ/bracketleftbigg\narctan/parenleftbiggµ−εn;γ\nΓn;γ/parenrightbigg\n+π\n2/bracketrightbigg\n(D6)\n(G)Fn;γ=1\n2/bracketleftbigg\nerf/parenleftbiggµ−εn;γ\nΓn;γ/parenrightbigg\n+1/bracketrightbigg\n(D7)\n(E)Fn;γ=1\nπ\narctan\nµ−εn;γ\nRe/bracketleftbigg/radicalig\nΓ2n;γ−(µ−εn;γ)2/bracketrightbigg\n+π\n2\n\n+(µ−εn;γ)\nπΓ2n;γRe/bracketleftbigg/radicalig\nΓ2n;γ−(µ−εn;γ)2/bracketrightbigg\n.\n(D8)\nFor the reduction factors, at zerotemperature RT,γ(p) =\n1, while for RΓ,γ(p) one has to consider the broadening\nto be the same for all the Landau levels, so Γ n;γ= Γγ.\nFrom equation ( 28) we get\n(L)RΓ,γ(p) =exp/parenleftbigg\n−2πpΓγ∂ψγ\n∂µ/parenrightbigg\n(D9)\n(G)RΓ,γ(p) =exp/bracketleftigg\n−/parenleftbigg\nπpΓγ∂ψγ\n∂µ/parenrightbigg2/bracketrightigg\n(D10)\n(E)RΓ,γ(p) =1\nπpΓγ(∂ψγ/∂µ)J1/parenleftbigg\n2πpΓγ∂ψγ\n∂µ/parenrightbigg\n,\n(D11)\nwhereJ1is the Bessel function of the first kind.\n[1] W. De Haas and P. 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Results presented ar e directly relevant to widely used\nFMR experimental techniques for extracting magnetic param eters from thin films, the results of\nwhich are often assumed to carry over to corresponding nanom eter-sized patterned devices. We\nshow that there can be significant variation in the FMR respon se with device size, and that the\nextent of the variation depends on the magnetic material pro perties. This explains, for example,\nwhy different experiments along these lines have yielded diff erent size-dependent trends from\ndamping measurements. Observed trends with increasing siz e and different material parameters are\nexplained through the evolution of three distinct eigen-mo des, demonstrating the respective roles of\ndemagnetization and exchange. It is also shown that there is a crossover of dominant eigen-modes\nin the response signal, accompanied by conjugating edge-ty pe modes, leading to evident effects\nin measured linewidth and damping. Among the sizes consider ed, in higher saturation magneti-\nzation, we observeasmuchas a40% increase inapparentdampi ng, duesolely todevicesizevariation.\nPACS numbers: 75.30.Ds, 76.50.+g, 75.75.-c, 75.78.Cd\nI. INTRODUCTION\nWhen a ferromagnetic material experiences a static\nmagnetic field, its ground state is additionally split by\nthe Zeeman energy, resulting in a condition for absorp-\ntionofelectromagneticpowerbyferromagneticresonance\n(FMR) given by ∆ EZ=gµBBeff=/planckover2pi1ω;µBis the Bohr\nmagneton; Beffis the effective magnetic induction; and\nωisthefrequencyoftheadditionalACfield, whichisusu-\nally in the gigahertz range.1–3For more than 60 years,\ntapping into such a phenomenon has proven to be a pow-\nerfultooltoextractinformationfrommagneticmaterials,\nincluding both static (e.g. magnetic anisotropy, effective\nmagnetization) as well as dynamic parameters (e.g. g-\nfactor, damping parameter α).4–10The damping parame-\nterα, which quantifies the magnetizationrelaxationrate,\nis crucial for magnetic systems driven by finite temporal-\nlengthinputs, suchasacurrentpulseoramodulatedfield\non a hard-disk drive (HDD). On a HDD, αdetermines\nthe time evolution ofmagnetic grainsin recordingmedia,\nand in Spin-Transfer Torque Magneto-Resistive Random\nAccess Memory (STT-MRAM), it also determines the\ncriticalcurrenttocontrolmagneticstates. Therefore, the\nperformance of such data storage devices relies strongly\non an accurate knowledge of damping. In FMR experi-\nments, the concept of apparent damping is often used for\nthe measured quantity to make the distinction between\nthat which is measured experimentally by inferring from\nlinewidth and resonance measurements and the actual\nintrinsic damping parameter αin the Landau-Lifshitz-\nGilbert (LLG)11equation. Apparent αis investigated\nhere directly using FMR analysis.\nFor some common FMR techniques, in order to obtaina strong enough signal in a measurement, the magnetic\nsample often needs to be orders of magnitude larger lat-\nerally than the device of interest. However, with the\nadvent of STT-MRAM12opening the way to magnetic\nstorage devices with characteristic dimensions less than\n20nm, there is a need to understand the fidelity of mea-\nsurements of dynamic parameters determined by FMR\non unpatterned thin films13,14. While recently developed\ntechniques such as Spin-Torque FMR15–17, making use\nof microwave spin-currents or even spin-orbit coupling\ninduced torques18, allow extracting magnetic properties\nat the device level, they are not suitable for all mag-\nnetic devices (ideal if designed to flow current, and con-\ntains few magnetic layers). Other techniques, such as\nFrequency- or Time- Resolved MOKE or Magnetic Res-\nonance Force Microscopy have also been used in recent\nyears to access structures smaller than typical thin films,\nwhich, aside from facing their own challenges19, are also\nnot as widespread. Thus, some of the FMR techniques\nin use are incapable of resolving nanometer sized device\nparameters. And, it is unclear down to what size will\nmeasurments remain valid. The effects of lateral size on\nthe FMR response havebeen investigatedexperimentally\ninpreviousstudies. Forinstance, Shaw et al.studied size\neffects in elliptical arrays and observed no change in ap-\nparentαfrom the patterning process14down to 50nm,\nwhile a later study by Noh et al.found that αchanges\nsignificantly with the device size.13It is worth pointing\nout that these studies use different materials. Thus, it\nis still an open question as to whether FMR measure-\nments on larger size films yield results which represent,\nwith fidelity, the device of interest, which is often much\nsmaller.2\nIn this work, we resolve this discrepancy through ex-\ntensive 3D finite element method (FEM) simulations,\ninvestigating the effects of lateral device size and mag-\nnetic material parameters on the FMR response. We\nshow that both results are indeed possible and demon-\nstrate why this is the case. In particular, we discuss\nresonance frequency, linewidth, and apparent αbehav-\nior with size variation. We also vary the magnetic\nmaterial properties using the saturation magnetization,\n0.6T≤µ0Ms≤2.4T, as it controls significant con-\ntributions to the resonance response, both through the\ndemagnetization field, and equally important, through\nthe exchange length lex=/radicalbig\n2A/µ0M2s;Ais the ex-\nchange stiffness; µ0is the magnetic permeability in free\nspace. The chosen range of µ0Mscovers most materials\nof interest from Permalloy ( µ0Ms∼1T) to CoFe alloys\n(µ0Ms∼1.8−2.2T)?. As will be shown, smaller lex\npermits higher order (i.e. more nonuniform) modal con-\ntributions into the response sooner with respect to size,\nand is important to consider.\nWe first describe the FEM model used and demon-\nstrate a validation of it, reproducing reported experi-\nmental and simulated data. Then, we discuss the res-\nonance behavior versus device size across different ma-\nterials. We especially emphasize the behavior of devices\nsmall enough for uniform precession modes to be domi-\nnant, as small as 10nm, then study the response through\nthe evolution of the coexisting eigen-modes when scaling\nup size laterally and varying magnetic material proper-\nties. In particular, our discussion of the corresponding\nresonance frequency, linewidth, and αresponse is given\nbased on three relevant spin-waveeigen-mode evolutions.\nFinally, we summarize our results.\nII. FMR MODEL VALIDATION\nOuranalysisutilizesanFEMformulationimplemented\nwithin the software package SpinFlow 3D.20The model-\ning procedure is as follows: first, a micromagnetic equi-\nlibriumstateiscomputedwiththeappliedDCfield HDC.\nThis is followed by a computation of the spin-waveeigen-\nmodes and associated eigen-frequencies in the vicinity\nof the known equilibrium state. With this information,\nthe FMR response to a small AC field hrfis deter-\nmined. More details of the software formulation can be\nfound elsewhere21. An important requirement of this ap-\nproach is to validate the model by simulating indepen-\ndently reported experimental and simulated data. For\nthis purpose, we compare our results with reported data\nfor Permalloy (Py) nanodisks from Shaw et al14. The\nsimulated device consists of a single magnetic layer sand-\nwiched between two nonmagnetic layers, as illustrated in\nFig. 1(a). HDCis applied along /vector y.\nNote that the sizes indicated are nominal sizes as de-\nfined in Ref. 14. Good agreement is achieved between\nour simulation results of the resonance frequencies and\nreported experimental data for all sizes considered (Fig.\nFigure 1. (Color online) (a) Device geometry used in our sim-\nulations: a magnetic layer (dark blue) is sandwiched betwee n\ntwo nonmagnetic layers (light green). For the model valida-\ntion, the geometry is elliptical, with the applied field alon g\nthe long axis of the ellipse, as in Ref. 14. (b) A typical FMR\nspectrum showing two visible resonance peaks simulated for\na circular disk with diameter d= 150nm, µ0Ms= 1.7T, and\nHDC= 0.05T.\n2), confirming the validity of our model. In the following\nsections, we present our analysis based on simulation re-\nsults using the same numerical procedure to investigate\nthe FMR responsein moredetail, varyinglateralsizeand\nmaterial parameters. All simulation parameters used are\nalso summarized in Table I.\n/s48/s50/s52/s54/s56/s49/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s50/s52/s54/s56/s49/s48/s102\n/s114/s101/s115/s32/s40/s71/s72/s122/s41\n/s32/s78/s73/s83/s84/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s78/s73/s83/s84/s32/s83/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s32/s79/s117/s114/s32/s83/s105/s109/s117/s108/s97/s116/s105/s111/s110/s40/s97/s41/s32/s53/s48/s32/s110/s109\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41/s40/s98/s41/s32/s49/s48/s48/s32/s110/s109\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41/s40/s99/s41/s32/s50/s48/s48/s32/s110/s109\nFigure 2. (Color online) Comparison of simulated resonance\nfrequencyresponse for Py devices with nominal size (a) 50nm ,\n(b)100nm, and(c) 200nm as per Ref. 14. Each figure includes\nexperimental data (black circles), reported simulations ( thin\nblue-line, NIST Simulation), and our simulation results (t hick\nred-line, Our Simulation).\nIII. RESULTS AND DISCUSSION\nFrom here out, we discuss the FMR response of cir-\ncular disk-shaped devices with varying lateral diameter\nd, where 10nm ≤d≤300nm. While magnetic devices3\nTable I. Parameter values in simulations\nParameter name Figure 2 Other figures\nµ0Ms 0.96T 0.6−2.4T\nexchange stiffness A11pJ/m 20pJ/m\nthickness t 10nm 2nm\nmesh size 5nm 2nm for φ >40nm\n1nm for φ≤40nm\nshape elliptical circular\nsize a.71×65nm210−300nm (diam)\nb.120×110nm2\nc.235×215nm2\nGilbert damping α0.01 0.01\nwith features as small as 10nm are challenging to fab-\nricate and access experimentally, numerical simulations\navoid this difficulty and enable us to observe behavior\nand address whether thin film measurements obtain rep-\nresentative device relaxation parameters down to suffi-\nciently small device sizes. Also note that in our FEM\nmodel, we do not take into consideration spin pumping\nnor inhomogeneous properties that would contribute to\nfield linewidth broadening. Thus, our results reveal con-\ntributions due solely to size variation.\nA typical frequency-swept FMR spectrum ˜ χ(f) is\nshown in Fig. 1(b). There are, in general, two observ-\nable peaks with significant amplitude in the frequency\nrange considered, indicating the presence of more than\none eigen-mode. We consider three relevant eigen-modes\nand the resulting interplay and evolution affecting both\nobserved peaks. Each resonance frequency fresis ob-\ntained by fitting each peak of the simulated FMR re-\nsponse to Lorentzians of the form\n˜χ(f) =2P\nπ∆f\n4(f+fres)2+∆f2(1)\nwherePis the peak amplitude, fis the frequency of the\napplied AC magnetic field hrf, and ∆fis the full width\nat half maximum. In Fig. 3, we plot fresversus disk\ndiameter for three distinct eigen-modes which contribute\nto the response within the frequency range considered.\nNote that µ0Ms= 1.7T, under an applied DC field of\n0.05T, enough to saturate the magnetization along /vector y. In\nall illustrations of mode profiles (shown as insets in Fig.\n3), the color scale corresponds to the amplitude of pre-\ncession - from largest (red) to smallest (blue). From the\nspatial distribution of precession amplitude, we identify\nuniform (U), edge(E) and center(C) modes, accordingly.\nWe observe that eigen-mode profiles generally evolve as\nthe diameter is increased, and are labelled, for instance,\nU-Etorepresentevolutionwith sizefromuniformtoedge\nmode. Next, we discuss the origin of the evolution of\nmodes as they are intimately tied to the behavior of fres.\nFigure 3. (Color online) Frequency response for the\n(a) Uniform-to-Edge (U-E), (b) Edge-to-Center (E-C) and\n(c) Edge-to-Edge (E-E) eigen-modes versus device size for\nµ0Ms= 1.7T, applied field HDC= 0.05T. Spatial mode pro-\nfiles of precession amplitude are shown as insets.\nA. Evolution of Modes\nFor devices with sizes on the order of lex, which we re-\nfer to as ”small” devices from hereon, the lowest energy\neigen-mode is the uniform precession mode21. Due to\nthe spatial distribution of the demagnetization field, the\nuniform-mode (U-mode) is seen to evolve into an edge-\nmode (E-mode) that can be observed as device size is\nincreased. We refer to this evolution as the U-E mode.\nToseethecorrelationmoreclearly, wealsoshowthe com-\nputeddemagnetizationfielddistributioninFig.4(a), cor-\nresponding to the magnetization equilibrium state for a\n150nm device. This distribution is typical for all sizes.\nNotice the inhomogeneity of the demagnetization field\nalong the edges in the direction of HDC, which is known4\nto create a potential well for spin-waves tending to lo-\ncalize them22. The modal evolution shown in Fig. 3(a)\nis a direct result of the demagnetization energy as well\nas the distribution. To further illustrate demagnetiza-\ntion edge localization, we also plot in Fig. 4(b) and (c),\nrespectively, computed demagnetizing field components\nHy\nDparallel to the DC field ( /vector y) andHz\nDperpendicular\nto the film plane ( /vector z) for different lateral sizes. The de-\nmagnetizing fields are clearly strongest at the edges, and\ncomparativelyweakintheinteriorregion. Note,however,\nthese observations are computed for the case of nearly\nuniform magnetization along ( /vector y). As size is increased\nsufficiently beyond lex, nonuniform affects manifest more\nstrongly, leading to more complex behavior.\nA second eigen-mode evolution is shown in Fig. 3(b),\nwhich evolves from an edge-mode into an eigen-mode\nwith dominant precession amplitude within the center\nof the device. This higher-order mode starts with edge\nprecession confinement along both /vector x(to lesser extent)\nand/vector y∼a four-fold symmetry. We refer to this distinct\nmodal evolution as the E-C mode (edge-to-center). This\nparticular mode shows the greatest nonuniformity, and\nconsequently involves more exchange energy in addition\nto the demagnization field. For a magnet with nonmag-\nnetic interfacial boundaries, exchange energy is known\nto be lowest at the edges, due to the free-spin boundary\ncondition.23In this case, the system develops more inte-\nrior motion for access to more exchange energy useful to\nlower its total magnetic energy arising from demagneti-\nzation field effects.\nThe third relevant eigen-mode is shown in Fig. 3(c),\ninwhichedge-dominatedprecessionsolelypersistsforthe\nentire frequency range considered. Therefore, it begins\nand ends as an E-mode, also localized along /vector yedges.\nWe refer to this as the E-E mode (edge-to-edge), and its\nevolutionary behavior is also explained by the demagne-\ntization field, as discussed above. Next, we analyse the\nimplications of the modal evolutions on fres.\nB. Resonance Frequency Behavior\nTo understand the resonance frequency response for\nnon-uniformhigherordereigen-modes,weconsideragen-\neralized form of the Herring-Kittel formula11acounting\nfor quantization in all three dimensions:\nfres=γ\n2π[(HDC+2Aq2\nMs)(HDC+2Aq2\nMs+4πMsFn,p)]1/2\n(2)\nwhereqis the wave vector given here by\nq2= (mπ\nd)2+(nπ\nd)2+(pπ\nt)2(3)\nThe parameters m,n, andpare quantization numbers\narising from the finite size of the device; tis the device\nthickness; dis the lateral dimension (i.e. diameter); and\nFn,pcontains nontrivial demagnetization field informa-\ntion, which also depends on the wave vector. Note that\nFigure4. (Color online)(a)Demagnetization field(amplitu de)\ndistributions in the middle plane (along /vector z) of the magnetic\nlayer, computed from the magnetization equilibrium distri bu-\ntion forµ0Ms= 1.7T, for a 150nm device. Color scale shows\n/bardbl/vectorHD/bardbl. Computed (b) in-plane and (c) out-of-plane demag-\nnetization field component along /vector yfor x=0 (center line) for\ndifferent sizes. (d) Resulting local ferromagnetic resonan ce\nusing Eq. (8). At both center and edge, the resonance fre-\nquency always increases with increasing size.\nalthough no simple quantization scheme can be used for\nstructures confined in all three dimensions11, the pre-\nsented form of the wave vector does allow us to qualita-\ntively understand the behavior of fresfor higher order\neigen-modes.\nIn the size regime where the U-E mode is mostly uni-\nform, the increase in fresis understood by considering\nthe decreasing (with increasing size) demagnetizing field.\nThis decreasing trend can be seen in Fig. 4(b) and (c).\nTo understand why the field decreases as size increases,\nconsider the exact solution for the demagnization field in\nthe case of uniform magnetization, given by11\nHD(x) =−1\n4π/integraldisplay\nM·nx′−x\n|x′−x|3dS′(4)\nConsider a rectangular film with uniform magnetization\nMin-plane along /vector y. In this case, only two of the six rect-\nangular boundary faces contribute to the integral in (4),\ni.e. the two faces with normal vectors nparallel to M.\nIn the center, the value is identically 0 due to symmetry,\nhowever, away from the center and only near the edges,\nnonzero values exist. Evaluating the field leads to terms\nof the following form\nHD∝Ms1\nd2dS′(5)\nBy keeping the film thickness tconstant, dS′scales only\nwithd(instead of d×t) and the result is that the de-\nmagnetization field scales as HD∝1/d. Thus, the de-\nmagnetization field decreases with increasing lateral size.5\nThe resonance frequency consequently increases because\nfres∝Heff=Hdc−|HD|where both predominantly lie\nalong/vector y. This accounts for the U-E mode increasing reso-\nnance frequency behavior in small device sizes. However,\nas size increases beyond ∼50nm, the nonuniform mode\nbehavior contributes more significantly.\nIn this case, equation (2) suggests exchange energy\nalso contributes to fresin higher-order modes, through\nthe wave-vector q. In particular, it follows from (2) that\nresonance frequencies generally decrease with increasing\ndevice size, as 1 /d2. This is in qualitative agreement\nwith trends shown in Figures 3(b) and (c) corresponding\nto higher order eigen-modes. This also applies to Fig-\nures 3(a), after the transition to an edge mode has taken\nplace. In the nonuniform modes, the gradient of the\neigen-modes also contains exchange information as they\nevolve with size. The modes indicate precession ampli-\ntude, propertional to normal components of the magne-\ntization (normal to /vector y). The total exchange energy given\nby\nEA=A\nM2s/integraldisplay\n(∇M)2dV (6)\nBy observing the trend in the gradient of the mode, the\ntrend of the exchange energy can also be seen. Fig-\nure 5(a) illustrates the qualitative trend of the exchange\nassociated with the E-C mode (the most nonuniform\nmode), indicating on the eigen-mode vector directions\n(with black arrows). As size increases, the complexity of\nthe mode reduces, and this generally reduces exchange\nenergy density. This behavior is equivalent to reducing\nthe quantization numbers mandn. For two ofthe device\nsizes in the region of sharp decrease in fres, the frequen-\ncies are plotted in Figure 5(b), along with the gradient\ndistributionsofthein-planenormalmodecomponent(x),\nand its maximum computed gradient indicated (in arbi-\ntray units). This demonstrates an overall decrease in the\nexchange energy through the E-C mode evolution, and\nthis contributes to a reduction in fresfor higher order\nmodes.\nC. Cross-over and Conjugation of Modes\nAnother important characteristic of the FMR spec-\ntrum, asidefrom the resonancefrequencytrend with size,\nis the relative amplitudes of the observed modes. The\namplitude of an FMR peak is proportional to the volume\nof the device participating in the precession. For exam-\nple, in the U-E mode, the edge precession is increasingly\nconfined to the edges with increasing size (as can be seen\nin the insets of Fig. 3) and the relative amplitude of the\npeak decreases, with additional contributions from a re-\nducing demagnetization field. In contrast, the amplitude\nof the E-C mode increases with size relative to the other\nmodes. Therefore, in smaller sizes, the U-E mode tends\nto dominate the signal, however, as size increases, the\nU-C mode begins to dominate. This leads to a cross-over\nFigure 5. (Color online) (a) Illustration of reduction in ex -\nchange energy in high frequency E-C mode, with increasing\nsize (50, 150, and 300nm shown) and (b) Computed gradients\nof in-plane normal modal component for 50nm and 100nm.\nMaximum gradient (in arbitrary units) shown beside the re-\nspective gradient distribution.\nfor the dominant mode of the FMR response signal as\nsize increases. The cross-over is illustrated in Fig. 6,\nseen clearly for µ0Ms= 1.7T and 2 .4T.\nNear the cross-over size, there also occurs a conjuga-\ntion orjoining ofthe U-Eand E-Eeigen-modes, asshown\nin Fig. 7, where µ0Ms= 1.7T. At the point of conjuga-\ntion, the mode profiles of the U-E and E-E modes be-\ncome similar or quasi-degenerate and resonate at similar\nfrequencies. Both the cross-over and the conjugation of\nmodes can have effects on the measured linewidth as will\nbe discussed in a later section.\nD. Effect of Different Ms\nOne key observation in this study is that the FMR\nresponse can be sensitive to magnetic material parame-\nters. For example, the effect of different Msonfrescan\nbe seen in Fig. 8. For the U-E mode, the device size at\nwhich an initially increasing frestransitions to decreas-\ningfres(due to the onset of the edge mode) is smaller in\nmaterials with larger Ms. For the smallest µ0Ms(0.6T),\na decrease in fresis not observed at all in our range of\nsizes. However, devices with higher Mshave shorter lex,\nand it follows that non-uniform modes may contribute\nmore significantly in smaller lateral sizes. Hence, the ef-\nfect of exchange, which tends to lower fresas discussed\nabove, appears earlier. Comcomitent with this, we also6\nFigure 6. (Color online) (a) Simulated FMR response versus\ndevice size for µ0Ms= 0.6T, (b) 1.0T, (c) 1.7T and (d) 2.4T.\nWhite arrow indicates the location of the change in domi-\nnant modes, crossing over from U-E mode to E-C mode with\nincreasing size.\nFigure 7. (Color online) Resonance frequency versus size fo r\nthe U-E and E-E modes for µ0Ms= 1.7T, zooming in to\nthe point of conjugation. Note the increasingly similar mod e\nprofiles of the U-E and E-E modes.\nobservethe occurence of both the modal conjugation and\ndominant mode cross-over at smaller sizes as Msis in-\ncreased. The effects on the cross-over is evident in Fig.\n6(c) and (d).\nE. Linewidth and Damping Behavior\nThe linewidth ∆ ffor each size is obtained by tak-\ning the full width at half maximum of each Lorentzian\npeak. For a multi-peak spectrum, we can identify sepa-/s52/s56/s49/s50/s49/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s56/s49/s50/s49/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s40/s100/s41/s40/s98/s41\n/s40/s99/s41/s102\n/s114/s101/s115/s32/s40/s71/s72/s122/s41/s32/s85/s45/s69/s32/s77/s111/s100/s101\n/s32/s69/s45/s69/s32/s77/s111/s100/s101\n/s32/s69/s45/s67/s32/s77/s111/s100/s101/s48/s77\n/s83/s32/s61/s32/s48/s46/s54/s32/s84\n/s48/s77\n/s83/s32/s61/s32/s49/s46/s48/s32/s84\n/s48/s77\n/s83/s32/s61/s32/s50/s46/s52/s32/s84\n/s68/s101/s118/s105/s99/s101/s32/s83/s105/s122/s101/s32/s40/s110/s109/s41/s48/s77\n/s83/s32/s61/s32/s49/s46/s55/s32/s84/s40/s97/s41\nFigure 8. (Color online) (top) Frequency response of three\nobserved eigen-modes for (a) µ0Ms= 0.6T, (b) 1.0T, (c) 1.7T\nand (d) 2.4T.\nrate linewidths for each peak. We typically see two sig-\nnificant peaks which we identify as ”low” and ”high” fre-\nquency peaks. The linewidth versus size behavior for the\nhigh frequency peak is mainly determined by the E-C\nmode. The linewidth for the low frequency peak can be\nmorecomplex becauseboth U-E andE-Emodes can con-\ntribute to the signal due to their conjugation. However,\nbelow the conjugation point, both the resonance frequen-\ncies and relative amplitudes of the U-E and E-E modes\nare very dissimilar, and the low frequency peak linewidth\nis controlled mainly by U-E. As size increases, the rela-\ntive amplitude of the U-E mode decreases and becomes\ncomparable to the amplitude of the E-E mode. In some\nparts of this regime, their resonance frequencies do not\ncoincide exactly. When the U-E and E-E fresmodes do\nnotexactlymatch, but arecloseenough, thereis somear-\ntifical broadening of the measured linewidth. Once their\nresonance frequencies coincide, this artificial broadening\nis lost. To illustrate this, we plot the linewidth of the\nlow frequency peak (which includes contributions from\nboth U-E and E-E modes, corresponding more closely to\na measured signal) and the linewidth of only the U-E\nmode in Fig. 9. The insets show spectra of the U-E and\nE-E modes in the size range where the transient broad-\nening may occur. Note the increased linewidth of the\nlow frequency peak at 200nm which can be attributed to\nmodal conjugation.\nFor experiments, however, looking at the dominant\npeak (i.e. with the highest intensity) may be more rele-\nvant because limitations in detection sensitivity may not\nbe enough to resolve peaks other than the dominant one.\nWithout the aid of rigorous calculation or simulation, it\ncould be difficult to trace the origin of a particular peak\nin the signal; our results show that the dominant peak\nmay be a different mode for different sizes. In the fol-\nlowing analysis of linewidth behavior versus size for the\ndominant peak, we use the field linewidth ∆ H, which is7\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s51/s53/s48/s52/s48/s48/s52/s53/s48/s53/s48/s48/s53/s53/s48/s54/s48/s48/s54/s53/s48\n/s102/s32/s40/s77/s72/s122/s41\n/s68/s101/s118/s105/s99/s101/s32/s83/s105/s122/s101/s32/s40/s110/s109/s41/s32/s76/s111/s119/s32\n/s32/s85/s45/s69\nFigure 9. (Color online) Linewidth versus size for the low\nfrequency peak (blue squares: with contributions from both\nU-E and E-E modes) compared to the linewidth of only the\nU-E mode (black circles). Broadening from conjugation is\nseen at 200nm. Inset shows the FMR spectrum for U-E (thin\nblack) and E-E (thick red) modes in the size range where\nbroadening may occur.\nobtained from ∆ fvia\n∆H=∆f\n(δf/δH)(7)\nThe denominator is calculated from the general Kittel\nequation3\nf=/radicalbig\n(H+H1)(H+H2) (8)\nwhereHis the applied DC field and H1andH2are\nstiffness fields. We plot the linewidth of the dominant\npeakfordifferent MsvaluesinFig.10. Weobserveaclear\nincreasingtrend, with slightlymore complex behaviorfor\nµ0Ms= 2.4T. Because the linewidth is given by\n∆H= ∆H0+4παfres\nγ(9)\nand is therefore proportional to fres, we find that the\nchangeindominantmode, whichisaccompaniedbyadis-\ntinct jump to higher fres, results in an abrupt increase in\n∆Hat around 200nm. Moreover, after the cross-over, if\nthe linewidth follows fres, the now-dominant E-C mode\nresonance frequency decreases, so the linewidth should\ndecrease according to Eq. 9, where ∆ H0is the inhomo-\ngeneous linewidth broadening due to inhomogeneities in\nthe device parameters, e.g. dead layers and edge dam-\nage, which are not considered here. However, linewidth\ndoes not decrease with device size, as does fres. This\nindicates that αis non-constant and is contributing to\nthe observed linewidth behavior.\nWe therefore extract the apparent damping αfor each\npeak - by fitting applied field-dependent linewidth data\nto Eq. 9 - to eliminate contributions from fres. The\nresults are shown in Fig. 11 for different Ms. For the/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s52/s54/s56/s49/s48/s72/s32/s40/s109/s84/s41\n/s68/s101/s118/s105/s99/s101/s32/s83/s105/s122/s101/s32/s40/s110/s109/s41/s32/s32/s32/s32/s32/s32/s32/s32\n/s48/s77\n/s83/s32/s40/s84/s41\n/s32/s48/s46/s54/s32/s32/s32 /s32/s49/s46/s55\n/s32/s49/s46/s48/s32/s32/s32 /s32/s50/s46/s52\nFigure 10. (Color online) Linewidths from FMR response\nversus device size for different MS.\nlow frequency peak, the increase in αis more significant\nin larger Ms. For the high frequency peak, αdecreases\nmonotonically for all Ms, approaching α= 0.01.\nTo further explain , we consider that each differen-\ntial volume in the device experiences a slightly differ-\nentHeffand therefore has a slightly different frescom-\nparedtoits neighbor. This differenceismorepronounced\nasMsincreases because shorter lexallows more non-\nuniformity and the effect of the demagnetization field\nis also stronger. The collective response - the sum of\nthese individual responses which do not necessarily over-\nlap - leads to broading. The low frequecy peak (mainly\nfrom the U-E mode) goes from uniform to non-uniform,\nleading to more broadening with size. However, for the\nthe higher frequency peak (E-C mode), which becomes\nmore uniform with increasing size, broadening generally\ndecreases.\nThe behavior of the linewidth for µ0Ms= 2.4T in\nlarger sizes is actually explained by the corresponding\ndecrease in αat 200nm as dominance shifts from the low\nfrequency peak to the high frequency peak. This also\ncontributes to the subsequent increase at 300nm, as can\nbe seen in Fig. 10.\nThe effects of size and material properties can also be\nseen clearly in Fig. 11. In the range considered, mate-\nrials with low Msgenerally show neglible variation in\ndamping, while for higher values of Ms, there is a clear\nvariation with size, indicating that the extent of the vari-\nation is sensitive to material properties, which in our\ncase isMs. In the present case, the characteristic length\nthat controls the FMR behavior with size is lex. Thus,\nour results suggest that the different findings on αver-\nsus size behavior13,14are not inconsistent, but stem pri-\nmarily from the use of different materials. Our results\nare consistent, quanlitatively, with both reports, where\nPy (µ0Ms= 1.0T) shows little variation, however CoFe\n(µ0Ms= 1.7T) does shows a visible increase with size.\nTherefore, since apparent αmay be non-constant with\nsize, care must be taken in using film measurements for\ndevice-related parameters. For a multi-peak FMR re-8\n/s48/s46/s48/s49/s48/s48/s48/s46/s48/s49/s50/s53/s48/s46/s48/s49/s53/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s49/s48/s48/s48/s46/s48/s49/s50/s53/s48/s46/s48/s49/s53/s48/s48/s77\n/s83/s32/s40/s84/s41\n/s32/s48/s46/s54\n/s32/s49/s46/s48\n/s32/s49/s46/s55\n/s32/s50/s46/s52/s76/s111/s119/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s80/s101/s97/s107\n/s72/s105/s103/s104/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s80/s101/s97/s107\n/s68/s101/s118/s105/s99/s101/s32/s83/s105/s122/s101/s32/s40/s110/s109/s41/s48/s77\n/s83/s32/s40/s84/s41\n/s32/s49/s46/s48\n/s32/s49/s46/s55\n/s32/s50/s46/s52\nFigure 11. (Color online) Apparent αfrom FMR response for\ndifferent Ms. Dashed line in (a) for µ0Ms= 2.4Tis a guide\nto the eye, highlighting the increased non-linearity aroun d\n200nm.\nsponse,αof the dominant peaks in the extremes of the\nsize spectrum considered here seem to converge to thesame value, however, they do so on opposite ends of the\nsize range. Take for example, 40nm where the low fre-\nquency peak is dominant, and 300nm where high fre-\nquencypeakisdominant. Bothvaluesapproach α= 0.01\nfor allMsvalues considered. Although the applicability\nof film measurements to the device level seems possible,\nthis, of course, assumes proper accounting of all other\ncontributionsto the linewidth. So, there canalsobe close\nagreementbetweendampingmeasurementsfromfilmand\nfrom devices below a certain size in some materials.24\nIV. CONCLUSION\nWe have investigated the effects of device size and\nmagnetic material parameters on the FMR response us-\ning a powerful 3D FEM tool. We identified three dis-\ntinct eigen-modes, the profiles of which evolve versus size\nand consequently lead to evident effects on the FMR re-\nsponse. Two of the three modes (U-E and E-E) were\nshown to conjoin at a given size and concomitant with\nthis conjugation, a cross-over in terms of mode domi-\nnance is also seen. These phenomena may affect mea-\nsured linewidth and damping. The amount of sensitivity\nto size was shown to increase as Msis increased and may\nbe the reason behind the discrepancy from different in-\nvestigations, which used different materials. Within the\nsize range of 10nm to 300nm, we have observed as much\nas a 40% increase in apparent α, due purely to size ef-\nfects. Our results suggests that an FMR measurement\ncan indeed lead to inflation of the intrinsic damping pa-\nrameter, depending on the material used.\n∗Electronic address: kwaku eason@dsi.a-star.edu.sg\n1J. H. E. Griffiths, Nature 158, 670 (1946).\n2C. Kittel, Phys. Rev. 71, 270 (1947).\n3C. 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Worledge,\nJournal of Applied Physics 111, 07C711 (2012)."    },    {        "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.  When  a LM is on resonance, the local reduction of magnetization in out -\nof-plane direction will generate a stray field, which will couple the altered magnetization to the magnet ic \ntip thus changing the cantilever oscillation amplitude and frequency. The change in cantilever oscillation \nis detected optically by laser fiber interferometry. Different LMs have different mode radi i. For this \nmicroscopy study, we focus on 𝑛=1 LM since  it gives the highest spatial resolution.  The 𝑛=1 LM \nresonance peak is fit to a Lorentzian line shape, from which the peak position and peak height are \nextracted, which correspond to the resonance field 𝐻r,1 and signal amplitude ∆𝐴.   \n \n     \n \n19 \n References:  \n  \n(1) Dieny, B.; Chshiev, M. Perpendicular magnetic anisotropy at transition metal/oxide interfaces and \napplications. Rev. Mod. Phys. 2017,  89, (2), 025008.  \n(2) Lee, K. S.; Lee, S. W.; Min, B. C.; Lee, K. J. 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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": "2312.02654v1.THz_Driven_Coherent_Magnetization_Dynamics_in_a_Labyrinth_Domain_State.pdf",        "content": "THz-Driven Coherent Magnetization Dynamics in a Labyrinth Domain State\nMatthias Riepp,1, 2,∗Andr´ e Philippi-Kobs,2, 3Leonard M¨ uller,2Wojciech Roseker,2Rustam Rysov,2\nRobert Fr¨ omter,4Kai Bagschik,2Marcel Hennes,5Deeksha Gupta,1Simon Marotzke,2, 3Michael Walther,2\nSaˇ sa Bajt,6, 7Rui Pan,2Torsten Golz,2Nikola Stojanovic,8Christine Boeglin,1and Gerhard Gr¨ ubel2,†\n1Institut de Physique et Chimie des Mat´ eriaux de Strasbourg,\nUMR 7504, Universit´ e de Strasbourg, CNRS, 67000 Strasbourg, France\n2Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany\n3Institut f¨ ur Experimentelle und Angewandte Physik,\nChristian-Albrechts-Universit¨ at zu Kiel, Leibnitzstraße 19, 24098 Kiel, Germany\n4Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany\n5Institut des NanoSciences de Paris, UMR7588 ,\nSorbonne Universit´ e, CNRS, 75005 Paris, France\n6Center for Free-Electron Laser Science CFEL, Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany\n7The Hamburg Centre for Ultrafast Imaging, 22761 Hamburg, Germany\n8Institute for Optical Sensor Systems, Deutsches Zentrum f¨ ur Luft- und Raumfahrt, Rutherfordstraße 2, 12489 Berlin, Germany\nTerahertz (THz) light pulses can be used for an ultrafast coherent manipulation of the mag-\nnetization. Driving the magnetization at THz frequencies is currently the fastest way of writing\nmagnetic information in ferromagnets. Using time-resolved resonant magnetic scattering, we gain\nnew insights to the THz-driven coherent magnetization dynamics on nanometer length scales. We\nobserve ultrafast demagnetization and coherent magnetization oscillations that are governed by a\ntime-dependent damping. This damping is determined by the interplay of lattice heating and mag-\nnetic anisotropy reduction revealing an upper speed limit for THz-induced magnetization switching.\nWe show that in the presence of nanometer-sized magnetic domains, the ultrafast magnetization\noscillations are associated with a correlated beating of the domain walls. The overall domain struc-\nture thereby remains largely unaffected which highlights the applicability of THz-induced switching\non the nanoscale.\nI. INTRODUCTION\nUnderstanding the magnetization dynamics driven by\nultrashort light pulses is of key importance for developing\nfaster and more energy efficient opto-magnetic memory\ntechnologies. A promising way to a controlled manipu-\nlation of the magnetization in ferromagnetic thin films\non ultrafast time scales is the use of light pulses with\nfrequencies in the terahertz (THz) regime ( νTHz≈0.1–\n10·1012Hz) [1–3]. In contrast to incoherent ultrafast\ndemagnetization induced by femtosecond optical laser\npulses with frequencies in the infrared (IR) regime ( νIR≈\n1014Hz) [4], the electric field component ETHzis capable\nof driving a coherent ultrafast demagnetization with sig-\nnificantly lower energy transfer to the sample [5]. More-\nover, the magnetic field component HTHzmay induce\ncoherent oscillations [6–9] and, at high intensities, even\na switching of the magnetization [10–13]. The possibility\nof exciting coherent magnetization dynamics at THz fre-\nquencies in ferromagnets, i. e., far from the ferromag-\nnetic precession resonance, was explained by the iner-\ntia of the magnetization [14–17]. As a consequence, the\nmagnetization may undergo nutation dynamics, i. e., a\ntrembling of the magnetization vector at THz frequen-\ncies respectively femtosecond time scales. Ultrashort\n∗E-mail: matthias.riepp@ipcms.unistra.fr\n†Present address: European X-Ray Free-Electron Laser Facility\nGmbH, Holzkoppel 4, 22869 Schenefeld, GermanyTHz light pulses therefore promise high-speed and low-\npower-consumption information writing in ferromagnets.\nSo far, experiments mainly addressed THz-driven ul-\ntrafast magnetization dynamics in homogeneously mag-\nnetized thin films, i. e., in the single-domain state [7, 8,\n18–23]. Information on the THz-driven magnetization\ndynamics in non-uniformly magnetized states, such as\nthe nanoscale multi-domain states in Co/Pt multilay-\ners with perpendicular magnetic anisotropy (PMA), is\nstill lacking. A dependence of the THz-driven coher-\nent magnetization dynamics on the magnetic anisotropy\nenergy (MAE) was discovered recently by investigating\nCo thin films with fcc, bcc and hcp crystal structure [20].\nIn this article, we present THz-driven coherent mag-\nnetization dynamics in the labyrinth domain state of a\nCo/Pt multilayer with PMA. We employ time-resolved\nXUV resonant magnetic scattering (tr-XRMS) at the\nfree-electron laser (FEL) FLASH to resolve these co-\nherent dynamics with femtosecond time and nanometer\nspatial resolution [24–28]. The magnetization shows dif-\nferent responses depending on the used THz pump flu-\nence. For low-fluence excitation with a filtered THz spec-\ntrum ( ν <6.0 THz), the magnetization undergoes an ul-\ntrafast quenching and recovery within 1 ps. For high-\nfluence excitation with the full THz spectrum, a step-\nlike quenching within 2 ps occurs followed by oscillatory\ndynamics in resonance with the THz fundamental fre-\nquency ν0= 2.5 THz. The data are consistent with in-\ncoherent and coherent ultrafast magnetization dynamics\ndriven by the ETHz- and HTHz-field components. How-arXiv:2312.02654v1  [cond-mat.mtrl-sci]  5 Dec 20232\nTHz focusing parabolaXUV undulator THz undulatore−Bending \nmagnet Back reflection\nfocusing mirrorTHz beamline \nwith delay stage\nSamplee−\nCCDFrequency ν (THz)100\n10−1\n10−2\n10−3Inorm\nν0\n0246 8 10 12141618\nFigure 1. Schematics of tr-XRMS at the BL3 instrument of FLASH Relativistic electron bunches consecutively\ntraverse the XUV and THz undulator producing intrinsically synchronized pump and probe pulses. In a custom-made end\nstation, time-delayed THz-pump and XUV-probe pulses are focused quasi-collinearly onto the sample via a parabolic mirror\nand a back-reflection multilayer mirror, respectively. Included is the polychromatic THz pump spectrum with a fundamental\nfrequency ν0= 2.5 THz measured by electro-optical sampling (EOS).\never, a time-dependent damping has to be introduced\nwhich is modeled by the interplay of lattice heating and\nPMA reduction. The oscillatory magnetization dynam-\nics are associated with correlated dynamics of the domain\nstate’s form factor, interpreted as a successive broadening\nand narrowing of the domain walls, whereas the overall\ndomain structure is conserved.\nII. RESULTS AND DISCUSSION\nExperimental details The THz-driven magnetiza-\ntion dynamics were studied by tr-XRMS at the BL3 in-\nstrument of FLASH (see Methods). The schematics of\nthe experiment are shown in Fig. 1. The planar electro-\nmagnetic THz undulator at BL3 with nine full periods\nwas tuned to generate pump pulses with a fundamental\nfrequency ν0= 2.5 THz which results in a pump-pulse\nduration of 3 .6 ps. Importantly, the pump pulses con-\ntain a broad frequency spectrum, in particular, also high-\nfrequency components reaching up to the IR regime (see\nFig. 1). We call this the unfiltered THz radiation. From\na pump-pulse intensity of 23 µJ measured by a radiome-\nter [29] and a beam size of 200 ×160µm2measured by\na fluorescent screen at the sample position, the calcu-\nlated pump fluence is FTHz= 92 mJ cm−2. This corre-\nsponds to electric and magnetic field amplitudes ETHz=\n4 MV cm−1andµ0HTHz= 1.4 T, respectively. Alterna-\ntively, a longpass filter that blocks frequency components\nν>∼6.0 THz was inserted in the THz beamline. For this\nfiltered THz radiation, the pump fluence is reduced at\nleast by a factor of four [30].\nA typical labyrinth domain state mz(r) in Co/Pt mul-\ntilayers with PMA is shown in Fig. 2 a. Here, mz=\nMz/Msis the z-component of the magnetization normal-\nized to its value at saturation. Note that for THz pump\npulses incident normally on such an OOP domain state,\nthe Zeeman torque T=m×His maximized. The scat-tered intensity I(q, t=−1 ps), obtained by tr-XRMS\nfrom the labyrinth domain state of the [Co/Pt] 8multi-\nlayer used in this experiment (see Methods), is shown\nin Fig. 2 b. In the kinematical limit using linearly polar-\nized light incident normally on a thin film with PMA, the\nscattered intensity is given by pure charge and pure mag-\nnetic scattering contributions I(q) =Ic(q) +Im(q) [31].\nRecently it was shown that the charge scattering contri-\nbution is orders of magnitude smaller as compared to the\nfirst-order magnetic scattering contribution in the here\ninvestigated region of q-space [32]. Hence, we assume\nIc(q)≈0. The Im(q) were then corrected by dark im-\nages, normalized to the average FEL-pulse intensity and\nmasked from parasitic scattering. We take the azimuthal\naverage of Im(q) which reduces the 2D to a 1D inten-\nsity distribution (see Fig. 2 c). By that, we treat the 2D\nlabyrinth domain state as a 1D chain of up- and down-\nmagnetized domains with average domain characteris-\ntics. For analysis of the resulting magnetic scattering\nintensity Im(q), we employ a Lorentzian empirical fitting\nfunction [32]\nIm(q) =e−2q/qw|{z}\nF(q)2\nm0+m1\u0010\nq−q1\nw1\u00112\n+ 1\n2\n| {z }\nS(q)2. (1)\nThe first term outside of the square brackets is the form\nfactor F(q)2which is associated with the magnetic unit\ncell in real space. It is determined by the domain wall\nparameter qwand accounts for the asymmetric shape of\nIm(q). The term in the square brackets is the magnetic\nstructure factor S(q)2corresponding to the spatial ar-\nrangement of magnetic domains, i. e., the basic magnetic\nlattice in real space. It consists of a linear superposition\nof random uniform spatial fluctuations m0and the first-\norder Lorentzian diffraction peak with amplitude m1,3\na\nc\ndb\nNorm. Int. (arb. units)0.02 nm−1200 nmmz (arb. units)\nmask\n0 0.02 0.04 0.06 0.08\nq (nm−1)00.511.52Norm. Int. (arb. units)\nBloch domain wall Domain DomainUp- Down-Im(q)\nF(q)2S(q)2\nz\nx yETHz\nHTHz\nFigure 2. Processing tr-XRMS data a Fourier-\ntransform holography image of a typical labyrinth domain\nstate mz(r) in Co/Pt multilayers showing up- and down-\nmagnetized domains as dark and bright contrast. Arrows in-\ndicate the propagation directions of the electric and magnetic\nfield components ETHz(t) = (0 , Ey(t),0) and HTHz(t) =\n(Hx(t),0,0). bNormalized magnetic scattering image\nIm(q, t=−1 ps) obtained by tr-XRMS from the labyrinth\ndomain state of the [Co/Pt] 8multilayer used in this exper-\niment. cCorresponding azimuthal average of the magnetic\nscattering intensity Im(q). Included are a fit to the data using\neq. (1) and its individual contributions, i. e., the form factor\nF(q)2and the structure factor S(q)2. The Im(q) and S(q)2\nare normalized to the maximum of Im(q) for clarity. d1D\nillustration of the individual contributions to Im(q) in real\nspace: Bloch domain walls correspond to the magnetic unit\ncell, up- and down-magnetized domains to the magnetic lat-\ntice.\nposition q1and linewidth w1. Let us emphasize that\neq. (1) is purely phenomenological. The same function-\nality, however, has been used and shown to fit scattering\ndata from tr-XRMS up to the fifth diffraction order with\nexcellent accuracy by substituting S(q)2with a sum of\nLorentzian functions [32].\nA fit of eq. (1) to Im(q, t=−1 ps) is shown in\nFig. 2 ctogether with the individual contributions F(q)2andS(q)2. An illustration of the individual contribu-\ntions in real space is given in Fig. 2 d. The exponen-\ntial form factor contribution with qw(t=−1 ps) =\n0.1446±0.0118 nm−1is interpreted as the domain wall\nwidth δm= 2πq−1\nw= 43.4±3.5 nm. Labyrinth domain\nstates in Co/Pt multilayers with PMA exhibit strong\nBloch domain wall character [33] with a width defined\nbyδB=πp\nAex(|K1+K2|)−1[34]. Using the measured\nK1= 19.6 kJ m−3andK2=−159.1 kJ m−3(see Meth-\nods), as well as an exchange stiffness Aex= 23.3 pJ m−1\nin Co/Pt multilayers with PMA and an individual Co-\nlayer thickness of 0 .8 nm [35], we obtain a calculated\nδB= 40.6 nm in good agreement with the δmdetermined\nby XRMS. The magnetic structure of the labyrinth do-\nmain state is characterized by q1(t=−1 ps) = 0 .0466±\n0.0004 nm−1corresponding to an average domain period\nξm= 2π/q1= 135 .3±1.2 nm and w1(t=−1 ps) =\n0.0199±0.0011 nm−1corresponding to a lateral corre-\nlation length λm= 2π/w 1= 316 .1±17.5 nm.\nTHz-driven magnetization dynamics In the fol-\nlowing, we discuss the time evolution of the amplitude\nm1of the magnetic structure factor S(q)2which cor-\nresponds to the z-component of the magnetization mz.\nRelative changes ∆ m1(t)−1 are presented for the case\nof the filtered ( ν<∼6.0 THz) and the unfiltered THz ex-\ncitation in Fig. 3 aandb, respectively, together with\ntheHTHz-(ETHz-)field traces measured by electro-optical\nsampling (EOS) before the respective measurements.\nHere, ∆ m1(t) =m1(t)/⟨m1(t <0)⟩t.\nThe response of the z-component of the magnetization\nto the filtered THz-pump pulses is an ultrafast quench-\ning by 16% within τd≈400 fs followed by an equally fast\nand full recovery (Fig. 3 a). A maximum degree of de-\nmagnetization of 16% agrees well with the observations\nin a 15 nm Co thin film pumped with a comparable flu-\nence and can be explained by ETHz-field driven ultrafast\ndemagnetization [8]. According to time-dependent den-\nsity functional theory (TD-DFT), the ETHz-field drives\na coherent displacement of the electrons accompanied\nby a very efficient spin–orbit-coupling-(SOC-)mediated\ndemagnetization [5]. Thereafter, one could expect a\nstep-like reduction of mzwith each half-cycle of the\nETHzfield, i. e., within τd= 0.5/ν0= 200 fs per demag-\nnetization step for monochromatic THz radiation with\nν0= 2.5 THz. We speculate that the differences in the\nultrafast response originate from the polychromaticity of\nthe THz radiation (0 THz < ν <∼6.0 THz) that causes\na more incoherent demagnetization driven by the dif-\nferent ETHz-field components. Employing the tranfer\nmatrix method, we obtain an absorbed fluence of only\nabout 0 .7 mJ cm−2for the highest frequency component\nν= 6.0 THz. It is known that incoherent ultrafast de-\nmagnetization driven by low-fluence IR laser pulses is\ngoverned by an efficient energy equilibration with the\nlattice on sub-picosecond time scales. Here, the efficient\nenergy transfer among sub-systems could explain why no\nfurther demagnetization steps within the 3 .5 ps pump-\npulse duration but an ultrafast recovery is observed.4\nHTHz (arb. units)\n−0.20.00.2HTHz (arb. units)\n−0.20.00.2Filtered\n−2−1 0 1 2 3 4\nDelay time t (ps)\n−2−1 0 1 2 3 4\nDelay time t (ps)a\nb−0.4−0.20.0Δm1 − 1 Δm1 − 1\n−1.0−0.8−0.6−0.4−0.20.0Unfiltered\nFigure 3. THz-driven magnetization dynamics\naHTHz-field trace determined from electro-optical sam-\npling (EOS) and transient z-component of the magnetiza-\ntion ∆ m1(t)−1 using the filtered THz-pump pulses (inci-\ndent fluence FTHz<23 mJ cm−2).bThe same as abut\nfor the unfiltered THz-pump pulses (incident fluence FTHz≈\n92 mJ cm−2). Gray-shaded areas are fit errors. Vertical dot-\nted lines are guides to the eye.\nFurthermore, the weak magnetic field of the low-fluence\nTHz-pump pulses could explain the absence of HTHz-\nfield induced coherent oscillations of mz. We show in the\nnext paragraph that ∆ m1(t)−1 can be modeled by the\nconvolution of low-fluence incoherent ultrafast demagne-\ntization and strongly damped coherent oscillations due\nto the presence of PMA.\nThe situation completely changes when exciting\nthe Co/Pt multilayer with the unfiltered THz-pump\npulses (Fig. 3 b). Now, mzundergoes a 3-step demag-\nnetization reaching a maximum degree of 75% after 2 ps.\nThe recovery is governed by magnetization oscillationswith an amplitude of about ±20% in resonance with\nthe THz fundamental frequency ν0= 2.5 THz. The in-\ncrease of the maximum degree of demagnetization can\nbe explained by the additional frequency components\nν > 6.0 THz and the associated increase of the pump\nfluence. Employing the transfer matrix method as be-\nfore, we obtain a 10 times higher absorbed fluence for\nν= 20.0 THz which is the highest frequency component\nwith intensity Inorm(ν)>0.01·Inorm(ν0). We note that\nthe pump spectrum contains even higher-frequency com-\nponents up to the IR regime. The step-like demagne-\ntization qualitatively agrees with the ETHz-field driven\ncoherent displacement of electrons accompanied by SOC-\nmediated demagnetization predicted by TD-DFT [5].\nHowever, demagnetization steps with a duration of about\n0.8 ps = 2 /ν0are much longer than predicted, presum-\nably due to a combination of demagnetization processes\ndriven by the various ETHz-field components. The high-\nfrequency components thereby facilitate substantial en-\nergy transfer to the electron- and the spin-system reach-\ning thermal equilibrium with the lattice on picosecond\ntime scales. An onset of the oscillatory magnetization\ndynamics at t= 2 ps is rather surprising, as typically, an\ninstantaneous response ( t= 0) is observed in THz-pump–\nprobe experiments (see, e. g., [7, 8, 22]). In compari-\nson to these publications, we have to consider that the\nCo/Pt multilayer exhibits PMA, i. e., an energetic mini-\nmum of aligning the magnetization along the z-direction.\nWe show in the following paragraph that ∆ m1(t)−1 can\nbe modeled by the convolution of high-fluence incoherent\nultrafast demagnetization and delayed coherent oscilla-\ntions due to a heat-induced reduction of PMA.\nPhenomenological model We model ∆ m1(t) as\na convolution of incoherent ultrafast demagnetization\n∆mi(t) and coherent magnetization oscillations ∆ mc(t)\nconsistent with, e. g., [7, 22, 30]\n∆m1(t)−1 = [(∆ mi(t)−1) Θ( t)]∗∆mc(t).(2)\nHere, Θ( t) is the Heaviside function accounting for the\ndemagnetization onset at t= 0. We treat the incoherent\ncontribution ∆ mi(t) as pure thermal demagnetization in-\nduced by an IR pump pulse with λi= 800 nm. Obviously,\nthis is an oversimplification as the filtered THz spectrum\ndoes not contain any IR components and the unfiltered\nTHz spectrum contains a broad frequency spectrum. In\nour approach we cast all ETHz-field induced contribu-\ntions, may they be coherent or incoherent electronic ex-\ncitations, in one ∆ mi(t) that is comparable to what is\nknown from IR-induced ultrafast demagnetization.\nThe incoherent contribution is simulated within the\nudkm1Dsim toolbox [36] that contains the microscopic\nthree temperature model (M3TM) [37] with heat diffu-\nsion along the sample z-direction (see Methods). We sim-\nulated ∆ mi(t) for a number of fluences and selected the\ntransients that match the experimentally observed max-\nimum degrees of demagnetization. This is the case for\na fluence Fi= 4 mJ cm−2andFi= 18 mJ cm−2when\nusing the filtered and the unfiltered THz-pump pulses,5\na b\nd c\nDelay time t (ps)−2−1 0 1 2 3 4\nDelay time t (ps)−2−1 0 1 2 3 4Dnorm\n0.00.20.40.60.81.0Δmc\n−0.20.00.2−0.20.00.2Δmi− 1\n−1.0−0.8−0.6−0.4−0.20.0\nData24 mJ cm−2\n20 mJ cm−2\n16 mJ cm−2\n14 mJ cm−2\n10 mJ cm−2\n4 mJ cm−2Filtered 18 mJ cm−2Unfiltered \nHTHz(t)\nΔm1 − 1\n−0.8−0.6−0.4−0.20.00.2\nFigure 4. Phenomenological model a Incoherent ultrafast demagnetization ∆ mi(t)−1 determined from M3TM simu-\nlations using Fi= 4–24 mJ cm−2. Magnetization transients that were found to match the experimental data are shown in blue\nand red. bTime-dependent damping Dnorm(t) as given by eq. (3). Details in the text. cCoherent magnetization oscillations\n∆mc(t) =HTHz(t)Dnorm(t).dFinal model for the transient z-component of the magnetization ∆ m1(t)−1 given by a convo-\nlution of the incoherent ( a) and coherent ( c) contributions.\nrespectively. The results from simulating ∆ mi(t) via\nthe M3TM are presented in Fig. 4 a. The electron- and\nphonon-temperature transients are provided in the ex-\ntended data figures.\nThe coherent contribution ∆ mc(t) is modeled via the\nproduct of the HTHz-field trace and a time-dependent\ndamping\nD(t) =e−\u0000\n1−kBTp(t)\nK1(t)V\u0001\nt, (3)\nwhere V= 10×10×10 nm3is the volume of a mag-\nnetic grain (macrospin approximation). The phonon-\ntemperature transients Tp(t) are known from the M3TM\nsimulations and the anisotropy transients are calculated\naccording to [38]\nK1(Tp(t)) =K1m(Tp(t))10. (4)\nWe use m(τ) = [1 −sτ3/2−(1−s)τ5/2]1/3, with the\nreduced temperature τ(t) =Tp(t)/TC, and s= 0.11 for\nfcc Co [39]. The Curie temperature TC= 840 K was\ndetermined by vibrating sample magnetometry after the\nexperiment (see Methods). The calculated K1(Tp(t)) are\nprovided in the extended data figures.\nIn the limit of low fluences, kBTp(t)≪K1(t)Vat all\ntimes, i. e., the pump-induced heating of the lattice istoo weak to induce a substantial reduction of PMA. In\nthis case, D(t) =Dnorm(t) becomes an exponential de-\ncay (Fig. 4 b) and the coherent oscillations ∆ mc(t) are\nstrongly damped (Fig. 4 c). In the limit of high fluences,\nD(t) diverges, which corresponds to the unphysical case\nof strongly amplified oscillations. In case of D(t)>1,\nwe therefore normalize eq. (3) to its value of minimum\nmagnetic anisotropy K1,minfort < t (K1,min) and set\nDnorm(t) = 1 for t > t (K1,min). In other words, Dnorm\ndynamically changes as K1decreases and reaches the\nregime of the undamped coherent oscillations ( Dnorm =\n1) when K1=K1,min(Fig. 4 b). The anisotropy K1de-\ncreases by about 75% within the first 2 ps while the coher-\nent oscillations ∆ mc(t) develop in amplitude (Fig. 4 c).\nThe convolutions of ∆ mi(t) and ∆ mc(t) are presented for\nthe filtered and the unfiltered THz radiation in Fig. 4 d.\nThe model perfectly reproduces the features of both\nmagnetization transients along the entire measured time\nrange. For the unfiltered THz radiation, larger devia-\ntions that exceed the experimental noise at t≈2 ps might\nbe explained by the strong electromagnetic field that is\npredicted to lead to non-linearities in the magnetization\nresponse [19].\nNote that for the case of a sample with negligible MAE,\nthe criteria D(t)>1 holds from the start ( t= 0) and6\nEquilibrium state:a b c\nd\nz z\nDW DW DW DWExcited state:\nDW DW DW DW0.150.20\n0.060.080.100.120.140.160.18qw (nm−1)\n−2 −1 0 1 2 3 4\nDelay time t (ps)0.0450.047\n0.0460.0470.0480.0490.050q1 (nm−1)\n−2 −1 0 1 2 3 4\nDelay time t (ps)0.0200.025\n−2 −1 0 1 2 3 4\nDelay time t (ps)0.0100.0150.0200.025w1 (nm−1)\nUnfilteredFiltered\nFigure 5. THz-driven domain dynamics a andbTransient position q1(t) and width w1(t) of the domain state’s\nstructure factor. cTransient domain-wall parameter qw(t) of the domain state’s form factor. Filtered and unfiltered scenarios\nare shown as blue and red data, respectively. Gray-shaded areas are fit errors. d1D illustration of the equilibrium ( t <0) and\nmaximum excited domain state.\nour model predicts an instantaneous (undamped) coher-\nent response to the HTHzfield as it was observed, e. g.,\nin [7, 8, 22]. Furthermore, it is consistent with the ob-\nservation of an increasing delay of the coherent response\nwith increasing MAE from fcc, bcc to hcp Co [20]. Even\nthough our Dnorm(t) is purely phenomenological, it is\nqualitatively in agreement with a time-dependent nuta-\ntion damping factor derived from combining the time-\ndependent non-equilibrium Green function with the con-\nventional Landau-Lifshitz-Gilbert (LLG) formalism [40].\nTheir generalized LLG equation contains a memory ker-\nnel that describes time retardation effects and originates\nfrom the fact that electron-spin can not follow instanta-\nneously a change in the orientation of the local magnetic\nmoments. It was even suggested in [20] that the coherent\nmagnetization dynamics could be fully described by one\ntime-dependent damping parameter that is qualitatively\nlinked to a stronger electron–phonon scattering at sub-\npicosecond time scales and weaker spin–lattice relaxation\nat longer time scales.\nTHz-driven domain dynamics Finally, we investi-\ngate the effect of the THz-pump pulses on the lateral\ndomain configuration, determined by the position q1and\nwidth w1of the structure factor as well as the domain-\nwall parameter qwof the form factor (Fig. 5 a–c).\nWhen using the filtered THz-pump pulses, constant fit\nparameters q1,w1andqware obtained which is consistent\nwith the fluence threshold for ultrafast domain dynamics\nobserved when using IR pump pulses [41]. The parame-\ntersq1,w1andqweven remain constant for t <2 ps when\nusing high-fluence THz pump pulses (unfiltered) which\ndemonstrates that both the domain structure and thedomain walls maintain their equilibrium size-distribution\non ultrafast time scales. This is qualitatively different\nto the ultrafast q1-shift by 3–6% to smaller values when\nusing high-fluence IR-pump pulses [41]. Originally ex-\nplained by a broadening of the domain walls due to lateral\nsuperdiffusive spin transport, more recent experiments\nsuggest ultrafast domain reconfigurations as an explana-\ntion, with a larger effect in low-symmetry systems like\nlabyrinth domain states [32, 42]. However, no such ultra-\nfast domain reconfigurations can be observed here, even\nfor high-fluence THz pump pulses. The absence of such\nultrafast domain dynamics but rather the existence of a\nwaiting time, that is determined by the time needed to\ncompensate PMA, was reported for stripe domain states\nbefore [43, 44]. For a compensated PMA and in the pres-\nence of small IP magnetic fields, the stripes were found to\nundergo a reorientation along the external field direction.\nUpon compensation of PMA after t≈2 ps, here, the do-\nmain wall parameter qwundergoes oscillatory dynamics\nthat are highly correlated with the magnetization dy-\nnamics in Fig. 3 b. Assuming that qwinversely relates to\nthe Bloch-wall width, this could be interpreted as a suc-\ncessive broadening and narrowing of the Bloch domain\nwalls between 43 nm and 89 nm at maximum. A slight\nincrease of q1within the error of the fit in combination\nwith a sharp drop of w1to almost half its equilibrium\nvalue reveals an increased long-range order during these\ncoherent oscillations from O=q1/w1≈2.3 toO ≈ 3.0 at\nmaximum. A situation where the domain-wall width in-\ncreases while the average domain period remains largely\nthe same is illustrated in Fig. 5 d. A high correlation\nbetween m1(t) and qw(t) for t >2 ps is naturally con-7\nvincing as, for oscillatory dynamics of the magnetization\nvector, a reduction of the z-component of the magnetiza-\ntion has to be associated with an increase of the x- and\ny-components and thus an increase of the domain-wall\ncontribution in tr-XRMS.\nIII. CONCLUSIONS\nIn conclusion, the magnetization of a Co/Pt multilayer\nwith PMA undergoes fluence-dependent dynamics upon\nexcitation by polychromatic THz pump pulses. These\ndynamics can be explained by a convolution of ultra-\nfast demagnetization and coherent magnetization oscil-\nlations with time-dependent damping. For low pump\nfluences (filtered), PMA causes a rapid alignment of mz\nalong the z-direction, i. e., strongly damped coherent os-\ncillations of mz. For high pump fluences (unfiltered),\nPMA undergoes a substantial reduction which enables\nundamped coherent oscillations of mzupon lattice heat-\ning. Our results demonstrate the existence of an upper\nspeed limit for THz-driven magnetization switching in\nferromagnets with PMA, i. e., a limit that is determined\nby the time needed to overcome the anisotropy energy\nbarrier. It will be interesting to see if theoretical calcu-\nlations can confirm a time-dependent nutation damping\nas the one proposed here. A reduction of the mzcom-\nponent during these coherent oscillations is associated\nwith an increase in the mx,ycomponents which, in tr-\nXRMS from a labyrinth domain state, is directly seen\nvia highly correlated dynamics of the domain-wall pa-\nrameter. The overall domain structure thereby remains\nlargely unaffected, showing no signs of spin superdiffusion\nor ultrafast domain rearrangements, which highlights the\napplicability of THz driven magnetization switching on\nthe nanoscale. Our results thereby provide a guidelinefor controlling the THz-driven magnetization dynamics\nby tailoring PMA and changing the pump fluence.\nACKNOWLEDGMENTS\nWe acknowledge DESY (Hamburg, Germany), a mem-\nber of the Helmholtz Association HGF, for the provision\nof experimental facilities. Parts of this research were car-\nried out at FLASH. We thank S D¨ usterer, M Temme and\nthe whole experimental team at FLASH for assistance\nin using the BL3 instrument. Beamtime was allocated\nfor proposal F-20160531. We thank E Jal, N Bergeard\nand B Vodungbo for many fruitful discussions as well\nas D Hrabovsky at the MPBT platform of Sorbonne\nUniversit´ e for his support with the VSM measurements.\nWe acknowledge funding by the Deutsche Forschungsge-\nmeinschaft (DFG) – SFB-925 – project ID 170620586,\nthe Cluster of Excellence ‘Advanced Imaging of Mat-\nter’ of the DFG – EXC-2056 – project ID 390715994,\nthe European Union’s Horizon 2020 research and inno-\nvation programme under the Marie Sk lodowska-Curie\ngrant agreement number 847471 and ANR-20-CE42-\n0012-01(MEDYNA).\nAUTHOR CONTRIBUTIONS\nM. R., A. P.-K., L. M., W. R., R. R., R. F., K. B.,\nM. W., R. P., T. G. and N. S. performed the time-resolved\nexperiments at FLASH and exploited the data. M. R.,\nA. P.-K. and K. B. grew the samples. M. R., A. P.-\nK., S. M. and M. H. performed the MOKE and VSM\nmeasurements. M. R. conducted the simulations and\nwrote the paper. All authors discussed and improved\nthe manuscript.\n[1] T. Kampfrath, K. 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With a beam size of 52 ×40µm2, the\ncalculated probe fluence is 2 .2 mJ cm−2. As expected for\nsuch a moderate fluence, no XUV-induced demagnetiza-\ntion nor XUV-induced permanent domain modifications\nwere observed [46, 47]. A THz beam with an about four\ntimes larger diameter than the XUV beam ensured homo-\ngeneous excitation of the probed area. Diagnostic tools\non the sample holder allowed for measuring coarse tempo-\nral as well as spatial overlap of the two beams at the sam-\nple position [33]. The scattered intensity was recorded by\na CCD with 2048 ×2048 pixels and a pixel size of 13 .5µm.\nA beamstop-photodiode was installed centimeters from\nthe detector to block the intense direct FEL beam and,\nat the same time, monitor FEL-intensity fluctuations for\nnormalization of the data [48]. The scattering statistics\nwere improved by binning 4 ×4 pixels and accumulating\n50 FEL pulse exposures in one exposure of the CCD.\nSAMPLE PROPERTIES\nThe sample used in this study was a ferromagnetic\nPt(2.0)/[Co(0.8)/Pt(0.8)] 8/Pt(5.0) multilayer grown on\na Si3N4(50.0) multi-membrane substrate using sputtering\ntechniques (numbers in nanometer). Structural investi-\ngations of Co/Pt multilayers that were fabricated in the\nsame way revealed polychrystallinity with pronounced\n(1 1 1) texture and a grain size of about 10 nm [49].\nThe first and second-order magnetic anisotropy con-\nstants K1,2were determined by magneto-optical Kerr ef-\nfect (MOKE) in polar and longitudinal geometry. Polar\nMOKE measurements revealed magnetic easy-axis be-\nhavior along the OOP direction with a coercive field\nµ0Hc≈25 mT and a saturation field µ0Hs≈150 mT.\nLongitudinal MOKE revealed magnetic hard-axis behav-\nior along the IP direction. K1,2were determined by fit-\nting the (inverted) hard-axis hysteresis loop with\nµ0HIP(m∥) =2K1\nMsm∥+4K2\nMsm3\n∥, (5)\nwhere Ms= 1.4·106A m−1is the saturation magne-\ntization in bulk Co at T= 0 K and m∥is the re-\nduced magnetization component parallel to HIP. A fit\nof eq. (5) to the data yields K1= 19 .6±4.7 kJ m−3Table I. Material-specific parameters used for the M3TM sim-\nulations (∗assumtion)\nCo Pt Si 3N4\nCe(J kg−1K−1) 0.0734 Te[50] 0 .0335 Te[50] 0 .0100 T∗\ne\nCp(J kg−1K−1) 421 [51] 133 [51] 700 [51]\nκe(W m−1K−1) 20∗20∗20∗\nκp(W m−1K−1) 100 [51] 71 .6 [51] 2 .5 [52]\nρ(kg m−3) 8860 [51] 21500 [51] 3190 [51]\nn+ik 2.53 + 4 .88i[53] 0 .60 + 8 .38i[53] 2 .00 [51]\nandK2=−159.1±3.7 kJ m−3. The MOKE measure-\nments and fit to the data are provided in the extended\ndata figures. Prior to the FEL beamtime, the sample\nwas exposed to alternating OOP magnetic field cycles\nwith decreasing amplitude and µ0Hmax= 1 T to gener-\nate a labyrinth domain state mz(r) close to the magnetic\nground state.\nAfter the experiment, the temperature dependence of\nthe saturation magnetization Ms(T) was measured em-\nploying vibrating sample magnetometry (VSM) in an ex-\nternal magnetic field µ0HIP= 500 mT. The temperature\nwas increased from T= 300 K to T= 950 K at a rate\n∆T= 10 K min−1. The Curie temperature TC≈840 K\nwas determined by a linear extrapolation of Ms(T) at\nhigh temperatures. The VSM measurement and the fit\nto the data are provided in the extended data figures.\nM3TM SIMULATIONS\nIncoherent ultrafast demagnetization is simulated\nwithin the udkm1Dsim toolbox [36] that contains the mi-\ncroscopic three temperature model (M3TM) as proposed\nby B. Koopmans et al. [37], including heat diffusion along\nthe sample z-direction\nCeρ∂Te\n∂t=∂\n∂z\u0012\nκe∂Te\n∂z\u0013\n−Gep(Te−Tp) +S(z, t)\nCpρ∂Tp\n∂t=∂\n∂z\u0012\nκp∂Tp\n∂z\u0013\n+Gep(Te−Tp) (6)\n∂mi\n∂t=Rm iTp\nTC\u0012\n1−coth\u0012miTC\nTe\u0013\u0013\n.\nThe first two differentials describe the electron- and\nphonon-temperature transients, respectively, where Ce\nand Cpare the heat capacities, κeand κpare the\nthermal conductivities, Gepis the electron–phonon cou-\npling parameter and ρis the density. The initial\nheating of the electron system is given by the laser\nsource term S(z, t). Instead of a spin-temperature tran-\nsient, the M3TM considers a magnetization transient\nthat depends on TeandTp, with a shape defined by\nR= 8asfGepkBT2\nCVatµBµ−1\natE−2\nD. Here, asf= 0.15 is\nthe spin-flip probability, kBis the Boltzmann constant,\nTC= 840 K is the Curie temperature, Vat= 4πr3\nat/310\nis the atomic volume with atomic radius rat= 1.35˚A,\nµat/µB= 1.72 is the atomic magnetic moment in units\nof the Bohr magneton and ED= 0.0357 eV is the De-\nbye energy of Co [37]. For the electron–phonon cou-\npling parameter we take a constant value of Gep=\n1.5·1018W m−3K−1in Co [54]. The udkm1Dsim tool-\nbox yields a reflectivity of 85 .6% and a transmission of\n4.5% at λi= 800 nm, calculated by the transfer matrix\nmethod including multilayer absorption.\nWithin the udkm1Dsim toolbox, in a first step,\nthe Pt(2.0)/[Co(0.8)/Pt(0.8)] 8/Pt(6)/Si 3N4(50) sample\nstructure is generated as a 1D amorphous multilayer with\nmaterial-specific properties for each subsystem (see Ta-\nble I). In a second step, the laser source term S(z, t) is de-\nfined as a delta-like pulse of high frequency ( λi= 800 nm)with fluence Fi= 4–24 mJ cm−2. Note that the influence\nof the pump-pulse duration of 3 .6 ps is taken into account\nvia the coherent contribution ∆ mc(t). In the final step,\ntheudkm1Dsim toolbox calculates spatio-temporal heat-\nmaps of the electron temperature, phonon temperature\nand magnetization for a certain delay range by solving\neq. (6) with an ODE solver. The Te(t),Tp(t) and ∆ mi(t)\nare obtained by taking the spatial average along the z-\ndirection. The Te(t),Tp(t) are provided in the extended\ndata figures.\nEXTENDED DATA FIGURES11\na\nbc\nData\nFitData\nFit\n-0.40.00.4ε (mrad)\n-1000 0 1000\nμ0HIP (mT)\n-404Φ (mrad)\n-300 0 300\nμ0HOOP (mT)\n200 300 400 500 600 700 800 900 10000.00.20.40.60.81.01.21.4\nMs (106 A/m)\nT (K)\nFigure 6. Static magnetic properties of the [Co/Pt] 8multilayer a Polar and blongitudinal MOKE at room\ntemperature. The solid line in bis a fit to the inverted data µ0HIP(ε) for small ε(details in the main article). cTemperature\ndependence of the sponatneous magnetization measured by VSM in external magnetic field µ0HIP= 500 mT. The Curie\ntemperature TC≈840 K is determined by a linear extrapolation at high temperatures.\nDelay time t (ps)−2−1 0 1 2 3 4b\nTp (K)\n300400500600700a\nTe (K)\n300700110015001900\n24 mJ cm−2\n20 mJ cm−2\n18 mJ cm−2\n16 mJ cm−2\n14 mJ cm−2\n10 mJ cm−2\n4 mJ cm−2\nDelay time t (ps)−2−1 0 1 2 3 4c\n05101520K1 (kJ m−3)\nFigure 7. Results from M3TM simulations of the [Co/Pt] 8multilayer a Electron-temperature transient Te(t) and\nbphonon-temperature transient Tp(t) for fluences Fi= 4–24 mJ cm−2. The transients are extracted from spatio-temporal heat\nmaps averaged along the sample z-direction using the udkm1Dsim toolbox. cFirst-order magnetic anisotroy transient K1(t)\ncalculated as described in the main article."    },    {        "title": "0801.3744v2.Attenuation_of_small_amplitude_oscillations_in_a_prominence_corona_model_with_a_transverse_magnetic_field.pdf",        "content": "arXiv:0801.3744v2  [astro-ph]  14 Aug 2008Attenuation of small-amplitude oscillations in\na prominence-corona model with a transverse\nmagnetic field\nR. Soler, R. Oliver and J. L. Ballester\nDepartament de F´ ısica, Universitat de les Illes Balears, E- 07122, Palma de\nMallorca, Spain\nAbstract\nObservationsshowthatsmall-amplitudeprominenceoscill ations areusuallydamped\nafter a few periods. This phenomenon has been theoretically investigated in terms\nof non-ideal magnetoacoustic waves, non-adiabatic effects b eing the best candidates\nto explain the damping in the case of slow modes. We study the a ttenuation of\nnon-adiabatic magnetoacoustic waves in a slab prominence e mbedded in the coro-\nnal medium. We assume an equilibrium configuration with a tra nsverse magnetic\nfield to the slab axis and investigate wave damping by thermal conduction and\nradiative losses. The magnetohydrodynamic equations are c onsidered in their lin-\nearised form and terms representing thermal conduction, ra diation and heating are\nincluded in the energy equation. The differential equations t hat govern linear slow\nand fast modes are numerically solved to obtain the complex o scillatory frequency\nand the corresponding eigenfunctions. We find that coronal t hermal conduction and\nradiative losses from the prominence plasma reveal as the mo st relevant damping\nmechanisms. Both mechanisms govern together the attenuati on of hybrid modes,\nwhereas prominence radiation is responsible for the dampin g of internal modes and\ncoronal conductionessentially dominatestheattenuation ofexternalmodes.Inaddi-\ntion, the energy transfer between the prominence and the cor ona caused by thermal\nconduction has a noticeable effect on the wave stability, radi ative losses from the\nprominence plasma being of paramount importance for the the rmal stability of fast\nmodes. We conclude that slow modes are efficiently damped, wit h damping times\ncompatible with observations. On the contrary, fast modes a re less attenuated by\nnon-adiabatic effects and their dampingtimes are several ord ersof magnitude larger\nthan those observed. The presence of the corona causes a decr ease of the damping\ntimes with respect to those of an isolated prominence slab, b ut its effect is still\ninsufficient to obtain damping times of the order of the period in the case of fast\nmodes.\nKey words: Sun: oscillations, Sun: magnetic fields, Sun: corona, Sun: p rominences\nPACS:52.35.Bj, 96.60.P-\nPreprint submitted to Elsevier 24 October 20181 Introduction1\nSolar prominences are large-scale coronal magnetic structures w hose material, 2\ncooler and denser than the typical coronal medium, is in plasma stat e. Promi- 3\nnences are supported against gravity by the coronal magnetic fie ld, which also 4\nmaintains the prominence material thermally isolated from the coron a. Small- 5\namplitude oscillations in solar prominences were detected almost 40 ye ars ago 6\n(Harvey, 1969). These oscillatory motions seem to be of local natu re and their 7\nvelocity amplitude is typically less than 2–3km s−1. Observations have also al- 8\nlowed to measure a wide range of periods between 30 s (Balthasar et al., 1993) 9\nand 12 h (Foullon et al., 2004). More recently, some high-resolution o bserva- 10\ntions of prominence oscillations by the Hinode/SOT instrument have b een 11\nreported (Okamoto et al., 2007; Berger et al., 2008; Ofman & Wang, 2008). 12\nFrom the theoretical point of view, the oscillations have been interp reted by 13\nmeans of the magnetoacoustic eigenmodes supported by the prom inence body. 14\nA recent example is the work by Terradas et al. (2008) in which the ob serva- 15\ntionsof Okamoto et al. (2007)areinterpreted asfast kink waves. The reader is 16\nreferred to Oliver & Ballester (2002); Ballester (2006); Banerjee et al. (2007) 17\nfor extensive reviews of both observational and theoretical stu dies. 18\nEvidence of the attenuation of small-amplitude prominence oscillation s has 19\nbeen reported in some works (Molowny-Horas et al., 1999; Terrada s et al., 20\n2002; Lin, 2004). A typical feature of these observations is that the oscillatory 21\nmotions disappear after a few periods, hence they are quickly damp ed by one 22\nor several mechanisms. The theoretical investigation of this phen omenon in 23\nterms of magnetohydrodynamic (MHD) waves has been broached b y some au- 24\nthors by removing the ideal assumption and by including dissipative te rms in 25\nthe basic equations. Non-adiabatic effects appear to be very efficie nt damping 26\nmechanisms and have been investigated with the help of simple promine nce 27\nmodels (Ballai, 2003; Carbonell et al., 2004, 2006; Terradas et al., 2005). Nev- 28\nertheless, other damping mechanisms have been also proposed, like wave leak- 29\nage (Schutgens, 1997a,b; Schutgens & Toth, 1999), dissipation b y ion-neutral 30\ncollisions (Forteza et al., 2007) and resonant absorption (Arregui et al., 2008). 31\nIn a previous work (Soler et al., 2007, hereafter Paper I), we have studied for 32\nthe first time the wave attenuation by non-adiabatic effects of a pr ominence 33\nslab embedded in the corona. In that work the magnetic field is paralle l to 34\nthe slab axis and it is found that the corona has no influence on the int ernal 35\nslow modes, but it is of paramount importance to explain the damping o f fast 36\nmodes, which are more attenuated than in simple models that do not c onsider 37\nthe coronal medium. Following the path initiated in Paper I, here we inv esti- 38\nEmail address: [roberto.soler,ramon.oliver,joseluis.ballester]@uib .es\n(R. Soler, R. Oliver and J. L. Ballester).\n2gate the wave damping due to non-adiabatic mechanisms (radiative lo sses and 39\nthermal conduction) in an equilibrium made of a prominence slab embed ded 40\nin a coronal medium, but now we consider a magnetic field transverse to the 41\nslab axis. This configuration and that studied in Paper I correspond to limit 42\ncases, since measurements with Zeeman and Hanle effects indicate t hat the 43\nmagnetic field lines are skewed to the long axis of prominences. On ave rage, 44\nthe prominence axis and the magnetic field form an angle of about 20 d eg. 45\nThus, the skewed case is relegated to a future investigation.46\nTheequilibriumconfigurationassumedherewasanalysedindetailbyJ oarder & Roberts 47\n(1992) and Oliver et al. (1993) in the case of ideal, adiabatic perturb ations. 48\nThemaindifferencebetweenbothworksisinthetreatmentofgravit y.Joarder & Roberts 49\n(1992) neglected the effect of gravity and so straight field lines wer e consid- 50\nered. On the other hand, Oliver et al. (1993) took gravity into acco unt and 51\nassumed curved field lines according to the Kippenhahn & Sch¨ ulter ( 1957) 52\nmodel modified to include the surrounding coronal plasma (Poland & A nzer, 53\n1971). Despite this difference, both studies agree in establishing a d istinction 54\nbetween different normal modes depending on the dominant medium s up- 55\nporting the oscillation. Hence, internal modes are essentially suppo rted by 56\nthe prominence slab whereas external modes arise from the prese nce of the 57\ncorona. In addition, hybrid (or string) modes appear due to the co mbined 58\neffect of both media.59\nThe investigation of the thermal attenuation of oscillations suppor ted by such 60\nequilibrium is unsettled to date and, indeed, this is the main motivation f or 61\nthe present study. However, two works (Terradas et al., 2001, 2 005) studied 62\nthe wave damping in an isolated prominence slab. Terradas et al. (200 1) con- 63\nsidered radiative losses given by the Newtonian law of cooling as dampin g 64\nmechanism and studied the attenuation in the Kippenhahn & Sch¨ ulte r (1957) 65\nand Menzel (1951) prominence models. Subsequently, Terradas e t al. (2005) 66\nconsidered amorecomplete energyequationincluding opticallythinra diation, 67\nplasma heating and parallel thermal conduction, and assumed stra ight field 68\nlines since gravity was neglected. The main conclusion of both works is that 69\nnon-adiabatic mechanisms are only efficient in damping slow modes wher eas 70\nfast modes remain almost undamped. Nevertheless, in the light of th e results 71\nof Paper I, the presence of the coronal medium can have an import ant reper- 72\ncussiononthewavedamping.Theinvestigationofthiseffectisthema inaimof 73\nthe present work. Therefore, we extend here the work of Terra das et al. (2005) 74\nby considering the presence of the corona and neglect the effect o f gravity as 75\nin Joarder & Roberts (1992) for simplicity.76\nThis paper is organised as follows. Section 2 contains a description of the 77\nequilibrium configuration and the basic equations which govern non-a diabatic 78\nmagnetoacousticwaves.Then,theresultsofthisworkareexten sively discussed 79\nin Sect. 3. Finally, our conclusions are given in Sect. 4.80\n32 Equilibrium and basic equations81\nThe equilibrium configuration (see Fig. 1) is made of a homogeneous pla sma 82\nslabwithprominenceconditions(density ρpandtemperature Tp),whoseaxisis 83\norientated along the z-direction, embedded in a coronal environment (density 84\nρcand temperature Tc). The system is bounded in the x-direction due to 85\nthe presence of two rigid walls representing the solar photosphere , but it is 86\nunlimited in the y- andz-directions. The width of the prominence slab is 2 xp87\nand the total width of the system is 2 xc. The magnetic field is transverse to 88\nthe prominence slab, /vectorB0=B0ˆex, withB0everywhere constant. 89\nFig. 1. Sketch of the equilibrium. The dark region represent s the prominence slab\nwhile the light region corresponds to the corona. The photos pheric walls are the two\nhatched areas on both sides of the corona.\nInordertofindthebasicequationsthatgovernnon-adiabaticmag netoacoustic 90\nwaves we follow the same process as in Terradas et al. (2005). We co nsider the 91\nusual MHD equations (Eqs. (1)–(6) of Terradas et al., 2005) in whic h non- 92\nadiabatic terms have been included in the energy equation,93\nDp\nDt−γp\nρDρ\nDt+(γ−1)[ρL(ρ,T)−∇·(/vector κ·∇T)] = 0, (1) 94\nwherep,ρandTare the gas pressure, density and temperature, respectively. 95\nThe quantity γis the adiabatic ratio, here taken γ= 5/3. The non-ideal 96\nterms in Eq. (1) are explained in detail in Carbonell et al. (2004). The rmal 97\n4conduction is represented by ∇·(/vector κ·∇T), where /vector κis the conductivity tensor 98\nwhich in coronal and prominence applications is usually approximated b y its 99\nparallel component to the magnetic field, κ/bardbl= 10−11T5/2Wm−1K−1. Radia- 100\ntive losses and heating are evaluated together through the heat- loss function, 101\nL(ρ,T) =χ∗ρTα−hρaTb, where radiation is parametrised with χ∗andα(see 102\nTable I of Paper I) and the heating scenario is given by exponents aandb 103\n(Rosner et al., 1978; Dahlburg & Mariska, 1988).104\nRegarding our equilibrium configuration, the reader must be aware t hat, al- 105\nthough there have been some attempts to construct a self-cons istent promi- 106\nnencemodelincludingbothmagnetostaticsandthermodynamics(e .g.Milne at al., 107\n1979; Low & Wu, 1981; Anzer & Heinzel, 1999), to date this task re mains to 108\nbe done. Here, we consider a simplified prominence-corona configur ation, but 109\nit includes the two basic ingredients observed in real prominences. F irst, the 110\nexistence of a steep temperature gradient between the prominen ce and the 111\ncorona and, second, the apparent thermal isolation of the promin ence mate- 112\nrial from the much hotter corona. The first point is addressed by c onsidering 113\nthat the temperature profile is a step function, and so the promine nce-corona 114\ntransition region (PCTR) has not been considered. This choice is sup ported 115\nby results of previous works (e.g. Oliver & Ballester, 1996) which sho wed that 116\nthe PCTR has a minor influence on the prominence oscillatory modes. O n 117\nthe other hand, to represent the thermal isolation we have neglec ted the heat 118\nflux due to thermal conduction at the boundary between the prom inence and 119\nthe corona. Therefore, we impose that both the prominence and t he corona 120\nare isothermal and thermally isolated, and so radiative losses and he ating are 121\nlocally balanced, i.e. L(ρ0,T0) = 0, where ρ0andT0are the local equilibrium 122\ndensity and temperature, respectively.123\nAssuming that the plasma is at rest in the equilibrium state (i.e. no flux o f 124\nmaterial) andconsidering small perturbations, we findthelinearised version of 125\nthe MHD equations (Eqs. (10)–(15)of Terradas et al., 2005). Acc ording to the 126\ngeometry of our model, we assume perturbations of the form f1(x)expi(ωt+ 127\nkyy+kzz) and exclude Alfv´ en waves from this analysis by considering only 128\nmotions and propagation in the xz-plane (vy= 0,ky= 0). Now we combine 129\nthe resultant expressions and eliminate all the perturbed quantitie s in favour 130\nof the velocity perturbations, vxandvz, and the temperature perturbation, 131\nT1. By this process, we obtain three coupled ordinary differential equ ations, 132\nc2\nsd2vx\ndx2+γω2vx+ikzc2\nsdvz\ndx−iωc2\ns\nT0dT1\ndx= 0, (2) 133\nv2\nAd2vz\ndx2+/bracketleftBigg\nω2−k2\nz/parenleftBigg\nv2\nA+c2\ns\nγ/parenrightBigg/bracketrightBigg\nvz+ikzc2\ns\nγdvx\ndx+ωkzc2\ns\nγT1\nT0= 0,(3) 134\n5κ/bardbl1\np0d2T1\ndx2−/parenleftBigg\nωT+iω\nγ−1/parenrightBiggT1\nT0\n−/parenleftbigg\n1+iωρ\nω/parenrightbiggdvx\ndx−ikz/parenleftbigg\n1+iωρ\nω/parenrightbigg\nvz= 0, (4)\nwherec2\ns=γp0\nρ0is the adiabatic sound speed squared whereas v2\nA=B2\n0\nµρ0is 135\nthe Alfv´ en speed squared. p0andB0denote the equilibrium gas pressure 136\nand magnetic field strength, respectively, and µis the magnetic permittivity 137\n(µ= 4π10−7in MKS units). Quantities ωTandωρare defined as follows, 138\nωρ≡ρ0\np0(L+ρ0Lρ), ω T≡ρ0\np0T0LT, 139\nLρ,LTbeing the partial derivatives of the heat-loss function with respec t to 140\ndensity and temperature, respectively,141\nLρ≡/parenleftBigg∂L\n∂ρ/parenrightBigg\nT, L T≡/parenleftBigg∂L\n∂T/parenrightBigg\nρ. 142\nEquations (2), (3) and (4) govern fast and slow magnetoacoustic waves to- 143\ngether with the thermal or condensation mode. In this work we do n ot study 144\nthe thermal wave since we pay our attention to the magnetoacous tic modes. 145\nTerradas et al.(2005)foundanapproximateanalyticalsolutionof Eqs. (2)–(4) 146\nby neglecting thermal conduction, a valid assumption in prominence p lasmas. 147\nHowever, thermal conduction has an important role in coronal con ditions and 148\ncannot be neglected in order to perform a realistic description of th e oscil- 149\nlatory modes supported by our equilibrium configuration. Hence, we solve 150\nthe full set of Eqs. (2)–(4) using the numerical code PDE2D (Sewe ll, 2003) 151\nbased on finite elements (see Terradas et al., 2005, for an explanat ion of the 152\nmethod). The jump conditions at the interface between the promin ence and 153\nthe corona are automatically well-treated by the code. These jump conditions 154\nare (Goedbloed & Poedts, 2004):155\n[/vector v] =/vector0,/bracketleftBig/vectorB/bracketrightBig\n=/vector0,[p] = 0, (5) 156\nwhere/vector vand/vectorBare the perturbed velocity and the magnetic field vectors, re- 157\nspectively. For a complete closure of the system we need to supply a physically 158\nconsistent set ofboundaryconditionsfortheperturbationsatt hephotospheric 159\nwalls,x=±xc. In this work, we consider two different sets of boundary con- 160\nditions,161\nvx=vz=T1= 0,atx=±xc, (6) 162\n6and163\nvx=vz=T′\n1= 0,atx=±xc, (7) 164\nwhere′indicates derivative with respect to x. Both sets consider line-tied 165\nconditions for the velocity perturbations, i.e. the disturbances ar e unable to 166\nperturb the dense photospheric plasma which acts as perfectly rig id wall. On 167\nthe other hand, sets (6) and (7) differ by the condition for T1, which has 168\ndifferent physical implications. Set (6) assumes that the perturba tion to the 169\ntemperature vanishes at x=±xcand this means that the photospheric walls 170\nare taken as isothermal. On the contrary, set (7) considers a zer o-temperature 171\ngradient for the perturbation between the corona and the photo sphere, so no 172\nperturbed heat flux is allowed at the boundaries. From our point of v iew, 173\nset (6) makes more physical sense than set (7), since one can exp ect that the 174\nmuch denser photospheric plasma can instantaneously radiate awa y any in- 175\ncoming perturbed heat flux from the corona. However, set (7) imp oses that 176\nthere is no heat exchange between the corona and the photosphe re, although 177\nthe temperature perturbation can have a non-zero value at the w alls. Regard- 178\ning these boundary conditions, Cargill & Hood (1989) performed a s tudy of 179\nthe thermal stability of wave and thermal modes in a Cartesian coro nal slab 180\nand pointed out that the solutions computed by assuming the bound ary con- 181\nditions given by set (6) are more thermally stable than those obtaine d for 182\nboundary conditions of set (7).183\nFor a fixed real kz, the numerical solution of Eqs. (2), (3) and (4) provides 184\nwith acomplex frequency, ω=ωR+iωI. Intheideal, adiabaticcase ωI= 0 and 185\ntherefore the solutions of Eqs. (2), (3) and (4) are those of Joa rder & Roberts 186\n(1992). Using the real and imaginary parts of the frequency, we c an com- 187\npute the oscillatory period, P, the damping time, τD, and the ratio of both 188\nquantities,189\nP=2π\nωR, τ D=1\nωI,τD\nP=1\n2πωR\nωI. (8) 190\n3 Results191\nUnlessotherwisestated,thefollowingequilibriumparametersareco nsideredin 192\nall computations: Tp= 8000K, ρp= 5×10−11kg m−3,Tc= 106K,ρc= 2.5× 193\n10−13kg m−3,B0= 5 G,xp= 3000 km and xc= 10xp. The coronal density is 194\ncomputed by fixing the coronal temperature and imposing pressur e continuity 195\nacross the interfaces. In addition, we assume an optically thin prom inence 196\nplasma (regime Prominence (1) of Paper I) and a constant heating p er unit 197\nvolume ( a=b= 0). In all the following expressions, subscript 0 indicates 198\n7local equilibrium values, while subscripts p and c denote quantities exp licitly 199\ncomputed with prominence and coronal parameters, respectively . 200\n3.1 Dispersion diagram and wave modes201\nSolutions of Eqs. (2)–(4) can be grouped in internal, external and hybrid 202\nmodes. Although there is an infinite number of harmonics for interna l and 203\nexternal modes, only two hybrid modes are possible: the hybrid slow mode 204\nandthehybrid fastmode(thisnomenclature is takenfromOliver et a l., 1993). 205\nFigure 2 shows the dimensionless real part of the frequency versu skzxpfor 206\nthe fundamental symmetric oscillatory modes (i.e. solutions with vxeven with 207\nrespectx= 0) and some of their harmonics, where we have assumed the 208\nboundary conditions given by Eq. (6). A similar diagram can be obtaine d for 209\nthe antisymmetric modes (i.e. solutions with vxodd with respect x= 0) and 210\nfor the other set of boundary conditions (Eq. (7)).211\nFig. 2. Dimensionless real part of the frequency versus kzxpfor the oscillatory\nsymmetric modes: hybrid slow (solid line at the bottom), fun damental internal slow\nand first harmonics (dotted lines), fundamental external sl ow and first harmonics\n(dashed lines), fundamental internal fast and first harmoni c (dash-dotted lines) and\nfundamental external fast (three dot-dashed line at the upp er left corner).\nThe behaviour of the real part of the frequency is the same as tha t explained 212\nby Oliver et al. (1993). The value of ωRfor both internal and external slow 213\nmodes and for the hybrid slow mode shows a very weak dependence o nkz214\nsince almost horizontal lines are seen in Fig. 2. On the contrary, bot h internal 215\nand external fast modes show a quasi-parabolic dependence on kz(this is 216\nalso applicable to the hybrid fast mode, present in the dispersion diag ram for 217\nantisymmetric modes). The reader is referred to Oliver et al. (1993 ) for more 218\nextensive details about the behaviour of ωR. 219\n8Fig. 3. Modulus of the eigenfunctions vx,vzandT1(in arbitrary units) versus the\ndimensionless distance to the slab axis corresponding to th e fundamental oscilla-\ntory modes for kzxp= 1. The solid line corresponds to the boundary conditions\nvx=vz=T′\n1= 0 atx=±xc, while the dotted line corresponds to the boundary\nconditions vx=vz=T1= 0 atx=±xc. The shaded region shows the location of\nthe prominence slab.\nNext, we focus on the fundamental modes and their eigenfunction svx,vzand 220\nT1are displayed in Fig. 3 for kzxp= 1 and for both sets of boundary con- 221\nditions. The spatial structure of the disturbances vxandvzis the one shown 222\nby Oliver et al. (1993). Hence, non-adiabatic effects do not modify t he spa- 223\ntial behaviour of velocity perturbations. Internal modes produc e large plasma 224\n9displacements inside the slab, external modes achieve large amplitud es in the 225\ncorona and the amplitude of hybrid modes is of the same order in both media. 226\nIt is worth to mention that the hybrid fast mode can be considered a s an 227\ninternal-like mode for large kzxpsince the amplitude of its perturbations in 228\nthe corona decreases as kzxpincreases. Regarding the temperature perturba- 229\ntion, it is larger for the slow modes than for the fast modes and, in ge neral, is 230\nlarger in the prominence than in the corona. Finally, the differences b etween 231\nthe eigenfunctions for the two sets of boundary conditions (Eqs.( 6)–(7)) are 232\nonly relevant for the hybrid slow mode.233\nFromtheobservationalpointofview, internalandhybridmodesco uldbemore 234\neasily observed than external modes by instruments focusing on p rominences, 235\nsince the amplitude of the latter ones is very small in the prominence b ody. 236\nFor this reason, the results corresponding to internal and hybrid modes are 237\nthe most interesting for prominence seismology. However, here we study the 238\nthree kinds of solutions in order to perform a complete description o f the 239\nfundamental wave modes supported by the equilibrium configuratio n. 240\n3.2 Mode coupling241\nOliver et al. (1993) showed that avoided crossings occur in the dispe rsion dia- 242\ngram when two solutions couple and interchange their magnetoacou stic prop- 243\nerties. Nevertheless, no avoided crossings seem to take place in ou r dispersion 244\ndiagram (Fig. 2) since the curves of ωRfor the internal fast modes and for 245\nthe slow modes cut each other. This fact can be understood by con sidering 246\nthat in the present, non-adiabatic case the complete dispersion dia gram is in 247\na three-dimensional space because the frequency has an imaginar y part. So, 248\nFig. 2 actually corresponds to a projection of the complete three- dimensional 249\ndispersion diagram on the kzωR–plane. 250\nUpon exploring the complete dispersion diagram, we have found that three 251\ndifferent couplings can take place:252\n(1) If the imaginary parts of the frequency of the coupling modes d iffer by 253\nseveral orders of magnitude, there is no avoided crossing betwee n the real 254\nparts. Hence, the coupling between modes is “weak” and only becom es 255\napparent by means of a slight mutual approach of the imaginary par ts of 256\nω(see Fig. 4, left panel). 257\n(2) If both imaginary parts of the frequency have a similar value, th e real 258\nparts show an avoided crossing and so a “strong” coupling takes pla ce 259\n(see Fig. 4, mid panel). 260\n(3) In very peculiar cases, an “anomalous” coupling takes place whe n the 261\nimaginary parts of ωof the two coupling modes repel each other (see 262\n10Fig. 4, right panel). This situation has important effects on the wave 263\nstability, as we explain in Sect. 3.5. 264\nThebehaviourofthemodecouplingwaspreviouslydescribedbyTerr adas et al. 265\n(2001) in the cases that we call “weak” and “strong” couplings (co mpare our 266\nFig. 4 with Fig. 12 of Terradas et al., 2001).267\nFig. 4. Three-dimensional dispersion diagrams (solid line s) close to a coupling be-\ntween a fast mode and a slow mode. Dashed and dotted lines are t he projections\nof the dispersion curves on the horizontal and vertical plan es. The left-hand side\npanel presents a “weak” coupling, the middle panel shows a “s trong” coupling and\nthe right-hand side panel displays an “anomalous” coupling .\n3.3 Periods and damping times268\nHereafter we restrict ourselves to the fundamental modes and c ompute the 269\noscillatory period, P, the damping time, τD, and the ratio of the damping 270\ntime to the period as functions of the dimensionless wavenumber, kzxp. We 271\nconsider values for kzxpbetween 0.01 and 3, which correspond to wavelengths 272\nbetween 5 ×103km and105km,approximately. These valuescover therangeof 273\ntypically observed wavelengths in prominence oscillations (Oliver & Balle ster, 274\n2002). The results of the computations are displayed in Fig. 5 consid ering the 275\ntwo sets of boundary conditions (Eqs. (6)–(7)).276\nThe periods obtained here agree with those provided and commente d by 277\nJoarder & Roberts (1992) and Oliver et al. (1993), therefore we t urn our at- 278\ntention to the damping times. Regarding slow modes, we see that the y are 279\nstrongly damped, with values of τD/Pclose to 1 for the three modes. How- 280\never, fast modes are much less attenuated and the obtained value s ofτD/Pare 281\nmuch larger than those observed. This fact involves an important d ifference 282\nwith the results of Paper I, in which fast waves were efficiently atten uated 283\nfor some values of the wavenumber. Such as happens with the perio d, the 284\ndamping time of slow modes is almost independent of kz. On the contrary, 285\nfast modes are less attenuated for small kzthan for large kz. This evidence 286\ncan be understood by means of the following arguments. Considerin gkz= 0 287\n11Fig. 5. Period (left), damping time (centre) and ratio of the damping time to the\nperiod (right) versus kzxpfor the fundamental oscillatory modes. The solid line\ncorresponds to the boundary conditions vx=vz=T′\n1= 0, while the dotted line\ncorresponds to the boundary conditions vx=vz=T1= 0.\nthen Eqs. (2)–(4) become,288\nc2\nsd2vx\ndx2+γω2vx−iωc2\ns\nT0dT1\ndx= 0, (9) 289\nv2\nAd2vz\ndx2+ω2vz= 0, (10) 290\n12κ/bardbl1\np0d2T1\ndx2−/parenleftBigg\nωT+iω\nγ−1/parenrightBiggT1\nT0−/parenleftbigg\n1+iωρ\nω/parenrightbiggdvx\ndx= 0. (11) 291\nEquations (9) and (11) are still coupled and govern slow and therma l waves, 292\nwhich are affected by non-adiabatic mechanisms through the terms withκ/bardbl, 293\nωTandωρin Eq. (11). On the contrary, Eq. (10) is now decoupled from the 294\nrest and governs fast modes alone, which become pure Alfv´ en wav es and are 295\nnot affected by non-adiabatic terms. Thus, for kz→0 fast waves tend to 296\nthe ideal, undamped behaviour. When kzis increased, fast modes are more 297\naffected by acoustic effects and their damping time decreases and s tabilises. 298\nThe little peaks shown in the bottom panels of Fig. 5, corresponding t o the 299\nexternal fast mode, are in fact the result of “strong” couplings w ith slow mode 300\nharmonics. The differences arising from the different boundary con ditions are 301\nonly of importance for the hybrid slow mode, as we indicated in Sect. 3 .1. 302\nWe see that the boundary condition T′\n1= 0 produces a substantially stronger 303\ndamping for the hybrid slow mode than the condition T1= 0. 304\nFinally, an approximate value to the frequency of internal and exte rnal slow 305\nmodes can be obtained by considering the approximation given in App. B of 306\nPaper I, namely307\nω≈Λkx, (12) 308\nwherekxis the wavenumber in the field direction and Λ is the modified sound 309\nspeed due to thepresence of non-adiabaticeffects, defined inPap er Iasfollows 310\nΛ2≡c2\ns\nγ\n(γ−1)/parenleftBig\nT0\np0κ/bardblk2\nx+ωT−ωρ/parenrightBig\n+iγω\n(γ−1)/parenleftBig\nT0\np0κ/bardblk2x+ωT/parenrightBig\n+iω\n. (13) 311\nThe value of kxis fixed by the equilibrium geometry, but for simplicity we 312\nconsider now the analytical approximations of the dominant wavenu mbers 313\ngiven by Joarder & Roberts (1992) in the adiabatic case and for the long 314\nwavelength limit, namely315\nkx≈π\n2xp(14) 316\nfor the fundamental internal mode, and317\nkx≈π\nxc−xp(15) 318\nforthefundamental external mode. Onemust bearinmind thatth ewavenum- 319\nbers in the present, non-adiabatic case are complex quantities, bu t we expect 320\n13that their real part is similar to that in the adiabatic case, such as ha ppens 321\nwith the value of the frequency. Applying now Eq. (12) to the intern al slow 322\nmode, i.e. considering prominence parameters in the expression for Λ (Eq. 13) 323\nand the approximation for kxgiven by Eq. (14), one obtains P≈23.50 min, 324\nτD≈73.05 min and τD/P≈3.11. On the other hand, if the process is re- 325\npeated for the external slow mode, this gives P≈6.20 min, τD≈6.70 min 326\nandτD/P≈1.08. We see that these approximate values reasonably agree with 327\nthose numerically obtained and represented in Fig. 5.328\n3.4 Importance of the damping mechanisms329\nFig. 6. Damping time versus kzxpfor the fundamental modes: a) hybrid slow, b)\ninternal slow, c) external slow, d) hybrid fast, e) internal fast, and f) external fast.\nDifferent linestyles represent the omitted mechanism: all me chanisms considered\n(solid line), prominence conduction eliminated (dotted li ne), prominence radiation\neliminated (dashed line), coronal conduction eliminated ( dot-dashed line) and coro-\nnal radiation eliminated (three dot-dashed line). Arrows i n panels dandepoint the\nlocation of thermal instabilities ( ωI<0) which appear if prominence radiation is\nomitted (dashed line).\nIn order to know which are the mechanisms responsible for the damp ing of 330\neach mode, we now follow the same procedure as in Paper I. We compa re the 331\ndamping time obtainedwhen considering all non-adiabaticterms (disp layed in 332\nthe middle column of Fig. 5) with the results obtained when a specific me cha- 333\nnism is removed from the energy equation (Eq. (1)). This analysis allo ws us to 334\nknow whether the omitted mechanism has a relevant effect onthe at tenuation. 335\nBeforeundertakingthisinvestigation,weneedtoknowifbothsets ofboundary 336\nconditions are adequate in the absence of thermal conduction. If one imposes 337\n14κ/bardbl= 0 in Eq. (4) then T1can be written as function of vzandv′\nx, 338\nT1=−T0(1+iωρ/ω)\nωT+iω/(γ−1)(v′\nx+ikzvz), (16) 339\nwhich can be substituted into Eqs. (2) and (3) in order to obtain two coupled 340\ndifferential equations involving the perturbed velocities alone,341\nc2\ns/parenleftbigg\n1+iω\nˆω/parenrightbiggd2vx\ndx2+γω2vx+ikzc2\ns/parenleftbigg\n1+iω\nˆω/parenrightbiggdvz\ndx= 0, (17) 342\nv2\nAd2vz\ndx2+/braceleftBigg\nω2−k2\nz/bracketleftBigg\nv2\nA+c2\ns\nγ/parenleftbigg\n1+iω\nˆω/parenrightbigg/bracketrightBigg/bracerightBigg\nvz\n+ikzc2\ns\nγ/parenleftbigg\n1+iω\nˆω/parenrightbiggdvx\ndx= 0. (18)\nHere ˆωis introduced to simplify the notation, 343\nˆω≡ωT+iω/(γ−1)\n1+iωρ/ω. 344\nNow, the system formed by Eqs. (17) and (18) is fully determined by assuming 345\nonly boundary conditions for vxandvz. Hence, the behaviour of T1at the 346\nboundaries cannot be imposed but is fixed by the conditions over the velocity 347\nperturbations. If one takes vx=vz=T′\n1= 0 as boundary conditions, then 348\nEq. (16) yields the constraint v′′\nx+ikzv′\nz= 0, which substituted in Eq. (17) 349\nautomatically gives the redundant condition vx= 0. On the other hand, if 350\none assumes vx=vz=T1= 0 atx=±xc, then Eq. (16) now imposes 351\nv′\nx= 0 at the boundaries. This last condition substituted in Eq. (18) give s 352\nthe extra condition v′′\nz= 0 over the system, which implies a new restriction 353\nthat is not generally satisfied by all solutions. Thus, T′\n1= 0 reveals itself 354\nas the “natural” boundary condition for the temperature pertur bation when 355\nthermal conduction is neglected. So, for the following investigation we restrict 356\nourselves to the boundary conditions vx=vz=T′\n1= 0 since the conditions 357\nvx=vz=T1= 0 are not consistent with the differential equations when 358\nthermal conduction is neglected.359\nThe results of the computations are displayed in Fig. 6. Although we h ave 360\nexplored a wide range of values of kz, the plots are only drawn again for 361\n0.01< kzxp<3 since we have found that the importance of the damping 362\nmechanisms does not show a strong dependence on kz. Regarding slow modes, 363\nwe clearly see that the damping of the internal mode is dominated by t he radi- 364\nation from the prominence plasma, as expected, while coronal cond uction has 365\n15a minor effect. On the other hand, the hybrid and external modes a re affected 366\nby coronal conduction together with prominence radiation. Both m echanisms 367\nhave a similar influence on the hybrid mode, while coronal conduction d omi- 368\nnates the attenuation of the external mode. This result for the h ybrid mode 369\nis coherent with the fact that its perturbations achieve large amplit udes both 370\nin the prominence and the corona (see top row of Fig. 3), so one exp ects that 371\nthe most relevant damping mechanisms of each medium govern toget her the 372\nattenuation of the hybrid mode. However, the result for the exte rnal mode is 373\na priorisurprising because its perturbations are very small in the prominen ce 374\n(see fifth row of Fig. 3) and one expects that the prominence-rela ted mecha- 375\nnisms have a minor effect on its damping. The following discussion attem pts 376\nto explain why prominence radiation affects so much the external mo de. 377\nThe equilibrium configuration assumed in the present work implies an ad di- 378\ntional complication with respect to the equilibrium considered in Paper I, in 379\nwhich magnetic field lines were taken parallel to the interface betwee n the 380\nprominence and the corona. Hence, both media were thermally isolat ed in the 381\nmodel of Paper I since there was no transfer of energy from one m edium to the 382\nother. However, in the present model thermal conduction conne cts both media 383\nsince field lines are transverse to the interfaces. This fact allows he at transfer 384\nbetween the prominence and the corona. So, some energy can flow along field 385\nlines and can be injected from the corona into the prominence, wher e the en- 386\nergy is efficiently radiated away by the plasma. In this way, the influen ce of 387\nprominence radiation on the damping of the external slow mode, and also the 388\nhybrid slow mode, is amplified by means of coronal thermal conductio n. 389\nNext we turn our attention to the fast modes. At first sight, the b ehaviour of 390\nthefastmodeswhenaspecificmechanism isremoved fromtheenerg yequation 391\nis absolutely different from that seen in the case of the slow modes an d needs 392\nmore extensive explanations. In Sect. 3.3, we commented that the damping 393\ntime of the fast modes is affected by the couplings with the slow modes . 394\nNow, we see that the nature of these couplings (being “weak”, “st rong” or 395\n“anomalous”) changes depending on which is the non-adiabatic mech anism 396\nomitted in the energy equation. These changes in the coupling natur e cause 397\nthe damping time of the hybrid fast mode and the internal fast mode to vary 398\nfrom small values to very large values depending on the proximity to t he 399\ncouplings. So, we see that the consideration of both prominence ra diation and 400\ncoronal conduction has the effect of smoothing the curves of τD. 401\nIn addition, the results corresponding to hybrid and internal fast modes show 402\nthe appearance of thermal instabilities in very localised values of kzxpwhen 403\nprominence radiation is neglected (dashed lines), since then the inte ractions 404\nbetween fast modes and external slow modes leads to “anomalous” couplings. 405\nAt these couplings, the value of ωIfor the fast modes is pushed towards neg- 406\native values (see the right-hand panel of Fig. 4). Such a situation h as very 407\n16important repercussions on the wave behaviour since for ωI<0 waves are 408\namplified in time. The location of these instabilities in panels dandeof Fig. 6 409\nhave been pointed by means of arrows.410\n3.5 Wave instabilities411\nWave instabilities discussed in Sect. 3.4 require a more in-depth invest igation. 412\nAccordingtoField(1965),thecriterionfortheappearanceofwav e instabilities 413\nis given by414\nκ/bardbl\nρ0k2\nx+LT+1\nγ−1ρ0\nT0Lρ<0, (19) 415\nwherekxis the wavenumber in the field direction. Results of Carbonell et al. 416\n(2004), see also Paper I, point out that the heating scenario used in our cal- 417\nculations (constant heating per unit volume) cannot lead to therma l desta- 418\nbilisation. So, we can affirm that instabilities described in Sect. 3.4 are n ot 419\ncaused by the heating mechanism. In addition, instabilities only appea r when 420\nradiativelossesareomitted. Insuchsituation,theinstabilitycriter ionbecomes 421\nκ/bardbl\nρ0k2\nx<0. (20) 422\nEquation (20) is never satisfied unless an additional source of heat ing is 423\npresent, which seems to be the present case. This extra energy s ource cor- 424\nresponds to heat injected from the corona into the prominence by thermal 425\nconduction, aswas commented in Sect. 3.4. Inthe absence of radia tion, promi- 426\nnence thermal conduction is the only mechanism that can dissipate t his extra 427\ninjected heat. One expects that in such situation the value of kxgrows in 428\norder to increase the efficiency of prominence conduction. Figure 7 shows the 429\neigenfunction of the temperature perturbation corresponding t o the internal 430\nfast mode for kzxp≈0.3 when all non-adiabatic mechanisms are considered, 431\npanela), and when radiative losses from the prominence plasma are omitted , 432\npanelb). For this value of kzxp, the wave becomes unstable ( ωI<0) if promi- 433\nnence radiation is omitted. We see that smaller spatial-scales (i.e. larg erkx) 434\nare obtained within the prominence when prominence radiation is not t aken 435\ninto account, as expected. Although the efficiency of prominence c onduction 436\nis increased in this way, it is still not enough to stabilise the perturbat ion. 437\nThislastdiscussionpointsoutthatprominenceradiativelossesareo fparamount 438\nimportance to stabilise the disturbances. The efficiency of prominen ce radia- 439\ntion can be quantified by means of the radiation time-scale for the pr ominence 440\n17Fig. 7. Modulus of the eigenfunction T1(in arbitrary units) versus the dimension-\nless distance to the slab axis corresponding to the fundamen tal internal fast mode\nforkzxp≈0.3 ifa) all non-adiabatic mechanisms are considered, and b) without\nprominence radiation. The shaded region shows the location of the prominence slab.\nplasma (De Moortel & Hood, 2004),441\nτr=γp0\n(γ−1)ρ2\npχ∗\npTαpp. (21) 442\nConsidering fixed equilibrium parameters, the value of τrchanges for different 443\noptical thicknesses of the prominence material (see regimes listed in Table I 444\nof Paper I). For Prominence (1) parameters (optically thin plasma) ,τr≈ 445\n309 s, whereas for Prominence (2) and Prominence (3) regimes (op tically thick 446\nand very thick plasma), τr≈2,876 s and τr≈47,822 s, respectively, and so 447\nprominence radiation is less efficient. Obviously, τr→ ∞if the radiative term 448\nis omitted. The coronal plasma is always taken optically thin. Figure 8 s hows 449\nthe damping time of the fundamental hybrid and internal fast mode s as a 450\nfunction of kzxpfor the different prominence optical regimes. We see that 451\nthe larger the optical thickness, the larger the damping time. This e ffect is 452\nespecially relevant at the coupling points with the external slow mode s, where 453\nthermal instabilities appear if radiative losses are completely inhibited . 454\n18Fig. 8. Damping time versus kzxpfor the fundamental a) hybrid and b) internal\nfast modes. The linestyles represent different optical thick nesses for the prominence\nplasma:Prominence(1)insolidline(thiscorrespondstoth esolidlinesinFig. 6 dand\ne), Prominence(2) indotted lineandProminence(3) indot-da shedline. Thedashed\nline corresponds to the results when the prominence radiati on is omitted (dashed\nlines in Fig. 6 dande). The boundary conditions considered are vx=vz=T′\n1= 0.\n3.6 Exploring the parameter space455\nIn this Section we investigate how the attenuation of oscillations is aff ected by 456\nchanging the equilibrium parameters. The motivation of this study is b ased 457\non the fact that the estimated values for prominence plasma param eters, 458\nsuch as temperature, density, magnetic field strength or optical thickness, 459\nvaries from one prominence to another, sometimes in a significant wa y (e.g. 460\nPatsourakos Vial, 2002). Thus, it is important for our investigation to ascer- 461\ntain the sensitivity of the damping time to the equilibrium parameters a round 462\nthe values considered in our previous calculations.463\nFirst, we plot in Fig. 9 the ratio of the damping time to the period cor-464\nresponding to the fundamental modes as a function of equilibrium ph ysical 465\nconditions, namely the prominence temperature, the prominence d ensity, the 466\nmagnetic field strength and the coronal temperature. The followin g ranges of 467\nvalues have been considered: 5000 K < Tp<15,000 K; 10−11kg m−3< ρp< 468\n10−10kg m−3; 1 G< B0<15 G; and 800,000 K < Tc<2,000,000 K. 469\nAt first sight, we notice that the attenuation of fast modes is much more 470\n19sensitive to the equilibrium conditions than the damping of slow modes. The 471\nattenuation of slow modes does not change in a significant way if the e qui- 472\nlibrium physical conditions are modified, since the obtained τD/Pare always 473\nsmall and of the same order of magnitude. On the contrary, fast m odes are 474\nhighly sensitive especially to the prominence density and the magnetic field. It 475\nis noticeable that small values of τD/Pare obtained for the fast modes when 476\nlarge densities and weak magnetic fields are considered. If the magn etic field 477\nstrength is increased or the prominence density is reduced, then τD/Pgrows 478\ndramatically. Additionally, fast modes are again strongly affected by the cou- 479\nplings with slow modes, a fact that shows up in the form of very localise d 480\nincreases and decreases of τD/P. 481\nFig. 9. Ratio of the damping time to the period for the fundame ntal oscillatory\nmodes as function of, from the left to the right, the prominen ce temperature,\nthe prominence density, the magnetic field strength and the c oronal tempera-\nture. Computations performed considering kzxp= 1 and the boundary conditions\nvx=vz=T1= 0 atx=±xc.\nOntheother hand, wehave studied theeffect ofconsidering adiffer ent heating 482\nscenario on the wave attenuation. In agreement with previous inve stigations 483\n20(Carbonell et al., 2004; Terradas et al., 2005), results do not show significant 484\ndiscrepancies if different heating mechanisms are assumed.485\nFinally, we have also varied the length of magnetic field lines (by modifyin g 486\nthe value of xc) and the prominence half-width, xp, in order to assess their 487\neffectonthedampingtime.Forrealisticvaluesofboth xcandxp,nosignificant 488\ninfluences appear inthe results with respect to thosepreviously dis cussed. It is 489\nworth to mention that prominence conduction becomes a relevant m echanism 490\nfor very a small, unrealistic prominence half-width ( xp/lessorsimilar10 km), and coronal 491\nradiation is only important for very large, and again unrealistic, lengt h of 492\nmagnetic field lines ( xc/greaterorsimilar106km). 493\n3.7 Comparison with Terradas et al. (2005)494\nThe final check of the importance of the coronal medium comes fro m the com- 495\nparison between our results and those obtained by Terradas et al. (2005) in 496\nthe case of an isolated prominence slab (see Fig. 10). Obviously, this com- 497\nparison can only be performed for internal modes, since external and hybrid 498\nmodes are not supported by an isolated slab. The boundary conditio ns as- 499\nsumed in the work of Terradas et al. (2005) are vx=vz=T′\n1= 0. According 500\nto the arguments given in Sect. 3.4, this condition for the perturba tion to 501\nthe temperature is the most suitable since thermal conduction is ne gligible 502\nin prominences. However, the line-tying condition at the edges of th e promi- 503\nnence slab seems not to be the most appropriate election in the light o f the 504\neigenfunctions plotted in Fig. 3. Hence, our results point out that t he interface 505\nbetween the prominence slab and the corona does not act as a rigid w all, and 506\nperturbations can be important in the corona even for internal mo des. 507\nContrary to what was shown in Paper I, in which only the fast mode wa s 508\naffected by the corona, in the present case both slow and fast mod es of the 509\nisolated slab differ from those of a prominence–corona equilibrium. A d ecre- 510\nment of the damping time is obtained for both waves in comparison with511\nthe solution of an isolated slab. The slow mode is less affected by the pr es- 512\nence of the corona but the fast mode damping time is reduced by an o rder of 513\nmagnitude, although it is still far from the observed values. As in the longi- 514\ntudinal magnetic field case, the consideration of the corona is of pa ramount 515\nimportance for a correct description of the behaviour of oscillation s and their 516\nattenuation, although its effect on the damping of fast modes is less noticeable 517\nthan in the longitudinal case.518\n21Fig. 10. Period (left), damping time (centre) and ratio of th e damping time to the\nperiod (right) versus kzfor the internal fundamental slow (upper panels) and fast\n(bottom panels) oscillatory modes. The solid lines are the s olutions of a promi-\nnence plus corona equilibrium whereas the dotted lines repr esent the solutions of\nan isolated slab (Terradas et al., 2005).\n4 Conclusions519\nIn this paper we have studied the wave attenuation in a system repr esenting a 520\nquiescent solar prominence embedded in the coronal medium. The pr ominence 521\nhas been modelled as a homogeneous plasma slab surrounded by a hom oge- 522\nneousmediumwithcoronalconditions. Magneticfieldlineshavebeena ssumed 523\ntransverse totheprominenceslabaxisandthewholesystem hasbe enbounded 524\nin the field direction by two photospheric rigid walls, in order to establis h a 525\nrealistic length for the field lines. The attenuation of the normal mod es of such 526\nequilibrium has been investigated by considering parallel thermal con duction, 527\nradiative losses and plasma heating as non-adiabatic mechanisms, an d focus- 528\ning our study on the fundamental oscillatory modes. The main conclu sions of 529\nthis work are summarised next.530\n(1) Slow modes are strongly attenuated by non-adiabatic mechanis ms, their 531\ndamping times being of the order of the corresponding periods. Fas t 532\nmodes are less affected and present greater damping times. 533\n(2) The most relevant damping mechanisms are prominence radiation and 534\ncoronal thermal conduction. The first one dominates the damping of in- 535\nternal modes, while the second one is responsible for the attenuat ion of 536\nexternal modes. The combined effect of both mechanisms governs the 537\ndamping of hybrid modes. Neither prominence conduction nor coron al 538\nradiation become of importance for realistic values of the length of m ag- 539\nnetic field lines and the prominence width. 540\n(3) The attenuation of slow modes is not affected by the value of the free 541\n22component of the wavenumber, kz. On the contrary, the behaviour of 542\nfast modes is strongly dependent on kz. 543\n(4) Thermal conduction allows energy transfer between the prom inence slab 544\nand the coronal medium. Prominence radiation has an essential role in 545\ndissipating the extra heat injected from the corona and stabilises t he 546\noscillations. Thermal instabilities appear if the radiative losses from t he 547\nprominence plasma are omitted or significantly reduced (e.g. caused by 548\nan increase of the optical thickness) since the plasma cannot dissip ate 549\nthe extra injected heat in an efficient way. 550\n(5) The damping time of fast modes is strongly sensitive to the equilibr ium 551\nphysical parameters while slow waves are less affected by the variat ion of 552\nthe equilibrium conditions. 553\n(6) The presence of the corona produces a decrement of the dam ping time 554\nof internal modes with respect to the solutions supported by an iso lated 555\nprominenceslab.Nevertheless, thiseffectisnotenoughtoobtaind amping 556\ntimes of the order of the period in the case of fast modes. 557\nConsidering the equilibrium parameters of Paper I, the efficiency of n on- 558\nadiabatic mechanisms on the damping of fast modes is smaller in the pre sent 559\ncase. This fact suggests that the orientation of magnetic field lines with re- 560\nspect to the slab axis hasa relevant influence on theattenuation of fast modes, 561\nthe configuration of Paper I and the present one being limit cases. M oreover, 562\nfast modes are strongly sensitive to the equilibrium physical conditio ns, and 563\nit is possible to obtain small values of the damping time by considering ex - 564\ntremeequilibriumparameters,suchasveryweakmagneticfieldsand verylarge 565\nprominence densities. In this way, fast modes show a wide range of t heoretical 566\ndamping times. On the other hand slow modes are always efficiently att enu- 567\nated, with damping times of the order of their periods. This result su ggests 568\nthat the attenuation of prominence fast waves may be caused by o ther damp- 569\ning mechanisms not considered here. Some candidates could be reso nant ab- 570\nsorption (Arregui et al., 2008) andion-neutral collisions (Forteza et al., 2007). 571\nAmong these mechanisms, resonant absortion may be a very efficien t damp- 572\ning mechanism if non-uniform equilibria are considered, e.g. models with a 573\ntransition region between the prominence and the corona. Other e ffects, as 574\nwave leakage, might only play a minor role in the damping of disturbance s. 575\nFinally, future studies should take into account the prominence fine structure 576\non the basis that small-amplitude oscillations are of local nature. The refore, 577\nthe investigation of the damping of fibril oscillations should be the nex t step. 578\nR. O. and J. L. B. want to acknowledge the International Space Sc ience Insti- 579\ntute teams “Coronal waves and Oscillations” and “Spectroscopy a nd Imaging 580\nof quiescent and eruptive solar prominences from space” for usef ul discus- 581\nsions. The authors acknowledge the financial support received fr om the Span- 582\nish MCyT and the Conselleria d’Economia, Hisenda i Innovaci´ o of the C AIB 583\nunder Grants No. AYA2006-07637 and PCTIB-2005GC3-03, resp ectively. Fi- 584\n23nally, R. S. thanks the Conselleria d’Economia, Hisenda i Innovaci´ o f or a 585\nfellowship.586\nReferences587\nAnzer, U. & Heinzel, P. 1999, A&A, 349, 974588\nArregui, I., Terradas, J. Oliver, R., & Ballester, J. L. 2008, ApJ, 68 2, L141 589\nBallai, I. 2003, A&A, 410, L17590\nBallester, J. L. 2006, Phil. Trans. R. Soc. A, 364, 405591\nBalthasar, H., Wiehr, E., Schleicher, H., & W¨ ohl, H. 1993, A&A, 277, 6 35 592\nBanerjee, D., Erd´ elyi, R., Oliver R., & O’Shea, E. 2007, Sol. Phys., 246 , 3 593\nBerger, T. E. 2008, ApJ, 676, L89594\nCarbonell, M., Oliver, R., & Ballester, J. L. 2004, A&A, 415, 739595\nCarbonell, M., Terradas, J., Oliver, R., & Ballester, J. L. 2006, A&A, 4 60, 573 596\nCargill, P. & Hood, A. W. 1989, Sol. 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L. 2007, A&A, 471, 1023 (Paper I ) 629\nTerradas, J., Oliver, R., & Ballester, J. L. 2001, A&A, 378, 635630\nTerradas, J., Molowny-Horas, R., Wiehr, E. et al. 2002, A&A, 393, 6 37 631\nTerradas, J., Carbonell, M., Oliver, R., & Ballester, J. L. 2005, A&A, 4 34, 741 632\nTerradas, J., Arregui, I., Oliver, R., & Ballester, J. L. 2008, ApJ, 67 8, L153 633\n25"    },    {        "title": "1303.3501v4.Microwave_assisted_switching_of_a_nanomagnet__analytical_determination_of_the_optimal_microwave_field.pdf",        "content": "arXiv:1303.3501v4  [cond-mat.mes-hall]  30 Aug 2013Microwave-assisted switching of a nanomagnet: analytical\ndetermination of the optimal microwave field\nN. Barros, M. Rassam, and H. Kachkachi\nPROMES-CNRS and Université de Perpignan Via Domitia,\n52 Avenue Paul Alduy, 66860 Perpignan Cedex, France\nAbstract\nWe analytically determine the optimal microwave field that a llows for the magnetization reversal of\na nanomagnet modeled as a macrospin. This is done by minimizi ng the total injected energy. The\nresults are in good agreement with the fields obtained numeri cally using the optimal control theory.\nFor typical values of the damping parameter, a weak microwav e field is sufficient to induce switching\nthrough a resonant process. The optimal field is orthogonal t o the magnetization direction at any time\nand modulated both in amplitude and frequency. The dependen ce of the pulse shape on the applied\nfield and damping parameter is interpreted. The total inject ed energy is found to be proportionnal to\nthe energy barrier between the initial state and the saddle p oint and to the damping parameter. This\nresult may be used as a means for probing the damping paramete r in real nanoparticles.\nPACS numbers: 75.10.Hk, 75.75.Jn, 84.40.-x\n1I. INTRODUCTION\nMagnetic recording is a key technology in the field of high den sity information storage. In\norder to increase thermal stability, small nanoparticles w ith high anisotropy may be used. How-\never, high fields are then needed to reverse the magnetizatio n but these are difficult to achieve\nin current devices. To overcome this so called magnetic reco rding trilemma, several solutions\nare being proposed. The mostly investigated route, and whic h already leads to industrial appli-\ncations, is the heat-assisted magnetic recording1. It consists in heating the particles by a laser\nwhich decreases the energy barrier between the two energy mi nima and thereby the switching\nfields. However, to avoid a loss of information, the heating m ust be very localized and followed\nby a very fast cooling, and as such these devices must be coupl ed to powerful heat dissipation\nsystems.\nAn alternative solution is to assist the switching by a micro wave (MW) field. In 2003\nThirion et al.2showed that the combination of a DC applied field (static field ) well below the\nswitching field with a small MW field pulse can reverse the magn etization of a nanoparticle.\nIndeed, in the presence of a MW field with appropriate amplitu de and frequency, the mag-\nnetization precession synchronizes with this field3–5. Then, energy is pumped into the system\nthus allowing the magnetization to climb up the energy barri er and cross the saddle point6–10.\nFurther experimental and theoretical studies proved that t his process is more efficient if the\nfrequency of the MW field is sligthly lower than the ferromagn etic resonance frequency of the\nnanoelements11–13. Moreover, the use of chirped MW fields was shown to be more effic ient to\nachieve switching7,11,14–16. This result is related to the anharmonicity of the energy we ll. Simi-\nlar results have been obtained in other areas of physics and c hemistry, like atomic or molecular\nspectroscopy17,18.\nIn a previous work19we developped a numerical method based on optimal control th eory\nwhich renders an exact solution for the MW field that is necess ary for the switching of a nano-\nmagnet within a given potential energy. The formulation of t his method consists in defining a\ncost functional and minimizing it using the conjugate gradi ent technique. Our results confirmed\nthat a weak MW field, modulated both in amplitude and frequenc y, can induce the switching of\nthe magnetization. Furthermore, the injected energy has be en found to increase with damping.\nThe aim of the present study is to provide analytical express ions and to compare them\nwith our numerical results, by using simple energy consider ations. Moreover, the analytical\ndevelopments presented here confirm the effects observed num erically and provide clear inter-\npretations for the underlying physical processes. In this w ork, our investigations are restricted\nto zero temperature, and as such the Landau-Lifshitz-Gilbe rt equation is used to describe the\n2magnetization trajectory. The additionnal effects of therm al fluctuations will be the subject of\na future study.\nIn the first part, we analytically determine the optimal MW fie ld and demonstrate its de-\npendence on the energy landscape (anisotropy, applied field ) and to the damping parameter.\nWe then investigate the trajectory of the magnetization in t he presence of the optimal field and\nshow that it can be described by the Landau-Lifshitz-Gilber t equation with a negative damping\nparameter. In the second part, the analytical results are co mpared directly with the results\nobtained numerically using the optimal control theory.\nII. ANALYTICAL CALCULATION OF THE OPTIMAL MICROWAVE FIELD\nWe consider a nanomagnet with spatially uniform magnetizat ion which can be modeled by a\nvectorM=MSm, whereMSis the saturation magnetization and /bardblm/bardbl= 1. This nanomagnet\nis characterized by a given anisotropy (uniaxial, biaxial, cubic...) and a damping parameter α.\nIn the presence of a static magnetic field H0lower than the Stoner-Wohlfarth switching field,\nthe potential energy surface presents several minima separ ated by saddle points.\nAt the initial time ti, we assume that the magnetization is in a minimum Mi=MSmi.\nAdding a microwave (MW) field H(t)can then induce switching to another (target) minimum\nMf=MSmf. Our aim is to find the optimal field Hopt(t)that achieves switching in a\ngiven time tfwhile minimizing the energy injected into the system. This c riterion is relevant\nfor experimental devices since it amounts to reducing both t he intensity and the duration of\napplied fields and the subsequent heating of the system, whic h can be of interest for magnetic\nrecording or biomedical applications. This approach is thu s complementary to other theoretical\nstudies which have focused on the reduction of the switching time9,20.\nFor the sake of simplicity, we introduce the normalized field sh0≡H0/Hanandh(t)≡\nH(t)/Han, whereHan≡2K/µ0Msis the anisotropy field and Kthe anisotropy constant of\nthe nanomagnet. We also define the normalized time τ≡γHant, whereγ= 1.76×1011(T.s)−1\nis the gyromagnetic factor. For instance, for a cobalt parti cle of 3 nm diameter with K≈\n2.2×105J.m−3andMs≈1.44×106A.m−1we haveµ0Han≈305mT andt/τ≈1.86×10−11s.\nA. Energy and time trajectory in the presence of a microwave fi eld\nIf only the static field is applied, the energy density of the s ystem (divided by 2K) reads\nE0(m,h0) =Ean(m)−m·h0whereEanis the anisotropy energy density. The normalized\neffective field is then defined by heff≡ −∂E0/∂m. If we add a MW field, the energy density\n3becomes\nE(m,h0,h(τ)) =Ean(m)−m·(h0+h(τ)) (1)\nand the normalized total effective field now reads\nζ(τ)≡ −∂E\n∂m=heff+h(τ). (2)\nThe time trajectory of the magnetization can be described by the driven Landau-Lifshitz-\nGilbert equation\n/parenleftbig\n1+α2/parenrightbigdm\ndτ=−m×ζ(τ)−αm×(m×ζ(τ)). (3)\nThis allows us to express the energy variation of the system a s follows\ndE\ndτ=−ζ(τ)·dm\ndτ−m·dh(τ)\ndτ=−heff·dm\ndτ−d\ndτ(m·h(τ)). (4)\nNext, we define the mobile frame (m,u,v)attached to the magnetization with u≡T/T\nandv≡m×T/T, whereT≡m×heffandT=/vextenddouble/vextenddoublem×heff/vextenddouble/vextenddouble. The MW field can then be\ndecomposed as h(τ) =hm(τ)m+hu(τ)u+hv(τ)v. In this frame, Eqs. (3) and (4) respectively\nbecome\n/parenleftbig\n1+α2/parenrightbigdm\ndτ=/parenleftbigg\n−1+αhu(τ)+hv(τ)\nT/parenrightbigg\nT−/parenleftbigg\nα+hu(τ)−αhv(τ)\nT/parenrightbigg\n(m×T),(5)\ndE\ndτ=−αT−hu(τ)+αhv(τ)\n1+α2T−dhm(τ)\ndτ. (6)\nWe note that the parallel component of the MW field hm(τ)has no direct effect on the\nmagnetization trajectory and that only its time derivative appears in the energy variation.\nB. Optimization of the MW field\nIn order to find the optimal MW field fulfilling the requirement s described earlier we proceed\nin two steps. First, we define the critical MW field which allow s us to maintain the precession\nof the magnetization by compensating the effects of damping ( sec. IIB1). This field represents\nthe lower limit for the optimal field sought. Using this resul t, we find the optimal MW field\nminimizing the injected energy (sec. IIB2) and check that it can induce switching of the\nmagnetization (sec. IIB3).\n41. MW field maintaining the precession: critical field\nIn order to induce switching the MW field must at least compens ate for the effect of damp-\ning, which tends to take the magnetization back to the initia l equilibrium position. If the\ncompensation is complete the energy variation of the system dE/dtvanishes at any time thus\nreflecting the conservation of energy. According to Eq. (6) a n infinity of MW fields leads to a\nfull compensation of damping. For instance, any field that is orthogonal to the magnetization\nso thathm(τ) = 0 and satisfying the equation −hu(τ)+αhv(τ) =−αTwill do.\nAmong these MW fields the critical field can be defined as the one that minimizes the power\ninjected in the system. The latter is proportional to the squ ared intensity of the MW field, i.e.\npi(τ) =h2(τ) =h2\nm(τ) +h2\nu(τ) +h2\nv(τ). Using the method of Lagrange multipliers we define\nthe functional\nL[hm(τ), hu(τ), hv(τ), λ(τ)] =p2\ni(τ)−λ(τ)dE\ndτ. (7)\nAssuming that dhm(τ)/dτdoes not depend explicitely on hm(τ),the minimization of this\nfunctional leads to\nhm(τ) = 0,\nhu(τ) =−α\n1+α2T, (8)\nhv(τ) =α2\n1+α2T.\nThe critical MW field thus reads\nhcrit(τ) =α\n1+α2[−T+αm×T] (9)\nand the injected power is then\npcrit\ni(τ) =α2\n1+α2T2. (10)\nIn the presence of this MW field the magnetization precesses a round the equilibrium position\nwith a constant angle. This critical field represents a lower limit. Indeed, if the injected power\nis smaller than pcrit\ni(τ)the magnetization goes back to the initial equilibrium posi tion.\n2. MW field minimizing the total injected energy: optimal fiel d\nIn order to minimize the total injected energy we have to make a few preliminary assumptions\nconcerning the shape of the MW field. Considering the result o f the previous section we limit\n5our search to the family of MW fields defined by\nhm(τ) =βmT\nhu(τ) =βuT\nhv(τ) =βvT\nwhereβm,βuandβvare constant parameters. In the presence of such a MW field the energy\nvariation reads\ndE\ndτ=−α−βu+αβv\n1+α2T2−dhm(τ)\ndτ. (11)\nThe total energy injected to the system can be defined as E=´τf\nτih2(τ)dt. Therefore,\nE=ˆτf\nτi/parenleftbig\nβ2\nm+β2\nu+β2\nv/parenrightbig\nT2dτ (12)\n=ˆτf\nτi/parenleftbig\nβ2\nm+β2\nu+β2\nv/parenrightbig/bracketleftbigg1+α2\n−α−βu+αβv/parenleftbiggdE\ndτ+dhm(τ)\ndτ/parenrightbigg/bracketrightbigg\ndτ\n=(1+α2)/parenleftbig\nβ2\nm+β2\nu+β2\nv/parenrightbig\n−α−βu+αβv[E(τf)−E(τi)+hm(τf)−hm(τi)].\nHence, the energy is minimal if βm= 0,βu=−2α/(1+α2)andβv= 2α2/(1+α2).\nConsequently, the optimal MW field is\nhopt(τ) =2α\n1+α2[−T+αm×T] = 2hcrit(τ). (13)\nThe optimal field is twice the critical field determined previ ously, see Eq. (9). It is orthogonal\nto the magnetization at any time and its magnitude reads\n/vextenddouble/vextenddouble/vextenddoublehopt(τ)/vextenddouble/vextenddouble/vextenddouble=2α√\n1+α2T. (14)\nThe total injected energy is then\nE= 4α[E(τf)−E(τi)]. (15)\nAccording to these results, both the optimal field magnitude and the total injected energy\nincrease with damping. This confirms the fact that the MW field must compensate for the\neffects of damping so as to induce switching.\n3. Trajectory of the magnetization in presence of the optima l MW field\nIn order to check whether the optimal MW field obtained in the p revious section induces\nswitching of the magnetization as required, we now investig ate the time trajectory of the mag-\nnetization. In the presence of this field, Eqs. (5) and (6) res pectively become\n/parenleftbig\n1+α2/parenrightbigdm\ndτ=−T+αm×T, (16)\n6dE\ndτ=α\n1+α2T2. (17)\nThe first equation is similar to the Landau-Lifshitz-Gilber t equation but with a negative\ndamping parameter: it describes an amplified precession. Th e precession frequency is equal to\nthe proper frequency of the magnetization. At any time the MW field is proportional to the\nderivative of the magnetization: hopt(τ) = 2αdm/dτ. This is in agreement with the results of\nSun et al.7.\nAt the minima and saddle points the effective field heffis parallel to the magnetization so\nthatT=0. Therefore, both the derivative of the magnetization and th e MW field vanish.\nConsequently, the optimal MW field can only induce the motion of the magnetization from\nan initial state miclose to an energy minimum, to a final state mfclose to a saddle point.\nA small amount of energy must thus be added (i) before the MW fie ld pulse, to drag the\nmagnetization away from the minimum and (ii) after the pulse , to cross the saddle point. The\nnature of this additionnal energy will be further discussed later on. Beyond the saddle point,\nthe damping takes up to lead the magnetization down to the sec ond energy minimum. If the\nenergy landscape is complex with several barriers successi ve pulses might then be necessary to\ninduce switching.\nAt both the initial and the final states the MW field is close to z ero. The difference E(τf)−\nE(τi)is thus close to the static energy barrier between the saddle point and the initial state\n△E0≡ E0(τf)−E0(τi)and Eq. (15) becomes\nE= 4α△E0. (18)\nThe total injected energy is therefore proportional to the e nergy barrier to be overcome.\nHence, if the static field is close to the switching field, a ver y weak MW field can induce\nswitching.\nC. Uniaxial anisotropy and longitudinal static field\nIn this section we study the trajectory of the magnetization in the presence of the optimal\nMW field for a nanoparticle with uniaxial anisotropy and a lon gitudinal static field.\nWe consider a nanomagnet with uniaxial anisotropy with easy axis in the zdirection. The\nanisotropy energy density is then Ean(mz) =−m2\nz/2. The static field is applied in the (−z)\ndirection with a magnitude 0≤h0<1. The magnetization is initially close to the metastable\nminimum and its zcomponent is thus m0≡mz(τi= 0)≈1. The static energy of the system\nis\nE0(mz) =−m2\nz\n2+h0mz. (19)\n7The saddle point then corresponds to mz=h0, so the static energy barrier between the\nlatter and the initial metastable state is △E0= (1−h0)2/2. For this system, the effective field\nreadsheff=−h0+mzez.Projecting Eq. (16) onto the zaxis then yields\ndmz\ndτ=α\n1+α2(m×T)·ez=−α\n1+α2(−h0+mz)/parenleftbig\n1−m2\nz/parenrightbig\n. (20)\nIn order to simplify the expressions we introduce the integr al\nI(mz) =ˆmz\n0−du\n(−h0+u)(1−u2)=1\n2(1−x2)ln/bracketleftBigg\n(1+mz)1−h0(1−mz)1+h0\n(−h0+mz)2/bracketrightBigg\n+Cte.(21)\nSolving Eq. (20) with the initial condition mz(t= 0) =m0leads to the equation\nI(mz)−I(m0) =α\n1+α2τ. (22)\nThis equation can be analytically solved for mzonly ifh0= 0(no static field). Otherwise,\nthe evolution of mzwith time can be obtained numerically, see Fig. 1. As predict ed previously,\nfor long times the magnetization goes towards to the saddle p oint but never reaches it since\nI(mz)diverges for mz=h0.\nFrom Eq. (14) we can express the optimal MW field intensity in t erms ofmzas follows\n/vextenddouble/vextenddouble/vextenddoublehopt(mz)/vextenddouble/vextenddouble/vextenddouble=2α√\n1+α2(−h0+mz)/radicalbig\n1−m2\nz. (23)\nThe time evolution of the MW field intensity is plotted in Fig. 1. We note that the pulses\nfollow neither a Gaussian nor a Lorentzian function. The pea k intensity is reached for mz=\n1\n4/parenleftBig\nh0+/radicalbig\n8+h2\n0/parenrightBig\nand is given by\nhoptmax=α\n2√\n1+α2/parenleftbigg\n−3h0+/radicalBig\n8+h2\n0/parenrightbigg/radicaltp/radicalvertex/radicalvertex/radicalbt\n1−/parenleftBig\nh0+/radicalbig\n8+h2\n0/parenrightBig2\n16. (24)\nFrom this, we can see that the peak intensity hoptmaxdecreases with h0(see Fig. 2). Indeed,\nfor higher magnitudes of the static field, the energy barrier △E0between the metastable state\nand the saddle point is lower, so that a lower energy is needed to reach the saddle point. Since\n0≤h0<1the peak intensity is limited as follows\nhoptmax<√\n2α√\n1+α2. (25)\nHence, for low values of the damping parameter α, the intensity of the optimal MW field is\nsmall. This fully confirms the results of our numerical study19. Using Eqs. (21) and (23), we\ncan also obtain analytically the pulse duration △τ, defined as the full width at half maximum\n△τ=1+α2\nαg(h0), (26)\n8200 400 600800 1000 1200 1400 1600\nNormalized time τ00.20.40.60.81Magnetization along the z axis mz00.010.020.030.040.050.06Optimal MW field magnitude || hopt(τ)||  h0 = 0.00\n h0 = 0.16\n h0 = 0.33\n h0 = 0.50\n h0 = 0.66\n h0 = 0.82\nFigure 1: (Color online) Optimal MW field intensity/vextenddouble/vextenddouble/vextenddoublehopt(τ)/vextenddouble/vextenddouble/vextenddouble(upper panel) and trajectory of the\nmagnetization mz(τ)(lower panel) for a longitudinal static field with magnitude h0. Parameters:\nα= 0.05,m0= 0.99998.\nwhereg(h0)is a cumbersome function of h0. This characteristic time increases with h0(see\nFig. 2). For high values of h0, the switching will thus require a very low field but a very lon g\ntime, as can be seen in Fig. 1. Moreover, the characteristic t ime decreases with damping.\n90 0.2 0.4 0.6 0.8 1\nStatic field magnitude h000.010.020.030.040.05MW field peak intensity\n0100200300400\nPulse durationAnalytical: MW field peak intensity hmax\nNumerical: MW field peak intensity hmax\nAnalytical: pulse duration ∆τ\nNumerical: pulse duration ∆τopt\nopt\nFigure 2: (Color online) Maximal peak intensity and peak dur ation of the optimal MW field for varying\nmagnitude of the static field h0.\nThe area below the curves/vextenddouble/vextenddouble/vextenddoublehopt(τ)/vextenddouble/vextenddouble/vextenddoubleis\nA=ˆt=∞\nt=0/vextenddouble/vextenddouble/vextenddoublehopt(τ)/vextenddouble/vextenddouble/vextenddoubledτ=ˆmz=h0\nmz=m0/vextenddouble/vextenddouble/vextenddoublehopt(mz)/vextenddouble/vextenddouble/vextenddoubledτ\ndmzdmz (27)\n= 2√\n1+α2[arccos(h0)−arccos(m0)] = 2√\n1+α2(θf−θi).\nwhereθiandθfare the polar angles of the magnetization respectively at th e initial state and\nsaddle point. This area is thus proportional to the “angular distance” that the magnetization\nmust cross to reach the saddle point. For increasing values o f the static field magnitude h0,\nthis area decreases since the saddle point comes closer to th e initial state.\nThezcomponent of hopt, given by Eq. (13), is\nhopt\nz(mz) =−2α2\n1+α2(−h0+mz)/parenleftbig\n1−m2\nz/parenrightbig\n. (28)\nFrom Eq. (23) we can see that the ratio/vextendsingle/vextendsingle/vextendsinglehopt\nz(mz)/vextendsingle/vextendsingle/vextendsingle//vextenddouble/vextenddouble/vextenddoublehopt(mz)/vextenddouble/vextenddouble/vextenddoubleisα√\n1+α2/radicalbig\n(1−m2z),\nwhose upper limit isα√\n1+α2. Consequently, for small values of the damping parameter α, the\ncomponent of the time-dependent field along the anisotropy e asy axis can be neglected and the\noptimal field lies in the xyplane.\nWe now define the precession phase of the magnetization as ϕ(τ) = arctan( my(τ)/mx(τ)).\n10Projecting Eq. (16) on the xandyaxes leads to the relation\nω(τ) =dϕ\ndτ=/vextenddouble/vextenddoubleheff/vextenddouble/vextenddouble\n1+α2=−h0+mz(τ)\n1+α2. (29)\nThis precession frequency is equal to the proper frequency o f the magnetization, obtained\nby solving Eq. 3 in the absence of a MW field. At the initial stat e, the precession frequency\nis close to the FMR frequency ωFMR= (1−h0)/(1+α2). It then decreases towards zero,\nfollowing the curvature of the energy well.\nFor small values of α, since the optimal field lies in the xyplane as shown previously, its\nphase can be defined by ˜ϕ(t) = arctan/parenleftBig\nhopt\ny(t)/hopt\nx(t)/parenrightBig\n. It can be shown that\ntan˜ϕ(τ) =mx(τ)+αmy(τ)mz(τ)\n−my(τ)+αmx(τ)mz(τ)≈ −mx(τ)\nmy(τ)= cotϕ(τ). (30)\nThis implies that the time-dependent field and the magnetiza tion are synchronized with\n˜ϕ(τ)≈ϕ(τ) +π/2. Hence, the frequency of the time-dependent field is equal to the proper\nprecession frequency of the magnetization.\nFinally, using Eqs. (20) and (23), the total injected energy can be computed. As shown\npreviously in Eq. (18), it is proportional to the damping par ameter and to the energy barrier\n△E0.\nE=ˆ+∞\n0/vextenddouble/vextenddouble/vextenddoublehopt(τ)/vextenddouble/vextenddouble/vextenddouble2\ndτ= 2α(h0−1)2= 4α△E0. (31)\nIII. COMPARISON WITH THE NUMERICAL RESULTS\nAs mentioned earlier, in Ref. 19 we developed a numerical met hod based on the theory of\noptimal control to determine the shape of the optimal MW field . It renders an exact solution\nfor the MW field that triggers the switching of a nanomagnet wi th a given anisotropy and\napplied field. The method consists in minimizing the cost fun ctional\nF[m(τ),h(τ)] =1\n2/bardblm(τf)−mf/bardbl2+η\n2τfˆ\n0dτh2(τ)\nalong the trajectory given by the Landau-Lifshitz-Gilbert equation (3), where mfis the target\nmagnetization (stable minimum), m(τf)is the magnetization reached at time τfandηis\na numerical control parameter. The numerical problem is the n solved using the modified\nconjugate gradient technique supplemented by a Metropolis algorithm.\nIn Ref. 19 we restricted the MW field along a polarization axis to comply with the exper-\nimental setup. In the present study, for a better comparison with the analytical results, the\nMW field is allowed to move in three dimensions during the opti mization.\n11Our model system is a particle with uniaxial anisotropy alon g thezaxis. Unless otherwise\nspecified, the numerical parameters used in the current stud y are: initial normalized time\nτi= 0, final normalized time τf= 800 (corresponding to a few ns in real time), number of\npointsN= 15000 , so the sampling time (τf−τi)/(N−1)is about 0.05; damping parameter\nα= 0.05; control parameter η= 0.01.\nA. Reference calculation\nA first numerical optimization was carried out for a static fie ld with magnitude h0= 0.5\napplied in the ( −z) direction. The results are in good agreement with the analy tical calculations\nof part II. As can be seen in Fig. 3, the optimal MW field is modul ated both in amplitude and\nfrequency. It is mainly in the xyplane and its magnitude is small (less than 0.03, corresponding\nto a few mT in real field) as expected since the damping paramet er is small. The pulse starts\nat about τ= 350 and progressively drives the magnetization away from the in itial equilibrium\nposition. The saddle point is reached at about τ= 650 (purple dotted line) but the MW field\npulse continues until τ= 700 (green dotted line), which allows the magnetization to cros s the\nsaddle point. Next, the damping takes up to lead the magnetiz ation to the more stable energy\nminimum, which is reached at about τ= 800 .\nFig. 4 shows that the MW field intensity obtained numerically is in good agreement with\nthe analytical result in Eq. (23). From τ≈570, the MW field intensity is slightly higher\nnumerically, which induces mzto decrease faster and the magnetization to finally cross the\nsaddle point. The total injected energy obtained numerical lyEnum=´τf\nτih2(τ)dt≈0.02548\nis slightly higher than the value predicted analytically Ean= 4α△E0= 0.02500 . This confirms\nthat the optimal MW field determined analytically represent s a lower boundary and that a\nsmall additionnal energy must be injected to achieve switch ing, as noticed previously. Never-\ntheless, the discrepancy between the numerical and analyti cal MW fields is very small, which\ncorroborates the relevance of the analytical model.\nFig. 5 confirms that the magnetization precession and the MW fi eld are synchronized, the\ninitial frequency being close to the FMR frequency. The time evolution of the frequency is\nsimilar to the evolution of mz(Fig.4), since both values are related by Eq. (29). Conseque ntly,\nafterτ≈570the numerical frequency is lower that the analytical freque ncy.\n12-0.02-0.010.000.010.02Optimal MW field hopt(τ) hx(τ)\nhy(τ)\nhz(τ)\n300 400 500 600 700 800 900\nNormalized time τ-1.0-0.50.00.51.0Magnetization m(τ)\nmx(τ)\nmy(τ)\nmz(τ)saddle point\nFigure 3: (Color online) Numerical results for the referenc e calculation: optimal MW field (upper\npanel) and magnetization time trajectory (lower panel). Th e purple and green dotted vertical lines\nrespectively indicate the crossing of the saddle point and t he end of the microwave pulse.\nB. Effect of the static field magnitude and direction\nThe MW field h(τ)has been optimized numerically for several magnitudes and o rientations\nof the static field h0. For each configuration, the energy barrier △E0and the total injected\nenergyE=´τf\nτi/vextenddouble/vextenddouble/vextenddoublehopt(τ)/vextenddouble/vextenddouble/vextenddouble2\ndτhave been computed numerically, and are reported in Fig. 6.\nAs shown in Eq. (18), the injected energy is found to be propor tional to the energy barrier and\nto4α.\n130.0000.0050.0100.0150.020||hopt(τ)||Numerical result\nAnalytical result\n300 400 500 600 700 800\nNormalized time τ0.0000.5001.000mzNumerical result\nAnalytical result\nsaddle point\nFigure 4: (Color online) Comparison between the analytical and numerical results for the reference\ncalculation: optimal MW field magnitude/vextenddouble/vextenddouble/vextenddoublehopt/vextenddouble/vextenddouble/vextenddouble(higher panel) and zcomponent of the magnetization\n(lower panel).\nIn the case of a static field applied along ( −z), the shape of the pulse can be directly\ncompared with the analytical results of sec. IIC, see Fig. 2. The numerical and analytical\nresults are in good agreement. As predicted analytically, t he maximal peak intensity hoptmax\ndecreases and the pulse duration △τincreases rapidly when the magnitude of the static field\nh0increases.\nC. Effect of damping\nAs predicted by Eqs. (18) and (31), for a given static energy b arrier△E0, the injected energy\nis proportional to the damping parameter α. This has been checked numerically by varying\nthe damping parameter from 0.015 to 0.30 (Fig. 7). In these ca lculations, the static field h0is\napplied in the ( −z) direction with the magnitude h0= 0.5.\nFig. 8 shows that for low values of α, the pulses height decreases but their duration increases,\nin agreement with Eqs. (24) and (26). For an undamped system, the switching should thus\nbe infinitely long, so our analytical and numerical methods a re not adapted to describe such a\nsystem.\nAs can be observed, for very low values of the damping paramet erα, a discrepancy between\n14400 500 600\nNormalized time τ0.000.100.200.300.400.50Normalized frequency ω\nNumerical: magnetization precession frequency\nNumerical: MW field precession frequency\nAnalytical result\nFigure 5: (Color online) Comparison between the analytical and numerical results for the reference\ncalculation: precession frequencies of the magnetization and MW field.\nthe analytical calculations and the numerical optimizatio n is observed. Indeed, since the optimal\npeak duration becomes very long, the number of numerical poi ntsNmust be increased, so the\nconjugate gradient algorithm becomes less efficient. Howeve r, as can be seen in Fig. 7, this\ndiscrepancy has a negligeable effect on the injected energy.\nIV. CONCLUSION\nWe analytically determined the optimal microwave field that allows for the switching of the\nmagnetization of a monodomain nanoparticle with uniaxial a nisotropy while minimizing the\ninjected energy. This study provides a clear interpretatio n of the results obtained numerically\nusing the optimal control theory19, especially the simple dependence of the pulse on the dampin g\nparameter.\nOur results confirm that the optimal MW field is modulated both in amplitude and fre-\nquency, since it is directly proportional to the derivative of the magnetization. It drives the\nmagnetization from an initial state close to the initial min imum to a final state close to a saddle\npoint. The time trajectory can then be described as an amplifi ed precession.\nIn order to cross the saddle point, a small additionnal energ y must be injected into the\nsystem. Our numerical results show that this energy can be ad ded by slightly increasing the\n150.0 0.2 0.4 0.6 0.8 1.0\nEnergy barrier ∆ε00.000.050.100.150.20Injected energy EE = 4α ∆ε0\nψ = 0°, h0 = 0.33\nψ = 30°, h0 = 0.33\nψ = 60°, h0 = 0.33\nψ = 90°, h0 = 0.33\nψ = 120°, h0 = 0.33\nψ = 150°, h0 = 0.33\nψ = 180°, h0 = 0.00\nψ = 180°, h0 = 0.16\nψ = 180°, h0 = 0.33\nψ = 180°, h0 = 0.50\nψ = 180°, h0 = 0.66\nFigure 6: (Color online) Total injected energy Ewith respect to the static energy barrier △E0for\nvarying magnitude and orientation of the static field h0.ψis the angle between the zaxis (anisotropy\naxis) and the static field.\nMW field intensity. In reality any source of noise, such as the rmal fluctuations, may suffice to\ninduce the saddle point crossing. Subsequently, the dampin g induces the relaxation to the final\nstate. We find that the injected energy is proportional to the damping parameter and to the\nenergy barrier between the initial state and the saddle poin t. For typical values of the damping\nparameter ( α <1), a weak MW field of a few mT is thus sufficient to induce switchin g.\nFor a nanomagnet with uniaxial anisotropy placed in a longit udinal static field, the shape\nof the MW field pulse has been obtained analytically. We have s hown that the optimal MW\nfield pulse becomes lower but more spread when the damping dec reases.\nIn the case of more complex energy landscapes (with biaxial o r cubic anisotropies) the\nswitching is likely to be triggered by a succession of MW field pulses. This hypothesis will\nbe later tested numerically. This study could then be extend ed to small nanomagnets where\nsurface effects can not be neglected using the effective one-s pin model (EOSP)21–23. More-\nover, the influence of temperature on the optimization could be investigated numerically using\nthe Langevin approach which introduces the temperature dep endence through an additionnal\nstochastic field24,25. In particular, we intend to investigate the conditions und er which the ther-\nmal fluctuations can favour the switching by assisting the ma gnetization in crossing the saddle\n160.00 0.05 0.10 0.15 0.20 0.25 0.30\nDamping parameter α0.000.050.100.15Injected energy ENumerical results\nE = 4α ∆ε0\nFigure 7: (Color online) Total injected energy Ewith respect to the damping parameter α.\npoint.\nThe optimal MW fields that we have found have an amplitude and a frequency which vary\nslowly and can be reproduced experimentally using a functio n generator. Consequently, our\ntheoretical results could be used to probe the damping param eter and assess the role of surface\neffects in real nanoparticles. The dependence of the MW field o n the energy landscape might\nbe used to address directly a given nanoparticle in a polydis perse assembly.\nACKNOWLEDGMENTS\nWe are grateful to our collaborators E. Bonet, C. Thirion (In stitut Néel, Grenoble, France)\nand V. Dupuis (LPMCN, Lyon, France) for instructive discuss ions on the microwave-assisted\nswitching of isolated nanoclusters. This work has been part ly funded by the collaborative\nprogram PNANO ANR-08-P147-36 of the French Ministry.\n1T. Thomson, L. Abelmann, and H. Groenland, in Magnetic Nanostructures in Modern Technology ,\nedited by B. Azzerboni, G. Asti, L. Pareti, and M. Ghidini (Spr inger Netherlands, 2008), pp.\n237–306, ISBN 978-1-4020-6338-1.\n170 500 1000 1500 2000\nNormalized time τ00.010.020.030.040.050.06Optimal MW field magnitude || hopt(τ)||α = 0.015\nα = 0.030\nα = 0.050 (reference calculation)\nα = 0.080\nα = 0.100\nα = 0.150\nAnalytical results\nFigure 8: (Color online) Optimal MW field pulse for several va lues of the damping parameter α. For\nα= 0.015, the final time has been increased ( τf= 1600 ) without changing the sampling time.\n2C. Thirion, W. Wernsdorfer, and D. Mailly, Nat. Mat. 2, 524 (2003).\n3G. Bertotti, C. Serpico, and I. D. Mayergoyz, Phys. Rev. Lett. 86, 724 (2001).\n4S. I. Denisov, T. V. Lyutyy, and P. Hänggi, Phys. Rev. Lett. 97, 227202 (2006).\n5G. Bertotti, I. D. Mayergoyz, and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems\n(Elsevier Science, 2009).\n6Z. Z. Sun and X. R. Wang, Phys. Rev. B 74, 132401 (2006).\n7Z. Z. Sun and X. R. Wang, Phys. Rev. B 73, 092416 (2006).\n8S. Okamoto, N. Kikuchi, and O. Kitakami, APPLIED PHYSICS LET TERS 93(2008), ISSN 0003-\n6951.\n9G. Bertotti, I. D. Mayergoyz, C. Serpico, M. D’Aquino, and R. Bo nin, J. Appl. Phys. 105, 07B712\n(2009).\n10T. V. Lyutyy, A. Y. Polyakov, A. V. Rot-Serov, and C. Binns, J. P hys.: Condens. Matter 21,\n396002 (2009).\n11K. Rivkin and J. B. Ketterson, Appl. Phys. Lett. 89, 252507 (2006).\n12G. Woltersdorf and C. H. Back, Phys. Rev. Lett. 99, 227207 (2007).\n13M. Laval, J. J. Bonnefois, J. F. Bobo, F. Isaac, and F. Boust, J. Ap pl. Phys. 105, 073912 (2009).\n14S. Okamoto, N. Kikuchi, and O. Kitakami, Applied Physics Let ters93, 142501 (2008).\n1815Z. Wang and W. Mingzhong, J. Appl. Phys. 105, 093903 (2009).\n16L. Cai, D. A. Garanin, and E. M. Chudnovsky, Phys. Rev. B 87, 024418 (2013).\n17B. Meerson and L. Friedland, Phys. Rev. A 41, 5233 (1990).\n18M. Jewariya, M. Nagai, and K. Tanaka, Phys. Rev. Lett. 105, 203003 (2010).\n19N. Barros, M. Rassam, H. Jirari, and H. Kachkachi, Phys. Rev. B 83, 144418 (2011).\n20A. Sukhov and J. Berakdar, Phys. Rev. B 79, 134433 (2009).\n21D. A. Garanin and H. Kachkachi, Phys. Rev. Lett. 90, 065504 (2003).\n22H. Kachkachi and E. Bonet, Phys. Rev. B 73, 224402 (2006).\n23R. Yanes, O. Chubykalo-Fesenko, H. Kachkachi, D. A. Garanin , R. Evans, and R. W. Chantrell,\nPhys. Rev. B 76, 064416 (2007).\n24W. F. Brown, Phys. Rev. 130, 1677 (1963).\n25C. Ragusa, C. Serpico, M. Repetto, M. d’Aquino, B. Xie, and G. Be rtotti, J. Appl. Phys. 103,\n07B119 (2008).\n19"    },    {        "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. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. \nHayakawa, F. Matsukura and H. Ohno , Nat. Mater. 9, 721 (2010).  \n7 J.-S. Kim et al. , Nat. Commun . 5, 3429 (2014).  \n8 L. Thomas, R. Moriya, C. Rettner and S. S. P. Parkin , Science  330, 1810 (2010).  12 \n                                                                                                                                         \n9 J.-S. Kim, M. Stark, M. Klaui, J. Yoon, C. -Y . You, L. Lopez -Diaz and E. Martinez , Phys. \nRev. B 85, 174428 (2012).  \n10 B. Kuanr, R. E. Camleym, and Z. Celinski, Appl. Phys. Let t. 87, 012502 (2005) . \n11 J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Le tt. \n89, 277201 (2002).  \n12 L. Kraus, G. Infante, Z. Frait and M. Vazquez, Phys. Rev. B 83, 174438 (2011) . \n13 C. T. Boone , J. A. Katine, J. R. Childress, V . Tiberkevich, A. Slavin, J. Zhu, X. Cheng and \nI. N. Krivorotov , Phys. Rev. Let t. 103, 167 601 (2009) . \n14 M. Sharma, B. K. Kuanr, M. Sharma and Ananjan Nasu, IEEE Trans. Magn 50, 4300110 \n(2014).  \n15 D.-H. Kim, H .-H. Kim, and Chun -Yeol You, Appl. Phys. Lett. 99, 072502 (2011).  \n16 D.-H. Kim, H. -H. Kim, Chun -Yeol You and Hyungsuk Kim, J. Magn etics 16, 206 (2011).  \n17 C. Kittel, Introduction to Solid State Physics, 7th ed., pp. 504, (1996) . \n18 M. J. Donahue and D. G. Porter: OOMMF User’ s Guide : Ver. 1.0, NISTIR 6376  (National \nInstitute of Standards and Technology, Gaithersburg, Maryland, United States, 1999 ). \n19 W. Wang, C. Mu, B. Zhang, Q. Liu, J. Wang, D. Xue, Comput. Mater. Sci. 49, 84 (2010) . \nSee: http://oommf -2dpbc.sourceforge.net.  \n20 J. Cho, J . Jung, K .-E. Kim, S .-I. Kim, S .-Y. Park, M .-H. Jung, C .-Y. You, J. Magn. Magn. \nMater. 339, 36 (2013).  \n21 C. Bilzer, T. Devolder, J -V . Kim, G. Counil, C. Chappert, S. Cardoso and P. P. Feitas , J. \nAppl. Phys. 100, 053903 (2006).  \n22 K.-S. Lee, D. -S. Han , S.-K. Kim, Phys. Rev. Let t. 102, 127202 (2009).  \n23 J. Yoon, C. -Y . You, Y . Jo, S. -Y. Park, M. H. Jung , J. Korean Phys. Soc. 57, 1594 (2010) .  \n24 Y . Shiota, F. Bonell, S. Miwa, N. Muzuochi, T. Shinjo and Y . Suzuki , Appl. Phys. Le tt. 103. \n082410 (2013),  \n25 I. Purnama, I. S. Kerk, G. J. Lim  and W. S. Lew , Sci. Rep. 5, 8754 (2015).  \n26 J. Ding, M. Kostylev, and A. O. Adeyeye, Phys. Rev. B. 84, 054425 (2011) . \n27 A. Aharoni, J. Appl. Phys. 83, 3432 (2011) . 13 \n Figure  Captions  \n \nFig. 1. Measurement geometry with SEM image s of the 100 -nm-width nanowires with a gap \nof 200 nm between nanowires . The longitudinal nanowire arrays are shown. After the \nnanowire patterns have been  defined by e -beam lithography, they are covered by co -planar \nwave guides.  \n \nFig. 2. (a) The measured FMR  spectrum of the CoFeB nanowire with H =0.194 T. The red  (lower \npeak)  and blue  (higher peak)  arrows indicate t he resonance frequencies of the  uniform FMR  mode and \nthe nanowire mode, respectively. (b) The measured FMR spectrum of the CoFeB thin  film with H \n=0.194 T. (c) Simulated FMR spectrum of the CoFeB nanowire with H= 0.200 T.  \n \nFig. 3. Measured and calculated FMR frequencies with the applied magnetic field for 100 -\nnm-width nanowire. The open black rectangles  are nanowire mode and open red circles are \nthe uniform FMR mode for CoFeB thin film. The closed black rectangles  are calculated by \nOOMMF and the closed blue circles  are theoretical ly calculated by Eq. (1)  using fitted \nparameters form un -patterned film . \n \nFig. 4. Variation of resonanc e frequencies with the applied magnetic field for the different \nPBC wire width for (a) longitudinal field and (b) transverse field.  Inset: The geometry of 2 -\ndimensional PBC micromagnetic simulation with nanowire width a and a gap of 200 nm \nbetween nanowire s. The black open rectangles, red open circles, green open upper triangles, \nblue open down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, \n100 nm, 125nm, and 150 nm, respectively.  \n \nFig. 5. Demagnetization factor with PBC wire widt h for (a) longitudinal and (b) transverse \nfield.  The black open circles, red open rectangles, blue open upper triangles represent as \ndemagnetization factors, Ny, Nz, and Nx, respectively.  \n \nFig. 6. Full width and half maxim a with the applied magnetic field for (a) longitudinal and (b) \ntransverse field.  The black open rectangles, red open circles, green open upper triangles, blue \nopen down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, 100 \nnm, 125nm, and 150 nm, respectively.  \n \nFig. 7. Damping constants with wavevector for transverse ( the black open rectangles ) and \nlongitudinal ( the red open circles) field with errors. The black line is the input value which is \ndetermined from un -patterned film. Inset presents the profile of the trans verse spin density as SWs.  \n Fig. 1 \n \n \n    \n \n      Fig. 2. \n \n \n \nFig. 3. \n \n \n \n  \nFig. 4 \n \n` \n \n \n \nFig. 5 \n \n \n \nFig. 6 \n \n \n \nFig. 7 \n \n \n"    },    {        "title": "1901.03394v1.Damping_and_softening_of_transverse_acoustic_phonons_in_colossal_magnetoresistive_La___0_7__Ca___0_3__MnO__3__and_La___0_7__Sr___0_3__MnO__3_.pdf",        "content": "arXiv:1901.03394v1  [cond-mat.str-el]  10 Jan 2019Damping and softening of transverse acoustic phonons in col ossal magnetoresistive\nLa0.7Ca0.3MnO 3and La 0.7Sr0.3MnO 3\nJoel S. Helton1,2,∗, Yang Zhao2,3, Dmitry A. Shulyatev4, and Jeffrey W. Lynn2,†\n1Department of Physics, United States Naval Academy, Annapo lis, MD 21402, USA\n2NIST Center for Neutron Research, National Institute of Sta ndards and Technology, Gaithersburg, MD 20899, USA\n3Department of Materials Science and Engineering,\nUniversity of Maryland, College Park, MD 20742, USA and\n4National University of Science and Technology “MISiS”, Mos cow 119991, Russia\nNeutronspectroscopy is usedtoprobetransverse acoustic p hononsnearthe(2 ,2,0) Braggposition\nin colossal magnetoresistive La 0.7Ca0.3MnO3and La 0.7Sr0.3MnO3. Upon warming to temperatures\nnearTc= 257 K the phonon peaks in La 0.7Ca0.3MnO3soften and damp significantly with the\nphonon half width at half maximum approaching 2.5 meV for pho nons at a reduced wave vector\nof/vector q= (0.2,0.2,0). Concurrently a quasielastic component develops that do minates the spectrum\nnear the polaron position at high temperatures. This quasie lastic scattering is ≈5 times more\nintense near Tcthan in La 0.7Sr0.3MnO3despite comparable structural distortions in the two. The\ndamping becomes more significant near the polaron position w ith a temperature dependence similar\nto that of polaron structural distortions. An applied magne tic field of 9.5 T only partially reverses\nthe damping and quasielastic component, despite smaller fie lds being sufficient to drive the colossal\nmagnetoresistive effect. The phonon energy, on the other han d, is unaffected by field. The damping\nin La0.7Sr0.3MnO3nearTcat a reduced wave vector of /vector q= (0.25,0.25,0) is significantly smaller but\ndisplays a similar trend with an applied magnetic field.\nI. INTRODUCTION\nPerovskite manganites of the form R1−xAxMnO3,\nwhereRis a trivalent rare-earth cation and Ais a di-\nvalent alkaline-earth cation, are important systems for\nexploring the interplay of magnetic, electrical, lattice,\nand orbital degrees of freedom [1, 2]. La 1−xCaxMnO3\nhas attracted particular attention [3, 4]. At half doping\nLa0.5Ca0.5MnO3displays an antiferromagnetic ground\nstate with CE-type charge and orbital order [5, 6] that\nresults in superlattice peaks with a /vectork= (0.25,0.25,0)\npropagation vector. At lower doping the ground state\nis a ferromagnetic metal, and colossal magnetoresistance\n(CMR) is observed at the combined ferromagnetic and\nmetal-insulator phase transition. Double exchange alone\nis insufficient to explain the magnitude of the CMR ef-\nfect [7], with a role played by polarons in the insulating\nphase where Jahn-Tellerinteractionscauselocal CE-type\ndistortions that trap charge carriers [8, 9].\nLa0.7Ca0.3MnO3displays a fairly significant CMR ef-\nfectandevidenceforbothstatic[10,11]anddynamic[12]\npolarons. Scattering near the /vector q= (0.25,0.25,0) polaron\nposition arises from correlated polarons and peaks near\nTc. Diffuse scattering from uncorrelated polarons rises\nrapidly in intensity as the temperature approaches Tc\nbut remains constant at higher temperatures, suggest-\ning that the number of polarons remains relatively con-\nstant above Tceven while their correlations decrease at\nhigher temperature. A polaron glass phase persists to\n400 K. Ridges of quasielastic magnetic scattering [13, 14]\nassociated with the magnetic part of diffuse polarons\nare only partially suppressed by a 10-T magnetic field,\nindicating that very large fields are needed to changethe number of polarons near Tc. An optical Jahn-Teller\nphonon mode displays anomalous and increasing damp-\ning with increasing temperature and collapses above\nTc[15]. La 0.7Sr0.3MnO3, with a higher transition tem-\nperature, displays a less pronounced CMR effect [16],\nthough there is evidence of significant structural distor-\ntions atTc[17, 18] and a polaronic metal phase [19]. Un-\nlike La 0.7Ca0.3MnO3, this compound displays only dy-\nnamic polarons demonstrating that the strength of the\ncolossal magnetoresistive effect is correlated with the po-\nlaron lifetime [19, 20].\nII. EXPERIMENT AND RESULTS\nLa0.7Ca0.3MnO3(LCMO) and La 0.7Sr0.3MnO3\n(LSMO) have an orthorhombic perovskite crystal\nstructure; however, the orthorhombic distortions are\nsmall relative to the resolution of neutron spectroscopy\nmeasurements and the crystallographic domains are\nequally populated so we index the samples with a\npseudocubic notation where a=b=c≈3.9˚A. Po-\nlaron scattering has been observed at a reduced wave\nvector of /vector q= (±0.25,±0.25,0) in materials such as\nLa0.7Ca0.3MnO3[10, 11] and the bilayer manganite\nLa2−2xSr1+2xMn2O7[21], consistent with the superlat-\ntice propagation vector of CE-type charge and orbital\norder [6]. The polaron scattering structure factor is\nparticularly intense for reduced wave vectors transverse\nto the (2, 2, 0) Bragg position: /vectorQ= (2.25,1.75,0) and\nequivalent. Both longitudinal and transverse phonons\nwith/vector q= (±0.25,±0.25,0) feature oxygen displace-\nments that partially match the distortions of CE-type\norder [22]. The structure factor for transverse acoustic2\nphonons is also quite large around the (2, 2, 0) Bragg\npeak. Therefore we have concentrated this study on the\nspectroscopy of transverse acoustic phonons at positions\nequivalent to /vectorQ= (2 +ξ,2−ξ,0) with ξbetween\n0.10 and 0.25. Because the sample contains multiple\ncrystallographic domains, at a reduced wave vector\nof (0.25, 0.25, 0) in the pseudocubic notation, only a\nportion of the sample will be oriented at the actual\npolaron position of /vector q= (0.5,0,0) in the orthorhombic\nnotation. A previous work found that the damping of\ntransverse acoustic phonons in LSMO [19] increased\nsignificantly upon approaching the nominal polaron\nposition with ξ=0.25 but was maximized at a larger\nvalue close to ξ= 0.3, possibly related to the averaging\nover multiple domains.\nNeutron spectroscopy measurements utilized the BT-7\nthermal triple-axis spectrometer [23] at the NIST Center\nforNeutronResearchwith Open-50′-50′-120′collimators,\na fixed final energy of 14 .7 meV, and pyrolytic graphite\nfilters after the sample. Constant- Qscansweremeasured\nfor transverse acoustic phonons at /vectorQ= (2−ξ,−2−ξ,0)\nin LSMO for ξ=0.10, 0.15, 0.20, 0.25, and 0.30 at tem-\nperatures of T=250 K, 275 K, 300 K, 325 K, 350 K,\nand 370 K. This LSMO sample had a mass of 3 g and\nTc= 351 K [24]. Constant- Qscans were measured for\ntransverse acoustic phonons at /vectorQ= (2+ξ,−2+ξ,0) in\nLCMOfor ξ=0.10,0.15,0.20,and0.25attemperaturesof\nT=100 K, 150 K, 200 K, 250 K, and 300 K. This LCMO\nsample has a mass of 1.5 g and Tc= 257 K (higher than\nmost other single-crystal samples of LCMO) [25]. Fur-\nther constant- Qscans were measured with the samples\nplaced in a vertical field superconducting magnet. For\nLCMOa10-Tmagnetwasusedand scansweremeasured\nat/vectorQ= (1.85,−2.15,0)and/vectorQ= (1.80,−2.20,0)at 265 K\nfor magnetic fields of 0 T, 3 T, 6 T, and 9.5 T as well\nas at 140 K in zero field. For LSMO a 7-T magnet was\nused and scans were measured at /vectorQ= (1.75,−2.25,0) at\n345 K for magnetic fields of 0.5 T, 3 T, and 6 T as well\nas at 250 K in zero field. A sloped background has been\nfit and then removed from data taken inside the magnet.\nError bars and uncertainties throughout this paper are\nstatistical in nature and represent one standard devia-\ntion.\nFigure 1 displays constant- Qscans of the transverse\nacoustic phonons in LSMO at ξ= 0.20 andξ= 0.25\n(the polaron position). For both positions a qualitative\nchangeis observedupon heating above 300K ( ≈0.85Tc).\nThephononsbecomebroadened,withextrascatteringin-\ntensity observed on both the high-energy and low-energy\nsides of the peak. The extra scattering on the low-energy\nsideofthepeakisparticularlypronouncedandextendsto\nthe lowest energy transfers measured. Maschek et al.[19]\nhave reported this behavior in LSMO modelled as a\nphonon peak that softens and damps with increasing\ntemperature alongside a quasielastic component arising/s48/s50/s53/s48/s53/s48/s48/s55/s53/s48/s49/s48/s48/s48/s49/s50/s53/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s50/s53/s48/s53/s48/s48/s55/s53/s48/s49/s48/s48/s48/s49/s50/s53/s48/s40/s98/s41/s81 /s61/s40/s49/s46/s56/s48/s44/s32/s45/s50/s46/s50/s48/s44/s32/s48/s41/s40/s97/s41\n/s81 /s61/s40/s49/s46/s55/s53/s44/s32/s45/s50/s46/s50/s53/s44/s32/s48/s41/s32/s50/s53/s48/s32/s75 /s32/s32/s32 /s32/s51/s50/s53/s32/s75\n/s32/s50/s55/s53/s32/s75 /s32/s32/s32 /s32/s51/s53/s48/s32/s75\n/s32/s51/s48/s48/s32/s75 /s32/s32/s32 /s32/s51/s55/s48/s32/s75/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s112/s101/s114/s32/s51/s32/s109/s105/s110/s41\n/s32/s40/s109/s101/s86/s41\nFIG. 1: Constant- Qscans of transverse acoustic phonons in\nLa0.7Sr0.3MnO3(Tc= 351 K) at a series of temperatures. (a)\nξ= 0.20 scan at /vectorQ= (1.80,−2.20,0).(b)ξ= 0.25 scan at\n/vectorQ= (1.75,−2.25,0).\nfrom dynamic polarons.\nFigure 2 displays constant- Qscans in LCMO at tem-\nperatures between 100 K and 300 K for phonons with\nξ=0.10to0.25(the polaronposition). Qualitativelysim-\nilar behavior is observed with phonon peaks that soften\nand damp upon warming through 200 K ( ≈0.78Tc).\nHowever,thequasielasticcomponentissignificantlymore\nintense compared to LSMO. For temperatures at 250 K\nand above the quasielastic component is sufficiently in-\ntense that it merges with the broadened phonon peak to\nyield intensity across a wide range of energy transfers on\nthe low-energy side of the peak that are comparable in\nintensity to the peak position.\nFigure 3 allows for a direct comparison of phonon\ndamping and softening as well as the quasielastic scatter-\ning component in LCMO and LSMO. Constant- Qscans\nare shown for both materials at /vectorQ= (1.85,−2.15,0);\nthisξ= 0.15 position is farther from the polaron po-\nsition than some other data but allows for easier dis-\ncernment between the phonon and quasielastic contribu-\ntions. At temperatures well below Tc, shown in Fig. 3(a),\nthe phonon is comparable in the two materials. The3\n/s48 /s51 /s54 /s57 /s49/s50 /s49/s53/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s48 /s51 /s54 /s57 /s49/s50 /s49/s53/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48\n/s48 /s51 /s54 /s57 /s49/s50 /s49/s53/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s48 /s51 /s54 /s57 /s49/s50 /s49/s53/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s32/s49/s48/s48/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s50/s53/s48/s32/s75\n/s32/s51/s48/s48/s32/s75/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s112/s101/s114/s32/s51/s32/s109/s105/s110/s41\n/s32/s40/s109/s101/s86/s41/s61/s48/s46/s49/s48/s40/s97/s41\n/s32/s40/s109/s101/s86/s41/s61/s48/s46/s49/s53/s40/s98/s41\n/s32/s40/s109/s101/s86/s41/s61/s48/s46/s50/s48/s40/s99/s41\n/s32/s40/s109/s101/s86/s41/s61/s48/s46/s50/s53/s40/s100/s41\nFIG. 2: Constant- Qscans of transverse acoustic phonons in La 0.7Ca0.3MnO3(Tc= 257 K) at a series of temperatures. (a)\nξ= 0.10 scan at /vectorQ= (2.10,−1.90,0).(b)ξ= 0.15 scan at /vectorQ= (2.15,−1.85,0). (c)ξ= 0.20 scan at /vectorQ= (2.20,−1.80,0).(d)\nξ= 0.25 scan at /vectorQ= (2.25,−1.75,0).Successive data sets are offset for clarity. The solid lines i n the first three panels are fits\nas described in the text.\natomic mass difference between La and Sr is significantly\nless than the difference between La and Ca leading to a\nsmaller average and greater variance of the A-site atomic\nmass in LCMO. Differing levels of A-site chemical dis-\norder [26] are also possible. Despite these differences\nthe phonon spectra at ξ= 0.15 are quite similar be-\ntween the two compounds. The phonon peak is slightly\nsofter and broader in LSMO, likely reflecting the higher\ntemperature. Close to Tcthe phonon has softened and\nbroadened to a comparable extent in both materials, as\nthe phonon peaks shown in Fig. 3(b) still largely over-\nlap. However the quasielastic intensity is far more pro-\nnounced in LCMO. In the LSMO data the quasielastic\ncontribution is visible only as a plateau in intensity be-\nlow 3 meV while in LCMO it presents as a peak with\nintensity comparable to the phonon. Fitting both data\nsets to a quasielastic component plus a phonon contribu-\ntion we find that the quasielastic dynamic susceptibility\nin LCMO at 265 K is a factor of 5 .3±1.2 more intense\nthan in LSMO at 350 K.\nThe constant- Qscans in LCMO were fit to the sum of\na quasielastic contribution and a phonon peak. Fits were\nperformed with the phonon modeled as both a Gaus-\nsian and a damped harmonic oscillator. Fit values for\nthe two cases were similar; values reported here used a\nGaussian line shape as it provided a slightly better fit\nto the data. For each /vectorQ-position the lowest tempera-\nture data ( T= 100 K for zero-field scans, T=140 K for\nscans measured in the magnet) were fit first with the\nquasielastic peak set at zero intensity. This fit was used\nto fix the phonon dynamic susceptibility, with the tem-\nperature dependence of the integrated phonon intensity\nset by the Bose factor. The quasielastic width in all fits\nwas fixed to a half width at half maximum (HWHM)of Γ = 3 .00 meV [12]. Fit values are shown in Fig. 4.\nThe fits for the ξ= 0.25 data at higher temperatures\ndid not reliably converge due to the quasielastic compo-\nnent overwhelming the phonon peak and are excluded\nfrom this figure. Figure 4(a) shows the phonon intrin-\nsic width (HWHM) after deconvolution with the instru-\nmental resolution displayed as a function of temperature.\nFits from the data taken at zero field in the magnet are\nalso included here, for ξ=0.15 and 0.20 at T=140 K\nand 265 K. The phonon peaks damp significantly for\nξ≥0.15,with thewidth increasingfromroughly0.5meV\nat low temperatures to roughly 2.5 meV (for ξ= 0.20)\nat temperatures exceeding Tc. When comparing to the\ndamping reported [19, 20] in LSMO, we find that the\ndamping near Tcis roughly comparable in the two mate-\nrials up to ξ= 0.15 (as was also demonstrated in Fig. 3)\nbut that by ξ= 0.20 LCMO displays a high tempera-\nture damping (Γ ≈2.5 meV) in excess of any damping\nobserved in LSMO (with a maximum reported value of\nΓ≈1.5meV). Thechangeinphononpeakposition(from\nT= 100 K) is displayed in Fig. 4(b) and shows that all\nphonon peaks soften with increasing temperature, with\na typical softening of about 0.6 meV at 300 K. Previ-\nous work on La 0.75(Ca0.45Sr0.55)0.25MnO3found trans-\nverse acoustic phonons near the polaron position that\ndisplayedasimilardampingandsoftenedbyabout1meV\nover this temperature range [27].\nFor the data at ξ= 0.25 the temperature dependence\nwas also measured for ¯ hω= 2 meV with smaller tem-\nperature increments. This is useful as a measure of the\nquasielastic intensity, as there will not be much contri-\nbution from the phonon at this energy. This temper-\nature dependence, after subtracting the background, is\nshown in Fig. 5(a). The quasielastic intensity begins to4\n/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s32/s76/s67/s77/s79/s44/s32 /s84 /s61/s49/s52/s48/s32/s75\n/s32/s76/s83/s77/s79/s44/s32 /s84 /s61/s50/s53/s48/s32/s75/s40/s97/s41\n/s81 /s61/s40/s49/s46/s56/s53/s44/s32/s45/s50/s46/s49/s53/s44/s32/s48/s41\n/s32/s76/s67/s77/s79/s44/s32 /s84 /s61/s50/s54/s53/s32/s75\n/s32/s76/s83/s77/s79/s44/s32 /s84 /s61/s51/s53/s48/s32/s75/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s40/s109/s101/s86/s41/s40/s98/s41\nFIG. 3: Constant- Qscans for both LCMO and LSMO at\n/vectorQ= (1.85,−2.15,0). (a) Scans performed at temperatures\nwell below Tc. The dashed line is a Gaussian fit to the LCMO\ndata. (b) Scans performed close to Tc. The dashed line is the\nsame function as in the top panel.\nrise when the temperature rises above 150 K, but the\nsharpest rise is observed just below Tc. The intensity in-\ncreases modestly upon warming from Tcto 315 K. Previ-\nously, the quasielastic scattering from dynamic polarons\nin LCMOhadbeen foundto persisttoatleast400K[12].\nThe equivalent data for La 0.70Sr0.30MnO3, measured at\n¯hω= 3 meV, is shown in Fig. 5(b). This rise is concen-\ntrated over a narrower temperature range, rising sharply\nabove 275 K.\nFigures 6 and 7 show the data for LCMO at ξ= 0.15\nand 0.20, respectively, measured in a superconducting\nmagnet at T=140 K (zero field) and T= 265 K (0 T,\n3 T, 6 T, and 9.5 T). In each figure the data and fits\nare shown in the top panel while the fit values are shown\nin the lower panel. The fit values for the quasielastic\nintensity and phonon width are shown together with the\naxes chosen so that the 140-K fit values (for which the\nquasielastic intensity is fixed to zero) occur at the same\nvertical position. At 265 K, the application of a 9.5-T\nmagnetic field decreases both the quasielastic intensity\nand the phonon damping by half compared to zero field.\nIn contrast, the field dependence of the phonon posi-\ntion fit value at /vectorQ= (1.85,−2.15,0) is shown in Fig. 8.\nThe phonon position softens by about 0.5 meV upon/s48/s49/s50/s51\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s40/s98/s41/s80/s104/s111/s110/s111/s110/s32/s119/s105/s100/s116/s104/s32/s40/s109/s101/s86/s41/s40/s97/s41\n/s32/s105/s110/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s40/s109/s101/s86/s41/s32 /s61/s48/s46/s49/s48\n/s32/s48/s46/s49/s53\n/s32/s48/s46/s50/s48\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFIG. 4: Fit parameters for La 0.7Ca0.3MnO3in zero field as\na function of temperature. (a) Phonon width (HWHM) after\ndeconvolution with the instrumental resolution. (b) Chang e\nin the phonon peak position from T= 100 K.\nwarming from 140 K to 265 K at zero field. However,\napplied fields up to 9.5 T do not reverse the softening,\nbut rather have no measurable effect on the position.\nFigure 9 shows the data for LSMO at ξ= 0.25 mea-\nsured at T=250 K (zero field) and T= 345 K (0.5 T,\n3 T, and 6 T). Near Tcthe low-field damping yields a\nphonon HWHM of about 1.75 meV which, while signifi-\ncant, is less than that observedin LCMO even at ξ=0.20.\nConversely, this phonon is fairly broad (1 meV) even in\nthe low-temperature data likely because of the elevated\ntemperature in comparison to 140 K used as the low-\ntemperature value in LCMO. The applied magnetic field\npartially reverses the damping much as was observed in\nLCMO. The lines in Figs. 6(b), 7(b), and 9(b) have all\nbeen chosen to represent a field-dependent damping such\nthat the width at 9.5 T is at the midpoint between the\nzero field width at the low and high temperatures.\nIII. DISCUSSION\nAcrossawide rangeof R1−xAxMnO3materialsatrend\nis observed where compounds with a high Tctypically\ndisplay a weaker CMR effect. Therefore La 0.7Sr0.3MnO3\n(Tc=351 K) [16] displays a significantly less pronounced5\nCMR effect than La 0.7Ca0.3MnO3(Tc= 257 K) [3].\nThis was historically taken as evidence that LSMO fea-\ntured weaker electron-lattice coupling and therefore dis-\nplayed weaker CE-type lattice distortions in the high-\ntemperature phase. However, more recent work has\nconfirmed that LSMO displays lattice distortions at the\ntransition temperature comparable to those observed in\nLCMO and other compounds with a significant CMR\neffect [18] as well as evidence of dynamic CE-type po-\nlarons [19]. Despite the comparable lattice distortions,\nwefindthatthequasielasticscatteringarisingfromshort-\nlifetime dynamic polarons near Tcis a factor of 5 more\nintense in LCMO than in LSMO. This quasielastic inten-\nsity rises sharply as the temperature approaches Tcand\nremains significant at temperatures above Tcup to at\nleast 400 K [12]. This is comparable to the temperature\ndependence of the diffuse scattering arising from uncor-\nrelated static polarons in this material [10]. We compare\nthe temperature dependence of the quasielastic scatter-\ning in LCMO with that in LSMO, and find that the rise\nin quasielastic intensity occurs over a larger temperature\nrange in LCMO. Figure 1 shows that the quasielastic in-\ntensity in LSMO is quite low up to 300 K ( ≈0.85Tc)\nand rises most sharply between 300 K and 325 K. The\nquasielastic scattering in LCMO, as observed in Fig. 2,\nrises steadily starting near 150 K ( ≈0.58Tc). Figure 5\nshows these temperature dependencies with equally wide\nscales and the transition temperatures aligned. Recently,\nLa0.8Sr0.2MnO3was also shown to feature a quasielas-\ntic component that appeared over a narrow temperature\nrange [20].\nThe transverse acoustic phonons in LCMO are found\nto damp with rising temperature for ξ=0.15 and\n0.20, approaching the polaron position. Similar to the\nquasielastic intensity, the damping increases upon warm-\ningthrough Tcbut remainsfairly constantasthe temper-\nature is increased further. LCMO undergoes significant\nlocal structural changes at Tc, with changes in the lat-\ntice parameters [28], increases in the O and Mn atomic\ndisplacements [29], and the width of the distribution of\nMn-O distances [30, 31]. It is natural to associate this\ndamping with increased scattering of phonons from the\nlocal distortions surrounding polarons. The temperature\ndependence of the phonon damping is particularly rem-\niniscent of the temperature dependence of the variance\nin Mn-O bond distances ascribed to polarons [30]. The\ntransverse phonons in LSMO are also found to damp\nwith increasing temperature in agreement with previ-\nous result s[19]. The high temperature damping is fairly\nsimilar between the two materials for ξ= 0.15. How-\never, the damping becomes more pronounced in LCMO\nat larger reduced wave vectors so that the damping near\nTcis larger in LCMO at ξ= 0.20 (2.5 meV) than in\nLSMO even at the largerreduced wave vector of ξ= 0.25\n(1.75 meV).\nThe application of a magnetic field lowers both the/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s76/s67/s77/s79\n/s61/s48/s46/s50/s53\n/s61/s50/s32/s109/s101/s86\n/s84\n/s99/s61/s50/s53/s55/s32/s75/s40/s97/s41\n/s84\n/s99/s61/s51/s53/s49/s32/s75/s76/s83/s77/s79\n/s61/s48/s46/s50/s53\n/s61/s51/s32/s109/s101/s86/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s112/s101/s114/s32/s49/s48/s32/s109/s105/s110/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s98/s41\nFIG. 5: (a) Quasielastic scattering intensity at ¯ hω= 2 meV\nandξ= 0.25 for La 0.70Ca0.30MnO3. (b) Quasielastic\nscattering intensity at ¯ hω= 3 meV and ξ= 0.25 for\nLa0.70Sr0.30MnO3. These data consist primarily of scattering\nfrom the quasielastic component, as the phonon contributio n\nwill be quite small at these energies.\nphonon damping and the quasielastic scattering; how-\never, a field of 9.5 T is sufficient to return these pa-\nrameters only half way to their low-temperature val-\nues. This is in keeping with previous information about\nthe effect of a magnetic field on the number of po-\nlarons present. Ridges of magnetic scattering arising\nfrom polaron-mediated spin correlations follows a simi-\nlar field dependence [14]. Near Tcan applied 9-T field\nlowered the variance in Mn-O bond lengths about half\nway back to its low-temperature value [30]. Correlations\nbetweenstaticpolaronsaresuppressedbyamuchsmaller\nmagnetic field of about 5 T [32], comparable to the fields\nunder which CMR is observed [3], and a field of only\n1 T will suppress these correlations in a bilayer mangan-\nite [33]. These facts have been used to argue that colos-\nsal magnetoresistance follows polaron correlations rather\nthan the number of polarons.\nTransverseacoustic phonons in LCMO at all measured\nwave vectors soften upon warming, with phonon peaks\ndecreasing in energy by about 0.6 meV from 100 K to\n300 K. We observe comparable softening in LSMO from\n250 K to 370 K, although a previous work had found sig-\nnificantly more softening in that material at wavevectors\nclosetothepolaronposition[19]. Thesofteningiscontin-6\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48\n/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s49/s46/s50/s53\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s40/s98/s41/s84 /s61/s49/s52/s48/s32/s75/s32/s40/s48/s32/s84/s41\n/s48/s32/s84/s32/s32 /s54/s32/s84\n/s51/s32/s84/s32/s32 /s57/s46/s53/s32/s84/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s112/s101/s114/s32/s56/s32/s109/s105/s110/s41\n/s32/s40/s109/s101/s86/s41\n/s40/s49/s46/s56/s53/s44/s32/s45/s50/s46/s49/s53/s44/s32/s48/s41/s40/s97/s41\n/s76/s67/s77/s79\n/s48/s72 /s32/s40/s84/s41\n/s80/s104/s111/s110/s111/s110/s32/s119/s105/s100/s116/s104/s32/s40/s109/s101/s86/s41/s32/s50/s54/s53/s32/s75\n/s32/s49/s52/s48/s32/s75\n/s32/s50/s54/s53/s32/s75/s81/s117/s97/s115/s105/s101/s108/s97/s115/s116/s105/s99/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\nFIG. 6: (a) Constant- Qscans in La 0.7Ca0.3MnO3at/vectorQ=\n(1.85,−2.15,0). Zero-field data were measured at 140 K and\n265 K. Data in an applied field were measured at T=265 K.\nSuccessive data sets are offset for clarity. The lines are fits as\ndescribed in the text. (b) Fit parameters for the quasielast ic\nintensity and phonon width (HWHM).\nuous, with no apparent anomaly at Tc. A slightly larger\nsoftening observed in La 0.75(Ca0.45Sr0.55)0.25MnO3like-\nwise featured no anomaly with temperature and could\nnot be explained by lattice thermal expansion without\na nonphysically large Gr¨ uneisen constant [27]. Most no-\ntably, the softening is field independent up to 9.5 T at\n265 K. Colossal magnetoresistancein the manganites has\noften been analyzed as a percolation effect [34–36] with\na precipitous breakdown of conducting pathways occur-\nring upon warming through Tc. The distribution width\nofMn-Obondshasbeenfoundtodisplayauniversalrela-\ntionship with magnetizationfor fields above2 T with two\ndistinct regimes; for sample magnetization below about\n65%ofsaturationthechangeinstructuraldistortionwith\nmagnetization was found to be small [30]. The magneti-\nzation of La 0.7Ca0.3MnO3at 265 K and 9 T was reported\nin that study to be about 70% of saturation, indicating\nthat our data are mostly still in the low-field regime.\nOne explanation for these results could be that phonon\nsoftening is driven by the relative fraction of insulating\nclusters in the sample while applied fields in this regime\nprimarily re-orient clusters of conducting and insulating\nphases with a relatively smaller effect on their relative\nfractions./s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48\n/s48/s50/s53/s48/s53/s48/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s32/s84 /s61/s49/s52/s48/s32/s75/s32/s40/s48/s32/s84/s41\n/s32/s48/s32/s84/s32/s32 /s32/s54/s32/s84\n/s32/s51/s32/s84/s32/s32 /s32/s57/s46/s53/s32/s84/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s112/s101/s114/s32/s56/s32/s109/s105/s110/s41\n/s32/s40/s109/s101/s86/s41\n/s40/s49/s46/s56/s48/s44/s32/s45/s50/s46/s50/s48/s44/s32/s48/s41/s40/s97/s41\n/s32/s50/s54/s53/s32/s75/s81/s117/s97/s115/s105/s101/s108/s97/s115/s116/s105/s99/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s48/s72 /s32/s40/s84/s41/s76/s67/s77/s79\n/s80/s104/s111/s110/s111/s110/s32/s119/s105/s100/s116/s104/s32/s40/s109/s101/s86/s41/s32/s50/s54/s53/s32/s75\n/s32/s49/s52/s48/s32/s75/s40/s98/s41\nFIG. 7: (a) Constant- Qscans in La 0.7Ca0.3MnO3at/vectorQ=\n(1.80,−2.20,0). Zero-field data were measured at 140 K and\n265 K. Data in an applied field were measured at T=265 K.\nSuccessive data sets are offset for clarity. The lines are fits as\ndescribed in the text. (b) Fit parameters for the quasielast ic\nintensity and phonon width (HWHM).\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s54/s46/s52/s54/s46/s53/s54/s46/s54/s54/s46/s55/s54/s46/s56/s54/s46/s57/s55/s46/s48/s55/s46/s49/s55/s46/s50\n/s32/s50/s54/s53/s32/s75\n/s32/s49/s52/s48/s32/s75/s80/s104/s111/s110/s111/s110/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s40/s109/s101/s86/s41\n/s48/s72 /s32/s40/s84/s41/s81 /s61/s40/s49/s46/s56/s53/s44/s32/s45/s50/s46/s49/s53/s44/s32/s48/s41\nFIG. 8: Phonon position fit value for La 0.7Ca0.3MnO3at/vectorQ=\n(1.85,−2.15,0). Application of fields up to 9.5 T have no\nmeasurable effect on the phonon position at 265 K.\nIV. SUMMARY\nTransverse acoustic phonons in La 0.7Ca0.3MnO3mea-\nsured near the (2, 2, 0) Bragg position damp and soften\nwith increasing temperature. The damping remains rel-7\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s49/s48/s48/s48\n/s48 /s50 /s52 /s54 /s56/s49/s46/s48/s48/s49/s46/s50/s53/s49/s46/s53/s48/s49/s46/s55/s53/s50/s46/s48/s48/s32/s84 /s61/s50/s53/s48/s32/s75 /s32/s40/s48/s32/s84/s41\n/s32/s48/s46/s53/s32/s84 /s32/s54/s32/s84\n/s32/s51/s32/s84/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s32/s112/s101/s114/s32/s56/s32/s109/s105/s110/s41\n/s32/s40/s109/s101/s86/s41/s40/s97/s41\n/s32/s51/s52/s53/s32/s75\n/s32/s50/s53/s48/s32/s75/s80/s104/s111/s110/s111/s110/s32/s119/s105/s100/s116/s104/s32/s40/s109/s101/s86/s41\n/s48/s72 /s32/s40/s84/s41/s40/s98/s41\n/s76/s83/s77/s79\n/s40/s49/s46/s55/s53/s44/s32/s45/s50/s46/s50/s53/s44/s32/s48/s41\nFIG. 9: (a) Constant- Qscans in La 0.7Sr0.3MnO3at/vectorQ=\n(1.75,−2.25,0). Zero-fielddatawere measuredat250 K.Data\ninanappliedfieldweremeasuredat T=345K.Successivedata\nsets are offset for clarity. The lines are fits as described in t he\ntext. (b) Fit parameters for the phonon width (HWHM).\natively constant above Tcand becomes more pronounced\nat wavevectors approaching the polaron position, with\na maximum measured width (HWHM) of 2.5 meV at\nξ= 0.20 above Tc. Applied magnetic fields up to\n9.5 T only partially reverse the damping, lowering the\nphonon width at high temperature. The temperature\nand magnetic-field dependence of the damping are simi-\nlar to those of the distribution width of Mn-O bond dis-\ntances measured by x-ray pair distribution function [30],\nstrongly suggesting that the phonon damping arises from\nscattering from local Jahn-Teller distortions surrounding\npolarons.\nIn addition to the phonons, a quasielastic component\narising from local distortions is observed and fit with\na width of Γ = 3 .00 meV [12]. When compared to\nLa0.7Sr0.3MnO3, a similar compound with a higher tran-\nsition temperature and significantly smaller CMR effect\nbut comparable lattice distortions, the quasielastic com-\nponent is about five time stronger in LCMO and fea-\ntures a temperature dependence that rises over a less\nconcentrated temperature range. The magnetic field de-\npendence of the quasielastic component closely matches\nthat of the phonon damping. Phonon softening is ob-\nserved at all wave vectors. While no anomaly is observedin the softening, the softening is likely too strong to be\nexplained solely by thermal expansion. An applied mag-\nnetic field does not affect the phonon energy, possibly\nsuggesting an effect that arises from relative concentra-\ntions of coexisting metallic and insulating phases and is\ninsensitive to details of percolation pathways.\nACKNOWLEDGEMENT\nJ.S.H acknowledges partial support from the Office of\nNaval Research through the Naval Academy Research\nCouncil.\nemails:∗helton@usna.edu, †jeffrey.lynn@nist.gov\n[1] A. P. Ramirez, J. Phys.: Condens. Matter 9, 8171 (1997).\n[2] B. Raveau and C. N. R. Rao (eds), Colossal Magnetore-\nsistance, Charge Ordering, and Related Properties of\nManganese Oxides (World Scientific, Singapore, 1998).\n[3] P. Schiffer, A. P. Ramirez, W. Bao, and S.-W. Cheong,\nPhys. Rev. Lett. 75, 3336 (1995).\n[4] D. M. Edwards, Adv. Phys. 51, 1259 (2002).\n[5] E. O. Wollan and W. C. Koehler, Phys. Rev. 100, 545\n(1955).\n[6] J. B. Goodenough, Phys. Rev. 100, 564 (1955).\n[7] A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys.\nRev. 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Sci. 112, 10869 (2016)."    },    {        "title": "1508.04043v1.Increased_magnetic_damping_of_a_single_domain_wall_and_adjacent_magnetic_domains_detected_by_spin_torque_diode_in_a_nanostripe.pdf",        "content": "Increased magnetic damping of a single domain wall and adjacent magnetic domains\ndetected by spin torque diode in a nanostripe\nSteven Lequeux,1Joao Sampaio,1Paolo Bortolotti,1Thibaut Devolder,2Rie Matsumoto,3\nKay Yakushiji,3Hitoshi Kubota,3Akio Fukushima,3Shinji Yuasa,3Kazumasa\nNishimura,4Yoshinori Nagamine,4Koji Tsunekawa,4Vincent Cros,1and Julie Grollier1\n1Unit\u0013 e Mixte de Physique CNRS/Thales and Universit\u0013 e Paris-Sud 11, 1 Ave. A. Fresnel, 91767 Palaiseau, France.\n2Institut d'Electronique Fondamentale, univ. Paris-Sud,\nCNRS UMR 8622, b^ at. 220, 91405 Orsay Cedex France.\n3National Institute of Advanced Industrial Science and Technology (AIST),\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan.\n4Process Development Center, Canon ANELVA Corporation,\nKurigi 2-5-1, Asao, Kawasaki, Kanagawa 215-8550, Japan.\nWe use spin-torque resonance to probe simultaneously and separately the dynamics of a magnetic\ndomain wall and of magnetic domains in a nanostripe magnetic tunnel junction. Thanks to the\nlarge associated resistance variations we are able to analyze quantitatively the resonant properties\nof these single nanoscale magnetic objects. In particular, we \fnd that the magnetic damping of both\ndomains and domain walls is doubled compared to the damping value of their host magnetic layer.\nWe estimate the contributions to damping arising from dipolar couplings between the di\u000berent layers\nin the junction and from the intralayer spin pumping e\u000bect. We \fnd that they cannot explain the\nlarge damping enhancement that we observe. We conclude that the measured increased damping is\nintrinsic to large amplitudes excitations of spatially localized modes or solitons such as vibrating or\npropagating domain walls.\nThe spin torque diode e\u000bect provides an e\u000ecient\ntool to access the resonant properties of sub-micrometer\nmagneto-resistive structures [1]. In particular, contrar-\nily to conventional ferromagnetic resonance techniques,\nits sensitivity allows probing the dynamics of individ-\nual magnetic solitons, such as vortices or domain walls,\nwhich are foreseen as information vectors in next gener-\nation magnetic devices [2{9]. By injecting a microwave\ncurrent through the stack at resonance, spin torque can\ninduce large magnetization precession of the free layer.\nMagneto-resistance then converts this precession into re-\nsistance variations. These resistance oscillations, multi-\nplied by the microwave current oscillating at the same\nfrequency, give rise to a recti\fed dc voltage. Most spin\ntorque diode studies have focused on the dynamical prop-\nerties of uniform magnetization con\fgurations [10{14]\nand few on the vibration modes of magnetic solitons [2{\n8]. Moreover, the latter studies were performed with\nmetallic samples, where the magneto-resistance ratios\nare typically restricted to a few %. These low magneto-\nresistance ratios strongly limit the amplitude of the out-\nput dc signal. In the case of domain wall (DW) vibra-\ntions, generally con\fned and of limited amplitude, it is\ntherefore very di\u000ecult to extract other quantitative pa-\nrameters than the resonance frequency of the wall [2{8].\nIn this Letter, we take advantage of the large\ntunnel magneto-resistance ratios provided by MgO-\nbased magnetic tunnel junctions to probe the dy-\nnamics of a DW and its neighboring domains by\nspin torque diode e\u000bect. The junctions stack, illus-\ntrated in Fig. 1(a), SiO 2//bu\u000ber/PtMn (15)/Co 70Fe30\n(2.5)/Ru (0.9)/CoFeB (3)/MgO (1.1)/Ni 80Fe20(5)/ Ru(10) (thicknesses in nanometers), gives rise to tunnel\nmagneto-resistance ratios of about 30%. A synthetic an-\ntiferromagnet (SAF) prevents the formation of a domain\nwall in the CoFeB reference layer. The injection of a DW\nin the free layer is facilitated by the arc shape geometry\nof our junctions, shown in Fig. 1(b). After the saturation\nof the free layer magnetization in the y direction with a\nstrong external \feld (300 Oe), the DW is nucleated when\nthe \feld is decreased below 75 Oe. In Fig. 1(c), micro-\nmagnetic simulations of the full stack realized using the\nOOMMF software[15] show that the DW is \frst nucle-\nated in the center of the wire, then displaced along the\nwire when the external \feld is decreased due to the stray\n\feld from the SAF. This scenario is in very good agree-\nment with the experimental measurement of the junction\nresistance as a function of the decreasing \feld applied\nalong the y direction shown in Fig. 1(d). The bottom\nand top black dashed lines, given for reference, corre-\nspond respectively to the resistance of the parallel (P)\nand anti-parallel (AP) states. Each change of the resis-\ntance value corresponds to a displacement of a DW in the\nPermalloy free layer. The \frst resistance plateau of 130\n\n for the \feld range between 75 and 26 Oe (blue square\ndot in Fig. 1(d)) corresponds to the magnetic con\fgura-\ntion labeled 1 in Fig. 1(c). For lower \felds, between 20\nand -35 Oe, the DW shifts to a strong and reproducible\npinning site due to the shape at the edge of the wire [16].\nThe resulting resistance plateau at 122 \n in Fig. 1(d)\ncorresponds to the magnetic con\fguration labeled 2 in\nFig. 1(c). For negative \felds larger than -40 Oe, the DW\nis expelled of the wire and the resistance value reaches\nthe P state at 116 \n (orange square dot labeled 3 in Fig.arXiv:1508.04043v1  [cond-mat.mtrl-sci]  17 Aug 20152\n(a) (d) \nRu Py \nCo70Fe30 CoFeB  \nPtMn  MgO  Iac Vdc \nx z y \n100 nm  \nx y \n3 \n1 \n2 (b) \nRu \n(c) \n-150-100 -50 050110120130140\nP state\n  resistance ()\nmagnetic field, H y (Oe)AP state2 \n3 1 \nFIG. 1. (a) Schematic of the spin torque diode measure-\nment set-up including the MgO-based magnetic tunnel junc-\ntion with the DW in the Permalloy free layer. (b) Scanning\nelectron microscope image of the sample. (c) Full-stack mi-\ncromagnetic simulations representing the di\u000berent steps in the\nDW displacement. (d) Resistance versus magnetic \feld mea-\nsurements. Black dashed lines: resistance values of the P and\nAP states. Black line: evolution of the resistance when a DW\nis nucleated (e.g. blue square at 26 Oe) in the middle of the\nwire (1) then pinned in the right edge (2) until its expulsion\n(e.g. orange square at -85 Oe) (3).\n1(d), and magnetic con\fguration labeled 3 in Fig. 1(c)).\nWe probe the resonant properties of our system by spin\ntorque diode in the con\fgurations with and without a do-\nmain wall (respectively blue square labeled 1 and orange\nsquare labeled 3 in Fig.1 (b)). We inject a microwave\ncurrent Ihfbetween the top and bottom electrodes, and\nsweep the frequency. We repeat this measurement for\ndi\u000berent values of applied \feld H y. In the absence of dc\ncurrent, the recti\fed voltage V dchas two main compo-\nnents [17]:\nVdc=1\n2@2V\n@I2\nIhf(t)2\u000b\n+@2V\n@I@\u0012hIhf(t)\u0012(t)i (1)\nThe \frst term is a purely electrical background sig-\nnal due to the bias dependence of the tunnel junction\nresistance, from which we extract the exact value of the\ninjected microwave current in the junction. The second\nterm is the spin torque diode contribution to the recti\fed\nvoltage and arises from magnetization oscillations. The\nparameter\u0012depends on the excited mode. It is related\nto the precession angle for quasi-uniform magnetization\noscillations of the domains, and to the domain wall posi-\ntion for DW vibrations in a pinning potential.\nFig. 2(a) and (b) show the recti\fed voltage response\nnormalized by the square of the microwave current am-\nplitude; V dc/I2\nhf, at two di\u000berent magnetic \feld values.\nFig. 2(a) corresponds to a measurement at 26 Oe, per-\nformed at 130 \n (state '1' in Fig. 1(d)), where the DW\nis positioned in the middle of the wire. Fig. 2(b) shows a\nmeasurement at -85 Oe, where the magnetization of the\nfree layer has a quasi-uniform con\fguration correspond-\ning to the parallel state (state '3' in Fig. 1(d)). The 26\nOe response curve shows two resonance signals at 1.24\nGHz (mode 1) and 2.15 GHz (mode 2) while the -85 Oe\nresponse shows only one at 2.3 GHz (mode 2). We sys-\n0.0 0.5 1.0 1.5 2.0 2.5 3.0-5000\n  Fit\n  data\nH= -85 OeH= 26 Oe\n  \nFrequency (GHz)-5000\n  Fit\n  data\n \n  Nomalized Signal (V/A²)Mode 1  Mode 2  \nMode 2  (a) \n(b) \n3 1 FIG. 2. Normalized recti\fed voltage V dc/I2\nhfas a function\nof frequency (a) Black line: measurement at 26 Oe with the\nDW positioned in the middle of the wire corresponding to a\nresistance of 130 \n. Blue line: \ft with Eq.(2). (b) Black\nline: measured at -85 Oe in the parallel state (DW expelled).\nOrange line: \ft with Eq. (2).\ntematically observed two resonance signals in the \feld\nrange between 50 and -35 Oe where the DW is present\nwhereas only the higher frequency mode (mode 2) was\nobserved between -40 and -150 Oe where the DW is ex-\npelled. The low frequency signal (mode 1) can therefore\nbe ascribed to DW oscillations while the higher frequency\nsignal can be attributed to magnetization precessions in\nthe domains. When the DW is not present mode 2 is not\nstrongly modi\fed, only its resonance frequency is shifted\nby the external \feld applied to allow the DW expulsion.\nThis means that the strongest vibrations of this mode\nare spatially located far enough from the DW not to be\nimpacted by the DW vibrations when the DW is still\npresent.\nAround the resonance frequency f0, the normalized re-\nsponse curve takes the shape of a linear combination of\nLorentzian and anti-Lorentzian pro\fles [1, 14]. We \ft the\nresonant signals of the mode 1 (DW) and 2 (domains) by\nthe following expression:\nVdc\nI2\nhf=A(f2\n0\u0000f2) +Bf2\n(f2\n0\u0000f2)2+ (\u0001f)2+C (2)\nwhere fis the frequency of the microwave current I hf\nand the free parameters of the \ft are the amplitudes A3\nand B of the anti-Lorentzian and Lorentzian pro\fles, a\nconstant shift C, the resonance frequency f0, and the\nlinewidth \u0001. The average value of \u0001 extracted from the\nmeasurements by \ftting with Eq. (2) (Fig. 2) at di\u000berent\nmagnetic \felds is about 0.4 \u00060.02 GHz both for mode 1\n(DW) and 2 (domains). The magnetic damping parame-\nter\u000bis related to the linewidth [11, 12]. Here, in order to\nestablish the correlation between \u0001 and the correspond-\ning damping \u000bfor the DW and domains resonant signals\nwe perform micromagnetic simulations of the 5 nm Py\nfree layer at zero external \feld using the OOMMF soft-\nware [15, 18]. We \fnd a resonant response for the DW\nand domains respectively at 1.7 (Fig. 3(a)) and 2.75 GHz\n(Fig. 3(b)) in good agreement with the experimental re-\nsults. To reconstruct the spin diode signal we sweep the\nfrequency of the microwave current I hfbetween 0.9 and 3\nGHz and extract the resulting magnetization oscillations.\nWe plot as a function of the frequency the average am-\nplitude of the product between the x component of the\nmagnetization oscillations Mxand Ihf= I0.sin(2\u0019ft+\u001e)\n(Fig. 3). The \u001eparameter (respectively equal to\u0019\n2and\n0 in Figs. 3(a) and (b)) is related to the resonant signal\nshape depending on the symmetry of the exciting force\n(Slonczewski or Field like torque [1]) and does not in\ru-\nence the linewidth. For a damping parameter \u000b=0.01,\nthe linewidth extracted from the simulated spin diode\nsignal by \ftting with Eq.2 is 0.15 \u00060.002 GHz both for\nmode 1 and 2 (Fig. 3).\nFor the particular case of the DW resonance, in a 1D\nmodel, the linewidth \u0001 is related to the damping param-\neter\u000bby:\n\u0001 =\r0\u000bHd\n2\u0019(3)\nwhere\r0is the gyromagnetic ratio, \u0001 the linewidth\nin Hz, andHdthe demagnetizing \feld, that we calculate\nto be equal to 0.45 MA/m in our samples where the\nmagnetization of the free NiFe layer is 0.47 MA/m.\nBased on this 1D model (Eq. 3), the expected linewidth\nfrom a damping of 0.01 is 0.15, in good agreement with\nthe linewidth extracted from micromagnetic simulations.\nThis result indicates that the 1D model is relevant to\nevaluate the damping extracted from the experimental\nlinewidth measurements for the DW. Moreover, the\nfact that the 1D model also allows to describe the\ndomains damping hints once again to the fact that the\nmagnetization oscillations in the domains are spatially\nlocalized and con\fned. This is con\frmed by the spatial\ndistribution of the modes calculated using the mode\nsolver from the SpinFlow3D package [19, 20] where\nthe full stack is considered [21], shown in the insets\nof Fig. 3(a) and (b). Indeed, the most important\ndynamic component of magnetization in the domains\nis con\fned and located in the left edge of the wire\n(inset in Fig. 3(b)). This edge mode is associated\n1.0 1.5 2.0-20-1001020304050\n \n  \n2.4 2.6 2.8 3.0-20-1001020304050\n \n   simulated spin diode signal (µV)\nFrequency (GHz)\n(a) (b) \nMode 1  \nMode 2  FIG. 3. Spin diode signal reconstructed from micromagnetic\nsimulations of the Py free layer at zero external \feld. (a)\nBlack line: Reconstructed spin diode signal in the 0.9-2.2\nGHz frequency range, showing the resonant response of the\nDW (mode 1). Red line: \ft with Eq. (2). (b) Black line:\nReconstructed spin diode signal in the 2.4-3 GHz frequency\nrange showing the domains resonant response (mode 2). Red\nline: \ft with Eq. (2). Insets: Spatial distribution of the am-\nplitude of the magnetization oscillations dynamic component\nfrom the SpinFlow3D mode solver associated with the two\nobserved modes. The color scale blue-green-yellow-red corre-\nsponds to the amplitude of the mode. The white arrows show\nthe direction of the magnetization.\nto the resonant response of the mode 2. For the\nmode 1, the resonant response is associated to a transla-\ntional vibration mode of the DW (inset in Fig. 3(a)) [19].\nBased on the previous micromagnetic simulations and\nthe 1D model expression (Eq. 3), the linewidth extracted\nfrom the measurements corresponds to a damping \u000bof\n0.026\u00060.001 both for DW and domains magnetization\noscillations. This value of \u000bis more than twice the value\nthat we have measured by FMR on the unpatterned\n\flm (0.01\u00060.0015) and that is typically observed in Py\n\flms (0.01 [11]) (see supplementary material). For each\nof the \fve samples we have measured we have obtained\nsimilar large damping values for the vibration modes, in\nthe range 0.019 to 0.028.\nSeveral mechanisms can be involved in this large in-\ncrease of the e\u000bective damping compared to classical\nFMR measurements on thin \flms. We consider possi-\nble e\u000bects at stake and estimate their relative contri-\nbutions. Surface roughnesses or edge defects are often\npointed out [22] but it has been recently shown by mi-\ncromagnetic simulations that the resulting damping is an\norder of magnitude smaller than what we observe [23, 24].\nAnother potential source of strong damping increase,4\nespecially in the case of a DW, has been pointed out by\nNdjaka et al. [25]. Indeed, in trilayer structures, the stray\n\feld generated by DWs in one ferromagnetic layer dipo-\nlarly couples to the other ferromagnetic layer, leading,\nin the case of very strong couplings, to DW and domain\nduplication [26, 27]. In the case of weak coupling only\na local perturbation of the magnetization in the refer-\nence layer is observed [25]. This is what we expect in our\nmagnetic tunnel junctions as the SAF prevents domains\nduplication. This magnetic shadow in the reference layer\ncoupled with the DW dynamic in the upper one results in\nan increased damping. To con\frm and quantify this be-\nhavior, we have performed micromagnetic simulations of\nthe full stack [21] using the OOMMF software [15]. Figs.\n4(a) and (b) show respectively the magnetization of the\nPy free and the CoFeB reference layer. As expected, we\nobserve that the magnetization of the reference layer re-\nveals a non-uniformity located just below the DW of the\nfree layer. We extract the e\u000bective DW damping \u000bDW\nfrom the DW damped oscillations (inset of Fig. 4(c)) as\na function of \u000bref, the damping of the SAF layers (Fig.\n4(c)) (see supplementary material).\nIn the simulations, we set the damping of the free layer\nto 0.01, and vary the damping of the SAF layers between\n0 and 0.1. Figure 4 (c) shows the dependence of \u000bDW\non\u000breffor two values of the Ruderman-Kittel-Kasuya-\nYosida (RKKY) interlayer exchange coupling in the SAF:\n\u00001:10\u00004(\flled symbols) and \u00003:10\u00004(open symbols)\nJ/m2and for three values of the reference CoFeB layer\nmagnetization: 0.82 MA/m, 1.03 MA/m and 1.2 MA/m.\nIn each case, we observe an increase of \u000bDWwith\u000bref,\nthus con\frming the dynamic coupling between the DW\nand its magnetic shadow. However, in our experiments,\nthe typical values of the RKKY exchange coupling, \u000bref\nand Ms for the CoFeB reference layer are respectively\n-0.1 erg/cm2, 0.01 and 1.2 MA/m [28, 29]. As shown\nin Figure 4 (c), the corresponding \u000bDWis about 0.0115,\nmeaning that the DW damping increase due to dipolar\ncoupling is one order of magnitude smaller than what we\nobserve.\nAn alternative phenomena that can lead to strong\ndamping increases is intralayer transverse spin pumping\n[23, 24, 30, 31]. In a conducting ferromagnet, the\nconduction electrons carry away the excess angular\nmomentum of a precessing magnetization. For spatially\ninhomogeneous magnetization dynamics, a nonuniform\nspin current is induced, resulting in a spatial dependance\nof the dissipative \row of conduction electrons in the\nmagnetic layer itself. This can give rise to an enhanced\ndamping in isolated magnetic nanostructures. This\nintralayer spin current results in the following additional\nnonlocal torque in the Landau-Lifshitz-Gilbert equation\n[24]:\u001bT(\u0000 !m\u0002r2@\u0000 !m\n@t), where\u001bTis the transverse spin\nconductivity. In Ref. [30], the increase of damping\ndue to intralayer spin pumping is estimated by \u0011(2\u0019\n\u0015)2,\n RKKY= -3.10-4J/m²   \n RKKY= -1.10-4J/m²\n0.00 0.02 0.04 0.06 0.08 0.100.0100.0120.0140.016\n  MsREF=0.82 MA/m\n MsREF=1.03 MA/m\n MsREF=1.2 MA/meffective DW damping (DW)\nreference layer damping (ref) \n0.6 1 \n(c) \n(a) \n(b) \nFree \nRef \n-1 1 \nExp. \nparameters  FIG. 4. OOMMF micromagnetic simulations of the Permalloy\nfree layer with the DW (a) and the CoFeB reference layer (b).\n(c) Dependence of the DW e\u000bective damping on the reference\nlayers damping. The \flled (resp. open) symbols correspond\nto an RKKY exchange coupling in the SAF of \u00001:10\u00004J=m2\n(resp.\u00003:10\u00004J=m2). The value of the CoFeB reference layer\nmagnetization is varied between 0.82 MA/m, 1.03 MA/m and\n1.2 MA/m. Inset: The black solid line is the Mxcomponent\nof the NiFe layer, describing the damped oscillations of the\nDW. The red solid line is the \ft (see supplementary material).\nwhere\u0011=g\u0016B~G0\n4e2MS, is a material dependent parameter\nrelated to \u001bTof Ref. [24]. The parameter \u0015is the\nwavelength for the magnetization pattern we consider.\nThis means that the more magnetization dynamics is\nspatially located the smaller is \u0015, and the more the\nincrease of damping is important. The spatial pro\fle of\nthe calculated modes shown in inset of Fig. 3 can give\nus a rough estimate for \u0015, typically 100 nm for both\nDW and domains. By taking the parameter \u0011equal to\n0.76 nm2for Permalloy with MS=0.47 MA/m and the\nconductivity G0= (5\u0016\ncm)\u00001[30], we \fnd that the\nassociated damping increase is 0.003. Recent analytical\ncalculations con\frm that for transverse domain walls\ndamping increases due to intralayer spin pumping are\nvery small [32].\nFinally, spin transfer torque can give rise to large am-\nplitude motion of magnetic objects. In our experiments,\nthe injected microwave current density is 1.43 105A/cm2,\nabout 10 times smaller than the critical current densi-\nties needed to depin the domain wall [16]. From the5\nmeasured recti\fed voltages in Fig. 2 we \fnd that the\namplitude of domain wall translation at resonance is 4\nnm, a value far from negligible compared to the domain\nwall width of 100 nm. It has been shown experimentally\nthat non-linear contributions, that are not accounted for\nin micromagnetic simulations, can give rise to very large\ndamping increases in nanostructures [33]. By elimination\nit seems that only these non-linear contributions appear-\ning for wide vibration amplitudes can be responsible for\nthe very large damping values that we measure. Our\nresults are in phase with the high damping values gen-\nerally derived from magnetic \feld or spin torque driven\ndomain wall motion over large distances [34]. They also\nshow that the damping of simple objects like transverse\ndomain wall in a standard material such as Permalloy are\nstill far from being understood.\nAs a conclusion, we have shown that by combining the\nspin diode e\u000bect with the large magneto-resistance ratios\nprovided by magnetic tunnel junctions, we can quan-\ntitatively analyze the resonance of individual magnetic\ncon\fned modes. The e\u000bective damping of the domain\nwall and edge modes extracted from the recti\fed spin\ndiode signal is more than twice increased compared to\nthe one extracted from FMR measurements on the ex-\ntended stack. We ascribe our observations to non-linear\ncontributions to the damping appearing for large ampli-\ntudes of vibration. Our results underline that damping\nmechanisms are still far from being elucidated in simple\nsystems. 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Klaeui, Materials Science\nand Engineering: R: Reports 72, 159 (2011)"    },    {        "title": "1509.00877v1.Energy_Dependence_of_Synchrotron_X_Ray_Rims_in_Tycho_s_Supernova_Remnant.pdf",        "content": "Accepted, ApJ, September 2, 2015\nPreprint typeset using L ATEX style emulateapj v. 5/2/11\nENERGY DEPENDENCE OF SYNCHROTRON X-RAY RIMS IN TYCHO'S SUPERNOVA REMNANT\nAaron Tran ( sÕ)1,4, Brian J. Williams1,5, Robert Petre1, Sean M. Ressler2, Stephen P. Reynolds3\n1X-ray Astrophysics Laboratory, NASA/GSFC, Code 662, Greenbelt, MD 20771, USA\n2Dept. Physics, University of California, Berkeley, CA 94720, USA\n3Dept. Physics, North Carolina State University, Raleigh, NC 27695, USA\nAccepted, ApJ, September 2, 2015\nABSTRACT\nSeveral young supernova remnants exhibit thin X-ray bright rims of synchrotron radiation at their\nforward shocks. Thin rims require strong magnetic \feld ampli\fcation beyond simple shock compres-\nsion if rim widths are only limited by electron energy losses. But, magnetic \feld damping behind\nthe shock could produce similarly thin rims with less extreme \feld ampli\fcation. Variation of rim\nwidth with energy may thus discriminate between competing in\ruences on rim widths. We measured\nrim widths around Tycho's supernova remnant in 5 energy bands using an archival 750 ks Chandra\nobservation. Rims narrow with increasing energy and are well described by either loss-limited or\ndamped scenarios, so X-ray rim width-energy dependence does not uniquely specify a model. But,\nradio counterparts to thin rims are not loss-limited and better re\rect magnetic \feld structure. Joint\nradio and X-ray modeling favors magnetic damping in Tycho's SNR with damping lengths \u00181{5% of\nremnant radius and magnetic \feld strengths \u001850{400\u0016G assuming Bohm di\u000busion. X-ray rim widths\nare\u00181% of remnant radius, somewhat smaller than inferred damping lengths. Electron energy losses\nare important in all models of X-ray rims, suggesting that the distinction between loss-limited and\ndamped models is blurred in soft X-rays. All loss-limited and damping models require magnetic \felds\n&20\u0016G, a\u000erming the necessity of magnetic \feld ampli\fcation beyond simple compression.\nSubject headings: acceleration of particles | ISM: individual objects (Tycho's SNR) | ISM: magnetic\n\felds | ISM: supernova remnants | shock waves | X-rays: ISM\n1.INTRODUCTION\nElectrons accelerated in the forward shocks of young\nsupernova remnants (SNRs) emit synchrotron radiation\nstrongly in the shock's immediate wake at radio wave-\nlengths and sometimes in X-rays. In a few cases, they\nquickly turn o\u000b downstream, producing a shell-like mor-\nphology of bright X-ray and radio rims/\flaments due to\nline-of-sight projection (Bamba et al. 2003; Reynoso et al.\n1997). Strong and time-variable synchrotron radiation\n(e.g., Uchiyama et al. 2007; Patnaude & Fesen 2007),\nin conjunction with multiwavelength spectral modeling\n(Aharonian et al. 2004; Acero et al. 2010; Ackermann\net al. 2013), suggests that electrons are accelerated to\nTeV energies in young SNRs. Although synchrotron\nemission due to accelerated electrons does not require ac-\nceleration of an unseen hadronic component, the prevail-\ning theory of di\u000busive shock acceleration (DSA) should\noperate on both positive ions and electrons. E\u000ecient\nhadron acceleration in supernova remnant shocks is a\nprime candidate source for galactic cosmic rays up to\nthe cosmic ray spectrum's \\knee\" at around 3 PeV (Vink\n2012). But many fundamental questions about shock ac-\nceleration remain unanswered. Under what conditions\ndo shocks accelerate particles e\u000eciently? How are mag-\nnetic \felds ampli\fed in such shocks? Reynolds (2008) re-\nviews relevant observations and open questions to date.\nThese questions are relevant to many astrophysical set-\ntings, such as Earth's bow shock (Ellison et al. 1990),\nstarburst galaxies (Heckman et al. 1990), jets of active\ngalactic nuclei (Chen et al. 2014), galaxy clusters (van\n4CRESST/University of Maryland, College Park, MD 20742\n5CRESST/Universities Space Research AssociationWeeren et al. 2010), and cosmological shocks (Ryu et al.\n2008).\nSpectral and spatial measurements of synchrotron rims\ncan constrain downstream magnetic \feld strength and\nstructure. If rim widths are set by electron energy losses,\npost-shock magnetic \felds must be ampli\fed to \u0018102\u0016G\nto account for the thinness of observed rims (Vink &\nLaming 2003; Bamba et al. 2003, 2005; V olk et al. 2005;\nParizot et al. 2006). In these models, the magnetic \feld\nis assumed advected downstream and nearly constant\nover rim widths. Alternately, the magnetic \feld strength\nmay be damped downstream of the shock and prevent\nelectrons from radiating e\u000eciently, so that thin rims re-\n\rect magnetic \feld variation rather than e\u000ecient par-\nticle acceleration and synchrotron cooling (Pohl et al.\n2005). Damping, in particular, may permit less extreme\nmagnetic \feld ampli\fcation. We refer to these as \\loss-\nlimited\" and \\damped\" models for rim widths. We shall\nsee that models can range continuously between these\ntwo cases, and that the distinction between the two can\nvary with observing frequency.\nThe possibility of damping in SNR shocks has not been\nfully tested. Marcowith & Casse (2010) compared phys-\nically motivated magnetic damping models to X-ray rim\nwidths and synchrotron spectrum cut-o\u000bs and thus sug-\ngested that only young SNRs (age .500 yr) can exhibit\nmagnetic damping in conjunction with e\u000ecient particle\nacceleration. Rettig & Pohl (2012) gave model predic-\ntions for several historical SNRs and proposed discrim-\nination based on \flament spectra { the expectation is\nthat damped spectra are softer, loss-limited harder.\nHydrodynamic models can reproduce X-ray rim pro-\n\fles reasonably well with both loss-limited and dampedarXiv:1509.00877v1  [astro-ph.HE]  2 Sep 20152 Tran et al. (June 4, 2021)\nmagnetic \feld models (Cassam-Chena \u0010 et al. 2007; Mor-\nlino & Caprioli 2012; Slane et al. 2014). However, loss-\nlimited models generally cannot reproduce thin radio\nrims. Radio-emitting GeV electrons do not lose substan-\ntial energy via radiation, so modeled intensities rise to a\nbroad maximum toward the remnant interior, then grad-\nually drop due to sphericity e\u000bects as the density drops\nin the interior. Reynolds (1988) empirically modeled ra-\ndio rims in the remnant of SN 1006 and concluded that\nsuch thin rims required sharp gradients in electron energy\ndensity (from, e.g., time-variable particle acceleration)\nor magnetic \feld strength. Cassam-Chena \u0010 et al. (2007)\nused a 1-D hydrodynamic model with nonlinear DSA to\njointly model radio and X-ray rims in Tycho's supernova\nremnant; neither loss-limited nor damped models could\nmatch radio pro\fles and radio/X-ray intensities simul-\ntaneously. Moreover, gradual X-ray spectral variation\nobserved downstream of the shock front was poorly re-\nproduced with both models (Cassam-Chena \u0010 et al. 2007).\nBut damped models were able generate limb-brightened\nradio rims at the forward shock, even if morphology was\nnot entirely consistent with observation. Cassam-Chena \u0010\net al. (2007) thus suggested that some combination of\nampli\fcation and magnetic \feld variation might explain\nradio morphology.\nRecently, Ressler et al. (2014) (hereafter, R14) sought\nto discriminate between damped and loss-limited rims by\nmeasuring rim width-energy dependence in X-ray ener-\ngies in SN 1006. In the simplest models, rim widths are\nexpected to be roughly energy-independent if rims are\ndamped, whereas widths should narrow with increasing\nenergy if rims are energy loss-limited. R14 included a\nvariety of e\u000bects which can blur this distinction. To fur-\nther test these models, we follow R14 by measuring X-\nray rim widths at multiple energies in Tycho's supernova\nremnant (hereafter, Tycho). Tycho exhibits an exten-\nsive shell of synchrotron-dominated thin rims around its\nperiphery (Figure 1); the rims show very little thermal\nemission, consistent with expansion into a low density\nISM (Williams et al. 2013). A deep 750 ks exposure of\nthe entire remnant from 2009 allows \fne sampling of the\nremnant rims. We build upon previous estimates of mag-\nnetic \feld strength and particle di\u000busion in Tycho that\ndraw from multiwavelength observations and various as-\nsumptions on CR acceleration (V olk et al. 2002, 2005;\nParizot et al. 2006; Morlino & Caprioli 2012; Rettig &\nPohl 2012, e.g.,).\nOur procedure closely follows that of R14. We \frst\nreview the model of R14 used to model rim pro\fles and\nwidths, then describe the procedure for selecting, mea-\nsuring, and \ftting rim widths to model width-energy de-\npendence. We explore degeneracies in model \ftting and\nconsider radio rim morphology as an additional discrim-\ninant between models.\n2.NONTHERMAL RIM MODELING\n2.1. Particle transport\nThe energy and space distribution of electrons at a\nsupernova remnant's forward shock controls the syn-\nchrotron rims we see in X-ray and radio. We assume\nthat di\u000busive shock acceleration generates a power law\ndistribution of electrons with an exponential cut-o\u000b at\nthe forward shock and model 1-D steady-state plane ad-\n0h24m40s25m00s20s40s26m00s\nRA (J2000)+64◦04/prime06/prime08/prime10/prime12/primeDec (J2000)\n116\n2345678910111213141517181920Figure 1. RGB image of Tycho with region selections overlaid.\nImage bands are 0.7{1 keV (red), 1{2 keV (green) and 2{7 keV\n(blue). Bold region labels (1, 16) indicate region selections shown\nin Figures 3, 4. Filament 1: Regions 1{3, \flament 2: regions 4{10,\n\flament 3: regions 11-13, \flament 4: regions 14{17, \flament 5,\nregions 18{20.\nvection and di\u000busion of the electron distribution f(E;x),\nwhereEis electron energy and xis distance downstream\nof the forward shock:\nvd@f\n@x\u0000@\n@x\u0012\nD@f\n@x\u0013\n\u0000@\n@E\u0000\nbB2E2f\u0001\n=K0E\u0000se\u0000E=E cut\u000e(x);\n(1)\nfollowing Berezhko & V olk (2004); Cassam-Chena \u0010 et al.\n(2007); Morlino et al. (2010); Rettig & Pohl (2012). The\nforward shock is located at x= 0 withx > 0 increas-\ning downstream of the shock; Dis the di\u000busion coe\u000e-\ncient,vdis \ruid velocity downstream of the shock, and\nthe constant b\u00114e4=9m4\nec7= 1:57\u000210\u00003in appropri-\nate CGS units arises from synchrotron power loss (i.e.,\n@E=@t =\u0000bB2E2, averaged over pitch angles). The ini-\ntial electron distribution is speci\fed by an arbitrary nor-\nmalization K0, DSA cut-o\u000b energy Ecut(given in Sec-\ntion 2.3), and spectral index s= 2\u000b+ 1. Zirakashvili\n& Aharonian (2007) derive an electron energy spectrum\nwith super-exponential cut-o\u000b e\u0000(p=pcut)2, but we use a\nsimple exponential cut-o\u000b for simplicity and consistency\nwith R14. We have not yet speci\fed the cause of the\ninjected spectrum cut-o\u000b (see Section 2.4), but the func-\ntional form of a power law with exponential cut-o\u000b is a\ngood approximation to predictions for DSA spectra lim-\nited by synchrotron losses, remnant age, or particle es-\ncape (Webb et al. 1984; Reynolds 1998, 2008). All con-\nstants and equations are given in CGS (Gaussian) units.\nOur presentation is somewhat abbreviated, but a fuller\nexposition and literature review are given by R14.\nFor Tycho, we adopted radio spectral index \u000b= 0:58\n(Sun et al. 2011) and hence electron spectral index\ns= 2\u000b+ 1 = 2:16. We assumed a remnant distance\n3 kpc (but cf. Hayato et al. 2010), which gives a shock\nradius of 1:08\u00021019cm from the observed angular radius\n24000(Green 2014) and further sets shock velocities for\neach rim pro\fle we consider. Tycho's forward shock ve-\nlocity varies with azimuth by up to a factor of 2 (KatsudaTycho's Synchrotron Rims (June 4, 2021) 3\net al. 2008); we linearly interpolated velocities reported\nby Williams et al. (2013) (rescaled to 3 kpc) to estimate\nindividual shock velocities for each region. We assume a\ncompression ratio of 4 as for a strong shock (unmodi\fed\nby cosmic-ray pressure), and take downstream velocities\nvdto be one-fourth the interpolated shock velocities.\nWe assume isotropic di\u000busion and only consider par-\nticle transport downstream of the forward shock. The\nvelocity is assumed constant, as is the magnetic \feld\nfor loss-limited model rims, in contrast to the expected\nSedov-Taylor similarity solution for velocity and density\nin an adiabatic blastwave. Our assumptions of constant\nvelocity, magnetic \feld, and plane \row should be reason-\nable as we generally consider synchrotron emission within\n10% of the shock radius, rs, from the forward shock,\nthough a few models are followed far enough inward that\nthis approximation begins to break down. We consider\npro\fle emission strictly upstream of both the contact dis-\ncontinuity and reverse shock (cf. Warren et al. 2005),\nand avoid regions where the contact discontinuity over-\nruns the forward shock. Our modeling neglects \row and\nelectron spectrum modi\fcation due to the nearby con-\ntact discontinuity and particle acceleration at the reverse\nshock. These e\u000bects could matter (and we do not quan-\ntify their importance), but we expect that at Tycho's age,\nX-ray emission is dominated by forward shock transport.\nGiven the uncertainty in shock morphology, our model\nseeks only to capture the most relevant physics. More\nsophisticated work may treat, e.g., sphericity, shock pre-\ncursors, anisotropic di\u000busion, and injection/acceleration\ne\u000eciency (e.g., Reville & Bell 2013; Bykov et al. 2014;\nFerrand et al. 2014, and references therein).\nTo determine rim pro\fles and widths, we compute the\nelectron distribution using Green's function solutions by\nLerche & Schlickeiser (1980) and Rettig & Pohl (2012),\nwith the caveat that D(x)B2(x) is assumed constant;\nwe discuss this assumption further below. The solutions\nare fully described in R14 using notation similar to ours.\nThe electron distribution may be integrated over the one-\nparticle synchrotron emissivity G(y) to obtain the \\to-\ntal\" emissivity:\nj\u0017(x)/Z1\n0G(y)f(E;x)dE (2)\nwherey\u0011\u0017=(c1E2B) is a scaled synchrotron frequency\nandG(y) =yR1\nyK5=3(z)dzwithK5=3(z) a modi\fed\nBessel function of the second kind (Pacholczyk 1970);\nthe constant c1= 6:27\u00021018in CGS units. Integrat-\ning emissivity over lines of sight for a spherical remnant\nyields intensity as a function of radial coordinate r:\nI\u0017(r) = 2Zp\nr2s\u0000r2\n0j\u0017\u0010\nrs\u0000p\ns2+r2\u0011\nds (3)\nwheresis the line-of-sight coordinate and rsis shock ra-\ndius. We take the full width at half maximum (FWHM)\nof the resulting intensity pro\fle as our metric for modeled\nrim widths. Using FWHM as opposed to, e.g., full width\nat three-quarters maximum, excludes measured rims and\nmodel parameters where X-ray intensity does not drop to\nhalf maximum immediately behind the rim. Our results\nthus focus on the most well-de\fned rims in Tycho rather\nthan the global shock structure.2.2. Magnetic \felds and damping\nWe consider two scenarios for post-shock magnetic\n\feld: (1) a constant \feld B(x) =B0corresponding to\nloss-limited rims, and (2) an exponentially damped \feld\nof form:\nB(x) = (B0\u0000Bmin) exp (\u0000x=ab) +Bmin (4)\nfollowing (Pohl et al. 2005). Here B0is the magnetic \feld\nimmediately downstream of the shock, i.e. B0=B(x=\n0), andabis ane-folding damping lengthscale; our use of\nB0for downstream magnetic \feld departs from typical\nnotation. A typical lengthscale for abis 1016to 1017cm\n(Pohl et al. 2005), corresponding to \u00180:1{1% of Tycho's\nradius. Hereafter, we report abin units of shock radius\nrsunless otherwise stated.\nThe distinction between damped and loss-limited rims\nis somewhat arbitrary; as ab!1 , model results con-\nverge to loss-limited rims. Moreover, rims much thin-\nner than the damping length are e\u000bectively loss-limited\nas electrons radiate in a nearly constant magnetic \feld.\nFurthermore, the dependence on observing frequency of\nelectron losses and di\u000busion means that a model may be\ndamping limited at one frequency and loss-limited at an-\nother. In the following analysis, we deem all \fts with\n\fniteabto be damped, but we compare rim widths and\ndamping lengths from such \fts further below to better\ndistinguish damped and loss-limited rim behavior, set by\na combination of ab,B0, and other model parameters.\n2.3. Di\u000busion coe\u000ecient\nMost previous work has assumed Bohm-like di\u000busion\nin plasma downstream of SNR shocks. Bohm di\u000busion\nassumes that the particle mean free path \u0015is equal to\nthe gyroradius rg=E=(eB), yielding di\u000busion coe\u000ecient\nDB=\u0015c=3 =cE=(3eB); herecis the speed of light, E\nis particle energy, eis the elementary charge, and Bis\nmagnetic \feld. Bohm-like di\u000busion encapsulates di\u000bu-\nsion scalings of D/E, introducing a free prefactor \u0011\nsuch that\u0015=\u0011rgallows for varying di\u000busion strength.\nHowever, Bohm di\u000busion at \u0011= 1 is commonly con-\nsidered a lower limit on the di\u000busion coe\u000ecient at all\nenergies.\nWe consider a generalized di\u000busion coe\u000ecient with ar-\nbitrary power law dependence upon energy following,\ne.g., Parizot et al. (2006):\nD(E) =\u0011cE\u0016\n3eB=\u0011hDB(Eh)\u0012E\nEh\u0013\u0016\n(5)\nwhere\u0016parameterizes di\u000busion-energy scaling and \u0011now\nhas units of erg1\u0000\u0016. The right-hand side of equation (5)\nintroduces\u0011h, a dimensionless di\u000busion coe\u000ecient scaled\nto the Bohm value at a \fducial particle energy Eh. Note\nthat\u0011hand\u0011are related as \u0011=\u0011h(Eh)1\u0000\u0016, and\u0011=\u0011h\nfor Bohm-like di\u000busion ( \u0016= 1). For subsequent analy-\nsis, we take \fducial electron energy E2=Ehcorrespond-\ning to a 2 keV synchrotron photon and report results in\nterms of\u00112=\u0011h. Although E2varies with magnetic \feld\nasE2/B\u00001=2and thus\u00112may vary around Tycho's\nshock for\u00166= 1, tying \u0011hto a \fxed observation energy\ngives a convenient sense of di\u000busion strength regardless\nof the underlying electron energies.\nThe solutions to equation (1) given by Lerche &\nSchlickeiser (1980) assume D(x)B2(x) constant to ren-4 Tran et al. (June 4, 2021)\nder equation (1) semi-analytically tractable. Although\nthis assumption has no obvious physical basis, it con-\ntains the qualitatively correct behavior Dconstant if B\nis constant and Dsmaller for larger B. We enforce it by\nmodifying the di\u000busion coe\u000ecient in the damping model\nas, following Rettig & Pohl (2012):\nD(E;x) =\u0011cE\u0016\n3eB0\u0014Bmin\nB0+B0\u0000Bmin\nB0e\u0000x=ab\u0015\u00002\n:(6)\nThis strengthens the spatial-dependence of the di\u000busion\ncoe\u000ecient as compared to the expected D(x)/1=B(x).\n2.4. Electron energy cut-o\u000b\nWe assume that the DSA process is limited by syn-\nchrotron losses at high energies and hence determine\ntheEcutby equating synchrotron loss and di\u000busive ac-\nceleration timescales. Here we ignore the possibility of\nage-limited or escape-limited acceleration (e.g., Reynolds\n1998). In the former case, low magnetic-\feld strengths\ncould mean that Tycho's age is less than a synchrotron\nloss time; the maximum energy is obtained by equating\nthe remnant age and acceleration timescale. In the lat-\nter case, the di\u000busion coe\u000ecient upstream may increase\nsubstantially above some electron energy due to an ab-\nsence of appropriate MHD waves. However, the mag-\nnetic \feld strengths we \fnd below justify the assumption\nof loss-limited acceleration. For low energies and small\nsynchrotron losses (cooling time longer than acceleration\ntime), electrons are e\u000eciently accelerated; near or above\nthe cut-o\u000b energy, electrons will radiate or escape too\nrapidly to be accelerated to higher energies and the en-\nergy spectrum drops o\u000b steeply. The cut-o\u000b energy is\ngiven as:\nEcut= (8:3 TeV)2=(1+\u0016)\u0012B0\n100\u0016G\u0013\u00001=(1+\u0016)\n\u0002\u0010vs\n108cm s\u00001\u00112=(1+\u0016)\n\u0011\u00001=(1+\u0016): (7)\nThis result is derived by Parizot et al. (2006) for \u0016= 1\nassuming a strong shock with compression ratio 4 and\nisotropic magnetic turbulence both upstream and down-\nstream of the shock.\nThe cut-o\u000b energy and accelerated electron spectrum\ndepend on the magnetic \feld, which varies in a damped\nmodel. Marcowith & Casse (2010) show that various\nturbulent damping mechanisms can modify the acceler-\nated spectrum, as particles traveling downstream may\nnot be e\u000bectively re\rected back across the shock and\nfurther accelerated. Nevertheless, we make the simpli-\nfying assumption that particle acceleration is controlled\nby di\u000busion at the shock and neglect spatially varying\ndi\u000busion and magnetic \felds in the acceleration process;\nequation (7) stands as evaluated with shock magnetic\n\feld strength B0. Cut-o\u000b energies of \u00181{10 TeV can\nbe plausibly achieved in the presence of damping due to\nAlfv\u0013 en and magneto-sonic cascades (Marcowith & Casse\n2010).\nAs the DSA imposed electron cut-o\u000b results in a cut-\no\u000b of SNR synchrotron \rux, the synchrotron cut-o\u000b fre-\nquency\u0017cut=cmE2\ncutBwithcm= 1:82\u00021018in cgs units\n(e.g. Pacholczyk 1970), which is the peak frequency emit-\nted by electrons of energy Ecut, provides an independentobservable to estimate shock di\u000busion and is given by:\n\u0017cut=cm(13:3 erg)4\n1+\u0016(100\u0016G)\u0000\n2657 erg2\u0001\u00001\u0000\u0016\n1+\u0016\n\u0002\u0012vs\n108cm=s\u00134\n1+\u0016\n(\u00112)\u00002\n1+\u0016: (8)\nThe cut-o\u000b frequency is independent of magnetic \feld\nBfor all values of \u0016(but recall that the electron en-\nergies associated with \u00112will depend on Bfor\u00166= 1).\nParizot et al. (2006) previously used measurements of\nsynchrotron cut-o\u000bs to estimate di\u000busion coe\u000ecients in\nTycho and other historical supernova remnants.\nWe point out that the electron spectrum softens down-\nstream of the shock due to synchrotron losses, but the\nlocal spectrum at any radial position will be a steeply\ncut-o\u000b power law (Webb et al. 1984; Reynolds 1998). No\nsteepening from E\u0000stoE\u0000(s+1)is observed because the\ncut-o\u000b limits the electron spectrum at high energies. A\nhomogeneous source of age tin which electrons are con-\ntinuously accelerated throughout, with an initial straight\npower law distribution to in\fnite energy, will produce a\nsteepened power law distribution above the energy at\nwhich the synchrotron loss time equals the acceleration\ntime (Kardashev 1962). If we model emission without\nan initial exponential cut-o\u000b (i.e., inject a straight power\nlaw) in a constant magnetic \feld, then integrated spectra\nin our model would steepen by about one power. But,\nthose assumptions do not apply to the current situation\nof continuous advection of electrons, with an energy dis-\ntribution which is already a cut-o\u000b power law, through\na region of non-constant magnetic \feld. See Reynolds\n(2009) for a fuller discussion of synchrotron losses in non-\nhomogeneous sources.\n2.5. Rim width-energy dependence\nX-ray rim widths are controlled by synchrotron losses,\nparticle transport, and magnetic \felds immediately\ndownstream of the shock, each of which in\ruences rim\nwidth-energy scaling di\u000berently. If di\u000busion is negligi-\nble and the downstream magnetic \feld is constant, loss-\nlimited rims narrow with increasing energy as more ener-\ngetic electrons radiate and cool more quickly. But di\u000bu-\nsion will dilute this e\u000bect: more energetic electrons may\ndi\u000buse further upstream or downstream than would be\nexpected from pure advection, smearing out rims and\nweakening energy dependence at higher energies. Mag-\nnetic \felds damped on a length scale comparable to \fla-\nment widths should also weaken rim width-energy depen-\ndence { if the magnetic \feld turns o\u000b, synchrotron radia-\ntion turns o\u000b regardless of electron energy. Additionally,\nonceB0varies, one observes electrons of di\u000berent energy\nat di\u000berent distances behind the shock.\nFigure 2 plots model X-ray and radio pro\fles for a\nrange ofB0andabvalues to illustrate how damping and\nmagnetic \feld strength impact width-energy dependence,\nwhich we will now explore. We discuss and incorporate\nmodel radio pro\fles into our analysis in Section 6.\n2.5.1. Undamped models\nFollowing R14, we parameterize rim width-energy de-\npendence in terms of a scaling exponent mEde\fned as:\nw(\u0017)/\u0017mE(9)Tycho's Synchrotron Rims (June 4, 2021) 5\n0.00.20.40.60.81.0IntensityB0=20µG\nab=∞\n(loss-limited)B0=30µG B0=100µG B0=400µG\n0.00.20.40.60.81.0Intensityab=0.050\n0.00.20.40.60.81.0Intensityab=0.020\n0.00.20.40.60.81.0Intensityab=0.005\n215 220 225 230 235 240\nRadial position (arcsec.)0.00.20.40.60.81.0Intensityab=0.002\n215 220 225 230 235 240\nRadial position (arcsec.)215 220 225 230 235 240\nRadial position (arcsec.)215 220 225 230 235 240\nRadial position (arcsec.)1.3750 GHz\n0.7 keV\n2 keV\n4 keV\nFigure 2. Radio and X-ray rim pro\fles for a range of magnetic \felds ( B0) and damping lengths ( ab, shaded regions) with \u0016= 1,\u00112= 1,\nandBmin= 5\u0016G \fxed. X-ray rim energies (0.7, 2, 4 keV) are representative of energy bands used in our rim width measurements. This\nplot summarizes several key features of our model: (1) radio pro\fles (dashed red) are strongly a\u000bected by damping for all values of B0\nbecause synchrotron losses are negligible at low electron energies. (2) X-ray pro\fles (solid green, black, blue) are in\ruenced by synchrotron\nlosses even in the presence of strong damping, which can be seen going from left to right (increasing B0). (3) When X-ray rim widths are\nsmaller than ab, \feld damping only weakly a\u000bects rim widths and width-energy dependence (top right panels). (4) Strongly damped X-ray\nrims can show signi\fcant width-energy dependence in our model due to synchrotron losses beyond ab. This occurs for smaller B0where\nthe contrast between B0andBminis less extreme (bottom left panels).\nwherew(\u0017) is \flament FWHM as a function of observed\nphoton frequency \u0017, and the exponent mE=mE(\u0017) is\nenergy dependent. We may refer to observed photons\ninterchangeably by energy or frequency \u0017, butEis re-\nserved for electron energy.\nTo better intuit the e\u000bects of advection and di\u000bu-\nsion on rim widths, we introduce advective and di\u000busive\nlengthscales for bulk electron transport. These depend\non electron energy; we write them in terms of the peak\nfrequency radiated by electrons of energy E,\u0017=cmE2B.\nlad=vd\u001csynch/vdB\u00003=2\n0\u0017\u00001=2(10)\nldi\u000b=p\nD\u001csynch/\u00111=2B\u0000(\u0016+5)=4\n0\u0017(\u0016\u00001)=4(11)\nThe characteristic time is the synchrotron cooling time\n\u001csynch = 1=(bB2E) withb= 1:57\u000210\u00003. For\u0016= 1,ldi\u000b\nis independent of \u0017and bothldi\u000bandladscale asB\u00003=2\n0.\nIf both di\u000busion and magnetic \feld damping are negli-\ngible and electrons are only loss-limited as they advectdownstream, mEattains a minimum value mE=\u00001=2\nas rim widths are set by lad> ldi\u000b. At higher energies\nwhereldi\u000b>lad, di\u000busion increases mEfrom\u00001=2 to a\nvalue between\u00001=4 and 1=4 for\u0016= 0 and 2 respectively\n(R14, Figure 3). The presence of an electron energy cut-\no\u000b decreases mEslightly in all cases due to the decreased\nnumber of electrons and hence thinner rims at higher en-\nergies (R14, Figure 5), but the qualitative behavior is the\nsame.\n2.5.2. Field damping e\u000bects\nWe expect magnetic damping to produce compara-\ntively energy-independent rim widths. If synchrotron\nrim widths are set by magnetic damping at some ob-\nservation energy, then rims will be damped at all lower\nobservation energies as well. Then rim widths will be\nrelatively constant (small jmEj) below a threshold en-\nergy and may decrease, or even increase once advection\nand/or di\u000busion control rim widths at higher photon en-6 Tran et al. (June 4, 2021)\nergies (advection: lad< ab; di\u000busion: ldi\u000b> lad;ab).\nThus, we intuit that rim widths should roughly scale\nasw\u0018min (ab;max (lad;ldi\u000b)). This is correct except\nfor one key region of parameter space: strong damp-\ning with weak magnetic \feld ampli\fcation, where syn-\nchrotron losses downstream of the FWHM create energy\ndependent widths even when ab\u001clad;ldi\u000b.\nThis counter-intuitive energy dependence for strongly\ndamped models occurs when rim brightness remains\nabove half-maximum within \u0018abof the shock. Farther\ndownstream, synchrotron losses in the reduced magnetic\n\feld\u0018Bmindrive intensity down to half-maximum; losses\nin a nearly constant \feld cause energy dependence de-\nspite magnetic damping at the shock. Synchrotron losses\nfar from the shock will only a\u000bect the spectrum at higher\nfrequencies; at lower frequencies, the spectrum will be\ne\u000bectively constant after the magnetic \feld's initial de-\ncay. Thus, there will be a characteristic frequency be-\nlow which the FWHM ceases to be de\fned. The energy\ndependence of rim FWHMs becomes large and diverges\nnear this frequency. The characteristic frequency and\nvalue ofmEare sensitive to our de\fnition of rim width\n(FWHM), but the physical behavior we describe (diver-\ngence ofmEat characteristic frequency, for appropriate\ndamping parameters) will occur regardless of our de\fni-\ntion of rim width.\n3.OBSERVATIONS\n3.1. Data and region selections\nWe measured synchrotron rim full widths at half max-\nimum (FWHMs) from an archival Chandra ACIS-I ob-\nservation of Tycho (RA: 00h25m19:s0, dec: +64\u000e08010:000;\nJ2000) between 2009 Apr 11 and 2009 May 5 (PI: J.\nHughes; ObsIDs: 10093{10097, 10902{10906); Eriksen\net al. (2011) present additional observation information.\nThe total exposure time was 734 ks. Level 1 Chan-\ndradata were reprocessed with CIAO 4.6 and CALDB\n4.6.1.1 and kept unbinned with ACIS spatial resolution\n0:49200. Merged and corrected events were divided into\n\fve energy bands: 0.7{1 keV, 1{1.7 keV, 2{3 keV, 3{4.5\nkeV, and 4.5{7 keV. We excluded the 1.7{2 keV energy\nrange to avoid Si XIII (He\u000b) emission prevalent in the\nremnant's thermal ejecta which might contaminate our\nnonthermal pro\fle measurements.\nWe selected 20 regions for pro\fle extraction around\nTycho's shock (Figure 1) based on the following criteria:\n(1) \flaments should be clear of spatial plumes of thermal\nejecta in Chandra images, which rules out, e.g., areas of\nstrong thermal emission on Tycho's eastern limb; (2) \fl-\naments should be singular and localized, so multiple \fla-\nments should either not overlap or completely overlap;\n(3) \flament peaks should be evident above the back-\nground signal or downstream thermal emission (rules\nout faint southern \flaments). We accepted several re-\ngions with poor quality peaks in the lowest energy band\n(0:7{1 keV) so long as peaks in all higher energy bands\nwere clear and well-\ft. We grouped regions into 5 \fla-\nments by visual inspection of the remnant. Within each\n\flament, we chose region widths to obtain comparable\ncounts at the thin rim peak. All measured rim widths are\nat least 100. The narrowest rim widths may be slightly\nover-estimated due to the Chandra point-spread func-\ntion (PSF) at\u001840o\u000b the optical axis, which has FWHM\u00141:400at 6:4 keV and\u0014100at 1:5 keV. For simplicity,\nwe neglect PSF e\u000bects in our analysis. Our observations\nof rim width-energy dependence are thus somewhat con-\nservative at the highest energy\n3.2. Filament spectra\nWe extracted spectra at and immediately behind thin\nrims in each region (\\rim\", \\downstream\" spectra respec-\ntively) to con\frm that rim width measurements are not\ncontaminated by thermal line emission. The two extrac-\ntion regions are determined by our empirical \fts of rim\npro\fle shape (Section 3.3). The rim section is the small-\nest sub-region containing the measured FWHM bounds\nfrom all energy bands. The downstream section extends\nfrom the interior thin rim FWHM to the intensity mini-\nmum behind the rim (speci\fcally, the downstream pro\fle\n\ft domain bound described in Section 3.3). To illus-\ntrate our selections, Figure 3 plots example rim pro\fles\n(4:5{7 keV) with the downstream and rim sections high-\nlighted.\nSpectra were binned to a minimum of 15 cts/bin. We\nextracted background spectra from circular regions (ra-\ndius\u00183000) around the remnant's exterior; for each re-\ngion's rim and downstream spectra, we subtracted the\nclosest background region's spectrum.\nWe \ft each region's rim and downstream spectra\nto an absorbed power law model (XSPEC 12.8.1,\nphabs*powerlaw ) between 0 :5{7 keV with photon index\n\u0000, hydrogen column density NH, and a normalization as\nfree parameters. Table 1 lists best \ft parameters and re-\nduced\u001f2values for all regions. Rim spectra are well-\ft\nby the power law model alone; the best \ft photon in-\ndices (2:4{3) and column densities (0 :6{0:8\u00021022cm\u00002)\nare consistent with previous spectral \fts to Tycho's non-\nthermal rims (Hwang et al. 2002; Cassam-Chena \u0010 et al.\n2007).\nDownstream spectra are poorly \ft by the absorbed\npower law model due to thermal contamination from\nSiXIII and S XVHe\u000bline emission at 1 :85 and 2:45\nkeV. To con\frm that thermal emission is dominated by\nthese two lines near the shock, we also performed \fts with\n(1) both lines excised (1 :7{2:0 keV, 2:3{2:6 keV counts\nremoved) and (2) with both lines \ftted to Gaussian pro-\n\fles. Fits with lines excised yield \u001f2\nredvalues between\n1{5. Fits with lines \ftted to Gaussian pro\fles yield \u001f2\nredvalues 0:83{1:6. In both \fts (lines excised or modeled),\nwe \fnd somewhat smaller best \ft column densities (0 :3{\n0:8\u00021022cm\u00001) but similar best \ft photon indices (2 :6{\n3:1), compared to those of the rim spectra. The consis-\ntent photon indices indicate that the same synchrotron\ncontinuum is present beneath thermal line emission.\nWe also \ftted \\rim\" spectra (Section 3.2) to the ab-\nsorbed XSPEC model srcut , modi\fed to \ft in log-\nfrequency space. srcut models a power law X-ray syn-\nchrotron spectrum set by a radio spectral index \u000bwith\nan exponential cut-o\u000b parameterized by cut-o\u000b frequency\n\u0017cut(Reynolds 1998; Reynolds & Keohane 1999). The \ft\nvalues of\u0017cut, in particular, permit an independent esti-\nmate of\u00112from equation (8). The radio spectral index is\n\fxed to\u000b= 0:58 (Sun et al. 2011) as done in our trans-\nport modeling, and we \ft for absorption column density\nNHand cut-o\u000b frequency \u0017cutin each region. Table 1\nlists best spectrum \ft parameters for each region. TheTycho's Synchrotron Rims (June 4, 2021) 7\n0 5 10 15 20 25\nRadial position (arcsec.)0.00.20.40.60.81.01.21.4Intensity (a.u.)×10−8\n1\n10−410−3Counts s−1keV−1\nχ2\nred=2.25\n−6−4−20246χ×10−3\n0.5 1.0 2.0 5.0\nEnergy (keV)0.51.01.52.02.53.03.54.0Ratio\n10−410−310−2Counts s−1keV−1\nχ2\nred=1.20\n−4−2024χ×10−3\n0.5 1.0 2.0 5.0\nEnergy (keV)0.51.01.52.02.5Ratio\n0 5 10 15 20\nRadial position (arcsec.)012345Intensity (a.u.)×10−8\n16\n10−410−3Counts s−1keV−1\nχ2\nred=2.04\n−4−2024χ×10−3\n0.5 1.0 2.0 5.0\nEnergy (keV)0.51.01.52.02.5Ratio\n10−410−3Counts s−1keV−1\nχ2\nred=1.13\n−2−1012χ×10−3\n0.5 1.0 2.0 5.0\nEnergy (keV)0.51.01.52.02.5Ratio\nFigure 3. Spectra and \fts from Regions 1 (top) and 16 (bottom) show varying rim morphology; Region 1 shows a rim where the 0 :7{1 keV\npeak could not be \ft. Left: 4 :5{7 keV pro\fles with downstream (blue) and rim (grey) sections highlighted. Intensity is in arbitrary units\n(a.u.). Middle: downstream spectra with absorbed power law \ft; Si and S lines at 1 :85, 2:45 keV are clearly visible. Right: rim spectra\nwith absorbed power law \ft show that rims in each region are likely free of thermal line emission.\nTable 1\nAbsorbed power law spectrum \ft parameters\nDownstream, power-law Rim, power-law Rim, srcut\nRegion NH \u0000\u001f2\nred(dof) NH \u0000\u001f2\nred(dof) NH\u0017cut\u001f2\nred(dof)\n(1022cm\u00002) (-) (1022cm\u00002) (-) (1022cm\u00002) (keV/h)\n1 0.53 2.72 2.25 (186) 0.72 2.84 1.20 (284) 0.62 0.30 1.17 (284)\n2 0.68 2.99 5.07 (178) 0.69 2.83 1.10 (202) 0.60 0.30 1.08 (202)\n3 0.67 2.96 1.97 (186) 0.77 2.80 1.15 (167) 0.68 0.33 1.14 (167)\n4 0.59 2.97 1.43 (163) 0.70 2.84 1.21 (278) 0.61 0.29 1.15 (278)\n5 0.62 2.93 4.76 (265) 0.73 2.88 1.18 (255) 0.64 0.27 1.11 (255)\n6 0.68 3.00 2.12 (200) 0.74 2.85 0.96 (231) 0.64 0.29 0.93 (231)\n7 0.65 3.02 1.00 (142) 0.82 2.97 1.14 (224) 0.71 0.23 1.14 (224)\n8 0.74 2.93 1.38 (170) 0.75 2.71 0.98 (198) 0.66 0.41 0.96 (198)\n9 0.78 3.03 1.12 (157) 0.82 2.83 0.90 (175) 0.73 0.30 0.88 (175)\n10 0.62 2.86 1.40 (220) 0.77 2.76 0.98 (164) 0.68 0.36 0.96 (164)\n11 0.67 2.94 2.56 (137) 0.69 2.60 1.10 (153) 0.61 0.55 1.07 (153)\n12 0.61 2.79 2.65 (137) 0.64 2.44 0.90 (172) 0.57 0.88 0.90 (172)\n13 0.61 2.98 3.12 (198) 0.67 2.73 1.12 (235) 0.59 0.38 1.09 (235)\n14 0.46 2.93 1.37 (148) 0.63 2.93 0.96 (167) 0.54 0.23 0.95 (167)\n15 0.43 2.92 1.33 (150) 0.65 2.84 1.05 (183) 0.57 0.29 1.03 (183)\n16 0.48 2.94 2.04 (189) 0.67 2.80 1.13 (182) 0.58 0.32 1.12 (182)\n17 0.48 2.86 1.70 (188) 0.68 2.83 0.97 (187) 0.59 0.30 0.94 (187)\n18 0.44 2.87 1.91 (200) 0.64 3.02 1.20 (220) 0.55 0.19 1.13 (220)\n19 0.40 2.84 1.31 (133) 0.66 2.78 1.01 (157) 0.57 0.34 0.99 (157)\n20 0.40 2.75 3.01 (140) 0.63 2.81 1.11 (192) 0.55 0.31 1.07 (192)\nNote . | Absorbed power law \ft parameters are photon index \u0000 and hydrogen column density NH.srcut \fts performed in log-frequency space;\nhis Planck's constant and \u0017cuta cut-o\u000b frequency. Horizontal rules group individual regions into \flaments.8 Tran et al. (June 4, 2021)\n\ftted cut-o\u000b frequency is typically 0 :3 keV=h, consistent\nwith \fts by Hwang et al. (2002), but Regions 11, 12 have\nunusually high cut-o\u000b frequencies 0 :55, 0:88 keV=h con-\nsistent with harder rim spectra.\nOur spectral \ftting con\frms that all selected region\nare practically free of thermal line emission, as already\nsuggested by visual inspection (Figure 1). Excluding 1 :7{\n2 keV photons in rim width measurements further lim-\nits thermal contamination as 1 :85 keV Si line emission\nis over a third of Tycho's thermal \rux as detected by\nChandra (Hwang et al. 2002).\n3.3. Filament width measurements\nWe obtained radial intensity pro\fles in \fve energy\nbands from\u001810{2000behind the shock to \u00185{1000in front\nfor each region. To increase signal-to-noise, we integrate\nalong the shock (5{2300) in each region. Plotted and\n\ftted pro\fles are reported in vignetting and exposure-\ncorrected intensity units; error bars were computed from\nraw counts assuming Poisson statistics. Intensity pro\fles\npeak sharply within \u00182{300behind the shock, demarcat-\ning the thin rims, then fall o\u000b gradually until thermal\nemission picks up further behind the shock.\nWe \ftted rim pro\fles to a piecewise two-exponential\nmodel:\nh(r) =8\n<\n:Auexp\u0010\nr0\u0000r\nwu\u0011\n+Cu; r\u0015r0\nAdexp\u0010\nr\u0000r0\nwd\u0011\n+Cd; r<r 0(12)\nwhereh(r) is pro\fle height and ris radial distance from\nremnant center. The rim model has 6 free parameters:\nAu;r0;wu;wd;Cu, andCd;Ad=Au+ (Cu\u0000Cd) en-\nforces continuity at r=r0. Our model is similar to that\nof Bamba et al. (2003, 2005) and di\u000bers slightly from that\nof R14. To \ft only the nonthermal rim in each intensity\npro\fle, we selected the \ft domain for each pro\fle as fol-\nlows. The downstream bound was set at the \frst local\ndata minimum downstream of the rim peak, identi\fed by\nsmoothing the pro\fles with a 21-point ( \u00181000) Hanning\nwindow. The upstream bound was set at the pro\fle's\nouter edge. Figure 4 illustrates the \ft domain selections\nfor two example regions.\nFrom the \ftted pro\fles we extracted a full width at\nhalf maximum (FWHM) for each region and each en-\nergy band after subtracting a constant background term\nmin(Cu;Cd). We could not resolve a FWHM in 8 of 20\nregions at 0.7{1 keV (Table 2); in these regions, either\nthe downstream FWHM bound would extend outside the\n\ft domain or we could not \fnd an acceptable \ft to equa-\ntion (12). We were able to resolve FWHMs for all regions\nat higher energy bands (1{7 keV).\nTo estimate FWHM uncertainties, we horizontally\nstretched each best \ft pro\fle by mapping radial coor-\ndinatertor0(r) =r(1 +\u0018(r\u0000r0)=(5000\u0000r0)) with\n\u0018an arbitrary stretch parameter and r0the best \ft\nrim center from equation (12), yielding a new pro\fle\nh0(r) =h(r0(r)). We varied \u0018(and hence rim FWHM) to\nvary each pro\fle \ft \u001f2by 2.7 and took stretched FWHMs\nas upper/lower bounds on reported FWHMs. This pro-\ncedure again follows R14.\n3.4. Filament model \fttingWe \ft model FWHM predictions given by equation (1)\nto measured rim widths as a function of energy by vary-\ning several physical parameters: magnetic \feld strength\nB0, normalized di\u000busion coe\u000ecient \u00112, di\u000busion-energy\nscaling exponent \u0016, and minimum \feld strength Bmin\nand lengthscale abfor a damped magnetic \feld. We\nmapped each width measurement to the lower energy\nlimit of its energy band; e.g., 0 :7{1 keV is assigned to\n0:7 keV and \ftted to model pro\fle widths at 0 :7 keV.\nWidth errors in our least squares \fts average the positive\nand negative errors on each FWHM measurement. For a\ngiven set of model parameters, we numerically computed\nintensity pro\fles and hence model FWHMs as detailed in\nSection 2.1; we then used a Levenberg-Marquardt \ftter\nto seek model parameters yielding best \ft FWHMs. To\nassist the nonlinear \ftting, we tabulated model FWHM\nvalues on a large grid of model parameters and used best\n\ft grid parameters as initial guesses for \ftting. We re-\nquired\u00112to be positive and deemed best \ft values with\n\u00112\u0015105andB0\u001510 mG to be e\u000bectively uncon-\nstrained. In subsequent analysis we focus on \fts with\n\u0016= 1 and\u00112= 1 \fxed, though we discuss the e\u000bects of\nvarying both \u0016and\u00112(and \ftting for \u00112) as well (recall\nthat for\u0016= 1,\u0011is energy-independent).\nThe purely loss-limited model (constant downstream\nmagnetic \feld B0) has three parameters \u0016,\u00112, andB0.\nTo make nonlinear \ftting tractable, we \fxed \u0016in all\n\fts and considered \u0016= 0, 1=3, 1=2, 1, 1:5, and 2. In\nparticular, nonlinear di\u000busion-energy scalings with \u0016=\n1=3 and 1=2 may arise from Kolmogorov and Kraichnan\nturbulent energy spectra respectively (Reynolds 2004).\nFor damped magnetic \feld \fts, we held the remnant\ninterior \feld strength Bminconstant at 5 \u0016G, slightly\nhigher than typical intergalactic values of \u00182{3\u0016G (Lyne\n& Smith 1989; Han et al. 2006). We stepped abthrough\n14 di\u000berent values between 0 :5 and 0:002 (sampling most\n\fnely between 0 :01 and 0:002). To ensure that damped\n\fts generate rims in\ruenced by magnetic damping, we ar-\nbitrarily require that the rim FWHM at 2 keV be strictly\ngreater than the \ft value of ab; we revisit this require-\nment below. For best \fts, we report the value of ab\nyielding the smallest \u001f2value, with the caveat that that\nourabsampling is relatively coarse.\nPredicted rim widths are subject to resolution error\nin the numerical integrals (discretization over radial co-\nordinate, line-of-sight coordinate, electron distribution,\nGreen's function integrals). We chose integration reso-\nlutions such that the fractional error in model FWHMs\nassociated with halving or doubling each integration res-\nolution is less than 1% for the parameter space relevant to\nour \flaments. The maximum resolution errors in a sam-\nple of parameter space are typically 0 :1{1%, but mean\nand median errors are typically an order of magnitude\nsmaller than maximum errors.\n4.RESULTS\n4.1. Rim widths\nMeasured rim widths decrease with energy in most re-\ngions and energy bands. Table 2 reports FWHM mea-\nsurements for all of our regions. We also report mEvalues\nfor all but the lowest energy band, computed point-to-Tycho's Synchrotron Rims (June 4, 2021) 9\n0 5 10 15 20 25\nRadial position (arcsec.)0.00.51.01.52.02.5Intensity (a.u.)×10−80.7–1 keV\n0 5 10 15 20 25\nRadial position (arcsec.)0123456×10−81–1.7 keV\n0 5 10 15 20 25\nRadial position (arcsec.)0.00.51.01.52.02.53.0×10−8 2–3 keV\n0 5 10 15 20 25\nRadial position (arcsec.)0.00.51.01.52.0×10−83–4.5 keV\n0 5 10 15 20 25\nRadial position (arcsec.)0.00.20.40.60.81.01.21.4×10−8\n14.5–7 keV\n0 5 10 15 20\nRadial position (arcsec.)0.00.20.40.60.8Intensity (a.u.)×10−70.7–1 keV\n0 5 10 15 20\nRadial position (arcsec.)0.00.51.01.52.0×10−71–1.7 keV\n0 5 10 15 20\nRadial position (arcsec.)0.00.20.40.60.81.01.2×10−7 2–3 keV\n0 5 10 15 20\nRadial position (arcsec.)0.00.20.40.60.8×10−73–4.5 keV\n0 5 10 15 20\nRadial position (arcsec.)0123456×10−8\n164.5–7 keV\nFigure 4. Best \ft pro\fles with measured FWHMs demarcated for each energy band in Region 1 (top) and Region 16 (bottom). Hereafter,\nwe use Regions 1 and 16 to illustrate results for pro\fles of di\u000bering quality and absolute width. We could not measure a 0 :7{1 keV FWHM\nin Region 1, re\rected in Table 2. Data in red were excluded from pro\fle \ft domains as described in text.\npoint between discrete energy bands as:\nmE(E2) =ln(w2=w1)\nln(E2=E1)(13)\nwherew1;w2andE1;E2are FWHMs and lower energy\nvalues for each energy band { e.g., mEat 1 keV is com-\nputed using FWHMs from 0 :7{1 keV and 1{1 :7 keV,\nwithE2= 1 keV and E1= 0:7 keV. Errors on mEare\npropagated in quadrature from adjacent FWHM mea-\nsurements.\nAlthough the measurement scatter is quite large,\nshown dramatically in the point-wise computed mEval-\nues, the mean rim width decreases consistently with in-\ncreasing energy. Furthermore, mean mEvalues are con-\nsistently negative and tend smoothly towards 0 (weaker\nenergy-dependence) with increasing energy. Errors on\nFWHM measurements are typically .10%, re\recting\nthe high quality of the underlying Chandra data. Scat-\nter in FWHM measurements may be attributed in part\nto (1) our measurement procedure, which depends on an\nempirical choice of pro\fle \ft function, and (2) variation\nin Tycho's rim morphology (e.g., Figure 4).\n4.2. Model \ft results\nBest \ft results for loss-limited and damped cases are\nsummarized in Figure 5 and Table 3 with \u0016= 1 and\n\u00112= 1 both \fxed. In all tables, we report values of \u001f2\nrather than \u001f2\nredto ease comparison of di\u000berent \fts, as\nwe often manually stepped parameters that were held\nconstant in \ftting.\nTable 4 reports model \fts with \u00112\fxed to values de-\nrived from srcut spectrum \fts for \u0017cutand hence \u00112\nfrom equation (8), with B0varying freely; cuto\u000b-derived\n\u00112values of\u001810 require increased magnetic \felds com-\npared to the \u00112= 1 case if rims are purely loss-limited.\nWe derive di\u000berent \u00112values for varying \u0016and \fnd that\n\fttedB0varies only weakly with \u0016; for example, Re-\ngions 1 and 16 have best \ft values of B0increasing over\n252{266\u0016G and 660{700 \u0016G respectively as \u0016ranges\nbetween 0{2, for loss-limited \fts.\nMagnetically damped rims are able to \ft width-energy\ndependence in our measurements at least as well as\npurely loss-limited rims for an acceptable range of ab\nvalues. As loss-limited models are a subset of dampingmodels, this is expected. But \fts with damping lengths\nsmall enough to exert in\ruence on rim widths are also\npermissible. In both loss-limited and damped \fts, we do\nnot observe systematic variation of \ft parameters with\nazimuth around the remnant.\n4.3. Fitting parameter degeneracy\nWe brie\ry explore the e\u000bects of varying \u00112and other \ft\nparameters, before focusing on results with \u0016=\u00112= 1\nfor simplicity. Fixing \u00112from measured \u0017cutvalues does\nnot yield any obvious insight. But, di\u000busion coe\u000ecients\ncomputed in this manner may give more credible derived\nestimates of B0if the DSA assumptions invoked are ac-\ncurate.\nLoss-limited model values of B0and\u00112may covary\nwithout strongly altering \ft quality in many regions. If\ndi\u000busion is the primary control on rim width { i.e., ldi\u000b>\nlad{ then\u00112andB0become degenerate as the product\n\u00111=2B\u00003=2\n0 exerts most control on rim width w\u0018ldi\u000b\n(equation (11)). If advection is the primary control on\nrim width (widths narrow rapidly with energy; i.e., mE\u0018\n\u00000:5), then\u00112\u001c1 becomes unimportant and \fts are\nwell-behaved with e\u000bectively one free parameter.\nWe also performed loss-limited model \fts with \u0016\fxed\nbetween 0 and 2 and both \u00112andB0free. Fits with\n\u0016.1 generally yield larger parameter values and errors\nfor both\u00112andB0, and they are more likely to be ill-\nconstrained (e.g., \u00112>103orB0>103\u0016G). Fits to the\nsame data with varying \u0016can yield\u00112varying by 1{2 or-\nders of magnitude. The \u001f2values for individual \fts are\nvariable and large ( \u001d1), so we cannot favor or disfavor\nparticular values of \u0016and\u00112. But, neglecting the mag-\nnitude of our \u001f2values, values of \u0016\u00151 are qualitatively\nfavored by \u001f2in most regions. This trend may be par-\ntially an artifact of the correlation between B0and\u00112as\nthey are not entirely independent parameters, but they\nare less correlated for larger \u0016; \fts at smaller \u0016may have\nfewer (non-integer) degrees of freedom, partially o\u000bset-\nting our observations.\nDamping modi\fes the degeneracy in B0and\u00112. For\nsmall values of ab(strong damping), increased di\u000bu-\nsion (\u00112) can cause rim widths at all energies to narrow\ncounterintuitively. Speculatively, spatial variation of the\ndi\u000busion coe\u000ecient may oppose downstream advection10 Tran et al. (June 4, 2021)\nTable 2\nMeasured full widths at half max (FWHMs) for all regions.\nFWHM (arcsec) mE(-)\nRegion Band 1 Band 2 Band 3 Band 4 Band 5 Bands 1{2 Bands 2{3 Bands 3{4 Bands 4{5\n(0.7{1 keV) (1{1.7 keV) (2{3 keV) (3{4.5 keV) (4.5{7 keV) (1 keV) (2 keV) (3 keV) (4.5 keV)\n1 8 :80+0:18\n\u00000:156:34+0:26\n\u00000:217:40+0:30\n\u00000:235:57+0:47\n\u00000:42\u00000:47\u00060:06 0:38\u00060:13\u00000:70\u00060:22\n2 4 :22+0:12\n\u00000:092:36+0:12\n\u00000:093:00+0:16\n\u00000:124:11+0:34\n\u00000:30\u00000:84\u00060:08 0:59\u00060:16 0:77\u00060:23\n3 2 :47+0:08\n\u00000:071:78+0:09\n\u00000:072:10+0:11\n\u00000:111:32+0:10\n\u00000:09\u00000:47\u00060:08 0:41\u00060:17\u00001:15\u00060:22\n4 5 :85+0:37\n\u00000:334:35+0:09\n\u00000:083:26+0:11\n\u00000:093:69+0:12\n\u00000:113:20+0:21\n\u00000:18\u00000:83\u00060:18\u00000:41\u00060:05 0:31\u00060:11\u00000:35\u00060:17\n5 4 :52+0:11\n\u00000:123:06+0:11\n\u00000:113:25+0:15\n\u00000:133:04+0:21\n\u00000:18\u00000:56\u00060:06 0:15\u00060:14\u00000:17\u00060:19\n6 2 :48+0:18\n\u00000:182:32+0:05\n\u00000:062:98+0:11\n\u00000:092:05+0:08\n\u00000:092:21+0:15\n\u00000:14\u00000:19\u00060:21 0:36\u00060:06\u00000:92\u00060:13 0:18\u00060:19\n7 2 :69+0:20\n\u00000:172:33+0:05\n\u00000:052:31+0:08\n\u00000:081:81+0:09\n\u00000:071:83+0:11\n\u00000:08\u00000:39\u00060:20\u00000:01\u00060:06\u00000:60\u00060:14 0:02\u00060:17\n8 2 :33+0:21\n\u00000:202:72+0:08\n\u00000:082:38+0:10\n\u00000:092:10+0:10\n\u00000:092:37+0:20\n\u00000:170:43\u00060:26\u00000:19\u00060:07\u00000:30\u00060:15 0:29\u00060:22\n9 2 :16+0:24\n\u00000:232:35+0:07\n\u00000:062:47+0:11\n\u00000:111:91+0:09\n\u00000:092:20+0:17\n\u00000:160:24\u00060:31 0:07\u00060:07\u00000:63\u00060:16 0:34\u00060:22\n10 2 :38+0:24\n\u00000:231:99+0:07\n\u00000:061:76+0:09\n\u00000:081:59+0:09\n\u00000:081:58+0:13\n\u00000:12\u00000:50\u00060:29\u00000:18\u00060:08\u00000:24\u00060:18\u00000:02\u00060:23\n11 3 :23+0:15\n\u00000:132:52+0:16\n\u00000:131:90+0:14\n\u00000:133:09+0:45\n\u00000:38\u00000:36\u00060:10\u00000:70\u00060:22 1:21\u00060:37\n12 3 :86+0:17\n\u00000:162:61+0:15\n\u00000:133:02+0:22\n\u00000:212:23+0:21\n\u00000:17\u00000:56\u00060:10 0:36\u00060:22\u00000:74\u00060:27\n13 2 :85+0:22\n\u00000:172:43+0:05\n\u00000:052:36+0:08\n\u00000:051:95+0:09\n\u00000:101:84+0:11\n\u00000:14\u00000:45\u00060:20\u00000:04\u00060:05\u00000:47\u00060:13\u00000:15\u00060:20\n14 2 :86+0:17\n\u00000:162:42+0:06\n\u00000:042:23+0:08\n\u00000:072:38+0:10\n\u00000:082:19+0:12\n\u00000:10\u00000:47\u00060:17\u00000:12\u00060:06 0:17\u00060:12\u00000:20\u00060:15\n15 2 :71+0:17\n\u00000:161:99+0:05\n\u00000:041:80+0:06\n\u00000:051:87+0:07\n\u00000:051:52+0:09\n\u00000:08\u00000:85\u00060:18\u00000:15\u00060:05 0:09\u00060:11\u00000:51\u00060:16\n16 1 :87+0:14\n\u00000:131:73+0:04\n\u00000:031:52+0:06\n\u00000:051:25+0:06\n\u00000:041:23+0:08\n\u00000:06\u00000:22\u00060:21\u00000:18\u00060:06\u00000:49\u00060:13\u00000:04\u00060:17\n17 1 :65+0:13\n\u00000:121:92+0:05\n\u00000:051:54+0:06\n\u00000:071:45+0:07\n\u00000:062:05+0:16\n\u00000:140:43\u00060:22\u00000:31\u00060:07\u00000:16\u00060:15 0:86\u00060:21\n18 4 :45+0:13\n\u00000:123:18+0:17\n\u00000:162:96+0:20\n\u00000:191:65+0:21\n\u00000:16\u00000:49\u00060:09\u00000:17\u00060:21\u00001:45\u00060:32\n19 2 :30+0:08\n\u00000:062:28+0:11\n\u00000:082:16+0:12\n\u00000:111:60+0:17\n\u00000:14\u00000:02\u00060:08\u00000:13\u00060:17\u00000:74\u00060:27\n20 4 :81+0:31\n\u00000:311:84+0:06\n\u00000:031:87+0:08\n\u00000:061:56+0:07\n\u00000:062:14+0:23\n\u00000:23\u00002:68\u00060:19 0:02\u00060:07\u00000:44\u00060:14 0:77\u00060:28\nMean 2:89\u00060:35 3:11\u00060:37 2:53\u00060:23 2:47\u00060:30 2:35\u00060:23\u00000:46\u00060:24\u00000:25\u00060:06\u00000:14\u00060:10\u00000:09\u00060:15\nNote . | Mean values computed for all regions; mean mEvalues are averages for region mEvalues (i.e., not computed from mean FWHMs).\nErrors on mean values are standard errors of the mean. Horizontal rules group individual regions into \flaments.\nTable 3\nBest width-energy \ft parameters, \u0016=\u00112= 1\nLoss-limited Damped\nRegionB0(\u0016G)\u001f2B0(\u0016G)\u001f2ab\n1 182 35.1 28 24.3 0.008\n2 312 80.9 25 74.4 0.003\n3 426 21.5 27 23.9 0.002\n4 284 50.9 28 29.9 0.004\n5 288 19.3 25 14.5 0.003\n6 410 173.7 68 65.2 0.004\n7 418 50.9 291 11.6 0.006\n8 388 47.7 138 10.0 0.005\n9 414 65.1 78 14.2 0.004\n10 466 17.1 29 1.5 0.002\n11 355 12.0 317 15.6 0.010\n12 317 9.6 26 10.4 0.003\n13 400 52.7 258 10.4 0.006\n14 383 102.9 60 12.7 0.004\n15 431 70.3 48 20.0 0.003\n16 493 15.0 434 4.0 0.006\n17 467 61.6 232 29.0 0.004\n18 283 28.1 24 27.5 0.003\n19 401 36.9 78 8.3 0.004\n20 463 121.8 58 93.8 0.003\nNote . | Fits for Regions 1{3, 5, 11, 12, 18, and 19 have 3 degrees\nof freedom; all others have 4. The choice of a best abvalue may be\nconstrued as removing one additional dof. Damped \fts require abto\nbe smaller than the FWHM at 2 keV in order to rule out e\u000bectively\nloss-limited \fts with large ab.Table 4\nBest model \fts for all regions, \u00112derived from srcut \fts,\u0016= 1\nLoss-limited Damped\nRegion \u00112B0(\u0016G)\u001f2B0(\u0016G)\u001f2ab\n1 12.0 256 29.1 19 25.5 0.008\n2 11.8 443 99.4 18 72.0 0.003\n3 10.6 599 31.8 19 29.3 0.002\n4 12.0 402 36.2 19 34.5 0.004\n5 12.9 419 32.5 19 19.1 0.004\n6 11.9 568 89.7 147 61.9 0.006\n7 15.3 617 11.7 473 11.1 0.009\n8 8.4 511 15.1 333 10.0 0.008\n9 11.2 573 23.3 66 13.7 0.005\n10 9.4 631 1.4 478 2.0 0.007\n11 6.1 452 18.1 21 17.5 0.003\n12 3.7 375 12.6 21 9.7 0.003\n13 8.4 529 13.8 368 9.9 0.008\n14 11.9 547 28.6 120 12.4 0.006\n15 9.5 598 23.1 426 20.0 0.007\n16 8.4 670 6.3 22 5.8 0.002\n17 9.0 637 32.6 32 29.4 0.003\n18 14.6 432 72.4 17 16.6 0.003\n19 8.9 542 12.8 209 7.7 0.006\n20 9.6 631 96.0 29 94.2 0.003\nNote . |\u00112values are computed from equation (8) and held \fxed\nin model \fts. All comments for Table 3 apply to srcut \fts as well.Tycho's Synchrotron Rims (June 4, 2021) 11\n5.05.56.06.57.07.58.08.59.0FWHM (arcsec.)1Loss-limited\nDamped\nData\n2.02.53.03.54.04.5\n2\n1.21.41.61.82.02.22.42.6\n3\n3.03.54.04.55.05.56.0\n4\n3.03.54.04.5FWHM (arcsec.)5\n1.82.02.22.42.62.83.03.2\n6\n1.61.82.02.22.42.62.83.0\n7\n1.82.02.22.42.62.83.0\n8\n1.61.82.02.22.42.62.8FWHM (arcsec.)9\n1.21.41.61.82.02.22.42.6\n10\n1.52.02.53.03.511\n2.02.53.03.54.0\n12\n1.61.82.02.22.42.62.83.0FWHM (arcsec.)13\n2.02.22.42.62.83.014\n1.41.61.82.02.22.42.62.83.0\n15\n1.01.21.41.61.82.016\n1 2 3 4 5\nEnergy (keV)1.21.41.61.82.02.2FWHM (arcsec.)17\n1 2 3 4 5\nEnergy (keV)1.52.02.53.03.54.04.5\n18\n1 2 3 4 5\nEnergy (keV)1.41.61.82.02.22.419\n1 2 3 4 5\nEnergy (keV)1.52.02.53.03.54.04.55.0\n20\nFigure 5. Rim width \fts as a function of energy for loss-limited and damped \fts with \u0016= 1 and\u00112= 1 \fxed for all regions, from best\n\ft parameters in Table 3. Damped model predictions (e.g., Region 5) are not given at low energies if FWHMs cannot be calculated for\nmodel pro\fles (i.e., modeled intensity behind thin rim exceeds half-maximum of rim peak within model domain of \u00182000). Ordinate ( y)\naxis limits vary between subplots and are o\u000bset from the origin to better show model predictions and data variation.\nthrough a decreased e\u000bective velocity vd\u0000(@D=@x ) in\nour transport equation:\n\u0014\nvd\u0000@D(x)\n@x\u0015@f\n@x\u0000D(x)@2f\n@x2\u0000@\n@E\u0000\nbB2E2f\u0001\n=K0E\u0000se\u0000E=E cut\u000e(x);(14)\nrequiring thinner rims as \u00112and hence @D(x)=@x in-\ncrease. Moreover, the e\u000bective velocity may impact rim\nwidths even if advection dominated. In practice, if \u00112varies freely in damped \fts, we observe that smaller val-\nues ofabpermit and favor smaller best \ft values of both\nB0and\u00112.\nBest \ftB0values are smallest for \u00112approaching 0\nand\u0016= 2; intuitively, \u0016>1 strengthens di\u000busion at en-\nergies>2 keV for \fxed \u00112, permitting a smaller best \ft\n\u00112and hence smaller B0. Fixing\u00112= 1 (Table 3), how-\never, ties the range of B0values to the range of observed\nrim widths. Our minimum loss-limited values of B0\nare consistent with prior estimates of \u0018200{300\u0016G for12 Tran et al. (June 4, 2021)\n220 225 230 235 240\nRadial position (arcsec.)0.00.20.40.60.81.0Intensity (a.u.)1\n220 225 230 235 240\nRadial position (arcsec.)0.00.20.40.60.81.0Intensity (a.u.)161.375 GHz\n0.01 keV\n0.1 keV\n0.7 keV\n1 keV\n2 keV\n3 keV\n4 keV\nFigure 6. Model predictions illustrate weak-\feld (Region 1) and\nstrong-\feld damping (Region 16) for damped best \ft parameters\nwith\u0016= 1 and\u00112= 1 \fxed. In X-ray energies (0 :7{4:5 keV) model\npro\fles are not strongly energy dependent, but weak-\feld damping\npro\fles evolve and show no measurable FWHM at su\u000eciently low\nenergies. Pro\fles are normalized to peak thin rim intensity, and\nshaded regions indicate damping lengthscale ab. Model parameters\nare given in Table 3.\nadvection-dominated transport (V olk et al. 2005; Pari-\nzot et al. 2006; Morlino & Caprioli 2012). If loss-limited,\nTycho's rims require strong magnetic \feld ampli\fcation\nto\u0018100\u0002typical galactic \feld values of \u00182{3\u0016G, versus\nthe expected 4\u0002ampli\fcation from a strong shock.\nMagnetic damping \fts permit much smaller values of\nB0, as expected. The minimum value of B0is around\n20\u0016G, which would require no \feld ampli\fcation beyond\nthat from a strong shock. Fixing Bmin= 2\u0016G instead\nof 5\u0016G permits smaller \ft B0values, though \ft values\nfor both\u00112andB0still display considerable scatter.\n5.DISCUSSION\n5.1. Damping is poorly-constrained by X-ray\nwidth-energy dependence\nWidth-energy dependence is not as sensitive a discrim-\ninant between damping and loss-limited models as may\nbe intuitively expected. Both loss-limited and damped\n\fts do not perfectly capture sharp drop-o\u000bs (especially\nbetween 0:7{1 keV and 1{2 keV), and the data show\nlarge scatter (Figure 5). In most regions, best \ft width-\nenergy curves have mE\u0018\u00000:2. Only sharp decreases\nin rim width ( mE\u0018\u00000:5) can disfavor damping as a\ncontrol on rim widths (e.g., Region 18).\nWhen width-energy dependence is weak, both loss-\nlimited and damped models give best \fts with similarpro\fles { ampli\fed magnetic \felds in both models cause\nrim intensities to drop sharply. When width-energy de-\npendence is stronger, damping pro\fles yield energy de-\npendent rims if magnetic \felds are small (say, .50\u0016G)\nand the damping lengthscale is much smaller than rim\nFWHMs. Figure 6 shows this contrasting behavior using\nbest \ft parameters for Regions 1 and 16.\nThe best damped \ft parameters for Regions 1{5, 10,\n12, and 18 predict a 1 :375 GHz radio pro\fle that does\nnot drop below 50% of peak intensity within \u00182000down-\nstream of the shock, which we treat as having no mea-\nsurable FWHM. These regions are all best \ft with \feld\nB0<40\u0016G. We refer to this model behavior as\n\\weak-\feld\" damping, associated with weak magnetic\n\felds and stronger emission intensity immediately down-\nstream of the thin rim. As observation frequency de-\ncreases, rim and trough contrast decreases, causing rim\nFWHMs to increase and eventually become unmeasur-\nable. Rim width-energy dependence strengthens dra-\nmatically (mE!\u00001 ), permitting model \fts to repli-\ncate strong width-energy dependence in observed rim\nwidths. Weak-\feld \ft parameters require energy losses\ndownstream of the shock to produce energy dependence\nin a damped \feld, as discussed in Section 2.5.2.\nDamping model \fts to all other regions predict consis-\ntently thin rims with measurable FWHMs at decreasing\nenergy. The width-energy dependence parameterized by\nmEtrends towards zero at low energy, indicating that\nrim widths are comparatively energy independent. At\nhigher energy, such rims narrow slightly with increas-\ning energy ( mE\u0018\u00000:2) to match the observed width-\nenergy dependence in X-rays. The gradual increase in\njmEjwith energy is expected in the damping model as\nadvection and/or di\u000busion take control of rim widths at\nincreasing energy, as discussed in Section 2.5. The best\ndamped \fts for these regions all have larger B0values\nthan in the \\weak-\feld\" damping case. We refer to these\n\fts as giving rise to \\strong-\feld\" damping, associated\nwith stronger magnetic \felds, weaker emission intensity\nbehind the thin rim, and clear rims with measurable\nFWHMs at low photon energies (at and below soft X-\nrays).\nWe are unable to determine whether weak- or strong-\n\feld damping is descriptive of Tycho's shock magnetic\n\feld, given the large \u001f2values on our model \fts. Sev-\neral regions may be well-\ft by \\weak-\feld\" and \\strong-\n\feld\" damping alike. But, the qualitative behavior of B0\nis suggestive; very strong \felds at the shock ( &100\u0016G)\nwith magnetic \feld damping on a scale comparable to\nrim FWHMs appear incompatible with signi\fcant energy\ndependence. The distinction between weak- and strong-\n\feld damping may be equally well described as strong\nversus weak damping. The dichotomy re\rects additional\ndegeneracy between abandB0{ for a given rim width,\nincreased damping length abrequires increased B0to re-\nproduce the same width, and vice versa.\nAlthough model \fts are poorly-constrained, we at-\ntempt to further explore how damping and synchrotron\nlosses set rim widths for damped \fts with \fnite ab.\nTable 5 compares rim widths at 2 keV to advection\nlengthscale ladand damping lengthscale abfrom the loss-\nlimited and damped \fts of Table 3. As discussed in\nSection 2.5, loss-limited rim widths are qualitatively set\nby either damping or the dominant transport process asTycho's Synchrotron Rims (June 4, 2021) 13\nTable 5\nLengthscale analysis\nMeasurements Loss-lim. \ft Damped \ft\nRegion vdw(2 keV) ladlad=ldiffladablad=abw(2 keV)=ab\n(108cm=s) (%rs) (%rs) (-) (% rs) (%rs) (-) (-)\n1 1.30 2.64 0.61 1.40 10.26 0.80 12.83 3.30\n2 1.29 0.99 0.27 1.39 11.54 0.30 38.48 3.28\n3 1.29 0.74 0.17 1.38 10.80 0.20 54.00 3.72\n4 1.28 1.36 0.31 1.38 10.08 0.40 25.20 3.40\n5 1.28 1.27 0.30 1.37 12.07 0.30 40.24 4.25\n6 1.28 1.24 0.18 1.37 2.61 0.40 6.53 3.10\n7 1.27 0.96 0.17 1.36 0.29 0.60 0.49 1.61\n8 1.27 0.99 0.19 1.36 0.90 0.50 1.80 1.98\n9 1.26 1.03 0.17 1.36 2.09 0.40 5.23 2.57\n10 1.26 0.73 0.14 1.35 9.21 0.20 46.05 3.66\n11 1.25 1.05 0.21 1.34 0.25 1.00 0.25 1.05\n12 1.24 1.09 0.25 1.33 10.80 0.30 35.98 3.62\n13 1.23 0.98 0.18 1.32 0.34 0.60 0.57 1.64\n14 1.14 0.93 0.17 1.23 2.84 0.40 7.10 2.32\n15 1.14 0.75 0.15 1.22 3.95 0.30 13.16 2.50\n16 1.13 0.63 0.12 1.21 0.14 0.60 0.24 1.06\n17 1.12 0.64 0.13 1.20 0.36 0.40 0.91 1.61\n18 1.15 1.32 0.28 1.23 11.29 0.30 37.64 4.42\n19 1.18 0.95 0.17 1.27 1.99 0.40 4.97 2.37\n20 1.17 0.78 0.14 1.26 3.08 0.30 10.28 2.59\nNote . | All lengthscales computed at \fducial energy 2 keV from best \fts with \u00112= 1 and\u0016= 1. Ratio of lad=ldiffis same for loss-limited and\ndamped \fts and depends only on plasma velocity vdand observation energy, as lad=ldiffis independent of B0for\u0016= 1.\nw\u0018min (ab;max (lad;ldi\u000b)). At 2 keV with \u00112= 1 and\n\u0016= 1 \fxed, the ratio lad=ldi\u000bis nearly constant; vari-\nation (1:2{1:4) arises solely from azimuthal variation in\nshock velocity. If rims are loss-limited and di\u000busion is\nnegligible, we anticipate rim widths w= 4:6lad, where\nthe factor 4 :6 may be derived assuming spherical sym-\nmetry and an exponential synchrotron emissivity (Ballet\n2006). At 2 keV, we \fnd that loss-limited rim widths\nw\u00185laddue to di\u000busion. Enforcing w(2 keV)=ab>1 for\ndamped \fts still permits lad=ab<1 asw(2 keV) .5lad,\nwith damping decreasing w(2 keV) below the expected\nloss-limited width at a given B0. We may impose a\ntighter or looser bound, but our results should not be\ngreatly a\u000bected given large uncertainty in our \fts and\nassociated degeneracy between B0andab.\nOur modeling can also reassess the signi\fcance of rim\nwidth-energy dependence in the remnant of SN 1006 pre-\nsented by R14. We \ft averaged \flament widths in SN\n1006 measured by R14 to our damping model using the\nsame procedure as for Tycho. We \fx Bmin= 5\u0016G, al-\nthough best \ft magnetic \felds for SN 1006 are smaller\nthan those of Tycho due to the much wider \flaments\nof SN 1006. A more extensive search of parameter space\nproduces acceptable damping (hybrid) models with more\nrapid shrinkage than found in R14. Damped \fts are com-\nparable to or better than loss-limited \fts in 3 of 5 \fla-\nments in SN 1006. Fits to two \flaments with strong en-\nergy dependence ( mE\u0018\u00000:5) favor a loss-limited model\nwith sub-Bohm di\u000busion ( \u00112\u001c1), though sub-Bohm\ndi\u000busion may be an unphysical result pointing to over-\nsimpli\fcations in the model. The width-energy depen-\ndence in SN 1006 ( mE\u0018\u00000:3 to\u00000:5) is overall slightly\nstronger than in Tycho, and the best damped \fts for SN\n1006 all fall into the \\weak-\feld\" case. The best \ft B0\nvalues are less than 40 \u0016G in the damped model, com-\npared to\u0018100{200\u0016G in the loss-limited model. If thin\nradio rims in SN 1006 (Reynolds & Gilmore 1986) indi-cate magnetic \feld damping on lengthscales comparable\nto X-ray \flament widths, some magnetic damping is not\nincompatible with rim width narrowing. A future study\nof SN 1006 with recent multi-frequency Karl G. Jansky\nVery Large Array data (PI: D. Green) may verify whether\nradio and X-ray rims can be jointly described by an am-\npli\fed and subsequently damped magnetic \feld.\n5.2. Model assumptions and correctness\nOur models adopted a distance dto Tycho of 3 kpc,\nbut estimates for Tycho's distance range between 2 :3{\n4 kpc (Hayato et al. 2010). As a larger remnant distance\nwould increase both physical \flament widths and shock\nvelocity estimates from proper motion, we may derive\nexpected scalings for modeled \u00112andB0as functions of\nassumed distance d. The advective lengthscale, equa-\ntion (10), may be rearranged to obtain:\nB0= (3:17\u0016G)\u0012vd\n108cm=s\u00132=3\n\u0002\u0012lad\n0:01 kpc\u0013\u00002=3\u0012h\u0017\n1 keV\u0013\u00001=3\n(15)\nor, more simply,\nB0/(vd)2=3(lad)\u00002=3\u0017\u00001=3: (16)\nBothladandvdscale linearly with remnant distance d\nand thus their e\u000bects cancel in determining the mag-\nnetic \feld. If di\u000busion is the primary control on \flament\nlengthscales, equation (11) yields:\n\u00112/(ldi\u000b)2B(\u0016+5)=2\n0\u0017\u0000(\u0016\u00001)=2(17)\nModel \fts with varying distance obey both scalings,\n\u00112/d2andB0constant. When comparing model \fts\nwith remnant distances of 3 kpc and 4 kpc, the devia-\ntion from the idealized scaling is .1% forB0and\u00181{14 Tran et al. (June 4, 2021)\n5% for\u00112. As varying remnant distance dleaves width-\nenergy scaling mEinvariant, the relative contributions of\nladandldi\u000bshould also be invariant. Then both length-\nscales should scale simultaneously with d, yielding the\nobserved behavior. In the damped model, a larger dis-\ntancedwill require larger physical damping lengths from\nmagnetic turbulence. But, \ftted values of abshould re-\nmain unchanged as we report abin units of shock radius\nrs.\nThe exponential variation in D(x) (equation (6)) may\nnot be physically reasonable; as noted above, the as-\nsumptionD(x)/1=B2(x) gives rise to sharp gradients\nin di\u000busion coe\u000ecient and even inverts the e\u000bect of \u00112on\nour model pro\fles (where larger \u00112can cause rim widths\nto narrow). Nevertheless, the modeled behavior appears\nphysically reasonable. Only X-ray pro\fles are impacted\nby the assumption on D(x) as radio pro\fles assume no\ndi\u000busion. And, model behavior driven by advection and\nmagnetic damping { namely, rim width-energy depen-\ndence in a damped magnetic \feld { occurs for D= 0, and\ncannot be an artifact of our assumption that D(x)B2(x)\nis constant.\nWe emphasize that additional counts from averaging\nmeasurements or selecting larger regions will likely not\nimprove our ability to constrain B0and\u00112from width-\nenergy modeling. This is easily seen from Figure 5 as well\nas large\u001f2values in Tables 3 that re\rect relatively tight\nerrors in our FWHM pro\fle measurements (Table 2).\n5.3. Other potential constraints\nRim width-energy dependence is a morphological man-\nifestation of spectral softening downstream of the for-\nward shock; previously, Cassam-Chena \u0010 et al. (2007) also\nsought to distinguish loss-limited and damped rims with\na careful spectral study and 1-D hydrodynamical model.\nAlthough knowledge of radial spectral variation, in prin-\nciple, fully determines rim pro\fles, there is not a clear\nrelationship between observed spectral variation and rim\nwidths alone. Our model, for example, replicates ob-\nserved rim width variation but underpredicts the ob-\nserved spectral variation. The more sophisticated model\nof Cassam-Chena \u0010 et al. (2007) similarly has trouble re-\nproducing the observed radial gradient in spectral pho-\nton index. Photon indices predicted in our model, for\nintegrated model rim spectra similar to those of Figure 3\nand Table 1, are also somewhat ill-constrained. Enforc-\ning limits on acceptable model photon indices (e.g., Fig.\n16 Cassam-Chena \u0010 et al. 2007) could help constrain shock\nparameters or identify discrepancies between model as-\nsumptions and measurements.\nEstimates of magnetic \feld strength that depend upon\nglobal models for Tycho's evolution (e.g., Morlino &\nCaprioli 2012) may not help constrain our rim model\nresults. In such models, assumptions on particle accel-\neration and magnetic \feld evolution downstream of the\nshock (e.g., further damping or ampli\fcation at the re-\nverse shock and contact discontinuity) would a\u000bect the\nspatially integrated spectrum of Tycho. We have focused\non e\u000bects immediately behind the forward shock, making\nas few assumptions as possible about particle accelera-\ntion and magnetic \felds throughout the remnant. It may\nnot make sense to extrapolate our results to estimate\nmagnetic \felds throughout the remnant or, similarly, to\nuse global models to constrain our results given all as-\n2’PO\nN\nM\nL\nK\nJ\nI\nH\nG\nF\nE\nD\nCBAFigure 7. Radio image of Tycho's SNR at 1 :375 GHz with linear\nscaling. Extraction regions (green) for joint radio and X-ray pro-\n\fle analysis overlay region selections for X-ray rim width analysis\n(Figure 1).\nsumptions involved.\n6.JOINT RADIO AND X-RAY MODELING\nThin radio rims spatially coincident with X-ray syn-\nchrotron rims may help constrain magnetic \feld damp-\ning. Radio synchrotron emission in the remnant interior\nmay arise from not only recently shock-accelerated elec-\ntrons but also long-lived electrons interacting with a tur-\nbulent \feld inside the remnant, as \u0018GeV electrons have\ncooling times of order 105{107yr in 10{100 \u0016G magnetic\n\felds. But simple models for synchrotron emissivity as\na function of density in Sedov and pre-Sedov dynami-\ncal stages predict radio synchrotron pro\fles rising grad-\nually to broad maxima well inside the shock radius (e.g.\nReynolds & Chevalier 1981; Reynolds 1988). Our steady-\nstate planar transport model predicts monotonically in-\ncreasing emission downstream of the forward shock in\na loss-limited model, until sphericity becomes important\nand our model is inapplicable. On scales of a few percent\nof the remnant radius, only magnetic \feld damping can\ncause emission to decrease.\nThe idea of modeling X-ray and radio pro\fles jointly\nwas pioneered by Cassam-Chena \u0010 et al. (2007), who\nfound that sharp radio rims in Tycho were not repro-\nduced by a loss-limited model. Cassam-Chena \u0010 et al.\n(2007) considered loss-limited and damping pro\fles con-\nsistent with physical constraints applied in a hydrody-\nnamical model. Here, we neglect physical constraints\nand estimate a damping length necessary to generate ra-\ndio rims with shape similar to observations.\nWe extract radio pro\fles from a 1 :375 GHz image of\nTycho taken with the Very Large Array (VLA) in A con-\n\fguration in March 1994 (PI: D. Mo\u000bett); see Reynoso\net al. (1997) for a detailed presentation. The half-power\nbeam width of\u00181:500just resolves thin radio rims and\nstructure near the forward shock; the image is sampled\nat 0:500. We also extracted 4{7 keV X-ray pro\fles in\nall regions from the previous archival Chandra observa-\ntion to jointly model radio and X-ray pro\fles, permit-Tycho's Synchrotron Rims (June 4, 2021) 15\nTable 6\nFit parameters from pro\fle shape comparison\nRegion B0abRegion B0ab\n(\u0016G) (-) ( \u0016G) (-)\nA 50 0.020 I 250 0.020\nB 200 0.050 J 300 1\nC 15 0.010 K 400 0.010\nD 100 0.050 L 250 1\nE 120 0.030 M 200 1\nF 300 0.025 N 800 0.010\nG 250 0.020 O 200 0.005\nH 300 0.020 P 150 0.012\nNote . | Damping lengths of 1indicate that a loss-limited \ft is\nfavored (ab>10% of shock radius rs).\nting somewhat \frmer discrimination of plausible model\nparameters. Figure 7 shows the extraction regions over-\nlaying the radio image and the previous X-ray pro\fle\nregions of Figure 1.\nWe compute model radio and X-ray measured pro\fles\nfrom the transport model of equation (1) for varying B0\nandab, similar to those shown in Figure 2. The pa-\nrametersBmin= 5\u0016G,\u00112= 1, and\u0016= 1 are held\n\fxed. Di\u000busion, in particular, is negligible for modeled\nradio emission as particle energies are 3 orders of mag-\nnitude lower than in X-ray. Neglecting di\u000busion also cir-\ncumvents the unphysical assumption that D(x)B2(x) is\nconstant, which was invoked to obtain Green's function\nsolutions to equation (1).\nWe align each set of model pro\fles at some B0and\nabto measurements by eye, varying relative amplitudes\nand translations in radio and X-ray independently to\nbest match the measured pro\fles. Although \\\ftting\"\nby eye does not quantitatively bound model parame-\nters, we can \fnd plausible values of B0andaband es-\ntimate the importance of magnetic damping throughout\nthe remnant. More involved nonlinear \ftting may be un-\nreliable as we cannot constrain spatially heterogeneous\nradio emission within the remnant. Moreover, our trans-\nport model neglects self-similar downstream evolution of\nshocked plasma (decaying velocity, density, and magnetic\n\feld) and is inaccurate further downstream than \u001810%\nof the shock radius.\nThe joint radio and X-ray pro\fle modeling contrasts\nstrongly with our previous width-energy \ftting from X-\nray measurements alone, which neglected pro\fle shape in\nfavor of more robust FWHM measurements. Manually\n\ftting pro\fles allows us to consider radio and X-ray \fla-\nments that do not have well-de\fned FWHMs, especially\nas radio rims do not fall below 50% of the peak emission.\nWe can also use pro\fles from regions not previously con-\nsidered due to the lack of an X-ray FWHM, especially in\nsofter (0:7{4 keV) X-rays.\nWe identify three classes of radio pro\fles: thin rims\nwith downstream troughs, plateaus, and continuous rises.\nRegions B, C, and D (southern limb) show plateaus in\nradio emission. Regions J, L, and M (around NW) show\ncontinuous rises in emission. All other regions have a\nradio rim within 1500of the forward shock, where the for-\nward shock in radio is assumed at zero intensity. Figure 8\nshows extracted radio and nonthermal X-ray (4{7 keV)\npro\fles for three well-\ft regions with our best manually\nselected model pro\fles, illustrating each of the three ra-\ndio pro\fle types observed.\n215 220 225 230 235 240 245\nRadial distance (arcsec)0.000.010.020.030.040.050.060.07Radio intensity (Jy/beam, arbitrary scaling)1.375 GHz\nab=0.05,B0=100µG\nab=0.05,B0=200µG\nab=0.05,B0=300µG\nab=0.07,B0=200µG\nab=0.03,B0=200µG\n215 220 225 230 235 240 245\nRadial distance (arcsec)0.00.51.01.52.0X-ray intensity (cts/cm2/s, arbitrary scaling)×10−8\nB 4-7 keV\n215 220 225 230 235 240 245\nRadial distance (arcsec)−0.0050.0000.0050.0100.0150.020Radio intensity (Jy/beam, arbitrary scaling)1.375 GHz\nab=0.01,B0=300µG\nab=0.01,B0=400µG\nab=0.01,B0=500µG\nab=0.02,B0=400µG\nab=0.005, B0=400µG\n215 220 225 230 235 240 245\nRadial distance (arcsec)0.00.51.01.52.02.53.03.54.04.5X-ray intensity (cts/cm2/s, arbitrary scaling)×10−8\nK 4-7 keV\n215 220 225 230 235 240 245\nRadial distance (arcsec)0.000.010.020.030.040.050.06Radio intensity (Jy/beam, arbitrary scaling)1.375 GHz\nab=∞,B0=150µG\nab=∞,B0=200µG\nab=∞,B0=300µG\nab=0.1,B0=200µG\nab=0.05,B0=200µG\n215 220 225 230 235 240 245\nRadial distance (arcsec)0.00.20.40.60.81.01.21.41.6X-ray intensity (cts/cm2/s, arbitrary scaling)×10−8\nM 4-7 keVFigure 8. Measured radio and X-ray pro\fles plotted with model\npro\fles for varying abandB0in each region, showing typical pa-\nrameters (and ranges) required to reproduce radio and X-ray rims\nsimultaneously in our model. Solid black curves in all regions plot\nour manually chosen best model pro\fles. Pro\fles are chosen to\nshow varying radio rim morphology, including plateaus (B), thin\nrims with troughs (K), and continuous rises (M); these pro\fles\nshow some of the best agreement of our selected regions, but cf.\nFigure 9. Pro\fle radial coordinates are shifted arbitrarily to aid\nvisual comparison. Negative radio intensity is unphysical and as-\nsociated with deconvolution of raw VLA visibilities.\nOur model requires damping length ab.0:1 to pro-\nduce a plateau or thin rim in radio emission. For regions\nwith thin radio rims, the best manually selected pro\fles\nhaveB0between 50{400 \u0016G, neglecting only Region N\nwhich could not be modeled simultaneously in both X-\nray and radio (Figure 9). The damping length abranges\nbetween 0:01{0:03, or 2{700. We list estimated best \ft\nparameters in Table 6.\nThe radio plateaus in regions B, C, D are compatible\nwith damping lengths between 0 :01{0:05. Continuous\nrises in radio emission are best modeled with ab&0:1 and\nB0\u0018200{300\u0016G. Although damping lengths ab&0:1\nare not physically meaningful well beyond the shock,\nthese large values of abyield practically constant mag-\nnetic \feld near the shock. Our results are also insen-\nsitive to the assumed value of Bmin; only model pro-\n\fles with small magnetic \felds ( B0.100\u0016G) and/or\nsmall damping length ( ab<0:01) are a\u000bected if we take\nBmin= 2\u0016G rather than 5 \u0016G.\nTycho's shock structure is more complex than assumed16 Tran et al. (June 4, 2021)\n215 220 225 230 235 240 245\nRadial distance (arcsec)0.0000.0050.0100.0150.0200.0250.030Radio intensity (Jy/beam, arbitrary scaling)1.375 GHz\nab=0.05,B0=250µG\nab=0.02,B0=250µG\nab=0.01,B0=250µG\n215 220 225 230 235 240 245\nRadial distance (arcsec)0.00.51.01.52.02.5X-ray intensity (cts/cm2/s, arbitrary scaling)×10−8\nI 4-7 keV\n215 220 225 230 235 240 245\nRadial distance (arcsec)−0.020.000.020.040.060.08Radio intensity (Jy/beam, arbitrary scaling)1.375 GHz\nab=0.005, B0=23µG\nab=0.01,B0=400µG\nab=0.005, B0=800µG\nab=0.01,B0=800µG\n215 220 225 230 235 240 245\nRadial distance (arcsec)0123456X-ray intensity (cts/cm2/s, arbitrary scaling)×10−8\nN 4-7 keV\nFigure 9. Extracted pro\fles poorly reproduced by our model,\ncompared to Figure 8. Region I shows irregularly shaped radio\nrim. Region N contains two superposed \flaments that cannot be\nmodeled by a single rim in both X-ray and radio; the narrow X-\nray rim requires atypically strong magnetic \felds ( \u0018800\u0016G), and\nthe emission plateau behind the radio rim requires small damping\nlengths (ab\u00180:005).\nby our transport model. Emission towards the remnant\ninterior clearly shows spatial structure (Figure 7). Figure\n9 shows two regions that were poorly described by our\nmodel. A majority of our regions have irregular rims;\nin at least 2{3 regions, this may be attributed to pro-\njection of multiple \flaments. Others (e.g., Region K)\nshow rims with slopes that cannot be matched by our\nmodels, whether too steep or shallow. Shape mismatch\nmay be attributed in part to point-spread mismatch, dif-\nfusion (e.g., \u001126= 1), shock precursors, or other e\u000bects.\nNevertheless, the conclusion that thin radio rims require\nmagnetic damping is supported by more sophisticated\nmodeling. As we have noted, hydrodynamic models with\ndi\u000busive shock acceleration (Cassam-Chena \u0010 et al. 2007;\nSlane et al. 2014) also cannot produce radio pro\fles with\nnarrow rims in a purely advected magnetic \feld. It is\nalso not clear that radio structure is due to a global ra-\ndial variation in magnetic \feld strength. E.g., Slane et al.\n(2014) suggest that Rayleigh-Taylor \fngers between the\nforward shock and contact discontinuity could locally\ncon\fne radio-emitting electrons, creating observed radio\nrim structure. But, if radio rims are due strictly to mag-\nnetic \feld e\u000bects, our conclusions on magnetic \feld drop-\no\u000b are independent of the causative physical mechanism\nso long as they yield magnetic \feld fall-o\u000b over 1% of\nremnant radius.\nWe may also use spatially overlapping radio and X-\nray region selections (Figure 7) to attempt to constrain\nX-ray width-energy \fts. All X-ray region selections (Fig-\nure 1) are associated with radio rims, except for Regions\n11 and 12 underlying Region J in radio. X-ray emission\nat regions B, C, D (radio plateaus) and L, M (radio rises)\nshowed rims, but their widths either could not be mea-\nsured, or could only be measured in the 4{7 keV band.If radio rims require damping at \u00181% remnant radius,\nwidth-energy \fts with \\weak-\feld\" damping lengthscales\nof\u00180:5% may be disfavored. Conversely, the radio and\nX-ray \flament of Regions 11,12 and J is the best remain-\ning loss-limited rim candidate, although width-energy \fts\nare equivocal towards damped and loss-limited models.\n7.CONCLUSIONS\nWe measured the widths of several thin synchrotron\n\flaments around Tycho's supernova remnant and found\nmoderate narrowing of rim widths throughout the rem-\nnant, corroborating rim narrowing observed by Ressler\net al. (2014) in the remnant of SN 1006. We con\frmed\nthat selected \flaments are dominated by nonthermal\nemission and have clearly measurable full widths at half\nmaximum in 4{5 energy bands. Both X-ray width-energy\n\fts and joint radio/X-ray pro\fle modeling require mag-\nnetic \felds &20\u0016G even with magnetic damping.\nA steady-state particle transport model with constant\nmagnetic \feld gives di\u000busion coe\u000ecients and magnetic\n\feld strengths broadly consistent with prior estimates\nfrom rim widths (e.g., Parizot et al. 2006; Rettig &\nPohl 2012) and radio and gamma ray measurements (Ac-\nciari et al. 2011; Morlino & Caprioli 2012). The same\nmodel with a damped magnetic \feld is equally capa-\nble of describing our measured data. At weak energy\ndependence the two models are indistinguishable and\nmagnetic damping \fts favor moderately ampli\fed mag-\nnetic \felds beyond simple compression, but lower than\nfor loss-limited models. At moderate energy dependence\n(mE\u0018\u00000:3), the damping model permits weak magnetic\n\felds and short damping lengths ( <1% remnant radius)\nto reproduce energy dependence, but we still cannot fa-\nvor either damped or loss-limited rims due to \ft uncer-\ntainty. The distinction between loss-limited and damped\nmodels is somewhat arti\fcial; only for large di\u000berences\nbetween the various transport lengthscales, a frequency-\ndependent occurrence, is one or the other mechanism\nclearly dominant.\nThin radio synchrotron rims, however, are not re-\nproduced in loss-limited models (Cassam-Chena \u0010 et al.\n2007). Assuming shocked electrons account for most ra-\ndio emission immediately downstream of Tycho's forward\nshock, we jointly model radio and X-ray pro\fles and \fnd\nthat damping lengths of 1{5% of the shock radius are\nrequired throughout most of the remnant; only a few\n(3/16) selected regions are plausibly consistent with a\nconstant advected magnetic \feld. Typical magnetic \feld\nstrengths range between 50{400 \u0016G. Although we can-\nnot bound damping lengths and \felds from our qualita-\ntive pro\fle comparisons, our results are physically rea-\nsonable and are likely good to order-of-magnitude. If\ndamping lengths inferred from radio rims are correct,\n\\weak-\feld\" damping is disfavored in explaining X-ray\nrim width-energy dependence, and damped rims require\nmagnetic \feld ampli\fcation to \u0018100\u0016G or more in Ty-\ncho.\nWe thank the anonymous referee for comments that\nhelped clarify and improve the manuscript. David Mof-\nfett kindly provided the VLA image used in this study.\nThe scienti\fc results reported in this article are based\non data obtained from the Chandra Data Archive. ThisTycho's Synchrotron Rims (June 4, 2021) 17\nresearch made extensive use of NASA's Astrophysics\nData System. 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N., & Aharonian, F. 2007, A&A, 465, 695"    },    {        "title": "0708.2467v1.Non_Riemannian_geometrical_asymmetrical_damping_stresses_on_the_Lagrange_instability_of_shear_flows.pdf",        "content": "arXiv:0708.2467v1  [physics.flu-dyn]  18 Aug 2007Non-Riemannian geometrical asymmetrical damping\nstresses on the Lagrange instability of shear flows\nby\nL.C. Garcia de Andrade\nDepartamento de F´ ısica Te´ orica – IF – Universidade do Esta do do Rio de Janeiro-UERJ\nRua S˜ ao Francisco Xavier, 524\nCep 20550-003, Maracan˜ a, Rio de Janeiro, RJ, Brasil\nElectronic mail address: garcia@dft.if.uerj.br\nAbstract\nIt is shown that the physical interpretation of Elie Cartan t hree-dimensional space tor-\nsion as couple asymmetric stress, has the effect of damping, p reviously Riemannian unstable\nCouette planar shear flow, leading to stability of the flow in t he Lagrangean sense. Actu-\nally, since the flow speed is inversely proportional to torsi on, it has the effect of causing\na damping in the planar flow atenuating the instability effect . In this sense we may say\nthat Cartan torsion induces shear viscous asymmetric stres ses in the fluid, which are able\nto damp the instability of the flow. The stability of the flow is computed from the sec-\ntional curvature in non-Riemannian three-dimensional man ifold. Marginal stability is ass-\nsumed by making the sectional non-Riemannian curvature zer o, which allows us to determine\nthe speeds of flows able to induce this stability. The ideas di scussed here show that tor-\nsion plays the geometrical role of magnetic field in hydromag netic instability of Couette flows\nrecently investigated by Bonnano and Urpin (PRE, (2007,in pr ess) can be extended and ap-\nplied to plastic flows with microstructure defects. Recentl y Riemannian asymmetric stresses\nin magnetohydrodynamics (MHD) have been considered by Bill ig (2004). PACS numbers:\n02.40.Hw:Riemannian geometries\n1I Introduction\nRecently two papers on the use of stresses magnetic[1, 2] or n onmagnetic [3] have been\nconsidered by Billig [1], Bonnano and Urpin [2] and L’ov et al [ 3]. In the first the Riemannian\ngeometry was used along asymmetric stresses in MHD equation s. In the second, Bonnano\net al showed that the presence of magnetic field could induce i nstabilities in Couette plane\nflows , and in the third also instabilities in polymer interac tions of fluids were investigated\nalso in MHD. All these three papers together motivated us to u se Elie Cartan idea [1] of\nassociating a non-Riemannian asymmetric connection known nowadays by the name of Cartan\ntorsion tensor [5, 6, 7], to moment or torque stresses to a Cou ette plane flow which has been\nsystemmatically used over the years to investigate its Lagr angean stability. Cartan torsion\nhas been also used in the fields of gravitation and cosmology, where torsion is in general\nassociated with a spin density, often called Einstein-Cart an (EC) theory [8], or in modern\nlanguage of theoretical physics, a torsioned high-energy m enbranes [9]. Analogies between\ntorsion in defects and in solids [10] and EC gravity has been p ut foward by Maugin [11] and\nKr¨ oner [12] and developed most recently by Epstein [13]. De spite Cartan first motivation to\napply torsion to physics was Einstein general relativistic cosmology, he argued that this study\ncould be done easily by investigating the mechanical torque stresses [4] in three-dimensional,\nwhich could be exactly due to asymmetries in coupled shear st resses. In this brief report,\nwe give an example that torsion can be applied also in physics of fluids, when the fluid, in\nthe case a Couette type flow, is viscous and sheared. Actually Kambe [14] has previously\ninvestigate Couette planar flows stability, and showed that Riemann curvature tensor would\nexist in the case of existence of pressure, even a constant on e. The stability of the Couette\nplanar flow is also obtained by him, using the symmetry in the c ase of symmetric scalar stress\npotential and by making use of the technique of Ricci section al curvature [15], where the\nnegative sectional curvature indicates instability of the flow, in the Lagrangean sense. This\nidea can be easily shifted to flows in the case they have strong shear and viscosity and are\nable to support these antisymmetric stresses. In this brief report, we also show that allowing\nthe presence of shearing stresses, and associating torsion to these stresses, the Couette flows\nare can be stable even in the Lagrangean sense. This can be exp lained by the fact, that as we\n2shall show, velocities involved in the fluid are inversely pr oportional to the flow torsion, and\nas torsion grows the velocity in the fluid decreases, which is a damping like physical effect in\nthe flow. Actually, Kondo [10] has shown that the antisymmetr y of stress tensor derivativative\nis associated with Cartan torsion. To test the range of this L agrangean stability, we assume\na marginal stability and put the sectional curvature for the torsioned connection to zero, and\nfind an algebraic equation to previously arbitrary velociti es in the fluid. Torsion would only\nallows for the instability if its sectional curvature is neg ative. A previously application of a\nnon-torsion free connection in fluids, in the context of soli tons, have been investigated by\nRicca [16]. Rakotomanana [17] in turn have investigated NR m anifolds with Cartan torsion, in\nthe context of the geometrical approach to the thermodynami cs of dissipating continua. The\nnon-Riemanniangeometry ofvortex acoustic flows havebeen a lso recently addressed [18]. The\npaper is organized as follows: Section II presents a brief re view of non-Riemannian geometry\nin the coordinate free language. Section III presents model , along with computation of the\nLagrange stability of non-Riemannian Couette flow. Section IV presents the conclusions.\nII Sectional non-Riemannian curvature\nIn this section, before we add we make a brief review of the diff erential geometry of surfaces\nin coordinate-free language. The Riemann curvature is defin ed by\nR(X,Y)Z:=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z (II.1)\nwhereXǫTMis the vector representation which is defined on the tangent s paceTMto the\nmanifold M. Here∇XYrepresents the covariant derivative given by\n∇XY= (X.∇)Y (II.2)\nwhich for the physicists is intuitive, since we are saying th at we are performing derivative along\nthe X direction. The expression [X,Y]represents the commutator, which on a vector basis\nframe/vector elin this tangent sub-manifold defined by\nX=Xk/vector ek (II.3)\n3or in the dual basis ∂k\nX=Xk∂k (II.4)\ncan be expressed as\n[X,Y] = (X,Y)k∂k (II.5)\nIn this same coordinate basis now we are able to write the curv ature expression (II.1) as\nR(X,Y)Z:= [Rl\njkpZjXkYp]∂l (II.6)\nwheretheEinstein summationconventionoftensor calculus isused. Theexpression R(X,Y)Y\nwhich we shall compute bellow is called Ricci curvature. The sectional curvature which is very\nuseful in future computations is defined by\nKRiem(X,Y) :=< R(X,Y)Y,X >\nS(X,Y)(II.7)\nwhereS(X,Y)is defined by\nS(X,Y) :=||X||2||Y||2−< X,Y >2(II.8)\nwhere the symbol <,>implies internal product. In the non-Riemannian (NR) case, the torsion\ntwo-form T(X,Y)is defined by\nT(X,Y) :=1\n2[¯∇XY−¯∇YX−[X,Y]] (II.9)\nwhere¯∇is the non-Riemannian connection [7] endowed with torsion. As in EC theory [5]\nthe geodesic equation does not depend on torsion; only Jacob i deviation equation depends\non torsion which is enough for investigate the role of torsio n on stability. Since the Jacobi\nequation is given by\nd2J\nds2= [||∇/vectort/vector eJ||2−KNR(t,/vector eJ)||J|| (II.10)\nwhere||/vector eJ||= 1and J is the Jacobi field, representing the separation betwee n geodesics,while\n/vectortis the geodesic tangent vector. Here KNR(X,Y)is given by\nKNR(X,Y) =KRiem(X,Y)+2< T(X,Y),¯∇YX > (II.11)\nHere as we shall see bellow the geodesic equation is ∇YY= 0simplified this expression. Note\nfrom this expression that the instability, or separation of the geodesics in the flow\nd2J\nds2≥0 (II.12)\nimplies that KRiem<0which is the condition for Lagrange instabillity.\n4III Couette shear flow stability in non-Riemannian\nbackground\nIn this section we shall consider the Couette constant press ure, planar shear flow [11]\nY= (U(y),0,0) (III.13)\nwith constant pressure p where U(y) =yand X is given by\nX=/vector vlel (III.14)\nwhereel:=exp[i(/vectorl/vector x)]and/vector x= (x,y,z)and/vectorl:= (lx,ly,lz)is the wave number. Here /vector vl\nrepresents an arbitrary velocity, which shall be determine d below in order to generate non-\nRiemannian stability of Couette shear flows. The hydrodynam ics inR3Euclidean space [12]\nis given by\n¯∇XY= (X.∇)Y+gradpXY (III.15)\nwhere the covariant derivative on the RHS of this equation [1 2] is\n(X.∇)Y=el(/vector vl.∇)(U(y),0,0) =el(vy\nlU′(y),0,0) (III.16)\nA simple computation led Kambe to the result\n∇2pXY=∇2pYX=−ilxvlyel (III.17)\nsinceU′(y) = 1. Note however that the equation (III.17) does not necessaril y implies that\npXY=pYX, since this is a sufficient but not necessary solution in the ma thematical language.\nThis can be easily seen by the argument that the equation (III .15) is equivalent to\n∇2[pXY−pYX] = 0 (III.18)\nand since ∇2=∇.∇we have that\n∇pXY−∇pYX=/vector c (III.19)\nwhere/vector cis an arbitrary constant. As we shall show this reasoning lea ds exactly to Cartan\ntorsion 2-form T(X,Y)which using the expression\npXY=−ilxvly(III.20)\n5implies that Cartan torsion vector can be expressed as\nT(X,Y) =grad(pXY−pYX) = [Tk\n21X2Y1]∂k (III.21)\nwhere the covariant components of torsion are\nTk\n21=c\nvxlU(y)=c\nvxly(III.22)\nAn example of totally skew-torsion [19] has shown also be pre sented in the cholesteric blue\nphase of liquid crystals. The torsion vector (III.22) in the N R sectional curvature obtains\n2< T(X,Y),¯∇YX >= (2π)3U′2m2x\nm2|vym|2(III.23)\nwhich comes from the relation\n2< T(X,Y),¯∇YX >=KRiem(X,Y)(2π)3U′2m2x\nm2|vym|2(III.24)\nwhich is the condition for marginal stability of\nKNR(X,Y) = 0 (III.25)\nMaking use of the expression for the covariant derivative in the Riemann-Cartan connection\n¯∇\n¯∇YX=imxemy /vector vm−mx\nm2em/vector m (III.26)\nSubstitution of the value of torsion form above and this cova riant derivative into expression\n(III.23) yields\nimxU(y)emU(y)< /vector vm,/vector c >−mx\nm2vmyU′(y)em< /vector m,/vector c > = (2π)3U′2m2x\nm2|vym|2(III.27)\nThus since U(y) =ythis equation is simply solved when we choose a direction for /vector vm. The\nsimplest choice is /vector c=c/vector m, where< /vector m, /vector m > =m2. Since [12] < vm, /vector m >= 0, which upon\nsubstitution into (III.27) yields\nvmy=−m2c\n(2π)3(III.28)\nPhysically this means that the curvature acts as a damping to the flow. Substitution of the\nvalue of the torsion scalar c into the value of torsion compon ent one obtains\nTk\n21=−1\n2(2π)3vym\nm2vlxy(III.29)\nA simple observation of this expression shows that the torsi on has the proper units of ∼L−1\nwhere L represents the length scale in the flow or liquid cryst al.\n6IV Conclusions\nOneofthemostimportantfeaturesoftheinvestigationofth estabilityofflowsintheEuclidean\nmanifold E3, in some detail the stability of an incompressible or volume preserving flow, using\nthe method of the sign of the Ricci sectional curvature. Also important is the issue of stability\nin plasma astrophysics as well as in fluid mechanics. In this p aper we discuss and present\nthe contribution of Cartan torsion tensor on damping the Rie mannian Lagrange instabilities of\nviscous Couette shear flows. We could say that contrary to the Riemannian case [12] where\nthe flow is unstable in the Lagrangean particle sense, but is n eutrally stable, here the Couette\nshear flow may be fully stable for an appropriate chice of velo cities. The role of torsion in\ncrystal curvature frustration [20] could also beinvestiga te concerning the stability ofgeodesics.\n7References\n[1] Y.Billig, Magnetic hydrodynamics with asymmetric stres s tensor, ArXiv:math-\nph/0401052v1.\n[2] A. Bonnano, Y. Urpin, Phys Rev E (2007) in press.\n[3] V. S. L’vov, A. Polyalov, I. Procaccia,V. Tiberkevich, Phy s. Rev. E 71 (2005) 016305.\n[4] E. Cartan, C.R. Acad. Sciences 174 (Paris) (1922) 593. L.C. Garcia de Andrade, C. A.\nSouza Lima jr, On the differential geometry of torque stresse s in cylindrical solids (1995)\nExtract Mathematicae 11, 1, 59.\n[5] E. Cartan, A. Einstein, Letters on Absolute Parallelism -(1929-1932), ed. R. De-\nbever,Princeton (1979).\n[6] E. Cartan, Riemann Spaces (2000) MIT Press, Boston.\n[7] E. Cartan, Riemannian Geometry on Orthonormal Frame (20 00) Princeton.\n[8] F. W. Hehl, Yu N. Obukhov, Foundations of Classical Elect rodynamics: Charge, Flux and\nMetric (2003), Birkhauser.\n[9] B. Muhopadhyaya, S. Sen, S, SenGupta, Phys Rev Lett. 89 (2 002) 121101.\n[10] K. Kondo, RAAG Memoirs of the Unified Problems in the scie nce and Engineering by\nMeans of Geometry (1955) vol. I.\n[11] G. Maugin, Material Inhomogeneous in Elasticity.\n[12] E. Kroener, Theory of Defects, Les Houches school, (197 0) Dunod, Paris.\n[13] M. Epstein, M. Elzanowski, Material Inhomogeneities a nd their Evolution: A Geometric\nApproach (2007) Springer in press.\n[14] T. Kambe, Geometrical Theory of Dynamical Systems and F luid Flows (2004) World\nScientific.\n8[15] I. Benn, W. Tucker, An introduction to spinors and geome try with applications to physics\n(1987) Adam Hilger, Bristol.\n[16] R. Ricca, Phys Rev A 43 (1990) 4281.\n[17] L.R. Rakotomanana, A Geometric Approach to the Thermome chanics of continua (2003)\nBirkhauser.\n[18] L.C. Garcia de Andrade,Phys Rev D 70 (2004) 640004-1\n[19] E. Dubois-Violette, E. Pansu, Which Universe is Blue Ph ase, in Geometry in Condensed\nMatter Physics (1990) World Scientific.\n[20] J.F. Sadoc, R. Mosseri, Geometric Frustration (2006) Ca mbridge university press. ,\n9"    },    {        "title": "1710.07717v1.Magnetic_field_amplification_in_supernova_remnants.pdf",        "content": "arXiv:1710.07717v1  [astro-ph.HE]  20 Oct 2017DRAFT VERSION OCTOBER 24, 2017\nPreprint typeset using L ATEX style emulateapj v. 01/23/15\nMAGNETIC FIELD AMPLIFICATION IN SUPERNOV A REMNANTS\nSIYAO XU1,2AND ALEX LAZARIAN1\nDraft version October 24, 2017\nABSTRACT\nBased on the new findings on the turbulent dynamo in Xu & Lazari an (2016), we examine the magnetic\nfield amplification in the context of supernova remnants. Due to the strong ion-neutral collisional damping\nin the weakly ionized interstellar medium, the dynamo in the preshock turbulence remains in the damping\nkinematic regime, which leads to a linear-in-time growth of the magnetic field strength . The resultant magnetic\nfield structure enables effective diffusion upstream and sh ock acceleration of cosmic rays to energies above the\n“knee”. Differently, the nonlinear dynamo in the postshock turbulence leads to a linear-in-time growth of the\nmagnetic energy due to the turbulent magnetic diffusion. Given a weak initia l field strength in the postshock\nregion, the magnetic field saturates at a significant distanc e from the shock front as a result of the inefficiency of\nthe nonlinear dynamo. This result is in a good agreement with existing numerical simulations and well explains\nthe X-ray spots detected far behind the shock front.\nSubject headings: supernova remnants - turbulence - magnetic fields - cosmic ra ys\n1.INTRODUCTION\nSupernova remnants (SNRs) are the most plausible sources\nof Galactic cosmic rays (CRs) (Blandford & Eichler 1987).\nMagnetic-field amplification is expected in SNRs to ensure\nan efficient diffusive shock acceleration (DSA) (Axford et a l.\n1977; Bell 1978), and is also supported by observational ev-\nidence, e.g., the magnetic fields on the order of 100µG near\nthe shock front inferred from narrow X-ray synchrotron rims\n(Bamba et al. 2003, 2005b,a; Vink 2012), the milligauss mag-\nnetic fields suggested by the rapid X-ray variability from co m-\npact sources in the downstream region (Patnaude & Fesen\n2007; Uchiyama et al. 2007; Uchiyama & Aharonian 2008).\nThe milligauss fields are also indicated from radio observa-\ntions (Longair 1994). Clearly, the amplified magnetic fields\nin SNRs cannot be accounted for by the shock compression\nof the interstellar field strength of a few microgauss.\nThere are extensive studies on the origin of the mag-\nnetic fluctuations for confining CRs near the shock, e.g.,\nthe non-resonant streaming instability for driving mag-\nnetic fields at length scales smaller than the CR Lar-\nmor radius (Bell 2004), an inverse cascade of Alfv´ en\nwaves excited at the Larmor radius to larger wavelengths\n(Diamond & Malkov 2007). More generally, turbulence is be-\nlieved to be an efficient agent to amplify magnetic fields via\nthe turbulent dynamo (Kazantsev 1968; Kulsrud & Anderson\n1992), which has also been invoked for explaining both\nthe preshock (Beresnyak, Jones, & Lazarian 2009, hereafter\nBJL09; Drury & Downes 2012; del Valle et al. 2016) and\npostshock (Balsara et al. 2001; Giacalone & Jokipii 2007;\nInoue et al. 2009; Guo et al. 2012; Fraschetti 2013; Ji et al.\n2016) magnetic fields. As turbulence is induced by the inter-\naction between SNR shocks and interstellar turbulent densi ty\nfluctuations, the turbulent dynamo is inevitable in SNRs wit h\nthe turbulent kinetic energy dominating over the pre-exist ing\nmagnetic energy.\nThe theoretical advances in magnetohydrodynamic (MHD)\nturbulence have been made since Goldreich & Sridhar (1995)\n1Department of Astronomy, University of Wisconsin, 475 Nort h\nCharter Street, Madison, WI 53706, USA; sxu93@wisc.edu, la zar-\nian@astro.wisc.edu\n2Hubble Fellowand later works (Lazarian & Vishniac 1999; Cho & Vishniac\n2000; Maron & Goldreich 2001) with the conceptual im-\nprovement by introducing the local system of reference. They\nbring new physical insights into the turbulent dynamo prob-\nlem. Within the framework of the Goldreich & Sridhar (1995)\nmodel of MHD turbulence, a detailed analytical study on the\nturbulent dynamo process in plasmas with arbitrary conduc-\ning and ionization degrees was carried out by Xu & Lazarian\n(2016) (hereafter XL16). A remarkable finding there is that\nthe kinematic dynamo in a weakly ionized medium leads to\na linearly growing field strength with time3, and the result-\ning characteristic scale of the magnetic field can significan tly\nexceed the viscous scale of turbulence. This new dynamo\nregime is referred to as the “damping regime”. Since the inte r-\nstellar media (ISM) that SNR shocks sweep through are fre-\nquently partially ionized (Draine 2011), the significant mo di-\nfications on the kinematic dynamo in the presence of neutrals\nshould be incorporated when studying the magnetic field am-\nplification in the preshock region and its implications on CR\nacceleration.\nFor CR acceleration at shocks, strong magnetic fields in\nboth the pre- and post-shock regions are necessary to trap\nand mirror CR particles to facilitate many shock crossings.\nThe amplification of the preshock magnetic field is cru-\ncial. It has been modelled in earlier studies (e.g. BJL09;\nDrury & Downes 2012; del Valle et al. 2016) in an ideal sit-\nuation with a fully ionized upstream plasma, and the amplifi-\ncation time is rather limited given a relatively thin precur sor\n(BJL09). To reach a more realistic and generalized descrip-\ntion, here we consider the partial ionization of the ISM and\nexamine its influence on the CR diffusion in the amplified\nmagnetic field in the CR precursor.\nIn the highly ionized postshock medium, the turbulent en-\nergy can cascade down to quite small scales. Very likely,\nthe turbulent dynamo starts with an equipartition between t he\nmagnetic and turbulent energies at an intermediate scale an d\nfall in the nonlinear regime. Different from the kinematic\ndynamo with a strong dependence on the microscopic mag-\n3This should not be confused with the linearly growing magnet ic energy\nwith time that characterizes the nonlinear turbulent dynam o regime (see the\nfollowing text).2\nnetic diffusion, the nonlinear dynamo, which is initiated b y\nthe equipartition between the magnetic energy and the local\nturbulent energy, is mainly subject to the turbulent magnet ic\ndiffusion, and consequently the magnetic energy has a linea r-\nin-time growth with a low dynamo efficiency. This theoreti-\ncal result in XL16 quantitatively agrees with earlier numer ical\nstudies, e.g., Cho et al. (2009), Beresnyak (2012).\nMotivated by the enhanced field strength in SNRs indicated\nby observations, in this work, we investigate the magnetic\nfield amplification in both the pre- and post-shock regions of\na SNR by applying the general turbulent dynamo theory de-\nveloped by XL16 and discuss its implications on the CR ac-\nceleration. In Section 2, we analyze the magnetic field ampli -\nfication in the weakly ionized preshock medium and its impli-\ncation on the CR diffusion upstream. In Section 3, we study\nthe magnetic field amplification and CR diffusion in the fully\nionized postshock medium. Discussions on an alternative ac -\nceleration mechanism of CRs in the postshock MHD turbu-\nlence are in Section 4. In Section 5, we summarize the main\nresults.\n2.MAGNETIC FIELD AMPLIFICATION IN THE\nPARTIALLY IONIZED PRESHOCK REGION\nThe turbulence in the preshock region can result from the\nCR pressure gradient in the shock precursor interacting wit h\nthe density inhomogeneities in the upstream ISM (BJL09).\nThe pre-existing interstellar field of a few µG is expected to\nbe amplified by the resulting turbulence via the small-scale\nturbulent dynamo process. The dynamo growth of magnetic\nfield is driven by the turbulent motions. They are essentiall y\nhydrodynamic over the length scales larger than the equipar -\ntition scale, where the turbulent and magnetic energies are\nin equipartition, and are assumed to follow the Kolmogorov\nscaling.\nIn the partially ionized ISM, the dynamo evolution of mag-\nnetic field in the linear regime has its time-dependence and\ngrowth rate strongly affected by the ion-neutral collision al\ndamping (Kulsrud & Anderson 1992; XL16). For the prop-\nagation of a strong shock wave through the partially ionized\nISM, it is necessary to take into account the partial ioniza-\ntion of the upstream medium for a realistic description of th e\nmagnetic field amplification in the preshock region.\n2.1. Damping effect on the kinematic dynamo\nThe damping rate of magnetic fluctuations due to ion-\nneutral collisions is given by (Kulsrud & Pearce 1969;\nKulsrud & Anderson 1992),\nωd=Ck2EM,C=ξn\n3νni, (1)\nwhereEMis the magnetic-fluctuation energy per unit mass,\nξn=ρn/ρis the neutral fraction with the neutral mass den-\nsityρnand the total mass density ρ, andνni=γdρiis the\nneutral-ion collision frequency with γdas the drag coefficient\n(see Shu 1992) and ρias the ion mass density.\nMagnetic fluctuations in strongly coupled neutrals and\nions are also subject to the neutral-viscous damping\n(Lazarian et al. 2004; Xu et al. 2015). The corresponding vis -\ncous scale is\nkν=L−1\n4V3\n4\nLν−3\n4n, (2)\nwhere the Kolmogorov scaling of turbulence is used, VLis the\nturbulent velocity at the injection scale L,νn=vth/(nnσnn)is the kinematic viscosity in neutrals, with the neutral num -\nber density nn, the neutral thermal speed vth, and the cross\nsection of a neutral-neutral collision σnn. If the ion-neutral\ncollisional damping rate at kν,ωd(kν) =Ck2\nνEM, is larger\nthan the viscous damping rate,\nωv(kν) =k2\nννn=L−1\n2V3\n2\nLν−1\n2n, (3)\nthat is\nEM>C−1νn, (4)\nion-neutral collisions dominate over the neutral viscosit y,\nleading to a damping scale of magnetic fluctuations larger\nthan the viscous scale.\nThe turbulent eddies at the damping scale kdare mainly re-\nsponsible for driving the dynamo growth of magnetic energy,\nat a rate comparable to the eddy-turnover rate,\nΓd=kdvd=L−1\n3VLk2\n3\nd, (5)\nwhere the turbulent velocity at kdis,\nvd=VL(kdL)−1\n3. (6)\nFrom the equalization between Γdandωdatkd, we find the\nexpression of kdas (Eq. (1) and (5))\nkd=C−3\n4L−1\n4V3\n4\nLE−3\n4\nM. (7)\nWe see that the damping scale increases with the growing\nmagnetic energy EM.\nThe dynamo is in the linear regime, i.e., kinematic dynamo,\nas long as the magnetic energy is below the turbulent kinetic\nenergy at kd,\nEM<1\n2v2\nd. (8)\nCombining the above relation with Eq. (6) and (7) yields\nEM<C\n4L−1V3\nL. (9)\nIt can be further rewritten as\nRene<ξn\n6Rt, (10)\nwith\nRene=EM\n1\n2V2\nL, Rt=ν−1\nni\nL/VL(11)\nas the ratio between EMand the turbulent energy at Land\nthe ratio between the neutral-ion collision time and the lar gest\neddy-turnover time, where the expression of Cin Eq. (1) is\nused. When the magnetic energy satisfies both conditions in\nEq. (4) and (9), one should consider the damping regime for\nthe kinematic dynamo growth of the magnetic field.\nAs illustrative examples, we use the typical ISM phases\n(see Draine 2011), e.g., the cold neutral medium (CNM)\nand molecular cloud (MC), as the preshock conditions for\nsupernova remnants interacting with atomic and molecu-\nlar clouds. Their typical parameters are listed in Table 1\n(see Draine & Lazarian 1998), including the number densi-\nties of the atomic hydrogen nHand electrons ne, the tem-\nperatureT. For the turbulence parameters, we adopt the\ncharacteristic scale ∼0.1pc of the density structure in\nthe CNM (Heiles & Troland 2003) and MC (Goodman et al.\n1998; Motte et al. 2007) as the injection scale Lof the up-\nstream turbulence. The turbulent velocity at Ldepends3\non the density perturbation ∆ρ/ρ and the shock speed vsh\n(Drury & Downes 2012). As an order-of-magnitude estimate,\nwe adopt\nVL∼∆ρ\nρvsh. (12)\nFor an illustrative purpose, we assume ∆ρ/ρ∼1and\nVLon the order of 103km/s in our following calcula-\ntions. It is worthwhile to notice that the density perturba-\ntion can be vastly different in different ISM phases, e.g., l o-\ncal density enhancements with ∆ρ/ρ≫1but a small vol-\nume filling factor in clumpy MCs (see Stutzki et al. 1988;\nHennebelle & Falgarone 2012), the low density contrast with\n∆ρ/ρ≪1in more diffuse medium. Thus the corresponding\nVLcan actually deviate significantly from vsh.\nWe find that given the magnetic field strength B0in the\nambient ISM, the initial magnetic energy is\nEM0=1\n2V2\nA0=B2\n0\n8πρ, (13)\nwhereVA0is the Alfv´ en speed. It satisfies\nC−1νn<EM0<C\n4L−1V3\nL, (14)\nor in terms of the ionization fraction ξi=ρi/ρ,\nξi<ξnEM0\n3γdρνn, ξi<ξnV3\nL\n12γdρLEM0, (15)\nfor both the CNM and MC, indicative of the dominant ion-\nneutral collisional damping and the kinematic dynamo growt h\nof magnetic energy according to the above analysis (Eq. (4)\nand (9)). In particular, we find Rene≈4×10−6,Rt≈185\nin the case of the CNM and Rene≈4×10−7,Rt≈6in\nthe case of the MC (Eq. (11)). Naturally, Reneis very small\nas the turbulence in the precursor carries substantial kine tic\nenergy originating from the supernova ejecta, while a large\nRtsuggests the weak coupling between neutrals and ions and\nthus strong damping. Here we also adopt mi=mn=mH\nfor the masses of ions and neutrals in the CNM and mi=\n29mH,mn= 2.3mHfor those in the MC (Shu 1992), γd=\n3.5×1013cm3g−1s−1(Draine et al. 1983), and σnn≈10−14\ncm2(Vranjes & Krstic 2013).\n2.2. Dynamo evolution of the magnetic field\nThe kinematic dynamo is described by the Kazant-\nsev theory, with EMdetermined by the integral of the\nKazantsev spectrum (Kazantsev 1968; Kulsrud & Anderson\n1992; Schekochihin et al. 2002; Brandenburg & Subramanian\n2005) over the wavenumbers smaller than kd,\nEM=1\n2/integraldisplaykd\n0M(k,t)dk\n=1\n2/integraldisplaykd\n0M0exp/parenleftbigg3\n4/integraldisplay\nΓddt/parenrightbigg/parenleftbiggk\nk0/parenrightbigg3\n2\ndk\n=1\n5M0k0exp/parenleftbigg3\n4/integraldisplay\nΓddt/parenrightbigg/parenleftBigkd\nk0/parenrightBig5\n2,(16)\nwhereM(k,t)is the growing Kazantsev spectrum of mag-\nnetic energy, and M0is the initial magnetic energy spectrum\ncentered about k0. By combining Eq. (5), (7), and (16), aftersome algebra we can reach the evolution law of the magnetic\nenergy,\n/radicalbig\nEM=/radicalbig\nEM0+3\n23C−1\n2L−1\n2V3\n2\nLt, (17)\nwhereEM0is the initial magnetic energy. This dynamo\nregime has been identified as the damping kinematic dynamo\nin XL16. Since B=√8πρEM, it shows that the magnetic\nfield strength grows linearly with time. The time evolution o f\nkdcan also be obtained by inserting the above equation into\nEq. (7),\nkd=/bracketleftBig\nk−2\n3\nd0+3\n23L−1\n3VLt/bracketrightBig−3\n2, (18)\nwherekd0is the initial damping scale corresponding to EM0.\nThe kinematic saturation happens when the magnetic en-\nergy grows to equipartition with the local turbulent energy ,\ni.e.,\nEM=1\n2v2\nd. (19)\nBy using Eq. (6) and (7), the above equation gives the critica l\ndamping scale kd,crcorresponding to the kinematic saturation,\nkd,cr=/parenleftBigC\n2/parenrightBig−3\n2L1\n2V−3\n2\nL. (20)\nFor the CNM and MC under consideration, there is\nkd,crL=/parenleftBig2L\nCVL/parenrightBig3\n2<1, (21)\nor equivalently,\nξi<ξnVL\n6γdρL. (22)\nIt implies that the kinematic saturation cannot be reached d ur-\ning the entire dynamo process.\nWhen the damping scale increases up to L, we have the\ntimescale of the damping kinematic dynamo in the partially\nionized gas (Eq. (18))\ntdyn=23\n3L1\n3V−1\nL(L2\n3−k−2\n3\nd0), (23)\nwhich is approximately 7.7largest eddy-turnover times. At\ntdyn, the magnetic energy is amplified to (Eq. (7), (17), and\n(23))\nEdyn=C−1LVL. (24)\nThe calculated tdynand the field strength corresponding to\nEdynin the CNM and MC are presented in Table 1. The re-\nsults here only serve as an order-of-magnitude estimate. Mo re\nrigorous calculations rely on a more realistic and detailed\nmodeling of the shock, as well as the ambient environment.\nWe caution that with the development of the strong magnetic\nfield in front of the shock, the shock Alfv´ en Mach number\nMsh,A=vsh/VAdecreases. In the linear kinematic dynamo\nregime, the magnetic back-reaction on the shock propagatio n\nis usually unimportant. But in the case when the Alfv´ en spee d\ncorresponding to the amplified magnetic filed at a later time\nof the dynamo approaches the shock speed (e.g. in the case\nof MC), the strong shock jump condition will not be satisfied.\nConsequently, the turbulence driving and the dynamo growth\nalso cease. In the self-regulated system in the realistic si tua-\ntion, the dynamo efficiency decreases with the arising of the\ndynamically important magnetic field.4\nTABLE 1\nPARAMETERS IN THE PRESHOCK REGION\nnH[cm−3]ne/nH T [K]B0[µG]k−1\nν[pc] k−1\nd0[pc]tdyn[yr]Bdyn[µG]\nCNM 30 10−3100 5 1 .3×10−71.2×10−4741.9 452 .6\nMC 300 10−420 5 1 .7×10−81.6×10−6749.7 7.7×103\nMoreover, for the amplification of the upstream magnetic\nfield, the turbulent dynamo can only operate within the pre-\ncursor crossing time tc=Lp/vsh. The maximum size of the\nCR precursor Lpis determined by the diffusion of the highest-\nenergy CRs,\nLp=κmax\nvsh, (25)\nwhere the CR diffusion coefficient κshould be computed self-\nconsistently as they interact with the magnetic fluctuation s\namplified by the CR-driven turbulence.\n2.3. Magnetic field structure and CR diffusion\nFig. 1(a) illustrates the magnetic energy spectrum M(k)\nduring the damping kinematic dynamo. The Kazantsev spec-\ntrum extends over large scales down to the spectral peak at\nthe damping scale, where the spectrum is cut off due to ion-\nneutral collisional damping. Since the magnetic energy is w ell\nbelow the turbulent kinetic energy over all scales, the non-\nlinear Lorentz back-reaction on turbulent motions is insig nifi-\ncant, and the entire dynamo process stays in the linear regim e.\nThe magnetic field resulting from the damping kinematic\ndynamo in the presence of ion-neutral collisions has a chara c-\nteristic scale as the increasing damping scale. It means tha t the\nmagnetic field has a tangled structure over larger scales, bu t\nis smooth over smaller scales with the magnetic fluctuations\nsuppressed. The bending of field lines by the super-Alfv´ eni c\nturbulent motions at kdgives rise to an effective mean free\npathλmfpof CRs equal to k−1\nd(see Brunetti & Lazarian\n2007). Note that the Larmor radius rLof CRs should be\nsmaller than k−1\nd. The corresponding diffusion coefficient is\nκ≈cλmfp\n3=c\n3kd, (26)\nwherekdis given by Eq. (18). Suppose that the damping\nscale, as well as λmfp, can reach L, thus the precursor has the\nmaximum thickness\nLp=cL\n3vsh, (27)\nand the crossing time is\ntc=c\n3vshL\nvsh=c\n3vshVL\nvshL\nVL. (28)\nIf we adopt VL∼vsh∼103km/s, then tcis equal to 100\nlargest eddy-turnover times, which is much larger than tdyn\n(Eq. (23)) and certainly sufficient for the full development\nof the precursor turbulence and the damping kinematic dy-\nnamo. More conservatively, the turbulent velocity should s at-\nisfyVL/vsh>7.7×10−2(see Eq. (23)) for the dynamo to\nproceed till the outer scale of the turbulence.\nAt the Bohm limit with rL=L, the maximum energy of\nCRs whose effective diffusion is governed by the amplified\nmagnetic field is\nECR, max=eBdynL, (29)whereeis the particle’s charge. Given the field strength in\nTable 1, it is 4.2×1016eV in the CNM and 7.1×1017eV in\nthe MC. It implies that the damping kinematic dynamo in the\nprecursor can easily generate the magnetic field required fo r\nthe acceleration of CRs up to the knee energy of ∼1015eV\nand beyond.\nThe magnetic field structure and the related CR diffusion\nin the preshock region are critical for the DSA. The limits\non the DSA arising from the partial ionization of the up-\nstream medium were studied by e.g. O’C Drury et al. 1996;\nMalkov et al. 2011, under the consideration of the ion-neutr al\ncollisional damping of Alfv´ en waves. In the situation with the\nprecursor turbulence, the turbulent dynamo increases both the\nstrength and length scale of the magnetic field4. Starting from\nthe weak interstellar field strength and in the presence of th e\nsevere damping, the turbulent dynamo generates the damping -\nscale magnetic field. Without the pitch-angle scattering, C Rs\nwith the Larmor radii smaller than the damping scale gyrate\nabout the field lines. The random change of the magnetic field\norientation over the distance equal to the damping scale en-\ntails an effective diffusion of CRs. Unlike the resonant in-\nteraction, both low- and high-energy CRs undergo the same\ndiffusion process in this dynamo-generated magnetic field5.\nThe arising of the precursor turbulence depends on the\ndensity fluctuations in the upstream medium, which have\nbeen commonly observed in the ISM (Armstrong et al. 1995;\nChepurnov & Lazarian 2010; Xu & Zhang 2017b), and es-\npecially in the cold and dense phases (Lazarian 2009;\nHennebelle & Falgarone 2012) but with a small volume fill-\ning factor (Tielens 2005; Haverkorn & Spangler 2013). In the\ncase that the SNR shock propagates through a relatively uni-\nform ambient medium, the above dynamo process would be-\ncome less efficient in the mildly turbulent precursor.\n3.MAGNETIC FIELD AMPLIFICATION IN THE FULLY\nIONIZED POSTSHOCK REGION\nSNR shocks efficiently heat the ISM gas, resulting in a\nhigh temperature in the postshock region. We assume that\nthe partially ionized gas passing through the shock becomes\nfully ionized downstream. Compared with the partially ion-\nized preshock medium, the magnetic fluctuations in the fully\nionized postshock medium are only marginally damped by the\nresistivity, with the resistive scale much smaller than the vis-\ncous scale (see the Appendix). On the other hand, the equipar -\ntition scale with the local turbulent energy in balance with the\n4Notice that with the magnetic field intensified through the tu rbulent dy-\nnamo, the cutoff scales of Alfv´ en waves increase (see e.g. X u et al. 2016).\n5The low-energy CRs can also be subject to additional confinem ent via\nthe scattering with e.g. the current-driven instability (B ell 2004) on small\nscales.5\nE(k)\nM(k)\nk-5/3\nk3/2\n1/Lkdkν\n(a) PreshockE(k)\nM(k)\nk-5/3k3/2\nk-1\n1/L 1/lAkνkR\n(b) Postshock\nFIG. 1.— Sketches of the magnetic (solid line) and turbulent (da shed line) energy spectra during the dynamo processes in the (a) preshock and (b) postshock\nregions. Fig. 1(a) is taken from Xu & Lazarian (2017).\ninitial magnetic energy is given by (Lazarian 2006)\nlA=LM−3\nA=LV−3\nLV3\nA\n= 1.3×1013/parenleftBigL\n0.1pc/parenrightBig/parenleftBigVL\n100km/s/parenrightBig−3\n/parenleftBigni\n10cm−3/parenrightBig−3\n2/parenleftBigB\n5µG/parenrightBig3\ncm,(30)\nwhereMAis the Alfv´ en Mach number, and the ion number\ndensityni=nH. Because it is larger than the viscous scale of\nturbulence (Eq. (A2)), the turbulent dynamo falls in the non -\nlinear regime in the presence of the significant Lorentz back -\nreaction on the turbulent motions over smaller scales. The\ninitial field strength adopted here is comparable to the inte r-\nstellar value. When the precursor dynamo is also taken into\naccount, a larger lAis expected, and the postshock dynamo\nstarts with a stronger magnetic field.\nThe magnetic energy spectrum M(k)during the nonlin-\near dynamo process is illustrated in Fig. 1(b). The Kazant-\nsev spectrum exists on scales above lAwhere the turbu-\nlence is super-Alfv´ enic. The trans-Alfv´ enic MHD turbu-\nlence is developed over smaller scales, with the same Kol-\nmogorov form for both the turbulent and magnetic energy\nspectra. Along with the magnetic energy growth, the equipar -\ntition scale lAshifts to ever larger scales and eventually\nthe MHD turbulence extends up to L. In the sub-viscous\nrange, i.e., k > k ν,M(k)is further prolonged to the re-\nsistive cutoff and follows the k−1profile as a result of the\nbalance between the magnetic tension force and the viscous\ndrag (Cho et al. 2002, 2003; Lazarian et al. 2004; XL16).\nThe numerical evidence for the above scalings of M(k)can\nbe found in dynamo simulations, e.g., Haugen et al. (2004);\nBrandenburg & Subramanian (2005).\nUnlike the kinematic dynamo which is only subject to the\nmicroscopic magnetic diffusion, e.g., ambipolar diffusio n in\na partially ionized medium, the nonlinear dynamo mostly\nsuffers the turbulent magnetic diffusion arising in the MHD\nturbulence. Since the turbulent diffusion rate is compara-\nble to the dynamo growth rate, i.e., the eddy-turnover rate\n(Lazarian & Vishniac 1999), the nonlinear dynamo is ineffi-\ncient. The evolution law of the magnetic energy in the non-\nlinear regime with the turbulent diffusion taken into accou ntTABLE 2\nPARAMETERS OF THE POSTSHOCK MEDIUM\nni[cm−3]B0[µG]L[pc]VL[km/s]\nModel 2 10 6 0 .1 450\nModel 3 19 6 0 .1 240\nModel 4 35 6 0 .1 130\nwas analytically derived by XL16,\nEM=EM0+3\n38ǫt, (31)\nwhere\nǫ=L−1V3\nL (32)\nis the constant turbulent energy transfer rate. It reveals a\nlinear-in-time growth of magnetic energy with the growth ra te\nas a small fraction of ǫ, which is consistent with direct numer-\nical measurements, e.g., Cho et al. (2009), Beresnyak (2012 ).\nWhen the dynamo saturation is achieved at L, the magnetic\nenergy is equal to the kinetic energy of the largest turbulen t\neddy,\nEM=1\n2V2\nL. (33)\nThe timescale of the nonlinear dynamo is (Eq. (31))\nτnl=38\n3ǫ(EM−EM0). (34)\nTo examine the applicability of the above analysis on the\nmagnetic field amplification in the postshock region, we next\ncarry out a comparison between our theoretical predictions\nand the numerical results in Inoue et al. (2009) (hereafter I 09).\nThey performed MHD simulations of SNR shocks propagat-\ning through the inhomogeneous ISM. We only consider paral-\nlel shocks (Model 2, 3, and 4 in I09) and disregard additional\nmagnetic field amplification due to the shock compression.\nThe parameters that we use are listed in Table 2.\nStarting from the initial field strength B0, the temporal evo-\nlution of the magnetic field is calculated according to Eq.\n(31). As shown in Fig. 2(a), our analytical results can well\nmatch the numerical findings. The saturated field strength is\ndetermined by the turbulent velocity VL(Eq. (33)). It ap-\nproaches 1mG given the settings of Model 2.6\nWe can also calculate the distance between the location\nwhere the magnetic field reaches saturation and the shock\nfront, which is given by\nlsat≈vdownτnl, (35)\nwhere the downstream bulk velocity is vdown= 1/4vshfor a\nstrong shock, and vsh= 1289 km s−1is the shock velocity\n(Model 2 in I09). The timescale of the nonlinear dynamo τnl\nis provided by Eq. (34). Fig. 2(b) shows that our estimate\n(the vertical dashed line) coincides with the peak position of\nthe magnetic field profile numerically produced in I09.\nBecause of the inefficiency of the nonlinear turbulent dy-\nnamo and its prolonged timescale, the peak field strength is\ndeveloped at a significant distance, rather than the immedi-\nate vicinity, behind the shock front. This feature naturall y\nexplains the location of the X-ray hot spots detected at more\nthan0.1pc downstream of SNR shocks (I09; Uchiyama et al.\n2007; Uchiyama & Aharonian 2008).\nThe above comparison confirms the predictive power of\nthe XL16 model for the nonlinear turbulent dynamo. It pro-\nvides an analytically tractable means of studying the evolu tion\nand distribution of the magnetic field in the postshock regio n.\nNotice that in the case of efficient precursor dynamo, start-\ning from a relatively high magnetization, the timescale of t he\npostshock nonlinear dynamo can be much shortened, and the\ngrowth of the magnetic energy is insignificant.\nThe amplified magnetic field affects the diffusion of CRs\nbehind the shock. For lower-energy CRs with ECR<\neB(lA)lA, whereB(lA)is the field strength reached at the\nenergy equipartition, their mean free path is independent o f\nenergy and given by lA(see BJL09). The corresponding diffu-\nsion coefficient is κ=clA/3. For higher-energy CRs with the\nLarmor radius exceeding lA, the larger-scale magnetic fluctu-\nations following the Kazantsev spectrum, i.e., B(k)∝k5/4,\nare too weak to confine the CRs as eB(k)k−1∝k1/4. But\nsincelAincreases with time, more energetic CR particles get\ninfluenced by the amplified magnetic field during the nonlin-\near turbulent dynamo. At the distance lsat(Eq. (35)) behind\nthe shock front, lAreaches the outer scale of turbulence L\nat the nonlinear saturation, where CRs with ECR< eB(L)L\nhaveκ=cL/3.\nAs regards the CR diffusion in the direction perpendicular\nto the mean magnetic field, lower-energy CRs with ECR<\neB(lA)lAare characterized by the fast superdiffusion, i.e.\n∝angbracketleftσ2\n⊥∝angbracketright ∝tα,α >1, where∝angbracketleftσ2\n⊥∝angbracketrightis the mean squared displace-\nment in the perpendicular direction (Lazarian & Yan 2014).\nThe superdiffusion dominates the perpendicular transport of\nCRs aslAincreases with the distance behind the shock front.\nThe higher-energy CRs undergo the normal diffusion with\n∝angbracketleftσ2\n⊥∝angbracketrightas a linear function of time.\n4.DISCUSSIONS\nCompared with earlier studies on the turbulent dynamo in\nthe shock precursor, e.g., BJL09, del Valle et al. (2016), ba sed\non the analytical results of turbulent dynamo in a partially ion-\nized medium (XL16), we have demonstrated that the mag-\nnetic field amplification in a partially ionized medium can\nbe drastically different from that in a fully ionized medium .\nIn the CNM and MC, because of the severe damping, the\ndynamo remains in the linear regime with a linear-in-time\ngrowth of field strength, rather than an exponential growth e x-\npected for the kinematic dynamo in a highly ionized medium\n(Balsara & Kim 2005), or a linear-in-time growth of magneticenergy in the nonlinear regime (BJL09). Through compar-\nison with the current-driven instability (Bell 2004), BJL0 9\nsuggested that the nonlinear turbulent dynamo is more favor -\nable in amplifying the upstream magnetic fields. We present a\nmore efficient dynamo regime arising in the partially ionize d\npreshock medium than the nonlinear dynamo. Furthermore,\ndifferent from the limited timescale considered in BJL09, t he\nextended CR precursor in our scenario leads to an ample time\nfor development of turbulence and magnetic field amplifica-\ntion6. Our results support the notion that the turbulent dy-\nnamo in the shock precursor is the dominant process of mag-\nnetic field amplification.\nIt is important to stress that only when the ionization frac-\ntion is sufficiently high can the kinematic dynamo be consid-\nered unimportant compared with the nonlinear dynamo stage.\nIn the new damping regime identified in XL16, depending on\nthe ionization fraction, the kinematic dynamo can be impor-\ntant through the turbulence inertial range up to the equipar -\ntition scale, or the outer scale as the case discussed in the\npaper. Hence in a weakly ionized medium, the kinematic dy-\nnamo can have astrophysically important applications. In t he\npreshock turbulence, the characteristic scale of the magne tic\nfield grows with the increasing damping scale of turbulence,\nindependent of the CR Larmor radius. The resulting damping-\nscale magnetic field regulates the diffusion behavior and th us\nthe acceleration of CRs from low energies up to very high en-\nergies. Different from the short-wavelength Bell mechanis m,\nthere is no need to invoke additional inverse cascade proces s\nfor transporting the magnetic energy to larger scales.\nIn the turbulent postshock medium, the observed year-scale\nX-ray variability of the compact hot spots suggests a largel y\namplified magnetic field of 1mG and very efficient particle\nacceleration in the emitting regions (Uchiyama et al. 2007;\nUchiyama & Aharonian 2008). Our results on the amplifica-\ntion of the postshock magnetic field show that the analytical\nexpectations in XL16 agree well with the existing numerical\nsimulations. Depending on the postshock turbulent velocit y,\nthe saturated field strength of the nonlinear dynamo can be on\nthe order of 1mG (Section 3). As pointed out by I09, the elec-\ntrons responsible for the hot spots are unlikely to be accele r-\nated at the shock front in view of the fast synchrotron coolin g.\nThe efficient in-situ acceleration is attributable to the re verse\nshock (Uchiyama & Aharonian 2008) or secondary shocks\narising from collisions of turbulent flows (I09). An alterna tive\nexplanation could be the adiabatic non-resonant accelerat ion\nin MHD turbulence proposed by Brunetti & Lazarian (2016)\n(see also Xu & Zhang 2017a). In MHD turbulence, magnetic\nfield lines can be stretched due to the turbulent dynamo and\nshrink via the turbulent reconnection, which happen throug h\nevery eddy turnover (Lazarian & Vishniac 1999). The CRs\nwith the Larmor radii much smaller than the size of the tur-\nbulent eddy are attached to field lines and confined within\nthe eddy during its eddy-turnover time, bouncing back and\nforth between converging magnetic fields in the turbulent re -\nconnection region. This process is similar to the first-orde r\nFermi acceleration (de Gouveia dal Pino & Lazarian 2005).\nThe resulting efficient acceleration within the eddy-turno ver\ntime may accommodate the year-scale variability in the syn-\n6We note that in our calculation for the maximum size of the pre cursor, we\nuse the diffusion length scale of the highest-energy CRs. Wh en confronting\nwith the precursor thickness indicated by X-ray observatio ns (see Vink 2012),\none should adopt that corresponding to the electrons with th e characteristic\nenergy of their energy spectrum for the observable emission .7\ntime [yr]0 1000 2000 3000 4000 5000B [µ G]\n1101001000\n(a) Temporal evolution of B\nposition [pc]1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2B [µ G]\n1101001000\n(b) Spatial profiles of B\nFIG. 2.— Comparisons between our analyses and the numerical res ults from I09. (a) Evolution of maximum (thick) and average ( thin) field strengths. (b)\nSpatial profile of the field strength for perpendicular (soli d) and parallel (dotted) shocks. On the original figures take n from I09, the overplotted circles in (a)\nrepresent our analytical field strength in comparison with d ifferent numerical models (Models 2 (green), 3 (blue), and 4 (black)) for a parallel shock. The\noverplotted vertical dashed line in (b) shows our estimated position for the saturation of the amplified magnetic field in the postshock region of a parallel shock\n(Model 2, dotted line).chrotron emission. A detailed study on this mechanism and it s\napplicability in CR acceleration in the postshock turbulen ce is\nthe subject of an upcoming paper.\nOur results on the magnetic field amplification and its im-\nplications on the diffusion of energetic particles in both t he\npre- and post-shock regions are important for further mod-\nelling the emission characteristics in comparison with mul ti-\nwavelength observations of SNRs (see e.g., Zeng et al. 2017) .\n5.SUMMARY\nBy applying the turbulent dynamo theory formulated by\nXL16, we have investigated the magnetic field amplification\nin SNRs.\nThe dynamo evolution of magnetic fields in the pre- and\npost-shock media are very different. The dynamo in the\nweakly ionized upstream medium, e.g., the CNM and MC,\nis characterized by a linear-in-time growth of field strengt h,\nthat is, the magnetic energy grows quadratically with time.\nIt is slower than the exponential growth in the linear regime\nwithout damping, but faster than the linear-in-time growth of\nmagnetic energy in the nonlinear regime. In the extended CR\nprecursor, the large damping-scale magnetic field formed at\nlater times of the damping kinematic dynamo is beneficial for\nconfinement of high-energy CRs. This finding is important\nfor a realistic treatment of the shock acceleration of CRs in\nthe partially ionized ISM.\nImportantly, we provide the criterion for the dominance of\nthe ion-neutral collisional damping over the neutral visco usdamping and thus the emergence of the damping regime of\nthe kinematic dynamo (Eq. (4)). Then the conditions for the\nentire dynamo process to be in the damping regime are given\nby Eq. (9) (or Eq. (10)) and Eq. (21).\nThe turbulent dynamo in the postshock region is in the\nnonlinear regime and drives a linear-in-time and inefficien t\ngrowth of magnetic energy. Provided a weak initial field\nstrength in the postshock region, the peak field strength at t he\ndynamo saturation can only be reached in the farther down-\nstream region. This explains the X-ray hot spots located far\nfrom the shock front. The consistency with the results of nu-\nmerical simulations (e.g., I09) shows that the XL16 analyti -\ncal model for the nonlinear turbulent dynamo can be used to\nquantify the evolution and distribution of downstream mag-\nnetic fields.\nThe postshock magnetic turbulence can serve as an alterna-\ntive source besides the shock front for efficient accelerati on\nof CRs. The corresponding acceleration mechanism deserves\nmore attention and detailed analysis in future by confronti ng\nwith updated observations.\nWe thank the anonymous referee for insightful comments.\nS.X. acknowledges the support for Program number HST-\nHF2-51400.001-A provided by NASA through a grant from\nthe Space Telescope Science Institute, which is operated by\nthe Association of Universities for Research in Astronomy,\nIncorporated, under NASA contract NAS5-26555. A.L. ac-\nknowledges the support from grant NSF DMS 1622353.\nAPPENDIX\nTHE VISCOUS AND RESISTIVE SCALES IN THE FULLY IONIZED POSTSH OCK REGION\nIn the presence of magnetic field, the ion viscosity is anisot ropic. Even for the initial magnetization as low as the level in the\nISM, the ion viscosity perpendicular to the field is very smal l (Simon 1955),\nνi,⊥=3kBTνii\n10Ω2\nimi= 2.0×107/parenleftBiglnΛ\n10/parenrightBig/parenleftBigT\n107K/parenrightBig−1\n2/parenleftBigni\n10cm−3/parenrightBig/parenleftBigB\n5µG/parenrightBig−2\ncm2s−1, (A1)\nwherec,me,e,lnΛ,kB,νii,Ωi,mi(=mH),niare the speed of light, the electron mass and charge, the Coul omb logarithm, the\nBoltzmann constant, the ion collision and cyclotron freque ncies, the ion mass (equal to the hydrogen atomic mass) and nu mber\ndensity. It corresponds to a small viscous scale\nkν=L−1\n4V3\n4\nLν−3\n4\ni,⊥= 2.5×10−5/parenleftBigL\n0.1pc/parenrightBig−1\n4/parenleftBigVL\n100km/s/parenrightBig3\n4/parenleftBigνi,⊥\n2.0×107cm2s−1/parenrightBig−3\n4cm−1. (A2)8\nOn the other hand, the resistivity is (Spitzer 1956),\nη=c2√mee2lnΛ\n4(kBT)3\n2= 3.0×102/parenleftBiglnΛ\n10/parenrightBig/parenleftBigT\n107K/parenrightBig−3\n2cm2s−1. 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R., Zhang, L., & Zhang, S. 2017, ApJ,\n834, 153"    },    {        "title": "2110.14643v2.Quantum_oscillations_in_interaction_driven_insulators.pdf",        "content": "SciPost Physics Submission\nQuantum oscillations in interaction-driven insulators\nAndrew A. Allocca1?and Nigel R. Cooper1,2\n1TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK\n2Department of Physics and Astronomy, University of Florence, 50019 Sesto Fiorentino, Italy\n?aa2182@cam.ac.uk\nMarch 10, 2022\nAbstract\nIn recent years it has become understood that quantum oscillations of the magnetization\nas a function of magnetic field, long recognized as phenomena intrinsic to metals, can also\nmanifest in insulating systems. Theory has shown that in certain simple band insulators,\nquantum oscillations can appear with a frequency set by the area traced by the minimum\ngap in momentum space, and are suppressed for weak fields by an intrinsic “Dingle damp-\ning” factor reflecting the size of the bandgap. Here we examine quantum oscillations of the\nmagnetization in excitonic and Kondo insulators, for which interactions play a crucial role.\nIn models of these systems, self-consistent parameters themselves oscillate with changing\nmagnetic field, generating additional contributions to quantum oscillations. In the low-\ntemperature, weak-field regime, we find that the lowest harmonic of quantum oscillations\nof the magnetization are unaffected, so that the zero-field bandgap can still be extracted by\nmeasuring the Dingle damping factor of this harmonic. However, these contributions dom-\ninate quantum oscillations at all higher harmonics, thereby providing a route to measure\nthis interaction effect.\nContents\n1 Introduction 2\n2 Rigid Band Insulator 3\n3 Excitonic Insulator 5\n3.1 Oscillations in the Linearized Theory 6\n3.2 Effects of nonzero temperature 8\n4 Kondo Insulator 9\n4.1 Oscillations in the Linearized Theory 11\n5 Discussion and Conclusion 13\nA Comparison with Previous Results 14\nB Evaluation of Oscillatory Functions 15\n1arXiv:2110.14643v2  [cond-mat.str-el]  9 Mar 2022SciPost Physics Submission\nC Excitonic Insulator Temperature Dependence 16\nReferences 17\n1 Introduction\nQuantum oscillations (QO) of observables as a function of applied magnetic field have been un-\nderstood as a phenomenon intimately tied to the idea of a Fermi surface ever since they were first\ndiscovered [1]. The well-established Lifshitz-Kosevich theory [2]of QO directly relates the fre-\nquency of the oscillations to extremal cross-sectional areas of the Fermi surface. This has allowed\nthe technique to become a useful tool for examining the geometry of Fermi surfaces in real mate-\nrials. However, this long-held understanding has been challenged recently by the measurement of\nquantum oscillations in insulators, notably the strongly-correlated Kondo insulators SmB6[3–5]\nand YbB12[6,7], and the insulating phase of WTe2[8], believed to be an excitonic insulator [9],\nwhich all lack a traditional notion of a Fermi surface entirely.\nMany theoretical works [10–29 ]have been put forward in response, seeking to understand\nthe phenomenon in these specific materials and how QO may arise in insulators more generally.\nIn this second direction, a direct calculation shows that generating QO in insulating systems is\nactually relatively straightforward—if the minimum band gap is both larger than the cyclotron\nenergy and traces out a nonzero area in the Brillouin zone, then oscillations can be found at\nthe frequency corresponding to this area as though it were a Fermi surface cross section [10,\n13]. Furthermore, it is found that these oscillations come with an intrinsic “Dingle damping”\nfactor–an exponential suppression of the form exp (\u0000B0=B), where Bis the applied magnetic field\nstrength. In metallic systems the Dingle factor accounts for the effect of disorder; B0is related\nto the finite quasiparticle relaxation time [30]and will vary between samples, but in an insulator\nB0is directly related to intrinsic properties the gapped band structure itself. This implies that for\nband insulators QO contain important information about electronic structure just as they do for\nmetals, and fundamental properties of the band structure may be extracted from careful analysis\nof the field dependence of oscillation amplitudes.\nThe question we examine here is whether this result also holds for systems where the band\nstructure is strongly affected by interactions, such as excitonic and Kondo insulators. In the mean\nfield descriptions of these systems at zero field, the mean field parameters obey self-consistent con-\nstraints and determine the form of the bands. When a magnetic field is applied these constraints\nnecessarily introduce B-dependence to these parameters, which causes the bands themselves to\nfluctuate with field and introduces additional contributions to QO not present in ‘rigid’ band in-\nsulators.\nTo analyze each of these systems we employ the following general procedure. First we analyze\nthe mean field description of the system at zero field, in particular identifying the self-consistent\nequations that the mean field parameters must obey. With the introduction of a magnetic field we\nassume that electronic dispersions are quantized into Landau levels and the mean field parameters\nacquire B-dependent oscillatory components, which for weak fields are small compared to the\nzero-field parts. We then linearize the self-consistent conditions around the zero-field values and\ndetermine the leading effect of the magnetic field on top of the rigid band case.\n2SciPost Physics Submission\nWe find very generally that when considering mean field theories the fundamental harmonic\nof QO of the magnetization is unaffected, and the oscillatory component of the mean field affects\nsecond and higher harmonics only. For both excitonic and Kondo insulator models these new\ncontributions to higher harmonics have exactly the same exponential sensitivity to the size of the\ngap as for the corresponding B=0 rigid band insulators, but have different overall dependence\non the field strength allowing them to be the dominant contribution to all harmonics to which\nthey contribute.\nThe remainder of the paper is organized as follows: We begin in Section 2 by examining QO\nfor the case of a rigid band insulator, which is the background around which we linearize in the\nfollowing sections. In Section 3 we analyze a model excitonic insulator, first applying the mean\nfield approximation at B=0, then considering the oscillations the mean field parameter acquires\nupon introduction of a magnetic field and it’s contributions to QO. In Section 4 we then do the\nsame for the case of a Kondo insulator using the mean field slave-boson formalism.\n2 Rigid Band Insulator\nWe first consider a spinless, two-dimensional band insulator at zero temperature described by the\nHamiltonian [10]\nH0=X\nk\t†\nk\u0012\n\u000fc\nkg\ng\u000fv\nk\u0013\n\tk. (1)\nHere and throughout the rest of our calculations we set ~h=1,candvlabel conduction and\nvalence bands, and \tk= ( c,k, v,k)T, with †\ni,kand i,kthe creation and annihilation operators\nfor electrons in band i. The conduction band dispersion is \u000fc\nk, which we take to be approximately\nparabolic in the region of interest, and we set the valence band dispersion to be \u000fv\nk=\u000f0\u0000\u0011\u000fc\nk,\nwith\u0011a dimensionless constant and \u000f0the shift of the valence band relative to the conduction\nband. The limit \u0011!0 yields the flat valence band of a heavy fermion system. We take \u000f0>0\nso the conduction and valence bands cross and the interband tunneling amplitude gopens a\nhybridization gap at the band crossing point. The single-particle energies of the system are then\nE\u0006\nk=1\n2\n\u000fc\nk+\u000fv\nk\u0006q\n(\u000fc\nk\u0000\u000fv\nk)2+4g2\n, (2)\nwhich are shown in Fig. 1. We assume a ground state with the lower band entirely filled and the\nupper band empty, so the system is an insulator. In writing this model we have implicitly assumed\nthat the physics of interest is captured by a two-band model. This is expected as long as any\nadditional bands are well separated in energy from the gap opening point.\nApplying a magnetic field Bperpendicular to the system quantizes \u000fc\nkinto discrete Landau\nlevels (LL), indexed by n=0, 1, 2, . . . , via the replacement \u000fc\nk!\u000fc\nn= (n+\r)!c, with cyclotron\nfrequency!cand phase shift \r. If\u000fc\nk=k2=2mcexactly, with effective conduction electron mass\nmc, then this replacement is exact, the cyclotron energy and phase shift are !c=eB=mcand\n\r=1=2, and\u0011=mc=mvrepresents the ratio of effective masses of the two bands. Otherwise\nthis substitution is an approximation valid for weak fields, with \r2[0, 1)in general. Within E\u0006\nk\nthis replacement gives the energies E\u0006\nn, and sums over momentum are replaced by sums over\nLL index,P\nk!n\bP1\nn=0, with n\b=B=\b0the degeneracy of each LL and \b0=h=ethe mag-\nnetic flux quantum. Because the hybridization is spatially homogeneous, after these replacements\nthe Hamiltonian only couples corresponding Landau levels in the conduction and valence bands,\nreflected in the form of E\u0006\nn.\n3SciPost Physics Submission\nFigure 1: An example band structure of the sort we consider. The solid lines show the energies E\u0006\nk\nin Eq. (2), while the dashed lines show \u000fc\nkand\u000fv\nk, the two bands in Eq. (1) prior to hybridization.\nWe have indicated the offset energy \u000f0and hybridization gap 2 g, and marked in red the point on\nthe lower band where the gap is minimized. Inset: a 3D view of the bands. The area traced by\nthe minimum gap, setting the QO frequency, is indicated with the red dashed line.\nAt zero temperature the free energy of the system is given by the sum over the energies of all\noccupied states, which for an insulator is the completely filled lower band. With a magnetic field\nthis energy is\n\nR(B) =n\b1X\nn=0E\u0000\nn, (3)\nwhere the subscript Rindicates the bands are rigid, unaffected by changing B. We can separate\n\nR(B)into constant \nR0and oscillatory ˜\nR(B)parts using the Poisson summation formula [30],\nwhich for a general function fis\n1X\nn=0f(n+\r) =Z1\n0dx f(x) +2Z1\n0dx1X\np=1f(x)cos(2\u0019p(x\u0000\r)). (4)\nThough this system lacks a Fermi momentum and Fermi surface, the momentum which mini-\nmizes the band gap characterizes the gapped band structure, and the area in momentum space\nthat it encircles, indicated in Fig. 1, functions in lieu of a Fermi surface for the purposes of QO.\nWe denote this momentum as k\u0003, defined through d\u0000\nE+\nk\u0000E\u0000\nk\u0001\n=dk\f\f\nk=k\u0003=0. From this we de-\nfine a corresponding (noninteger) reference value of the LL index n\u0003through\u000fc\nk\u0003=\u000fc\nn\u0003, giving\nn\u0003=\u000f0=!\u0003\u0000\r, where we put !\u0003= (1+\u0011)!c. For weak magnetic fields we have n\u0003\u001d1, which\nallows us to find an approximate analytic form of ˜\nR(B),\n˜\nR(B)\u00192jgjn\b\n\u00191X\np=1cos(2\u0019pn\u0003)\npK1\n2\u0019p2jgj\n!\u0003\n\u0018vtjgj!\u0003\n2n\b\n\u00191X\np=1cos(2\u0019pn\u0003)\np3=2e\u00002\u0019p2jgj\n!\u0003, (5)\nwhere Kiare the modified Bessel functions of the second kind. The final expression here uses the\nasymptotic form Ki(x)\u0018p\n\u0019=2xexp(\u0000x)forx\u001d1, which further refines the weak field regime\nto the condition !\u0003\u001c2jgj, cyclotron energy much smaller than the band gap. This result is very\nsimilar in form to the results in Ref. [31]which examines a system with a superconducting gap. It\n4SciPost Physics Submission\ncan be derived from the T!0 limit of the expression in Ref. [10]as shown in Appendix A. We see\nthat for weak magnetic fields the harmonics of the band insulator free energy are exponentially\nsuppressed by powers of exp\n\u00002\u00192jgj\n!\u0003\n, which we identify as the Dingle factor in an insulating\nsystem. This factor will function as a small parameter when considering additional oscillatory\ncontributions in the following sections.\n3 Excitonic Insulator\nWe now consider the case of an excitonic insulator [32–35 ]. This type of system is formed from the\ncondensation of excitons with binding energy greater than the inherent band gap of the system,\nso that the band gap is (predominantly) generated by electron-electron interactions. In the mean\nfield approximation there is a single parameter controlling the insulating properties of the system,\nthe exciton condensate amplitude, which allows for a very simple treatment of QO in the weak\nfield regime.\nTo describe this type of system we start from a two-band, two-dimensional model Hamiltonian\nfor spinless electrons with an interband interaction,\nH=X\nk\t†\nk\u0012\n\u000fc\nk0\n0\u000fv\nk\u0013\n\tk\u0000VX\nk,k0 †\nc,k v,k †\nv,k0 c,k0, (6)\nwhere Vis the strength of the short-range exciton pairing potential. We decouple the interaction\nvia a mean field approximation neglecting fluctuations, defining the exciton condensate order\nparameter \u0001=VP\nk¬\n †\nv,k c,k¶\n, whereh\u0001\u0001\u0001i denotes the expectation value in the state with a\nfilled valence band and empty conduction band. Though generally complex, we can choose \u0001to\nbe purely real and positive by adjusting the phases of  c,v. We then obtain the excitonic insulator\nHamiltonian\nHX=X\nk\t†\nk\u0012\n\u000fc\nk\u0000\u0001\n\u0000\u0001\u000fv\nk\u0013\n\tk+\u00012\nV(7)\nwith\u0001obeying the BCS-type gap equation\n1\nV=X\nk1r\u0000\n\u000fc\nk\u0000\u000fv\nk\u00012+4\u00012. (8)\nNote that the fermionic part of Eq. (7) is the same as Eq. (1) with g=\u0000\u0001.\nThis two-dimensional model and our main results can in principle be extended to three dimen-\nsions by including additional dispersion along kz, the direction of the magnetic field, as described\nin e.g. Ref. [30]. Such an extension should not change the fundamental nature of our results. We\nalso note that the role of fluctuations about the mean field order could be considered by extending\nthe mean-field theory (7) via standard means [36–38 ].\nWe now consider applying a perpendicular magnetic field B, which quantizes the electronic\ndispersion into Landau levels as discussed in Section 2. Because we assume \u0001to be spatially ho-\nmogeneous, we still have coupling only between corresponding Landau levels in the two bands. In\ncontrast to the rigid band insulator, however, the value of the gap \u0001acquires magnetic field depen-\ndence because of its relationship to the electronic energies through Eq. (8). We put \u0001!\u0001(B) =\u00010+˜\u0001(B),\nwhere \u00010is the constant value of the order parameter at zero field, solving Eq. (8), ˜\u0001(B)is the\npart of the order parameter that varies with changing field, and we assume that\f\f˜\u0001(B)\f\f\u001c\u00010.\n5SciPost Physics Submission\n3.1 Oscillations in the Linearized Theory\nThe free energy of an excitonic insulator at zero temperature, which we denote \nX, is the sum\nover energies of all states in the lower band, which has the same form as Eq. (3), plus the energy\nof the mean field parameter, the second term in Eq. (7). Unlike for the band insulator, the full\ndependence of \nXonBis partially implicit through ˜\u0001(B). To find the first corrections on top of\nthe band insulator result and we expand around \u0001=\u00010, keeping terms up to second order in\noscillatory quantities, assumed to be small:\n\nX(B,\u0001)\u0019\nXR+@\nXR\n@\u00010˜\u0001+1\n2@2\nXR\n@\u00012\n0˜\u00012=\nXR0+˜\nXR+@˜\nXR\n@\u00010˜\u0001+1\n2@2\nXR0\n@\u00012\n0˜\u00012, (9)\nwhere we identify \nXR=\nX(B,\u00010). The function ˜\nXR(B)is the oscillatory part of \nXRand has\nthe same form as ˜\nR(B)given in Eq. (5), but with the replacement g!\u0000\u00010. The mean field\ngap\u00010is, by definition, the value for which the action is stationary with respect to variation in \u0001\n(Eq. (8) is equivalent to @\nXR0/@\u00010=0), so the only term remaining at first order in oscillatory\nquantities is the rigid band contribution, ˜\nXR. Therefore, the next largest contribution comes at\nsecond order, given by the final two terms. This is a general implication of mean field theory,\nindependent of the choice of system or mean field being considered.\nWe now consider these next largest terms. For both terms we need the form of ˜\u0001(B), which\ncan be evaluated by analyzing the gap equation. For B6=0 this has the same form as in Eq. (8)\nbut with the replacements noted above, i.e. \u000fc\nk!\u000fc\nn= (n+\r)!c,\u0001!\u00010+˜\u0001(B), etc. We begin\nby expanding to first order in ˜\u0001(B)\n1\nV\u0019n\b1X\nn=01Ç\u0000\n\u000fc\nn\u0000\u000fv\nn\u00012+4\u00012\n0\u0000n\b1X\nn=04\u00010\u0000\n\u000fc\nn\u0000\u000fv\nn\u00012+4\u00012\n03=2˜\u0001(B)\u0011\u000b(B) +\f(B)˜\u0001(B). (10)\nIn the second equality we define the two sums as the functions \u000b(B)and\f(B). We then rewrite\nthese functions in terms of their constant ( \u000b0and\f0) and oscillatory ( ˜\u000band˜\f) parts, which can be\nevaluated with the Poisson summation formula, and keep terms only to first order in oscillations,\ngiving\n1\nV\u0019\u000b0+˜\u000b(B) +\f0˜\u0001(B). (11)\nBecause the left hand side is a constant, we must have\n1\nV=\u000b0 (12)\n˜\u0001(B) =\u0000˜\u000b(B)\n\f0. (13)\nCalculating the explicit forms of \u000b0,˜\u000b(B), and\f0(see Appendix B) verifies that Eq. (12) is exactly\nEq. (8), the zero-field gap equation determining \u00010, and gives the explicit form for ˜\u0001(B)via\nEq. (13),\n˜\u0001(B) =2\u000101X\np=1cos(2\u0019pn\u0003)K0\n2\u0019p2\u00010\n!\u0003\n\u0018vt\u00010!\u0003\n21X\np=1cos(2\u0019pn\u0003)\nppe\u00002\u0019p2\u00010\n!\u0003, (14)\nwhere!\u0003= (1+\u0011)!cas before and the second expression is the asymptotic form for weak fields,\n!\u0003\u001c2\u00010. As for the oscillatory part of the free energy, we see that the pthharmonic comes with\nppowers of the Dingle factor.\n6SciPost Physics Submission\nWith explicit forms for ˜\u0001(B)and ˜\nXR, the last quantity we need to evaluate is the second\nderivative of the B=0 free energy \nXR0with respect to \u00010. Using the gap equation to simplify,\nwe find\n1\n2@2\nXR0\n@\u00012\n0=2n\b\n!\u0003. (15)\nPutting all of the terms together we find the dominant contribution to the free energy at second\norder in small oscillatory quantities is\n@˜\nXR\n@\u00010˜\u0001+1\n2@2\nXR0\n@\u00012\n0˜\u00012\u0018\u0000\u00010n\b\n2cos(4\u0019n\u0003)e\u00004\u00192\u00010\n!\u0003, (16)\nwhere we have kept only the oscillatory terms at lowest order in the Dingle factor and discarded\na term that is smaller by a factor of !\u0003=2\u00010\u001c1. We see that this contributes to the second\nharmonic of QO. Comparing this to the p=2 term of the rigid band contribution, there is a clear\ndifference in the overall field dependence–the prefactor of the mean field term goes as B, whereas\nthe rigid band term goes as B3=2, so the rigid band term is smaller by a factor of1\n2\u0019Ç\n!\u0003\n\u00010\u001c1 at\nsmall fields. Therefore, for weak fields the oscillations of the mean field order parameter provide\nthe dominant contribution to the second harmonic of the free energy.\nThe contribution from the mean field is likely dominant for all higher harmonics as well.\nIn addition to other terms, such as those acquired by calculating ˜\u0001at higher orders than the\nlinearized framework presented here, we can write down several terms that have a leading B\ndependence at a lower power than the corresponding term in ˜\nXR. First, in the term in Eq. (9)\nproportional to ˜\u00012, cross terms between the qharmonic of one factor and the p\u0000qharmonic of\nthe other give contributions to the pthharmonic that also have ppowers of the Dingle factor. All\nsuch terms have a coefficient that goes as B, making them larger than the corresponding term of\n˜\nXR, going as B3=2. Additionally, there will be a term contributing to the pthharmonic of the free\nenergy the form\n@p\u00001˜\nXR\n@\u0001p\u00001\n0˜\u0001p\u00001\u0018B2\u0000p\n2e\u00002\u0019p2\u00010\n!\u0003, (17)\nwith ˜\u0001given by Eq. (14). We see that this goes as B2\u0000p=2, which for small Bis larger than B3=2\nfor all p\u00152, and is larger than Bforp>2 (it is B1=2forp=3), as shown in Fig. 2. There is no\nreason why the mean field contributions such as these should exactly cancel for any harmonic p\nabove the first–we have shown this explicitly for p=2–so for weak fields these mean field terms\nwill dominate for all harmonics p\u00152.\nWe pause to emphasize an important feature of the result that we have found: There is only a\nsingle dimensionless parameter, !\u0003=\u00010, that controls the size of the quantum oscillations (in both\nthe Dingle damping factor and its multiplicative prefactors) arising from the contributions of both\nthe rigid band and self-consistent mean-field parts of the free energy. Rewriting !\u0003=\u00010=B=B0\nso that B0=mc\u00010=((1+\u0011)e), the value of B0, proportional to the product of the hybridization\ngap and the cyclotron mass, can be used to characterize individual materials. Indeed, fitting\nmeasurements of quantum oscillations to the form of the Dingle factor–as done with metallic\nsystems to extract mean free paths–would here allow for a direct experimental determination of\nthis quantity for a given material. For B\u0018B0the rigid band and mean field contributions to\nthe higher harmonics ( p\u00152) are of the same size; below this the mean field part dominates\nand above this point our approximations begin to break down. Consequently, we see that the\noscillations of the gap that we analyze here cannot be ignored whenever they are present–a system\n7SciPost Physics Submission\nα=3/2α=1α=1/2\n0.00 0.05 0.10 0.15 0.20 0.25 0.30\nB/B 0(B/B 0)α\nFigure 2: Demonstrating the size of the leading dependencies B\u000bthat we find for various terms,\nfor small B=B0=!\u0003=(2\u00010). The rigid band case has \u000b=3=2 for all harmonics, while the contri-\nbutions from the mean field to the second harmonic are larger, with \u000b=1. There are contributions\nto the third harmonic with \u000b=1=2, which is even larger still.\nwith an interaction generated gap is never accurately described by just the corresponding rigid\nband structure.\nWe now briefly comment on how our results relate to those in Refs. [28]and[29], which\nalso analyzes oscillations in an excitonic insulator. There are several key differences between\nwhat is done there and what we present here, but there is no obvious disagreement. First, they\nfocus on electronic transport via thermally activated electrons and holes and not thermodynamic\nproperties which are our focus. Second, the calculations there consider a rigid band structure of\nhybridized particle and hole bands as in Section 2–the fixed hybridization set equal to the excitonic\ncondensate order parameter at B=0–and define the gap for B6=0 as the energy between the\nhighest energy Landau level in the lower band and the lowest energy Landau level in the upper\nband. The resulting oscillations of the gap are then more akin to the framework of Ref. [10]than\nwhat we find here.\nBecause we consider the free energy, and can therefore discuss only thermodynamic quantities,\nour result cannot be directly compared to those of Refs. [28]and[29]. Importantly, our generic\nconclusion that there can be no interaction contribution to the first harmonic does not apply, and\nin general one should indeed expect an additional contribution to the fundamental frequency os-\ncillation of non-thermodynamic quantities. How such contributions would compare to the results\nof Refs. [28]and[29]is left as future work.\n3.2 Effects of nonzero temperature\nAll of our calculations so far have been performed for a model at exactly zero temperature. We\nshow in Appendix C that our results apply for a range of nonzero temperatures provided that\nT\u001c\u00010, well below the transition temperature into the excitonic insulator phase. It is clear that\nthe temperature dependence of quantum oscillations in this model must be quite distinct from the\n8SciPost Physics Submission\ntypical Lifshitz-Kosevich form, given by the factor\nRT=p\u0012\nsinh (p\u0012), (18)\nwhere plabels the harmonic and \u0012=2\u00192kBT=!\u0003, with kBBoltzmann’s constant. Even for the\nrigid bandstructure the oscillations show a temperature dependence that departs from the LK\nform [10]. We expect further corrections beyond this, arising from the effects of self-consistency\nof the gap \u00010at a function of temperature. Computing this dependence would require considering\ntemperatures of the order of the gap, at which point thermal occupation of the upper band would\nbecome relevant. This is a regime for which we do not have any analytic results. We note however,\nthat the behaviour is potentially quite rich. The typical expectation is for temperature to reduce\nquantum oscillation amplitudes due to thermal broadening of occupied states near the relevant\nenergy. However, here the fact that \u00010diminishes with increasing temperature has the potential\nto counteract that, by lessening the damping from the Dingle factor. We note that a temperature-\ndependent gap would force one to reconsider the validity of analyzing just the “weak field regime”\nthat we have used so far. For a fixed magnetic field strength and increasing temperature, the ratio\n!\u0003=\u00010grows as \u00010diminishes, so for any field strength the regime !\u0003\u0018\u00010becomes relevant\nat some temperature. This is certainly a rich avenue for future work, but is beyond the present\nscope.\n4 Kondo Insulator\nWe now look to the case of Kondo insulators, a class of strongly-correlated, heavy fermion sys-\ntem[39]with narrow band gaps first identified over 50 years ago [40]. We begin with an Anderson\nmodel in two dimensions [41–44 ], describing the coupling of a light conduction band to a heavy\nvalence band, localized by strong interactions. The Hamiltonian is\nH=\u0000tX\nhi ji,\u001b\u0000\nc†\ni\u001bcj\u001b+h.c.\u0001\n\u0000tdX\nhi ji,\u001b\u0000\nd†\ni\u001bdj\u001b+h.c.\u0001\n+X\ni,\u001bVi\u0000\nc†\ni\u001bdi\u001b+d†\ni\u001bci\u001b\u0001\n+X\ni\n\u000fdnd\ni+Und\ni\"nd\ni#\n. (19)\nThe first line describes the two species of electrons (conduction cand heavy dbands) hopping on\na lattice with amplitudes tandtdrespectively, with td<0 andjtd=tj=\u0011\u001c1. The next term\ndescribes interband transitions with amplitude Vi, which opens the gap in the spectrum. The final\ntwo terms are written in terms of the d-electron densities, nd\ni,\u001b=d†\ni,\u001bdi,\u001bandnd\ni=nd\ni,\"+nd\ni,#. The\n\u000fdterm gives the shift of the heavy electron band relative to the conduction band. The Uterm is\na Hubbard interaction between d-electrons, forbidding double occupancy in the large Ulimit. For\nU!1 this condition can be enforced with the slave-boson formalism: put d†\ni\u001b=f†\ni\u001bbi, where\nfi\u001bis a new fermionic degree of freedom, which we refer to as f-electrons, and biis the slave\nboson. Each site contains either a boson or a single f-electron, and the Hubbard term is replaced\nbyP\ni\u0015i(P\n\u001bf†\ni\u001bfi\u001b+b†\nibi\u00001), where\u0015iis a Lagrange multiplier field enforcing the constraint.\nWe assume that the interband interaction is spatially uniform, Vi=V, and now employ the\nmean field approximation \u0015i!h\u0015ii\u0011\u0015,bi!hbii\u0011b, and b†\ni!hb†\nii\u0011b. In the continuum\nlimit, which is a valid approximation when considering weak magnetic fields, we obtain the mean\n9SciPost Physics Submission\nfield Hamiltonian\nHK=X\nk,\u001b\t†\nk\u001b\u0012\n\u000fc\nkbV\nbV\u000ff\nk\u0013\n\tk\u001b+\u0015\u0000\nb2\u00001\u0001\n. (20)\nHere the f-band dispersion is \u000ff\nk=\u000fd+\u0015\u0000\u0011b2(\u000fc\nk\u00004t), and the limit of immobile heavy fermions,\ni.e. infinite f-band mass, corresponds to \u0011!0. We can identify what we called \u000f0in the case of\nthe rigid band insulator with \u000fd+\u0015+4\u0011t b2and what we called gwith bV.\nAs written here we are considering an even parity coupling between the bands. We could\nconsider an odd parity coupling instead, ˆV(k) =Vd(k)\u0001ˆ\u001bwith d(k) =\u0000d(\u0000k), which results in\nnontrivial topological properties [45,46 ]. This choice does not affect the nature of the results we\npresent here, however; the zero-field gap appearing in our final results would have a different\nform reflecting its origins from an odd parity coupling, but the overall form of the expressions in\nterms of the size of the gap would remain the same. We continue with the simpler case of even\nparity coupling considered thus far.\nFor the Kondo insulator there are two self-consistent equations allowing us to determine the\ntwo mean field parameters band\u0015, contrasting with the excitonic insulator considered in Section 3\nwhich has only one. The first equation is simply the constraint imposed by the Hubbard interaction,\nwhich in the mean field approximation becomes\nX\nk,\u001b\nf†\nk\u001bfk\u001b\u000b\n\u0011nf\n0=1\u0000b2, (21)\nwhere we have defined nf\n0, the total f-electron density at B=0. The second constraint follows\nfrom the equation of motion for the boson field, which in the mean field approximation becomes\na demand that the energy be stationary with respect to variation of b,\nX\nk,\u001bV\n2\u0000\nc†\nk\u001bfk\u001b+f†\nk\u001bck\u001b\u0001\n\u0000\u0011b(\u000fk\u00004t)f†\nk\u001bfk\u001b·\n=C0+\u0011b\nKf\n0+4t nf\n0\n=\u0000\u0015b. (22)\nHere we have defined two additional functions,\nC0\u0011V\n2X\nk,\u001b\nc†\nk\u001bfk\u001b+f†\nk\u001bck\u001b\u000b\n(23)\nKf\n0\u0011\u0000X\nk,\u001b\u000fc\nk\nf†\nk\u001bfk\u001b\u000b\n, (24)\nC0the interband correlation energy and \u0011Kf\n0the kinetic energy of the f-electrons.\nWe now consider applying a perpendicular magnetic field Bto the system. As discussed in\nSection 2, this can be done by replacing energies with their Landau quantized versions, and sums\nover momentum with sums over LL index. We note here specifically that for a generic anisotropic\nKondo system the hybridization gap does not necessarily open at a fixed energy unless the f-band\nis completely immobile, \u0011=0. Therefore, the conclusions we arrive at only generically apply for\n\u00116=0 in the case of an isotropic system.\nWe assume that the effect of a nonzero field on the mean field parameters is to induce an\noscillatory component for each above the value determined at B=0, as we did for the case of the\nexcitonic insulator. Explicitly, we put\nb!b(B) =b0+˜b(B),\u0015!\u0015(B) =\u00150+˜\u0015(B), (25)\n10SciPost Physics Submission\nwith ˜band˜\u0015the components of the order parameters that vary with changing field and vanish\nforB=0. We assume these vanish continuously as the field is switched off, so we can consider a\nregime where\f\f˜b\f\fand\f\f˜\u0015\f\fare small compared to the zero-field parts.\n4.1 Oscillations in the Linearized Theory\nThe free energy of the Kondo system at zero temperature, \nK, is given by the sum over energies of\nthe lower band, plus the final term in Eq. (20), giving an additional contribution from the mean\nfields. Because we must include spin when discussing a Kondo system, the band contribution to\n\nKis the same as Eq. (3) but with an additional sum over the spin degree of freedom, amounting\nto a factor of 2. As in the case of the excitonic insulator, the free energy has an implicit dependence\nonBthrough the mean field functions ˜b(B)and˜\u0015(B), so we expand around (b,\u0015) = ( b0,\u00150)and\nkeep terms up to first order in oscillatory quantities to find the first corrections on top of the band\ninsulator result,\n\nK(B,b,\u0015)\u0019\nKR+@\nKR\n@b0˜b+@\nKR\n@\u00150˜\u0015+1\n2@2\nKR\n@b2\n0˜b2+1\n2@2\nKR\n@\u00152\n0˜\u00152+1\n2@2\nKR\n@b0@\u00150˜b˜\u0015\n\u0019\nKR0+˜\nKR+@˜\nKR\n@b0˜b+@˜\nKR\n@\u00150˜\u0015+1\n2@2\nKR0\n@b2\n0˜b2+1\n2@2\nKR0\n@\u00152\n0˜\u00152+1\n2@2\nKR0\n@b0@\u00150˜b˜\u0015. (26)\nWe define \nKR=\nK(B,b0,\u00150)to be the free energy evaluated with rigid bands, which we then\nseparate into constant \nKR0and oscillatory ˜\nKRparts. The form of ˜\nKRis identical to Eq. (5) up\nto an overall factor of 2 due to spin, the replacement g!b0V, and n\u0003= (\u000fd+\u00150+4\u0011t b2\n0)=!\u0003\u0000\r\nwith!\u0003= (1+\u0011b2\n0)!c. The vanishing of the first derivatives of \nKR0with respect to b0and\n\u00150is synonymous with working at the level of mean field theory, and as a result we see that\nthe contributions to the free energy from magnetic-field-induced oscillations of the mean field\nparameters enter at second order in small oscillations. We now seek to determine the size of these\nterms and their dependence on B, we did for the excitonic insulator.\nWe begin by examining ˜b(B)and ˜\u0015(B). We can evaluate the forms of these functions by\nanalyzing the constraint equations, which for nonzero field have the same form as Eq. (21) and\nEq. (22) but with the standard replacements we have made throughout,\nnf(B) =1\u0000b(B)2(27)\nC(B) +\u0011b(B)\u0000\nKf(B) +4t nf(B)\u0001\n=\u0000\u0015(B)b(B), (28)\nnow with nf,C, and Kffunctions of Bboth explicitly and through their dependence on b(B)\nand\u0015(B). We now expand these functions around the rigid band case up to first order in small\noscillatory quantities. For nfwe have\nnf(B,b,\u0015)\u0019nf\n0+˜nf\nR(B) +@nf\n0\n@b0˜b(B) +@nf\n0\n@\u00150˜\u0015(B), (29)\nwhere nf\n0is the f-electron density for B=0, equal to the constant part of nf(B,b0,\u00150), and we\ndefine ˜nf\nR(B)to be the oscillatory part of nf(B,b0,\u00150). The same expansion can be done for C(B)\nandKf(B), letting us similarly define the quantities ˜CR(B)and˜Kf\nR(B).\n11SciPost Physics Submission\nWe now expand Eqs. (27) and (28) up to first order in small oscillations. The terms at zeroth\norder are precisely Eqs. (21) and (22). We are left with the oscillatory components, obeying\n\u0012\n˜nf\nR(B)\n˜CR(B) +\u0011b0˜Kf\nR(B)\u0013\n=\u0000\u0012\nubu\u0015\nvbv\u0015\u0013\u0012˜b(B)\n˜\u0015(B),\u0013\n(30)\nwith\nub=2b0+@nf\n0\n@b0(31)\nu\u0015=@nf\n0\n@\u00150(32)\nvb=\u00150+@C0\n@b0+\u0011\nKf\n0+b0@Kf\n0\n@b0+4t(1\u00003b2\n0)\n(33)\nv\u0015=b0+@C0\n@\u00150+\u0011b0@Kf\n0\n@\u00150. (34)\nThis system of equations can be inverted in general, and doing so gives ˜band˜\u0015as linear combi-\nnations of ˜nf\nRand ˜CR+\u0011b0˜Kf\nR. The task is then to evaluate these quantities, for which we have\nexplicit expressions.\nUsing Eqs. (21), (23) and (24) with the standard replacements for the case of B6=0, and eval-\nuating all quantities at (b,\u0015) = ( b0,\u00150), we arrive at expressions for nf(B,b0,\u00150),C(B,b0,\u00150),\nandKf(B,b0,\u00150), from which we can extract the function ˜nf\nR,˜CR, and ˜Kf\nRusing the Poisson sum-\nmation formula.\nUsing the same methods we employed in Section 3 (see Appendix B), we find\n˜nf\nR(B)\u0019\u00008b0Vn\b\n!\u00031X\np=1sin(2\u0019pn\u0003)K1\n2\u0019p2b0V\n!\u0003\n\u0018\u0000vt8b0V\n!\u0003n\b1X\np=1sin(2\u0019pn\u0003)\nppe\u00002\u0019p2b0V\n!\u0003(35)\nfor the oscillatory part of the f-electron density,\n˜CR(B)\u0019\u00008b0V2n\b\n!\u00031X\np=1cos(2\u0019pn\u0003)K0\n2\u0019p2b0V\n!\u0003\n\u0018\u0000Vvt8b0V\n!\u0003n\b1X\np=1cos(2\u0019pn\u0003)\nppe\u00002\u0019p2b0V\n!\u0003(36)\nfor the oscillatory part of the interband correlation, and\n˜Kf\nR(B)\u0019\u0000\u000fd+\u00150+4\u0011t b2\n0\n1\u0000\u0011b2\n0˜nf\nR(B) (37)\nfor the oscillatory part of the f-electron kinetic energy. The asymptotic forms in second lines\nof Eqs. (35) and (36) apply in the regime where !\u0003\u001c2b0V. We see that, as has been true\n12SciPost Physics Submission\nfor all oscillatory quantities we have evaluated thus far, ˜nf\nR,˜CR, and ˜Kf\nRall have a leading B1=2\ndependence and the pthharmonic is accompanied by ppowers of the Dingle factor. From Eq. (30),\n˜band˜\u0015are linear combinations of these functions, so it follows that they share the same B1=2\ndependence and the same Dingle factor structure, which are also true of the derivative of ˜\nKR\nappearing in Eq. (26).\nUsing these insights we can draw important conclusions about the additional oscillatory con-\ntribution of the free energy Eq. (26) without explicitly inverting Eq. (30), calculating the constants\nub,u\u0015,vb, and v\u0015, or taking the second derivatives of \nKR0. All of the oscillatory quantities com-\nprising these additional terms go as B1=2and their lowest harmonics ( p=1) are proportional to a\nsingle power of the Dingle factor. The largest terms they contribute to Eq. (26) are second order\nin oscillatory quantities, so they have a coefficient that is linear in Band contribute at the same\norder as the p=2 term of ˜\nKR, which has a coefficient going as B3=2. Therefore, for weak fields\nthese new terms are the dominant contribution to the second harmonic of the free energy, and the\nsame sort of argument as at the end of Section 3 suggests that this is true for all higher harmonics\nas well. We also note that, also as for the excitonic insulator case, the behavior of these functions\nis determined by a single dimensionless parameter !\u0003=b0V, so that when these oscillations are\npresent they provide a non-negligible contribution.\n5 Discussion and Conclusion\nWe have shown that the field-induced oscillatory components of the mean field parameters in\nexcitonic and Kondo insulator models yield qualitatively similar contributions to the oscillatory\npart of the free energy, and both systems differ from the band insulator in similar ways. In both\ncases oscillations of the mean field order parameters generate the dominant contributions to the\nsecond and higher QO harmonics for weak fields, which should have observable consequences in\nmeasurements of, e.g. the de Haas-van Alphen effect. In particular, our results demonstrate that\nmeasuring the field dependence of these higher harmonics allows one to distinguish between a\nsimple band insulator and an insulating system with bands that are strongly affected by interac-\ntions. Additionally, since both the rigid band insulator and mean field contributions to the free\nenergy, and therefore all thermodynamic quantities, are parametrized by the same dimensionless\nparameter, the oscillations of the self-consistent mean field parameters are always relevant when\npresent and produce a distinct functional dependence on the magnetic field strength to second\nharmonic and higher oscillations.\nImportantly, however, several features of the free energy are entirely insensitive to the mean\nfield parameters acquiring weak magnetic field dependence. First, the lowest QO harmonic is\nunchanged from the behavior predicted by a rigid band model, which is guaranteed since the\nmean field state is defined as the saddle point of the free energy. Second, there are no changes to\nthe nature of the Dingle factor–exponential sensitivity to the size of the B=0 gap is the same as\npredicted from the theory of QO in a rigid band insulator. Thus, our results demonstrate that for\ninteracting insulators the non-rigidity of the band structure with changing magnetic field strength\ndoes not preclude the use of the Dingle damping of QO as a means to measure properties of the\ngapped band structure at zero field.\nThough we have focused here on the free energy, it is worth also considering other experi-\nmentally accessible quantities. First, the vanishing of mean field contributions to the fundamental\nfrequency oscillation only applies to the free energy and thermodynamic quantities. In general,\n13SciPost Physics Submission\nother quantities like the conductivity may have additional contributions at first order from the\neffects we study here. Second, there have been a number of works examining the specific heat\nand thermal transport measured in certain Kondo insulators, which are more akin to what would\nbe expected in metals and have been attributed to neutral in-gap states such as excitons [17]or\nimpurity bands [25], or neutral Fermi surfaces resulting from fractionalized electronic degrees\nof freedom [11, 12, 19, 21, 22, 27 ]. Our work here suggests that replacing rigid band structures\nwith mean-field bands dependent on Bin those models that rely on band geometry may have\nqualitatively important effects.\nOur results emphasize that QOs of the magnetization provide rich detail on the nature of the\nelectronic state of band insulators. Measurements of the Dingle damping factor are particularly\nvaluable. For materials that fall in the category of conventional band-insulators–including those\nwhere the band-gap includes self-consistent mean-field contributions–there should be agreement\nbetween the Dingle damping factor of the first harmonic and the electronic hybridization gap in\nthe zero field limit. Disagreement would be an indication of the relevance of physics beyond what\nis captured by the mean-field models we have considered here.\nAcknowledgements\nWe thank Johannes Knolle for helpful discussions. We would also like to acknowledge Brian\nSkinner, Trithep Devakul, and Yves Hon Kwan for their insightful comments and questions.\nFunding information This work is supported by EPSRC Grant No. EP /P034616 /1 and by a\nSimons Investigator Award.\nA Comparison with Previous Results\nHere we confirm that our T=0 result for the free energy agrees with the T!0 limit of Eq.(8) in\nRef.[10]. The system considered therein assumed an infinite valence band mass, corresponding\nto\u0011=0 here. In the notation used here, setting the chemical potential to lie inside the gap, and\ncorrecting for a missing factor of 2 and alternating sign in that equation, the oscillatory part of\nthe free energy obtained there is\n˜\nR(T) =2T n\b1X\np=1(\u00001)p\npcos\n2\u0019p\u000f0\n!c1X\nn=0exp\n\u00004\u00192pT\n!c\nn+1\n2\n\u0000pg2\n!cT1\nn+1\n2\n. (38)\nDefine the dimensionless quantity tn=T(n+1=2)=!, so that in the T!0 limit the sum over n\nbecomes an integral over t,\n˜\nR(T!0)!2!cn\b1X\np=1(\u00001)p\npcos\n2\u0019p\u000f0\n!cZ1\n0dtexp[p f(t)], (39)\nwhere\nf(t) =\u00004\u00192t\u0000g2\n!2\nct. (40)\n14SciPost Physics Submission\nOne may recognize the resulting integral as being proportional the modified Bessel function of the\nsecond kind, K1(2\u0019pg=!c)g=(\u0019!c). Alternatively, the integral can be evaluated by the method\nof steepest descent to directly find the form for g\u001d!c. The saddle point is given by\nf0(t\u0003) =0)t\u0003=g\n2\u0019!c(41)\nletting us to approximate f(t)as\nf(t)\u0019f(t\u0003) +1\n2f00(t\u0003)(t\u0000t\u0003)2(42)\ninside the integral, which is then of Gaussian form and can be evaluated to give\n˜\nR(T!0) =vtjgj!c\n2n\b\n\u00191X\np=1(\u00001)p\np3=2cos\n2\u0019p\u000f0\n!c\ne\u00002\u00192jgj\n!c. (43)\nRecalling that n\u0003=\u000f0=!\u0003\u0000\rand!\u0003= (1+\u0011)!c, this exactly matches Eq. (5) for \u0011=0 and\n\r=1=2.\nB Evaluation of Oscillatory Functions\nFunctions written as a sum over Landau level indices can be divided into oscillatory and non-\noscillatory parts using the Poisson summation formula. As a demonstration of the general proce-\ndure, here we provide the explicit calculation of the functions \u000b0,˜\u000b(B), and\f0, which then give\n˜\u0001(B)as in Eq. (13).\nIntroduce the notation E(n+\r)\u0011\u000fc\nn\u0000\u000fv\nn= (n+\r)!\u0003\u0000\u000f0= (n\u0000n\u0003)!\u0003, with n\u0003=\u000f0=!\u0003\u0000\r.\nThen we have\n\u000b(B) =n\b1X\nn=01q\nE(n+\r)2+4\u00012\n0\n=n\bZ1\n0dx1q\nE(x)2+4\u00012\n0+2n\bZ1\n0dx1X\np=1cos(2\u0019p(x\u0000\r))q\nE(x)2+4\u00012\n0. (44)\nThe first term in the second equality is what we call \u000b0and the second term is ˜\u000b(B). Putting\n!\u0003x=k2=2mcin\u000b0we find\n\u000b0=Z1\n0dk\n2\u0019k1r\u0000\n\u000fc\nk\u0000\u000fv\nk\u00012+4\u00012\n0, (45)\nand we see that setting this equal to 1 =Vas in Eq. (12) is precisely equivalent to the B=0\ngap equation, Eq. (8), at least for the isotropic, parabolic dispersion implicitly assumed with this\nchange of variables.\nNow for ˜\u000b(B), the change of variables z=x\u0000\u000f0=!\u0003=x\u0000n\u0003\u0000\rgives\n˜\u000b(B) =2n\b\n!\u0003Z1\n\u0000n\u0003\u0000\rdz1X\np=1cos(2\u0019p(z+n\u0003))È\nz2+2\u00010\n!\u00032. (46)\n15SciPost Physics Submission\nWe now assume that many Landau levels are occupied, i.e. \u000f0\u001d!\u0003, implying n\u0003\u001d1, which\nallows us to extend the lower limit of integration to \u00001. Rewriting the cosine as a sum of\nexponentials we then have\n˜\u000b(B)\u0019n\b\n!\u00031X\np=1e2\u0019ipn\u0003Z1\n\u00001dze2\u0019ipz\nr\nz2+2\u00010\n!\u0003+c.c., (47)\nwhere c.c. means the complex conjugate of the given term, and we see that the integral has become\na Fourier transform which gives a modified Bessel function of the second kind. Combining the two\nterms we then arrive at\n˜\u000b(B)\u00194n\b\n!\u00031X\np=1cos(2\u0019pn\u0003)K0\n2\u0019p2\u00010\n!\u0003\n. (48)\nWe now need\f0, the non-oscillatory part of\n\f(B) =\u0000n\b1X\nn=04\u00010\u0000\nE(n+\r)2+4\u00012\n0\u00013=2\n=\u0000n\bZ1\n0dx4\u00010\u0000\nE(x)2+4\u00012\n0\u00013=2\u00002n\bZ1\n0dx1X\np=14\u00010cos(2\u0019p(x\u0000\r))\n\u0000\nE(x)2+4\u00012\n0\u00013=2, (49)\nwhich is the first term in the second line. Making the same changes of variables as above, then\nsimilarly extending the lower limit of integration to \u00001 we find\n\f0=\u00004\u00010n\bZ1\n\u00001dz1\n\u0010\nz2+2\u00010\n!\u00032\u00113=2=\u00002n\b\n\u00010!\u0003. (50)\nCombining Eqs. (48) and (50) as in Eq. (13) we find precisely the form of ˜\u0001(B)in Eq. (14).\nC Excitonic Insulator Temperature Dependence\nHere we find the leading nonzero temperature corrections to the T=0 results presented in the\nmain text for the excitonic insulator. At nonzero Tthe mean field free energy is\n\nX=\u00012\nV\u0000TZ1\n\u00001d\u000fg(\u000f)ln\u0000\n1+e\u0000\u000f=T\u0001\n(51)\ng(\u000f) =1\nNX\nk,\u000bA(\u000f\u0000E\u000b(k)), (52)\nwhere g(\u000f)is the density of states, written in terms of the spectral density Awhich is simply a\n\u000e-function in the absence of disorder. Including a nonzero magnetic field via the prescriptions\nalready discussed and employing the Poisson summation formula we find\ng(\u000f) =2n\b\n!\u0003vt\u000f2\n\u000f2\u0000\u00012\u0002(\u000f2\u0000\u00012)(\n1+21X\np=1cos\u0014\n2\u0019p\u0012\n2p\n\u000f2\u0000\u00012+\u000f0\n!\u0003+\r\u0013\u0015)\n, (53)\n16SciPost Physics Submission\nwhere \u0001=\u0001(B,T)is the full field- and temperature-dependent gap function, and \u0002(x)is the\nHeaviside theta function, which equals 1 for x>0 and vanishes otherwise. Here the theta function\ngives the gap in the spectrum–there are no states for energies with j\u000fj<j\u0001j. Using this form of\nthe density of states in Eq. (51) and changing to a new integration variable \u0018defined through\n\u000f=p\n\u00182+\u00012we obtain\n\n(B,T) =\u00012\nV\u00002n\bT\n!\u0003Z1\n0d\u0018ln\u0014\n2\u0012\n1+cosh\u0012p\n\u00182+\u00012\nT\u0013\u0013\u0015\n\u0002(\n1+21X\np=1cos\n2\u0019p2\u0018+\u000f0\n!\u0003+\r)\n. (54)\nAs noted above, the gap \u0001is itself a function of temperature, and for B=0 obeys\n1\nV=\u0017Z1\n0d\u0018tanhp\n\u00182+\u00012\n2T\n2p\n\u00182+\u00012, (55)\nwhere\u0017is the density of states in two dimensions. We can expand around T=0 to give\ntanh\u0012p\n\u00182+\u00012\n2T\u0013\n\u00191\u00002e\u0000p\n\u00182+\u00012=T, (56)\nallowing us to separate the gap equation into a temperature independent ( T=0) part, determin-\ning the zero-temperature value of the gap, \u0001(T=0), and a nonzero temperature part providing\na correction to \u0001that is exponentially small for temperatures T\u001c\u0001(T=0). (The same can be\ndone for B6=0 as well.) This defines what we mean by the low-temperature regime.\nReturning now to the free energy, we can approximate the temperature dependent factor in\nthe low temperature regime,\nTln\u0014\n2\u0012\n1+cosh\u0012p\n\u00182+\u00012\nT\u0013\u0013\u0015\n=Æ\n\u00182+\u00012+2 ln\u0010\n1+e\u0000p\n\u00182+\u00012=T\u0011\n\u0019Æ\n\u00182+\u00012\u00002Te\u0000p\n\u00182+\u00012=T. (57)\nWith this we then separate Eq. (54) into two terms. It is straightforward to confirm that the T-\nindependent term reproduces what is found for the T=0 free energy of the excitonic insulator\nafter applying the Poisson summation formula. The second term then contains the entirety of\nthermally activated contribution to the free energy, which we see is exponentially suppressed–\nthe largest this term can be is Texp(\u0000\u0001(T=0)=T)\u001c1. 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Coleman, Topological Kondo Insulators , Annual Review\nof Condensed Matter Physics 7(1), 249 (2016), doi:10.1146 /annurev-conmatphys-031214-\n014749.\n20"    },    {        "title": "1010.0268v2.Ferromagnetic_resonance_study_of_Co_Pd_Co_Ni_multilayers_with_perpendicular_anisotropy_irradiated_with_Helium_ions.pdf",        "content": "Ferromagnetic resonance study of Co/Pd/Co/Ni multilayers with perpendicular\nanisotropy irradiated with Helium ions\nJ-M. L. Beaujour, A. D. Kent,1D. Ravelosona,2and I. Tudosa and E. E. Fullerton3\n1Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA\n2Institut d'Electronique Fondamentale, UMR CNRS 8622,\nUniversite Paris Sud, 91405 Orsay Cedex, France\n3University of California, San Diego, Center for Magnetic Recording Research, La Jolla, CA 92093-0401, USA\n(Dated: April 24, 2022)\nWe present a ferromagnetic resonance (FMR) study of the e\u000bect of Helium ion irradiation on the\nmagnetic anisotropy, the linewidth and the Gilbert damping of a Co/Ni multilayer coupled to Co/Pd\nbilayers. The perpendicular magnetic anisotropy decreases linearly with He ion \ruence, leading to\na transition to in-plane magnetization at a critical \ruence of 5 \u00021014ions/cm2. We \fnd that the\ndamping is nearly independent of \ruence but the FMR linewidth at \fxed frequency has a maximum\nnear the critical \ruence, indicating that the inhomogeneous broadening of the FMR line is a non-\nmonotonic function of the He ion \ruence. Based on an analysis of the angular dependence of the\nFMR linewidth, the inhomogeneous broadening is associated with spatial variations in the magnitude\nof the perpendicular magnetic anisotropy. These results demonstrate that ion irradiation may be\nused to systematically modify the magnetic anisotropy and distribution of magnetic anisotropy\nparameters of Co/Pd/Co/Ni multilayers for applications and basic physics studies.\nPACS numbers:\nCo/Ni multilayers are of great interest in informa-\ntion technology and in spin-transfer devices because they\ncombine high spin polarization with large perpendicu-\nlar magnetic anisotropy (PMA) [1{3]. Perpendicular\nanisotropy was predicted in Co/Ni multilayers and has\nbeen shown experimentally to be a function of layer com-\nposition and thin \flm growth conditions [4]. Recently\nit has been shown that the coercivity and perpendicu-\nlar anisotropy of a multilayer can be tailored by Helium\nion irradiation [5], making it possible to modify \flms af-\nter growth to tune their magnetic properties. This is\nof great interest for applications and basic physics stud-\nies, as in many cases the perpendicular anisotropy of a\nstructure sets importance device metrics, and ion irradi-\nation o\u000bers the possibility of changing these properties,\nboth locally and globally, after device fabrication. For\ninstance, in spin-transfer magnetic random access mem-\nories (STT-MRAM) the current threshold for switching\nis proportional to the PMA [2, 6].\nLight ion irradiation has been used to vary the mag-\nnetic properties of multilayer \flms in many earlier studies\n[7{11]. For instance, the coercivity of Co/Pt multilayers\nwas found to decrease with ion dose [8]. This behavior\nwas attributed to interface mixing and strain relaxation\nreducing the PMA. Very recently, it was reported that\nthe coercive \feld of Co/Ni multilayers decreases linearly\nwith increasing He+irradiation \ruence up to F= 1015\nions/cm2, suggesting changes in the magnetic anisotropy\nof the \flm [10]. The e\u000bect of ion irradiation on the FMR\nlinewidth has also been studied in Au/Fe multilayer \flms\nwith PMA [11]. The PMA is reduced by He+irradi-\nation and the authors explained this by a reduction of\nthe inhomogeneous contribution to the FMR linewidth.\nIn a recent paper, we presented a FMR study of the\nanisotropy and the linewidth of a Co/Ni multilayer \flmexposed to a relatively high He+irradiation \ruence ( F\n= 1015ions/cm2) [12]. In addition to a strong decrease\nof the PMA, the contribution to the linewidth from spa-\ntial variation of the anisotropy, was reduced compared\nto that of a non irradiated Co/Ni multilayer. Further-\nmore, a correlation between the anisotropy distribution\nand the linewidth broadening from two-magnon scatter-\ning (TMS) mechanism was observed. However, a system-\natic study of the e\u000bect of the He+irradiation on the FMR\nspectra as a function of \ruence has yet to be reported.\nIn this paper, we present a FMR study of a Co/Ni\nmultilayer coupled to Co/Pd bilayers exposed to Helium\nion irradiation of \ruence up to 1015ions/cm2. The PMA\nand the contributions to the FMR linewidth, including\nthose from Gilbert damping ( \u000b), are studied as a function\nof \ruence.\nThe samples had the following layer structure: jj3 Taj1\nPdj0.3 Coj1 Pdj0.14 Coj[0.8 Nij0.14 Co]\u00023j1 Pdj0.3 Coj1\nPdj3 Tajj(layer thickness in nm) and was fabricated by dc\nmagnetron sputtering. The Co/Ni multilayer is embed-\nded between Co/Pd bilayers to enhance the overall PMA\nof the \flm and to have resonance frequencies in which the\nfull angular dependence of the FMR response could be\ninvestigated in a 1 T electromagnet. The substrate was\ncleaved into several pieces that were then exposed to dif-\nferent doses of Helium ion irradiation of energy 20 keV\nwith \ruence in the range 1014\u0014F\u00141015ions/cm2.\nFMR measurements were conducted at room tempera-\nture using a coplanar waveguide (CPW). Details of the\nexperimental setup can be found in [13]. The \feld swept\nCPW transmission signal ( S21) was recorded as a func-\ntion of frequency for dc magnetic \felds normal to the \flm\nplane and as a function of the out-of-plane \feld angle at\n20 GHz. The magnetization density of the \flm at room\ntemperature was measured with a SQUID magnetome-arXiv:1010.0268v2  [cond-mat.mes-hall]  5 Oct 20102\n03 06 09 00.40.60.8K\nK0\n5 1 001020H\n /s615490Hres  (T) 0 \n1 \n2.5 \n F\nield angle /s61542H  (deg)20 GHz 5 \n7.5 \n10HH\n-0.40.00.40.80\n102030f  (GHz)(a) \n(b) /s615490Hres    (T) \n/s61549 0Meff (\nc)  \nF (1014 ions/cm2)K (105 J/m3)21\nFIG. 1: a) Angular dependence of the resonance \feld for dif-\nferent irradiation \ruence ( \u00021014ions/cm2). The solid lines\nare guides to the eye. b) Frequency dependence of the reso-\nnance \feld when the applied \feld is normal to the \flm plane\n(\u001eH= 90o) at selected \ruences. The solid lines are the \fts\nto Eq. 1. The zero frequency intercept gives the e\u000bective de-\nmagnetization \feld, \u00160Me\u000b. c) The second and fourth order\nperpendicular anisotropy constants, K1andK2, versus \ru-\nence.\nter:Ms'4:75\u0002105A/m. Within the measurement\nuncertainty, Msremains unchanged after irradiation.\nFig. 1a shows the out of plane angular dependence\nof the resonance \feld at 20 GHz for di\u000berent \ruences.\nFor the non-irradiated \flm, the resonance \feld when H\nis normal to the \flm plane ( \u001eH= 90o) is smaller than\nthat when the \feld is in the \flm plane ( \u001eH= 0o). This\nshows that the magnetic easy axis is normal to the \flm\nplane. As the \ruence increases, Hresat\u001eH= 90oin-\ncreases whereas that at \u001eH= 0odecreases. In the high\n\ruence range, F>5\u00021014ions/cm2,Hresis the larger\nwhen the \feld is normal to the \flm plane, i.e. the mag-\nnetic easy axis is in the \flm plane. Fig. 1b shows the\nfrequency dependence of the resonance \feld for di\u000berent\n\ruences when the dc \feld is normal to the \flm plane.\nThis data is \ftted to the resonance condition [14]:\nf=1\n2\u0019\r\u00160(Hres\u0000Me\u000b); (1)\nwhere\ris the gyromagnetic ratio. \u00160Me\u000b, the e\u000bec-\ntive easy plane anisotropy, is given by: Me\u000b=Ms\u0000\n2K1=(\u00160Ms), whereK1is the second order anisotropy\nconstant. We \fnd that \u00160Me\u000bis negative at low \ru-\nence which implies that the PMA is su\u000ecient to over-\ncome the demagnetizing energy and hence the easy axis\nis normal to the \flm plane. As the \ruence is further\nincreased,\u00160Me\u000bbecomes positive. These results con-\n\frm that there is a re-orientation of the easy axis, as was\ninferred indirectly through magnetic hysteresis loop mea-\nsurements in Ref. [10]. \u00160Me\u000bchanges sign for \ruence\nbetween 5 and 7.5 \u00021014ions/cm2. Therefore, by expos-\ning the \flm to a speci\fc \ruence, it is posible to engineer\n01000\n1000\n3 06 09 001000\n102030020406080 \n \nF=0Δ\nHα       ΔHinh         ΔHtot  \nF=5/s615490ΔH  (mT)f\n/s61472 ( GHz )  F=10F\nield angle /s61542H  ( deg )HH /s615490ΔH  (mT) \n \nF=7.5H\nFIG. 2: On the left, the linewidth as a function of frequency\nfor the \flm irradiated at 7.5 \u00021014ions/cm2. The solid line\nis a linear \ft to the experimental data. On the right, the\nangular dependence of the linewidth at 20 GHz for a selec-\ntion of \ruences. The solid lines represent the \fts to the total\nlinewidth \u0001 Htot= \u0001H\u000b+\u0001Hinh, , where the intrinsic damp-\ning and the inhomogeneous contribution are represented by\nthe dashed line and the dotted line respectively.\nthe anisotropy so that the PMA \feld just compensates\nthe demagnetization \feld.\nThe second order perpendicular anisotropy constant\nK1decreases linearly with \ruence (Fig. 1c). The \flm\nirradiated at 1015ions/cm2has an anisotropy constant\n40% smaller than that of the non-irradiated \flm. The\n4thorder anisotropy constant K2is determined from\nthe angular dependence of Hresfor magnetization angles\n45\u0014\u001e\u001490o[13].K2is smaller than K1by a factor 10,\nand is nearly independent of \ruence.\nThe FMR linewidth \u00160\u0001Hwhen the dc \feld is ap-\nplied normal the \flm plane was measured as a function\nof frequency. Fig. 2 shows \u00160\u0001Hversusffor the \flm\nirradiated at F= 7:5\u00021014ions/cm2. The linewidth in-\ncreases linearly with frequency, characteristic of Gilbert\ndamping, an intrinsic contribution to the linewidth \u0001 H\u000b\n[15]:\n\u0001H\u000b=4\u0019\u000b\n\u00160\rf: (2)\nFrom a linear \ft to the experimental data, the magnetic\ndamping constant is estimated from the slope of the line:\n\u000b= 0:037\u00060:004. The \flms irradiated at F= 0;1 and\n10\u00021014ions/cm2shows a similar frequency dependence\nof the linewidth and have about the same damping con-\nstant,\u000b\u00190:04. At intermediate \ruence F= 2:5, 5\u00021014\nions/cm2, the linewidth is enhanced and is frequency in-\ndependent, i.e. the linewidth is dominated by an inhomo-\ngeneous contribution, \u0001 Hinh. The angular dependence\nof the linewidth measured at 20 GHz is shown in Fig.\n2 for \flms irradiated at selected \ruences. For the non-\nirradiated \flm and the \flm irradiated at 1015ions/cm2,\nthe linewidth is practically independent of the \feld an-3\n05 1 01 53060901200\n5100.030.04µ0 Δ/s61512/s61472 \n ⊥ \n(mT)F\n  (1014 ions/cm2)ΔK1  ( 105 J/m3 )H\n0.00.10.20.30.4 \nα-dampingF\n (1014 ions/cm2)\nFIG. 3: The \ruence dependence of the linewidth at 20 GHz\nwhen the dc \feld is normal to the \flm plane (squares). The\nsolid circles represent the \ruence dependence of the distribu-\ntion in the PMA constant K1determined from \ftting \u0001 Hvs.\n\u001eH. The inset shows the Gilbert damping constant \u000bas a\nfunction of \ruence.\ngle from about 30oup to 90o. For the \flm irradiated at\n5\u00021014ions/cm2, \u0001His clearly angular dependent and\nshows a minimum at an intermediate \feld angle.\nThe angular dependence of the linewidth was \ft to\na sum of the intrinsic linewidth \u0001 H\u000band an inhomo-\ngeneous contribution \u0001 Hinhfor magnetization angles\n45o\u0014\u001e\u001490o, an angular range in which TMS does\nnot contribute to the linewidth [13]. The inhomogeneous\nlinewidth is given by:\n\u0001Hinh:(\u001eH) =j@Hres=@K 1j\u0001K1+j@Hres=@\u001ej\u0001\u001e;(3)\nwhere \u0001K1is the width of the distribution of anisotropies\nand \u0001\u001eis the distribution of the angles of the magnetic\neasy axis relative to the \flm normal. The computed\nlinewidth contributions are shown for the \flm irradiated\natF= 5\u00021014ions/cm2in Fig. 2. Note that the intrin-\nsic contribution \u0001 H\u000bis practically independent of \feld\nangles, as expected when the angle between the magne-\ntization and the applied \feld is small. For this sample,\nthe maximum angle is about 5oand it is due to the fact\nthat the resonance \feld ( Hres'0:6 T) is much larger\nthan the e\u000bective demagnetization \feld ( Me\u000b'0). Theinhomogeneous contribution from the distribution in the\nanisotropy \feld directions does not signi\fcantly a\u000bect the\n\ft. For the \flm irradiated at the lower and upper \ruence\nrange, the angular dependence of the intrinsic linewidth\nis computed \fxing the value of \u000bto that obtained from\nthe \ft of the frequency dependence of the linewidth. For\nthe other \flms ( F=2.5 and 5\u00021014ions/cm2),\u000bwas\na \ftting parameter.\nThe \ruence dependence of \u0001 K1and the linewidth at\n20 GHz are shown in Fig. 3. The inset shows the Gilbert\ndamping constant as a function of \ruence. The linewidth\nat 20 GHz when the \feld is normal to the \flm plane is\na non monotonic function of \ruence. \u0001 Hincreases as\nthe \ruence increases, reaching a maximum value at F\n\u00195\u00021014ions/cm2. Then, as the \ruence is further in-\ncreased, \u0001Hdecreases and falls slightly below the range\nof values at the lower \ruence range. Interestingly, the\nlarger linewidth is observed just at the \ruence for which\n\u00160Me\u000b= 0. The magnetic damping is practically not\na\u000bected by irradiation within the error bars: \u000b\u00190:04.\nThe distribution of PMA constants, \u0001 K1, shows a similar\n\ruence dependence as the total linewidth, with a maxi-\nmum at F\u00195\u00021014ions/cm2, clearly indicating that\nthis is at the origin of the \ruence dependence of the mea-\nsured linewidth. The distribution in PMA anisotropy is\nalmost zero when the \ruence is above 7 \u00021014ions/cm2.\nThe largest value of \u0001 K1corresponds to variation of K1\nofabout 8%, which is much larger than that of non irradi-\nated \flm and the highly irradiated \flm, \u0001 K1=K1\u00192%\nand 0.3% respectively.\nIn summary, irradiation of Co/Pd/Co/Ni \flms with\nHelium ions leads to clear changes in its magnetic char-\nacteristics, a signi\fcant decrease in magnetic anisotropy\nand a change in the distribution of magnetic anisotropies.\nImportantly, this is achieved without a\u000becting the \flm\nmagnetization density and magnetic damping, which re-\nmain virtually unchanged. It would be of interest to have\na better understanding of the origin of the maximum in\nthe distribution of magnetic anisotropy at the critical\n\ruence, the \ruence needed to produce a reorientation of\nthe magnetic easy axis. Nonetheless, these results clearly\ndemonstrate that ion irradiation may be used to system-\natically tailor the magnetic properties of Co/Pd/Co/Ni\nmultilayers for applications and basic physics studies.\n[1] S. Mangin et al: , Nat. Mater. 5, 210 (2006).\n[2] S. Mangin et al: , Appl. Phys. Lett. 94, 012502 (2009).\n[3] D. Bedau et al: , Appl. Phys. Lett. 96, 022514 (2010).\n[4] G. H. O. Daalderop et al: , Phys. Rev. Lett. 68, 682\n(1992); S. Girod et al: , Appl. Phys. Lett. 94, 262504\n(2009).\n[5] C. Chappert et al: , Science 280, 1919 (1998).\n[6] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[7] A. Traverse et al: , Europhysics Letters 8, 633 (1989).\n[8] T. Devolder, Phys. Rev. B 62,5794 (2000).\n[9] J. Fassbender etal: , J. Phys. D: J. Appl. 37, R179 (2004).[10] D. Stanescu et al: , J. Appl. Phys. 103, 07B529 (2008).\n[11] C. Bilzer et al: , J. Appl. Phys. 103, 07B518 (2008).\n[12] J.-M. L. Beaujour et al: , Phys. Rev. B 80, 180415(R)\n(2009).\n[13] J.-M. L. Beaujour etal: , Eur. Phys. J. B 59, 475 (2007).\n[14] S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon,\nOxford, 1966).\n[15] Spin Dynamics in Con\fned Magnetic Structures II\n(edited by B. Hillebrands and K. Ounadjela (Springer,\nHeidelberg, 2002))."    },    {        "title": "2011.08136v2.Switchable_Damping_for_a_One_Particle_Oscillator.pdf",        "content": "Switchable Damping for a One-Particle Oscillator\nX. Fan,1, 2,\u0003S. E. Fayer,2T. G. Myers,2B. A. D. Sukra,2G. Nahal,2and G. Gabrielse2,y\n1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA\n2Center for Fundamental Physics, Northwestern University, Evanston, Illinois 60208, USA\n(Dated: January 1, 2021)\nThe possibility to switch the damping rate for a one-electron oscillator is demonstrated, for an\nelectron that oscillates along the magnetic \feld axis in a Penning trap. Strong axial damping can be\nswitched on to allow this oscillation to be used for quantum nondemolition detection of the cyclotron\nand spin quantum state of the electron. Weak axial damping can be switched on to circumvent the\nbackaction of the detection motion that has limited past measurements. The newly developed switch\nwill reduce the linewidth of the cyclotron transition of one-electron by two orders of magnitude.\nI. INTRODUCTION\nA single isolated trapped particle is an ideal system to\ntest predictions of the Standard Model of particle physics\n(SM) involving magnetic moments and charge-to-mass\nratios of the electron, positron, proton and antiproton [1{\n6]. After decades of agreement, the measured electron's\nmagnetic moment in Bohr magnetons \u0016=\u0016B[2, 3] now\ndisagrees with the SM prediction [7, 8] by 2.4 standard\ndeviations. This discrepancy at the 3 \u000210\u000013level has\nstimulated many new theoretical investigations [9{15].\nThe electron measurements were done with a sin-\ngle electron in the magnetic \feld and electrostatic\nquadrupole potential of a Penning trap [2, 3]. Quantum\njump spectroscopy of fully-resolved quantum levels of the\ncyclotron and spin motion with an electron cooled to 0.1\nK was key [16]. The detection backaction uncertainty\nthat remains must be reduced to improve the accuracy\nof the measurements to investigate the intriguing discrep-\nancy between prediction and measurement. This work is\na demonstration of a 200 MHz electronic switching sys-\ntem at cryogenic temperatures developed to accomplish\nthis. It will make it possible to resolve the quantum en-\nergy levels of the electron's axial motion for the \frst time\n[17, 18]. The harmonic oscillation along the magnetic\n\feld direction is a detection motion used for quantum\nnondemolition (QND) detection [19{21] of the quantum\ncyclotron and spin states [22]. The cryogenic switching is\nbetween weak axial damping to circumvent all detection\nbackaction, and strong axial damping to make a one-\nparticle signal large enough to observe.\nA recent quantum calculation establishes the theoret-\nical basis for circumventing detector backaction [17, 18].\nThe calculation shows that reduced axial damping makes\nit possible to resolve a cyclotron resonance for each indi-\nvidual axial quantum state. The backaction e\u000bect is cir-\ncumvented insofar as the cyclotron excitations that take\nplace during the fraction o\u000b the time that the backaction\n\ructuations take the axial motion into its ground state\ncan be resolved.\n\u0003xingfan@g.harvard.edu\nygerald.gabrielse@northwestern.eduSection II is a brief summary of the detector backaction\nthat must be eliminated; it does not repeat the deriva-\ntions and calculations recently reported [17, 18]. Sec. III\npresents the detection circuity and calculations of its ef-\nfect upon the electron axial motion. Sec. IV provides the\n\frst demonstration of the circuitry being used to change\naxial damping and expected lineshape. An overview of\nthe experimental apparatus within which this circuit was\nused are in [23{25]. Sec. V presents a summary and con-\nclusions.\nII. QND DETECTION AND BACKACTION\nThe QND detection of the quantum cyclotron and spin\nstates (described elsewhere in detail [17, 18]) is summa-\nrized brie\ry here to provide the context needed to discuss\nthe detector.\nThe energy eigenstates of the electron in the Penning\ntrap are direct products of independent cyclotron, spin\nand axial eigenstates, designated by their quantum num-\nbersjnc;ms;nzi. For an electron cooled to 0 :1 K, only\nthree cyclotron and spin states are populated. Two are\ncyclotron ground states (with cyclotron and spin quan-\ntum numbers nc= 0 andms=\u00061=2), both stable for\nyears or more. One is a spin down cyclotron excited\nstate (with ms=\u00001=2 andnc= 1) that has a damp-\ning time due to cavity-inhibited spontaneous emission of\ntypically 5 seconds [26, 27]. The energy eigenvalues for\nthese eigenstates are\nE(nc;ms;nz) =~!c\u0000\nnc+1\n2\u0001\n+~!sms+~!z\u0000\nnz+1\n2\u0001\n+~\u000ec\u0000\nnc+1\n2\u0001\u0000\nnz+1\n2\u0001\n+~\u000esms\u0000\nnz+1\n2\u0001\n;\n(1)\nwhere ~is the reduced Planck constant, and the last line\nis due to a magnetic bottle gradient [22] that is intro-\nduced to couple the cyclotron and spin states to the axial\nstates. The magnetic bottle parameters, \u000ecand\u000es, are\nthe shifts of axial frequency for one-quantum cyclotron\nand spin transitions, respectively [17, 22]. It is a QND\ncoupling that does not change the energy eigenstates for\nthe system [16].\nThe magnetic bottle gradient is essential for detecting\nthe quantum spin and cyclotron state of the electron be-arXiv:2011.08136v2  [quant-ph]  31 Dec 20202\ncause it couples the axial frequency to the cyclotron and\nspin energy. From Eq. (1) we see that axial frequency\nshifts,\n\u0001!z= (nc+1\n2)\u000ec+ms\u000es (2)\nreveal changes in the cyclotron and spin quantum num-\nbers. Measuring the axial frequency is thus a QND de-\ntection that does not itself change the quantum state.\nCompared to the spin and cyclotron frequencies in the\nbest measurement [2],\n!s=(2\u0019) = 150:5 GHz (3)\n!c=(2\u0019) = 150:3 GHz (4)\n!a=(2\u0019) = (!s\u0000!c)=(2\u0019) = 173 MHz (5)\nthe corresponding magnetic bottle shifts are very small,\nwith\n\u000es=(2\u0019) = 3:872 Hz (6)\n\u000ec=(2\u0019) = 3:868 Hz (7)\n\u000ea=(2\u0019) = (\u000es\u0000\u000ec)=(2\u0019) = 0:004 Hz (8)\nThe frequencies for spin and cyclotron di\u000bers only by\n1 part per 1000 since g=2\u00191:001. If these tiny shifts\nare detected, the QND coupling keeps the axial detec-\ntion motion (and the transistor ampli\fer to which it is\ncoupled) from altering the quantum state of the spin and\ncyclotron motion.\nHowever, the QND coupling does not prevent a back-\naction that shifts the cyclotron and anomaly frequen-\ncies in proportion to the axial energy. This parallel de-\ntection backaction is an unavoidable consequence of the\nQND coupling. The physical reason is that axial mo-\ntion through the magnetic bottle gradient changes the\nmagnetic \feld in which the cyclotron and spin motions\nevolve. The backaction shifts from Eq. (1) are\n\u0001!c= (nz+1\n2)\u000ec (9)\n\u0001!a= (nz+1\n2)\u000ea: (10)\nWith the\u000ecneeded to detect the quantum states (above),\nthe backaction shifts are large. The range of excited axial\nstates is given by the size of the average axial quantum\nnumber for a Boltzmann distribution,\n\u0016nz=kBT=(~!z)\u001910 (11)\nfor!z=(2\u0019) = 200 MHz and a detector temperature of\nT= 0:1 K. The backaction cyclotron linewidth, greater\nthan \u0016nz\u000ec, limited the accuracy of past electron measure-\nments.\nSince the magnetic bottle shift \u000eccannot be reduced\nwithout making it impossible to detect the cyclotron and\nspin state, it is advantageous to \fnd a way around the\nbackaction linewidth. The cyclotron width produced by\nthe axial ground state ( nz= 0) is the axial decoherence\nwidth \u0016nz\rz, where\rzis the damping rate for the axial\noscillation energy due to the detection circuit. The pro-\nposal in References [17, 18] is to make \u0016 nz\rzmuch lessthan the cyclotron frequency shift (Eq. (9)) between the\naxial ground state and the \frst excited state,\n\u0016nz\rz\u001c\u000ec (12)\nIn this \\strongly dispersive regime\" [28], the broad cy-\nclotron resonance of width greater than \u0016 nz\u000ecis resolved\ninto individual resonance lines, each of which corresponds\nto a particular axial state and quantum number. Mea-\nsuring the resonance line that corresponds to the axial\nground state, with nz= 0, will make it possible to de-\ntermine the cyclotron frequency that is shifted only by\nthe zero point motion of the axial oscillation (see also\nSec. IV).\nTo realize this proposal for circumventing the detector\nbackaction requires that the axial damping caused by the\ndetector\rzbe reduced during the time that one-quantum\ncyclotron transitions are driven. However, since the in-\nduced signal across a detection circuit is proportional to\nthe axial damping rate, this rate must be switched back\nto a higher value during the time of the measurement\nthat the cyclotron quantum state is read out. The axial\ndamping rate was \rz=(2\u0019) = 1 Hz in the best measure-\nment [2], so it requires about two orders of magnitude\nof reduction to achieve Eq. (12). The rest of this paper\nis a proposal for switching the axial damping constant,\na \frst demonstration of such switching, and a conclud-\ning estimate of the improvement in electron and positron\nmagnetic moment measurement that should now be pos-\nsible.\nIII. DETECTION CIRCUITRY\nA. Impedance and Damping\nThe axial motion of a trapped electron along the direc-\ntion of a strong magnetic \feld within surrounding trap\nelectrodes is represented in Fig. 1(a). This motion in-\nduces a 200 MHz electrical current in the frequency de-\npendent impedance Z(!) of an attached electrical circuit\nthat can be switched between two circuit states. The\nelectronic switch between the two circuit states is a high\nelectron mobility transistor (HEMT) in series with ca-\npacitorCtuning . This switch is shown within the dashed\nrectangle in the \fgure, and its operation will be described\npresently (Sec. III C).\nThe \frst state of the circuit (Fig. 1(b)) is a purely re-\nsistive circuit impedance Rthat pertains at the circuit's\nresonant frequency, which is made equal to the electron\noscillation frequency. This resistance is due to the un-\navoidable loss in the circuit and the e\u000bective input resis-\ntance of the HEMT, rather than from a physical resistor\nthat soldered into the circuit. The value of Ris largest\nwhen the losses are the smallest. The largest possible R\nis desired because this maximally damps this motion and\nproduces the largest possible oscillating voltage that can\nbe Fourier transformed to determine the axial oscillation\nfrequency (Sec. III B).3\nThe second state of the circuit is e\u000bectively a capaci-\ntance and resistance in parallel (Fig 1(c)) that is intended\nto couple as weakly as possible to the electron motion.\nThe high frequency current that the electron motion in-\nduces in the circuit \rows primarily through the capac-\nitor. Very little signal voltage is produced so that this\ncircuit state is not useful for detection. Because very lit-\ntle power is dissipated, there is also very little damping of\nthe electron motion. This circuit state is extremely use-\nful for spectroscopy just because the electron motion is\nso weakly coupled to the circuit that the electron motion\nis perturbed very little.\nWith no detection circuit attached, the harmonic os-\ncillation of a particle displaced from equilibrium by zis\ndescribed by the familiar equation of an undamped har-\nmonic oscillator with\nd2z\ndt2+!2\nzz= 0: (13)\nThe axial oscillation frequency, !z, is determined by the\nDC bias voltages applied to the trap electrodes and by\nthe geometry of the trap electrodes [26, 29, 30]. It was at\n!z=(2\u0019) = 200 MHz in the most recent experiment [3].\nUsing complex circuit notation, the real displacement z\nis given by\nz= Re(z[!]ei!t); (14)\nwherez[!] is complex. Eq. (13) becomes ( \u0000!2+\n!2\nz)z[!] = 0. The familiar solution is a free oscillation\nat frequency !zwhose \fxed amplitude and phase is de-\nscribed by the complex z[!z].\nThe detection circuit is attached to the trap electrodes\nas shown in Fig. 1(a). The top electrode is the detec-\ntion endcap of the trap. The ring electrode to which the\ncircuit is attached, and other electrodes, are grounded\nfor frequencies at or near the axial frequency, !z. The\nDC lines needed to bias the trap electrodes, and those\nfor biasing the two high electron mobility transistors\n(FHX13LG from Fujitsu) are omitted because they are\ndesigned to not a\u000bect the RF behavior of the circuit.\nThese bias lines for the HEMTs, and all leads entering\nthe cryostat, are carefully \fltered to keep external noise\nfrom heating the trapped electron.\nThe oscillation of a particle with charge einduces an\noscillatory displacement current that \rows through the\nelectrodes and circuit,\nI=e\u0014\n2z0dz\ndt; (15)\nwherez0is a trap dimension and \u0014is the fraction of\ninduced charge determined by the trap geometry [26]. In\ncomplex circuit notation, with I= Re[I[!]ei!t],\nI[!] =i!e\u0014\n2z0z[!]: (16)\nThe imaginary number indicates that the current is 90\ndegrees out of phase with the electron oscillation that\ninduces it.\nGDSCtuningCtrapL2RlossswitchHEMTPenningtrapoutputamplifier HEMTGDSL1VGSfor γztuning R(a)\n(b)(c)switch biased off(VGS = -1V, open)switch biased on(VGS = 0V, closed)R’C’𝒞ℛoff(≫10kΩ)ℛon   (~10Ω)orswitch HEMT drain-to-source modelDSFIG. 1. (a) The electron trap and RF electronic circuit used\nto detect its axial motion. The switch HEMT model is used\nin Sec. III D. The electron sees the e\u000bective circuits when the\nHEMT in dashed box is biased o\u000b (b) or on (c).\nAn oscillatory voltage V[!] is induced between the de-\ntection endcap and the ring when the induced current\n\rows through the circuit. This creates an electrical force\non the a particle of charge e,\u0000e\u0014V[!]=(2z0) that opposes\nthe motion. The circuit presents a complex impedance,\nZ(!), across the two trap electrodes to which it is con-\nnected, whereupon V[!] =I[!]Z(!). The equation of\nmotion for the particle with mass mand the circuit to-\ngether is\n\"\n\u0000!2+!2\nz+i!Z(!)\nm\u0012e\u0014\n2z0\u00132#\nz[!] =F[!]\nm;(17)\nor\n\u0014\n\u0000!2+!2\nz+i!\rzZ(!)\nR\u0015\nz[!] =F[!]\nm; (18)\nwhere a driving force F= Re[F[!]ei!t] is included, and\nthe constant,\n\rz=1\nm\u0012e\u0014\n2z0\u00132\nR (19)\nis the damping rate of electron motion coupled to a re-\nsistive impedance R. We will take Rto be the maximal\nresistive impedance of detection circuit. In Eq. (18), the\ncircuit impedance Z(!) is scaled by this maximal resis-\ntance. The damping rate for general impedance is given\nby\rz\u0002(Re[Z(!z)]=R), and the suppression parameter\n\u0011=R\nRe[Z(!z)](20)\nde\fnes how much the damping is suppressed for general\nZ(!) from its maximum value R.4\nB. Maximal Coupling\nThe maximum coupling of the electron axial motion\nand the detection circuit takes place when the HEMT\nswitch within the dashed box in Fig. 1(a) is biased \\o\u000b,\"\nwith a gate-to-source voltage of VGS=\u00001 V typically.\nThe e\u000bective impedance of the HEMT and Ctuning to-\ngether is then large enough that it can be neglected\nin determining the impedance that the detection circuit\npresents to the electron axial motion.\nThe detection circuit is constructed so that the reac-\ntance of the inductor L=L1+L2cancels the reactance of\nthe capacitance of the circuit at the oscillation frequency\nof the electron oscillation, !z. Near to this frequency,\nthe circuit acts as a pure resistance, Z(!z) =R. When\nthe induced current \rows through the resistance, the I2R\nloss removes energy from the axial motion of the electron.\nSinceZ(!z) =R, the damping rate from Eqs. (18-19) is\n\rz. Experiments have shown that R> 60 k\n su\u000eces so\nthat the one-particle signal can be Fourier transformed to\nascertain that oscillation frequency[2]. Shifts in this mea-\nsured frequency signal one-quantum transitions of the\nspin and cyclotron motions[16, 17].\nIf there is no driving force (i.e. F= 0), an initial dis-\nplacement ~z0of the particle from equilibrium damps ex-\nponentially as\nz= Re\u0002\n~z0ei!zt\u0003\ne\u00001\n2\rzt: (21)\nThe amplitude and energy of this free oscillation damp\nwith time constants 2 =\rzand 1=\rz. A resonant driving\nforceF0cos (!zt) applied for a time t\u001d(\rz)\u00001sets up a\nsteady state\nz=F0\nm\rz!zsin (!zt): (22)\nThe driving force equals the damping force for this steady\nstate. The steady-state signal, because it does not damp\nout, can be averaged to determine the electron's axial\noscillation frequency as accurately as needed. In princi-\nple, increasing the drive force Fmakes the amplitude of\nthe driven particle to be arbitrarily large. In practice,\nthe oscillation amplitude and hence the particle velocity\nmust be kept small enough so that the particle oscillation\nremains in the harmonic potential region near the center\nof the trap.\nThe maximum power dissipated by the current \rowing\nthrough the resonant detection circuit is\nP=m\rz_z2(23)\nFor constant oscillation amplitude, Pis proportional to\n\rz. The cryogenic detection circuit is designed to make R\nas large as possible, thus maximizing both the damping\nrate\rzand the signal power. The circuit with its HEMT\nampli\fer is designed to maximize the signal power sent\nout of the dewar to be Fourier-transformed to determine\nthe electron axial oscillation frequency !z.The detection circuit resonant at the electron oscil-\nlation frequency acts as a simple resistance Rthrough\nwhich the induced current \rows. This current would pri-\nmarily \row through the capacitance between the trap\nelectrodes, Ctrap= 8:2 pF except that a parallel induc-\ntorL=L1+L2= 70 nH cancels the reactance of the\ncapacitor to prevent this. (The cancelled capacitance also\nincludes small contributions from the distributed capaci-\ntance in the inductor, and from the input capacitance of\nthe ampli\fer HEMT that is outside the dashed box in the\ncircuit \fgure.) The inductor is tapped such that L1= 55\nnH andL2= 15 nH to match the impedance of the tuned\ncircuit and HEMT ampli\fer. The tapped inductor is a\n78 \n coaxial transmission line resonator, tapped at 15 cm\nout of its total length of 20 cm. The mutual inductance\nbetweenL1andL2is negligible in such a resonator.\nThe resistance Ris due to the RF losses that cannot\nbe avoided in the circuit, Rloss, along with a contribution\nfrom the input impedance of the ampli\fer HEMT and the\nswitching circuit. No explicit resistive element is added\nto the circuit. Since R=Q!zL, the e\u000bective value of\nthis resistance is determined by the measured resonance\nfrequency,!z, the measured inductance L=L1+L2;and\nthe quality factor of the LCR circuit that is determined\nby the observed resonance width. For these demonstra-\ntion experiments, we typically observed Q= 800 and\nR= 83 k\n.\nC. Minimal Coupling\nWhile one-quantum cyclotron and spin transitions are\nbeing driven to make a magnetic moment measurement,\nthe highRand high\rzare not desirable. These would\ncause the backaction of the strongly coupled detector\nupon the electron. As a result, the electron's axial ampli-\ntude \ructuates within a magnetic \feld gradient, causing\n\ructuations of the electron cyclotron and spin frequen-\ncies.\nWhat is desirable during the time that these quantum\ntransitions are being driven is the smallest possible cir-\ncuit resistance and axial damping rate, \rz. Heater or\nvaractor based tuners have been used previously in dif-\nferent environments[31]. However, in a 0.1 K apparatus\nat 5 Tesla, switching the resistance of the circuit from a\nhigh value to a low one is not a trivial task. The solu-\ntion explored here is the use of an electrical HEMT switch\n(within the dashed box in Fig. 1(a)) that can be switched\nrapidly and reliably with the low heat dissipation needed\nto operate at 0.1 K.\nA HEMT gate-to-source voltage of VGS= 0 V makes\nthe HEMT act like a small resistance. It is in se-\nries with a capacitor called Ctuning . Its major e\u000bect\nis tuning the detection circuit's resonant frequency to\na much lower value. For example, for a typical value\nCtuning = 22 pF,!0\n0=(2\u0019) = 172 MHz, with a paral-\nlel resistance R0= 3:4 k\n and quality factor Q0= 45.\nThe current induced by the electron oscillation at !z5\n0.2pF0.2pF100pF100pFnetwork analyzer drivenetwork analyzerdetectionresonance circuit Z(ω)50ΩVoutVin50Ω\nFIG. 2. Circuit to measure the impedance Z(!) of the reso-\nnance detection circuit\nthus sees an e\u000bective detection circuit that is a resistor\nR0in parallel to a capacitor C0(Fig. 1(c)). The e\u000bec-\ntive damping resistance Re[ Z(!z)] is reduced by approx-\nimately 300 and the damping rate \rzis reduced by a\nfactor of\u0011=R=Re[Z(!z)]\u0019300. In Sec. III D we ex-\nperimentally determine the parameters of this e\u000bective\ncircuit.\nD. Optimization of the Tuning Capacitor\nThe value of the capacitor Ctuning must be experimen-\ntally optimized, since at 200 MHz the stray capacitance\nand inductance modify the nominal component values.\nWhen the HEMT switch is biased o\u000b, Ctuning must be\nsmall enough so that its reactance keeps the biased-o\u000b\nHEMT drain-to source resistance from unacceptably low-\nering theR,\rzand the detection sensitivity. At the same\ntime,Ctuning must be large enough that its reactance\nand the low drain-to-source resistance of the biased-on\nHEMT will reduce \rzas much as possible.\nWith a low loss capacitor substituted for trap elec-\ntrodes for the optimization measurements, the e\u000bective\nresistance on resonance was about 83 k\n. The circuit\nwas located in a dewar that could be cooled as low as 3.1\nK by a pulse tube refrigerator in a 0 to 6 Tesla solenoid\nmagnet. A network analyzer injects a small drive into\nthe dewar and circuit through the weak coupling of a\ncapacitive divider made with 0.2 pF and 100 pF capac-\nitors (Fig. 2). The detector of the network analyzer is\nsimilarly weakly coupled to the circuit. The weak cou-\nplings increase the impedance of the 50 \n input and out-\nput lines to Z0\n0\u0019(100 pF=0:2 pF)2\u000250 \n = 12:5 M\n.\nThe loss and phase shift of the cables are calibrated by\nshorting them by a straight connector. As in the 2 port\nshunt-through measurement [32], when jZ(!)j\u001cZ0\n0, the\ntransmission Vout=Vinis a function of the impedance of\nthe resonance circuit Z(!) as\nVout\nVin=Z(!)\nZ0\n0+ 2Z(!): (24)\nThus, the impedance Z(!) can be calculated from the\ntransmission amplitude and phase.\n213\n2142152162170.10.2121020100)Ω)]  (kωRe [Z( = 2.1 pFtuningC  (off)  \n = -1.0 VGSV  (on) \n = 0 VGSV=38\nη(a)213\n2142152162170.10.2121020100)Ω)]  (kωRe [Z( = 180 pFtuningC  (off)  \n = -1.0 VGSV  (on) \n = 0 VGSV=330\nη(b)frequency (MHz)\n1\n23456102030210210×2 (pF)\ntuningC55606570758085)ΩR (k20\n30401002003004001000η\nsuppression factor  (c)FIG. 3. Measured Re[ Z(!z)] for a transmission line resonator\nwith the HEMT switch on and o\u000b, for Ctuning values of 2.1\npF (a) and 180 pF (b). (c) For various Ctuning , the measured\nparallelRfor the HEMT switched o\u000b are the black dots,\nwith a higher values desired. The measured reduction of the\ndamping resistance as the HEMT is switched on is shown in\ngray, with the largest value desired. The dashed lines shows\nthe requirements for Rand\u0011respectively.\nFigure 3 shows the optimization for T= 3:1 K and 5\nT.Z(!) is measured from Eq. (24). The reduction of the\ndamping resistance as the HEMT is switched on, at the\nfrequency!z= (2\u0019)\u0002215:3 MHz, is given by Eq. (20)\nwithR= Re[Z(!z)]o\u000b,\n\u0011=Re[Z(!z)]o\u000b\nRe[Z(!z)]on: (25)\nFor a small Ctuning = 2:1 pF in (a), the resonance shifts\nslightly downward as the HEMT is switched on, and the\ndamping of an electron oscillation at 215.3 MHz is re-\nduced by\u0011= 38. For a large Ctuning = 180 pF in (b), the\nresonance shifts downward much more when the HEMT\nis switched on, and the damping of an electron oscilla-\ntion at 215.3 MHz is reduced by \u0011= 330. In this case,\nmost of the current induced by an electron oscillation at\nthis frequency goes though the capacitance in Fig. 1(c)\nrather than through a damping resistance.\nThe o\u000b-state resonance shapes for these value of\nCtuning and others shown in (c) are \ft to a Lorentzian to\nget the quality factor Qneeded to deduce the R=Q!zL\n(black dots in (c)). The gray dots in (c) are the reduction\nfactor\u0011determined from traces like those as in (a) and6\n(b).\nThe impedance between the detection electrode and\nground can be analytically calculated using the electron-\nics model in Fig. 1 (a). Ctrap,L1,L2are independently\ndetermined from the resonance shape and the dimension,\nandCtuning is a controlled variable here. The drain-to-\nsource impedance of the HEMT is modeled by a capaci-\ntorCand a resistorRo\u000borRonin parallel (depending on\nVGS, thus the switch state). We can calculate theoreti-\ncalRand\u0011from this model for these given parameters.\nThe measured Rand\u0011in Fig. 3 (c) are \ftted by this\nmodel. As discussed above, the only free parameters are\nC,Ro\u000b, andRon. The best \ft values are Ro\u000b= 65 k\n,\nRon= 9:6 \n, andC= 1:8 pF.\nAs discussed earlier, when the HEMT switch is biased\no\u000b, a large Ris needed to e\u000bectively detect quantum\ntransitions made by a trapped electron, and to e\u000bectively\ndamp the electron's axial motion. Past experiments have\ndemonstrated that 60 k\n su\u000eces. When the switch is bi-\nased on, a reduction of the axial damping rate by a factor\nof\u0011= 100 (or greater) is desired from Eq. (12). The R\nvalue stays quite high over most of the range of capacitor\nvalues used. The factor \u0011increases to a saturation value\nasCtuning increases, leading us to choose Ctuning = 22 pF\nfor experiments going forward with this detection circuit.\nIV. DEMONSTRATION AND IMPLICATIONS\nA switchable detection circuit attached to a Penning\ntrap with a 5.3 Tesla \feld provides the \frst demon-\nstration of switching the damping resistance the circuit\npresents to the electrons. The large signal from the\ncenter-of-mass motion of order of a thousand electrons\nis used to demonstrate that the circuit can make the de-\nsired damping resistance changes. The Penning trap and\nthe detector system are cooled to an ambient tempera-\nture of 0.1 K by a dilution refrigerator [24]. The e\u000bective\ncircuit resistance Ris thereby kept as low in tempera-\nture as possible to minimize the thermal Johnson noise\nin the resistor that heats the electron at the same time\nas it damps its 200 MHz motion. Cold damping is im-\nportant for both a high frequency electron oscillator and\nlow frequency oscillator with a large mass [33, 34]. To\nthis end, the ampli\fer HEMT used here is operated with\na power dissipation of only about 120 \u0016W that still heats\nthe circuit to a temperature of 5{10 K.\nThe resistance Rused for this demonstration was only\n36 k\n for unrelated reasons related to loading positrons\ninto the trap. The behavior of the switchable detection\ncircuit was characterized by applying an RF drive to an-\nother electrode and measuring the transmitted amplitude\nfrom the detector HEMT, S21, the well known Sparam-\neter that characterizes the transmission of a signal am-\nplitude through the circuit. Figure 4 shows S21as a\nfunction of gate-source bias voltage, VGS, on the switch\nHEMT. No particle is in the trap in this measurement.\nSince only the relative change of S21is meaningful, the\n180\n181 182 183 184 185 186 frequency (MHz)\n00.20.40.60.8121S    (off)    = -0.14 VGSV (on)  \n = -0 VGSVFIG. 4. Demonstration of the tuning of the quality factor\nof a detection circuit connected to a Penning trap. The top\ncurve is for the HEMT biased o\u000b (with a gate-to-source bias\natVGS=\u00000:14 V). Each line corresponds to di\u000berent S21\nwithVGSreduced by 0.005 V in each step.\nvalue is scaled so that the peak value is 1. The e\u000bective\nresistanceRcan be tuned from 36 k\n to around 100 \n.\nThis \fne control of the resistance Rmakes it possible\nto observe the change of the axial damping rate \rzwith\nthe trapped particles (Eq. (19)). When a cloud of N\nelectrons is trapped, a dip appears at the axial frequency\n!z=(2\u0019) on the noise resonance [29]. The full-width-at-\nhalf-maximum (FWHM) of the dip is N\rz, the damping\nrate that we seek to be able to change. Figure 5 illustrates\nthe change of the dip width for the center-of-mass motion\nof aboutN\u00191500 electrons. The particles are trapped\nin a well-tuned harmonic trap and their magnetron mo-\ntion [29] is cooled to center them in the trap. Since the\namplitude of the detected signal is also proportional to\nR, the signals are smaller for lower R. The baseline volt-\nages in the graphs are shifted so that the baseline volt-\nages are the same among three. The measured damping\nwidth can then be compared to the Rdeduced from the\nS21measurements in Fig. 4 and R=Q!zL.Ris about\n36 k\n, 19 k\n, and 7 :6 k\n from left to right, respectively.\nThe damping rate \rzclearly reduces as Ris reduced.\nFigure 6 summarizes the change of dip widths for four\ndi\u000berent sizes of clouds. Ris set by adjusting the gate\nvoltageVGSon the switch HEMT as in Fig. 4. The data\nwithRlower than 7 k\n cannot be detected reliably be-\ncause the noise resonance is too small to observe the dips.\nAll data are taken with the same trap voltages. Results\nof linear \ftting on each cloud is also shown in the graph.\nThe ability of tuning \rzby changing the Ris demon-\nstrated.\nThe tuning of axial damping rate \rzdemonstrates\nthat the HEMT switch is compatible with the Pen-\nning trap. The measurements with coaxial transmission\nline resonator (Sec. III) shows that the HEMT based\nswitch has low enough loss to detect a single particle7\n2\n−1−012voltage (normalized) = -0.135 V GSV2\n−1−012 (kHz)\ncenterf - f = -0.085 V GSV2\n−1−012 = -0.050 V GSV\nFIG. 5. Change of the dip width of a cloud with N\u00191500\nparticles. The gate voltage of the switch HEMT VGSis ad-\njusted to set the Rto be about 36 k\n, 19 k\n, and 7 :6 k\n from\nleft to right respectively. The change of the dip width reveals\nthe change oft the electron damping rate. The noise becomes\nrelatively larger for smaller Rsince the detection sensitivity\nis also proportional to R.\n0\n510152025303540)\nΩR (k01234567dip FWHM (kHz)N=7600N=5900\nN=1500\nN=500\nFIG. 6. Change of dip width of 4 di\u000berent sizes of clouds. The\nRis set by adjusting the gate voltage on the switch HEMT\nVG. Linear \fttings on each cloud are also shown.\nand high enough suppression on \rz. The suppression of\n\rzis demonstrated with trapped electrons. With these\ndemonstrations, the newly developed detector is able\nto reduce\rzenough while maintaining single-particle-\ndetection sensitivity.\nAccording to recent calculations [17], the demonstrated\nswitchable detection circuit should dramatically change\nthe cyclotron resonance lineshape that must be observed\nto measure the electron and positron magnetic moments.\nThe dashed lineshape is what would be observed if the\ndetection circuit was not switched, as in completed ex-\nperiments [2, 35]. The dramatically di\u000berent series of\nresonances (solid) is what is expected when the detec-\ntion circuit demonstrated here is switched on. The broad\nand asymmetric resonance (dashed) turns into a series of\nextremely narrow and symmetric peaks, each of which\ncorresponds to an individual quantum state of the axial\nmotion. The linewidth is reduced by about two orders\n0\n50 100 150 200 relative frequency detuning in ppt\n00.20.40.60.811.2probability (normalized)) = 1 Hzπ/(2zγwithout switch) = 0.01 Hz\nπ/(2zγwith switch = 0zn = 1\nzn = 2\nzn=1.5pptcω/cωΔFIG. 7. Lineshape of cyclotron transition for traditional mea-\nsurement without the switch (dashed) and with the demon-\nstrated switch (solid) for \u0016 nz= 10 and\u000ec=(2\u0019) = 4 Hz[2]. See\n[17] for the details of the calculation.\nof magnitude. The details of how the detection circuit\nis used to observe cyclotron resonance is well beyond the\nscope of this report and is discussed in [17].\nV. CONCLUSIONS\nA 200 MHz detection circuit that can be switched be-\ntween high and low resistive impedance levels has been\ndeveloped for use at cryogenic temperatures as low as\n0.1 K. The switchable detection and damping circuit is\ndemonstrated by using it to change the damping rate\nfor the axial, center-of-mass motion of trapped electrons.\nThe change in the damping rate for a single electron will\nbe about a factor of 100 for the demonstrated circuit. Ac-\ncording to a recent calculation, being able to switch the\ndamping rate by this factor will make it possible to evade\nthe detector backaction that limited the accuracy of ear-\nlier measurements by producing broad and asymmetric\ncyclotron resonances. The switchable detection circuit\nthus promises to revolutionize electron and positron mag-\nnetic moment measurements made to test the most pre-\ncise predictions of the standard model of particle physics.\nACKNOWLEDGEMENTS\nThis work was supported by the NSF, with X. 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Lett. 97, 030801 (2006)."    },    {        "title": "1507.02653v4.Resonant_absorption_of_kink_magnetohydrodynamic_waves_by_a_magnetic_twist_in_coronal_loops.pdf",        "content": "arXiv:1507.02653v4  [astro-ph.SR]  17 Aug 2016Resonant absorption of kink magnetohydrodynamic\nwaves by a magnetic twist in coronal loops\nZanyar Ebrahimi∗, Kayoomars Karami†\nDepartment of Physics, University of Kurdistan, Pasdaran St., Sa nandaj, Iran\nNovember 9, 2021\nAbstract\nThereisampleevidenceoftwistedmagneticstructuresinthesolarc orona. Thismotivates\nus to consider the magnetic twist as the cause of Alfv´ en frequenc y continuum in the coronal\nloops, which can support the resonant absorption as a rapid dampin g mechanism for the\nobserved coronal kink magnetohydrodynamic (MHD) oscillations. W e model a coronal loop\nwith a straight cylindrical magnetic flux tube which has constant but different densities in\nthe interior and exterior regions. The magnetic field is assumed to be constant and aligned\nwith the cylinder axis everywhere except a thin layer near the bound ary of the flux tube\nwhich has an additionalsmall magnetic field twist. Then, we investigat eanumber of possible\ninstabilities that may arise in our model. In the thin tube thin boundary approximation, we\nderive the dispersion relation and solve it analytically to obtain the fre quencies and damping\nrates of the fundamental ( l= 1) and first/second overtone ( l= 2,3) kink ( m= 1) MHD\nmodes. We conclude that the resonantabsorption by the magnetic twist can justify the rapid\ndamping of kink MHD waves observed in coronal loops. Furthermore , the magnetic twist\nin the inhomogeneous layer can cause deviations from P1/P2= 2 and P1/P3= 3 which are\ncomparable with the observations.\nKey words: Sun: corona — Sun: magnetic fields — Sun: oscillations\n1 Introduction\nThe first identification of transverse oscillations of coron al loops was reported by Aschwanden\net al. (1999) and Nakariakov et al. (1999) using the Transiti on Region and Coronal Explorer\n(TRACE) observations of 14 July 1998 in the 171 ˚A Fe IX emission lines. Nakariakov et al.\n(1999) for a loop with length of (130 ±6)×103km and width of (2 .0±0.36)×103km reported\na spatial oscillation with period of 4 .27±0.13 min and decay time of 14 .5±2.7 min. They\nsuggested resonance of the global mode as the cause of such fa st damping. On 17 April 2002,\nthe vertical polarization of coronal loops oscillations wi th period of 3.9 min and decay time of\n11.9 min were identified by Wang & Solanki (2004) using the TRA CE observations in the 195\n˚A Fe XII emission line. According to Roberts et al. (1984), th e goal of coronal seismology is to\ndeduce the properties of the solar corona using observed par ameters of oscillations and waves.\nFor instance, Nakariakov & Ofman (2001) applied a new method for determination of the local\nmagnetic field strength base on the observed length, density and frequency of an oscillating\n∗E-mail: Z.Ebrahimi@uok.ac.ir\n†E-mail: KKarami@uok.ac.ir\n1coronal loop. For reviews on coronal seismology, see e.g. De Moortel (2005), De Moortel &\nNakariakov (2012), Andries et al. (2009) and Ruderman & Erd´ elyi (2009).\nThe theory of resonant absorption of MHD waves was first estab lished by Ionson (1978) as\na conceivable mechanism for heating of the solar corona. Sin ce then, many theoretical works\nhave been done to develop this theory (see e.g. Davila 1987; S akurai et al. 1991a,b; Goossens et\nal. 1992; Steinolfson & Davila 1993). In this mechanism, ene rgy of the global mode oscillations\nis transferred to the local Alfv´ en perturbations within a r esonance layer inside the loop. The\nnecessary condition for this process is a gradient of Alfv´ e n frequency in this layer which varies\nbetween the interior and exterior Alfv´ en frequencies of th e loop. For a good review on the\nresonant absorption see also Goossens et al. (2011).\nHeatingofthecoronalloopsbytheresonantabsorptionofMH Dwaves wasstudiedbyErd´ elyi\n& Goossens (1995). Solving the visco-resistive MHD equatio ns of motion, they concluded that\nunder coronal conditions both viscus and ohmic dissipation mechanisms are important. Erd´ elyi\n& Goossens (1996) showed that the equilibrium plasma flow in c oronal flux tubes affects the\nresonant absorption rate due to driving waves.\nRuderman & Roberts (2002) investigated the resonant absorp tion of kink mode oscillations\nin a coronal loop with radial density inhomogeneity in the ze ro-beta approximation. They\nconcluded that the oscillations of coronal loops are cohere nt only in the presence of small scale\ninhomogeneities in density. Goossens et al. (2002) showed t hat the damped quasi-modes give\nan accurate description of rapid damping of the observed cor onal loop oscillations if the length\nscale of inhomogeneity changes from loop to loop. Also dampi ng of quasi-modes is completely\nconsistent with large estimates of the Reynolds numbers in t he corona (1014).\nVan Doorselaere et al. (2004) investigated the oscillation s of kink modes in coronal loops\nby the LEDA numerical code (van der Holst et al. 1999). Taking into account the large\ninhomogeneity length scales, Van Doorselaere et al. (2004) showed that the rapid damping\nof oscillating coronal loops can be justified by resonant abs orption without resorting to the\nReynolds numbers smaller than the classical values. They co ncluded that the numerical results\nof dampingrates can deviate from the analytical ones obtain ed in thin boundaryapproximation,\nby up to 25% .\nTerradas et al. (2006a) investigated the temporal evolutio n of resonant absorption in a one-\ndimensional cylindrical coronal loop. They found that when the coronal loop is excited by the\nexternal perturbation, the first stage of the loop oscillati on has the leaky behavior. After that,\nthe loop oscillates like the kink mode and then it is dissipat ed by the mechanism of resonant\nabsorption. Terradasetal. (2006b) developedtheirprevio usworkandfoundthatconsideringthe\ncurvature of coronal loops enhances the efficiency of resonan t absorption slightly. They showed\nthat there are two kink modes with polarizations mostly in th e horizontal/vertical directions\nwith respect to the photosphere. These modes show both reson ant and leaky behavior at a same\ntime.\nThe effect of longitudinal density stratification on the reson ant absorption of MHD waves\nin coronal loops with radial density gradients has also been studied in the literature (see e.g.\nAndries et al. 2005; Karami et al. 2009). Karami et al. (2009) showed that in the zero-\nbeta approximation, when the stratification parameter incr eases, both the period and damping\ntime of the kink and fluting modes decrease but the stratificat ion does not affect the ratio of\nfrequencies to damping rates. They further showed that the r atio of fundamental period to first\novertone one decreases from its canonical value P1/P2= 2 when the stratification parameter\nincreases.\nBesides the above considerations, there are observational evidences for twisted magnetic\nfields in coronal loops. Chae et al. (2000) reported the trace s of rotational motions in coronal\n2loops and suggested that the existence of an azimuthal magne tic field component that encircles\nthe axis may be required to guide the rotational motions. Cha e & Moon (2005) considered a\nmodel of twisted fluxtubein which the force of pressuregradi ent is balanced by the tension force\nof the azimuthal magnetic field component. They suggested th at the constriction of plasma can\nbe used to determine the magnetic twist in coronal loops. Kwo n & Chae (2008) measured the\nmagnetic twist of 14 coronal loops. The magnetic twist value φtwistof the loops was in the range\n[0.22π,1.73π].\nMany theoretical works have also been done on the effect of twis ted magnetic fields on the\nMHD waves in coronal loops (see e.g. Bennett et al. 1999; Erd´ elyi & Carter 2006; Erd´ elyi &\nFedun 2006, 2007, 2010; Carter & Erd´ elyi 2007, 2008; Ruderm an 2007, 2015; Karami & Barin\n2009; Karami & Bahari 2010, 2012; Terradas & Goossens 2012). Ruderman (2007) considered\na straight flux tube in the zero-beta approximation with a mag netic twist inside the loop pro-\nportional to the distance from the tube axis. Using the asymp totic analysis in the limit of\nsmall twists, Ruderman obtained an analytical solution for perturbations inside the loop and\nshowed that the magnetic twist does not affect the standing kin k modes. Karami & Bahari\n(2012) extended the work of Ruderman (2007) to a magneticall y twisted flux tube containing\nboth core and annulus regions. They showed that the frequenc ies and the period ratio P1/P2of\nthe fundamental and first overtone nonaxisymmetric kink and fluting modes are affected by the\ntwist parameter of the annulus. Terradas & Goossens (2012) s tudied the effect of magnetic twist\non the kink oscillations of coronal loops with a piecewise pa rabolic twist profile. They solved\nthe MHD equations using the PDE2D (Sewell 2005) code. Terrad as & Goossens (2012) in the\nlimit of small twists showed that the magnetic twist changes the polarization of the transverse\nmotions of standing kink oscillation along the flux tube but d oes not affect its frequency. Ru-\nderman (2015) investigated the effect of a continues twisted m agnetic field on the propagating\nkink modes in the thin tube approximation. He showed that the re are two propagating kink\nwaves with the same longitudinal wave numbers but opposite p ropagation directions, which have\ndifferent frequencies. Ruderman (2015) called these waves “a ccelerated” and “decelerated” kink\nwaves which have larger and smaller frequencies with respec t to the well known kink frequency,\nrespectively.\nKarami & Bahari (2010) studied the effect of twisted magnetic fi eld on the resonant absorp-\ntion of MHD waves due to the radial density structuring in cor onal loops. They showed that\nby increasing the twist parameter, the frequency, the dampi ng rate and their ratio for both the\nkink and fluting modes increase. Also the magnetic twist caus es the ratio of fundamental period\nto first overtone one for kink and fluting modes to be smaller th an 2.\nThe main goal of the present work is to study the resonant abso rption of kink MHD waves\nby magnetic twist to explain the rapid damping of oscillatin g coronal loops and departure of the\nperiod ratios P1/P2andP1/P3from their canonical values reported by observations. To th is\naim, in section 2 we introduce the coronal loop model and find t he solutions of the equations\nof motion. In section 3, we investigate a number of possible i nstabilities that may arise in\nour model. In section 4, we use the appropriate connection fo rmulae to obtain the dispersion\nrelation. In section 5, we solve the dispersion relation, an alytically. Section 6 gives the summary\nand conclusions.\n32 Model and equations of motion\nAs a simplified model for a coronal loop, we consider a straigh t cylindrical flux tube with length\nLand radius R. The background density profile is assumed to be\nρ(r) =/braceleftbiggρi,0< r≤R,\nρe, r > R,(1)\nwhereρiandρeare the constant densities of the interior and exterior regi ons of the flux tube,\nrespectively. We define the density ratio ζ≡ρi/ρein the rest of the paper.\nWe further assume that the background magnetic field to have a small twist in a thin layer\nand to be constant and aligned with the loop axis everywhere e lse, i.e.\nB=\n\nB0ˆz, 0< r < a,\nBϕ(r)ˆϕ+B0ˆz, a≤r≤R,\nB0ˆz, r > R.(2)\nHere, we set a parabolic profile for Bϕas follows\nBϕ(r) =Ar(r−a), (3)\nwhereAis a constant. Note that the jump in the value of Bϕ(r) across r=Rimplies a\ndelta-function current sheet there.\nThe magnetohydrostatic equilibrium equation takes the for m\nd\ndr/parenleftBigg\nP+B2\nϕ+B2\nz\n2µ0/parenrightBigg\n=−B2\nϕ\nµ0r, (4)\nwhereµ0is the magnetic permeability of free space. Then, by integra ting Eq. (4) and using the\ncontinuity of the total (magnetic plus gas) pressure at r=aandr=R, we can find the gas\npressure as follows\nP(r) =\n\nB2\n0\n2µ0β, 0< r < a,\nB2\n0\n2µ0β−A2\n12µ0(r−a)\n×(9r3−11ar2+a2r+a3), a ≤r≤R,\nB2\n0\n2µ0β−\nA2\n12µ0(3R4−8aR3+6a2R2−a4), r > R,(5)\nwhere parameter β≡P\nB2\n0/(2µ0)is the ratio of the plasma pressure to the magnetic field press ure,\ninside the loop.\nThe linearized ideal MHD equations for incompressible plas ma are given by\n∂δv\n∂t=−∇δP\nρ+1\nµ0ρ{(∇×δB)×B+(∇×B)×δB}, (6)\n∂δB\n∂t=∇×(δv×B), (7)\n∇·δv= 0, (8)\n4whereδv,δBandδPare the Eulerian perturbations of velocity, magnetic fields and plasma\npressure; Bandρare the background magnetic filed and the mass density, respe ctively. Also t-,\nφ- andz-dependency of the perturbations are supposed to be of the fo rm exp[i(mφ+kzz−ωt)]\nwherekz=lπ\nLis the longitudinal wave number. Here mandlare the azimuthal and longitudinal\nmode numbers, respectively, and ωis the mode frequency. So the perturbations are of the form\nδv(r,φ,z,t) =δv(r)exp[i(mφ+kzz−ωt)],\nδB(r,φ,z,t) =δB(r)exp[i(mφ+kzz−ωt)],\nδP(r,φ,z,t) =δP(r)exp[i(mφ+kzz−ωt)]. (9)\nFollowing Bennett et al. (1999), substituting the perturba tions (9) into Eqs. (6) and (7) and\ndoing some algebra we get\nd2\ndr2δp+/braceleftbiggC3\nrDd\ndr/parenleftbiggrD\nC3/parenrightbigg/bracerightbiggd\ndrδp\n+/braceleftbiggC3\nrDd\ndr/parenleftbiggrC1\nC3/parenrightbigg\n+1\nD2(C2C3−C2\n1)/bracerightbigg\nδp= 0,\nξr=D\nC3d\ndrδp+C1\nC3δp,(10a)\n(10b)\nwhereδp=δP+B·δB/µ0andξr=−δvr/iωare the Eulerian perturbation of total pressure\nand the Lagrangian displacement in the radial direction, re spectively and\nD=ρ(ω2−ω2\nA),\nC1=−2mBϕ\nµ0r2/parenleftBigm\nrBϕ+kzBz/parenrightBig\n,\nC2=−/parenleftbiggm2\nr2+k2\nz/parenrightbigg\n,\nC3=D2+D2Bϕ\nµ0d\ndr/parenleftbiggBϕ\nr/parenrightbigg\n−4B2\nϕ\nµ0r2ρω2\nA.(11)\nHere, the Alfv´ en frequency, ωA, is defined as\nωA(r)≡1√µ0ρ/parenleftBigm\nrBϕ+kzBz/parenrightBig\n. (12)\nPutting Eqs. (1)-(3) into (12) gives the profile of Alfv´ en fr equency as follows\nωA(r) =\n\nB0kz√µ0ρi, 0< r < a,\n1√µ0ρi/parenleftBig\nmA(r−a)+kzB0/parenrightBig\n, a≤r≤R,\nB0kz√µ0ρe, r > R.(13)\nHere we use the twist parameter defined as α≡Bϕ(R)\nB0. For a special value of α=αc, the Alfv´ en\nfrequency is continues at the tube boundary ( r=R). Ifα/ne}ationslash=αcthere would be a gap in the\nAlfv´ en frequency profile across the boundary. It is straigh tforward from Eq. (13) to show that\nαc=Rkz\nm/parenleftBig/radicalbig\nζ−1/parenrightBig\n. (14)\n5Figure 1: Azimuthal component of the background magnetic fie ld, Alfv´ en frequency of the\nfundamental ( l= 1) kink ( m= 1) mode, background Alfv´ en speed and background plasma\npressure versus the fractional radius r/R. The loop parameters are: L= 105km,R/L= 0.01,\na/R= 0.95,ζ= 2,ρi= 2×10−14g cm−3,β= 0.1 andB0= 100 G. Here α=αc= 0.013 and\nvAi=B0/√µ0ρi= 2000 km s−1.\nFigure 1 shows the azimuthal component of the background mag netic field, the Alfv´ en frequency\nof the fundamental ( l= 1) kink ( m= 1) mode, the background Alfv´ en speed, vA(r) =B/√µ0ρ,\nandthebackgroundplasmapressureforthetwist parameter α=αc= 0.013. Notethat although\nforα > α c, there is a sudden drop in the Alfv´ en frequency ωA(r) at the tube boundary, the\ninterior Alfv´ en speed vA(r) in the present model is still smaller than the exterior one w hich is\nlikely to occur under coronal conditions.\nFrom Eqs. (10a)-(12), it is clear that when ω2=ω2\nA, i.e.D= 0, the equations of motion\nin the inhomogeneous layer ( a < r < R ) become singular in the presence of magnetic twist.\nTherefore, resonant absorption can occur not only by the rad ial density inhomogeneity (like\nprevious works, see e.g. Ruderman & Roberts 2002; Karami et a l. 2009; Karami & Bahari 2010;\nRuderman 2011; Ruderman & Terradas 2013; Soler & Terradas 20 15) but also by the magnetic\ntwist (present work).\nIn the untwisted regions, 0 < r < a andr > R, Eqs. (10a) and (10b) are reduced to the\nfollowing equations\nd2δp\ndr2+1\nrdδp\ndr−/parenleftbigg\nk2\nz+m2\nr2/parenrightbigg\nδp= 0,\nξr=1\nρ(ω2−ω2\nA)dδp\ndr.(15a)\n(15b)\nSolutions for equations (15a) and (15b) in the interior (0 < r < a) and exterior ( r > R) regions\n6are obtained as\nδp(r) =/braceleftbiggAiIm(kzr),0< r < a,\nAeKm(kzr), r > R,\nξr(r) =\n\nAikz\nρi(ω2−ω2\nAi)I′\nm(kzr),0< r < a,\nAekz\nρe(ω2−ω2\nAe)K′\nm(kzr), r > R.(16a)\n(16b)\nHereImandKmare the modified Bessel functions of the first and second kind, respectively.\nAlso “′” onImandKmrepresents a derivative with respect to their arguments. Th e constants\nAiandAeare determined by the appropriate boundary conditions.\n3 Stability constraints on the model\nIn what follows, we are interested in investigating a number of possible instabilities that may\narise in our model.\n3.1 Kink instability\nIf the magnetic twist value in the loop, which is defined as φtwist≡L\nRBϕ\nBz= 2πNtwistexceeds\na critical value φc, then the loop becomes kink unstable (see e.g. Shafranov 195 7; Kruskal et\nal. 1958). Here, Ntwistis the number of twist turns in the loop. For force-free magne tic fields\nof uniform twist, a critical twist of φtwist/greaterorsimilar3.3π(or 1.65 turns) was found to lead to kink\ninstability (Hood & Priest 1979), while the critical value r anges between 2 πand 6πfor other\ntypes of magnetic fields (see e.g. Aschwanden 2005; Priest 20 14). Numerical MHD simulations\nof an increasingly twisted loop system demonstrated linear instability of the ideal MHD kink\nmode for twist angles in excess of ≈4.8π(or 2.4 turns) (Miki´ c et al. 1990). Baty & Heyvaerts\n(1996) examined the kink instability of a radially localize d twist profile and obtained φc= 5π.\nIn our model to avoid the kink instability, following Hood & P riest (1979) we consider φc= 3.3π\n(or 1.65 turns). This yields an upper limit for the twist para meter as αmax= (R/L)φc.\n3.2 Kelvin-Helmholtz instability\nThe Kelvin-Helmholtz instability (KHI) can occur during th e kink oscillations of coronal loops.\nThe kink wave, whose initial energy is mostly transverse, co nverts into azimuthal waves locally\nresembling torsional Alfv´ en waves, which are finely locali zed around the tubes boundary layer\n(Verth et al. 2010; Arregui et al. 2011). Azimuthal motions a re thus amplified and introduce\nvelocity shear, making them prone to be unstable to the KHI (H eyvaerts & Priest 1983; Soler\net al. 2010). Terradas et al. (2008) using numerical simulat ions showed that azimuthal shear\nmotions generated at the loop boundary during kink oscillat ions can give rise to a KHI, but\nthis phenomenon has not been observed to date. The KHI can als o extract the energy from the\nresonant layer and convert it into heat through viscous and o hmic dissipations at the generated\nvortices and current sheets (Antolin et al. 2015). Soler et a l. (2010) pointed out that the\npresence of a small azimuthal component of the magnetic field can suppress the KHI in a stable\ncoronal loop. They showed that the required twist is small en ough to prevent the development\nof the pinch instability. A weak twist of magnetic field lines is very likely and realistic in the\ncontext of coronal loops.\n73.3 Resistive kink instability\nThe effect of resistive diffusion on the ideal kink mode yields a s o-called resistive kink instability\nwhich is a reconnecting process. The resistive kink becomes important when the loop is twisted\ntoo much (see e.g. Biskamp 2000; Wesson 2004; Priest 2014). F rom Eq. (7), it is possible to\nhave\n∇×(δv×B) = (B·∇)δv−B(∇·δv) = 0, (17)\nat aspecificlocation r=rs. Inthiscase, thediffusionterm η∇2δBneglected fromtherighthand\nside of Eq. (7) becomes important and consequently the field l ines diffuse through the plasma\nand reconnect. Here ηis the magnetic diffusivity. For an incompressible plasma, in serting\nEq. (8) into (17) gives rise to k·B/vextendsingle/vextendsingle\nr=rs= 0, where kandBare the wave vector and the\nbackground magnetic field, respectively. Using Eq. (2), the necessary condition for the resistive\nkink instability takes the form\nk·B/vextendsingle/vextendsingle\nr=rs=Bϕ(rs)\nrs+kzBz(rs) = 0. (18)\nIn Fig. 2, we plot k·Bfor the fundamental ( l= 1) kink ( m= 1) mode. Figure illustrates that\nthe term k·Bcannot be zero and consequently the resistive kink instabil ity cannot occur in our\nmodel.\n3.4 Tearing mode instability\nThe tearing mode instability can occur in a thin current shee t when the driving force of the\ninflow exceeds the opposing Lorentz force (see e.g. Furth et a l. 1963; Goldstone & Rutherford\n1995; Magara & Shibata 1999; Tenerani et al. 2015). As a conse quence of non-zero resistivity,\nthemagnetic fieldlines tear and reconnect inthecurrent she et. According toFurth et al. (1973),\nthe smallest growth time of the tearing mode in the cylindric al flux tube is given by about\nτt∼τ2/5\nAτ3/5\nd, (19)\nwhereτA=ls/vAiandτd=l2\ns/ηare the Alfv´ en and magnetic diffusion time scales, respectiv ely.\nHerelsis a half-thickness of the current sheet and vAi=B0/√µ0ρiis the interior Alfv´ en speed.\nIn our model, due to having a rotational discontinuity of the background magnetic field at the\ntube boundary, we have a delta-function current sheet. Ther efore, from Eq. (19) when ls→0\nthe growth time of the tearing process goes to zero. It should be noted that the delta-function\ncurrent sheet in our model is used to approximate a finite widt h current layer. In reality, the\nthickness of such current sheets in the solar corona takes pl ace in the range of macroscopic\nvalues (10 km, for example), see e.g. Magara & Shibata (1999) . For the present model, if\nwe take T= 5×106K,η= 109T−3/2= 0.09 m2s−1(Priest 2014), vAi= 2000 km s−1\nthen for 2 l0≃(1−10) km (Magara & Shibata 1999) we estimate the tearing growth time as\nτt/Pkink∼(3−123). Here Pkink= 2π/ωkink≃87 s is the period of the fundamental kink mode\noscillation (see Eq. 36). Therefore, by choosing an appropr iate thickness for the current sheet,\nthe tearing mode instability can be avoided in our model duri ng the resonant absorption of the\nkink oscillations.\nIt is worth noting that considering a continuous background magnetic field like the profile\nused in the work of Hood et al. (2016), allows us to have a twist in the tube without having\nthe delta-function current sheet. This will make it easier t o avoid the growth of the tearing\ninstability. However, for a certain density distribution ( like the one we use in this paper) the\nmodel of Hood et al. (2016) does not yield a monotonic functio n of the background Alfv´ en\n8Figure 2: Variation of k·Bfor the fundamental ( l= 1) kink ( m= 1) mode versus the fractional\nradiusr/R. Hereα= 0.1 and the other auxiliary parameters as in Fig. 1.\nfrequency. As a result it may give rise to two resonant layers . In this case, in order to use the\nconnection formulae (see section 4) at the resonance layers , we need to solve the MHD equations\nin the twisted regions around the resonance points. However , this is beyond the scope of this\npaper and we leave it for future work.\n4 Connection formulae and dispersion relation\nHere, we do not solve Eq. (10a) in the inhomogeneous layer ( a≤r≤R), where the singularity\noccurs due to the existence of the magnetic twist. Instead, t he solutions inside and outside of\nthe tube can be related to each other by the connection formul ae introduced by Sakurai et al.\n(1991a). To check the validity of the connection formulae, t he radius SAof the region around\nthe resonance point that connects the solutions of the pertu rbations of the interior and exterior\nof the flux tube, must obey the following condition (see Gooss ens et al. 2011)\nSA≪h≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndrω2\nA(r)\nd2\ndr2ω2\nA(r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=rA. (20)\nIn our work, we set SA=R−a\n2. In Fig. 3, we plot SAandhversus the twist parameter\nα=Bϕ(R)\nB0. Figure clears that for the range of magnetic twist consider ed in this study, the\ncondition (20) holds. Therefore, in the thin boundary appro ximation we can use the connection\nformulae around the inhomogeneous layer. According to Saku rai et al. (1991a) the jumps across\nthe boundary (resonance layer) for δpandξrare\n[δp] =−iπ\n|∆|2Bϕ(rA)Bz(rA)fB(rA)\nµ0ρirAB2(rA)CA(rA),\n[ξr] =−iπ\n|∆|gB(rA)\nρiB2(rA)CA(rA),(21a)\n(21b)\nwhere\nCA=gBδp(r)−2fBBϕBz\nµ0rAξr(r),\nfB=m\nrBϕ+kzBz,\ngB=m\nrBz−kzBϕ,\n∆ =−d\ndrω2\nA(r),(22)\n9Figure 3: Variations of h,SA= 0.025 and thickness of the resonance layer δAversus the twist\nparameter αfor the fundamental ( l= 1) kink ( m= 1) modes. Here h,δAandSAare in units\nof the loop radius R= 1000 km. Auxiliary parameters as in Fig. 1.\nandrAis the location of the resonance point. Notice that the ideal MHD solutions for the\nuntwisted regions, i.e. inside (0 < r < a) and outside ( r > R) the flux tube, are valid in the\njump conditions (21a) and (21b) only if the following condit ion is established (see Stenuit et al.\n1998; Goossens et al. 2011)\nδA≤SA=R−a\n2, (23)\nwhereδA=/vextendsingle/vextendsingleω\n|∆|(ν+η)/vextendsingle/vextendsingle1/3= (vAiR)1/3/vextendsingle/vextendsingleω\n|∆|(1\nRv+1\nRm)/vextendsingle/vextendsingle1/3is a measure of the thickness of the\nresonance layer (see Sakurai et al. 1991a). Here ν=vAiR\nRv,η=vAiR\nRmandRvare the kinematic\nviscosity, magnetic diffusivity and viscus Reynolds number, respectively. The classical values of\nviscus and resistive Reynolds numbers of the solar corona ar e about 1014and 1013, respectively\n(see Colub & Pasachoff 1997). In Fig. 3, we also plot δAfor the aforementioned values of the\nReynolds and Lundquist numbers. Figure clears that the cond ition (23) is respected.\nSubstituting the ideal solutions (16a) and (16b) in the jump conditions (21a) and (21b) and\neliminating AiandAe, one can find the dispersion relation as\nd0(˜ω)+d1(˜ω) = 0, (24)\nwhere\nd0(˜ω) =kz\nρe(˜ω2−ω2\nAe)K′\nm(kzR)\nKm(kzR)\n−kz\nρi(˜ω2−ω2\nAi)I′\nm(kza)\nIm(kza),(25)\nand\nd1(˜ω) =\niπ\n|∆|/bracketleftBigg\ngB\nρiB2\n0−kz\nρe(˜ω2−ω2\nAe)2fBBϕ\nµ0ρirAB0K′\nm(kzR)\nKm(kzR)/bracketrightBigg\n×/bracketleftBigg\ngBIm(kzrA)\nIm(kza)−kz\nρi(˜ω2−ω2\nAi)2fBBϕB0\nµ0rAI′\nm(kzrA)\nIm(kza)/bracketrightBigg\n.(26)\nHere ˜ω=ω−iγ, in which ωandγare the mode frequency and the corresponding damping rate,\nrespectively.\n10Using Eq. (13) in the last relation of Eq. (22) we get\n∆ =−B2\n0\nµ0ρi2mα\nR(R−a)/parenleftbigg\nmαr−a\nR(R−a)+k/parenrightbigg\n, (27)\nwhich shows that in the limit of a→R, i.e. when the twisted annulus region is removed, then\n∆ goes to infinity. As a result, in Eq. (26), d1= 0. In this case there is no resonant absorption\nand the dispersion relation (24) is the same as Eq. (25) in Ben nett et al. (1999).\nIn thelimit of thin tubeapproximation, usingthefirstorder asymptotic expansions for Im(x)\nandKm(y) one can get\nIm(kzrA)\nIm(kza)≃/parenleftBigrA\na/parenrightBigm\n,\nI′\nm(kzrA)\nIm(kza)≃m\nkzrA/parenleftBigrA\na/parenrightBigm\n,\nI′\nm(kza)\nIm(kza)≃m\nkza,\nK′\nm(kzR)\nKm(kzR)≃ −m\nkzR.(28)\nUsing the above approximations, the dispersion relation (2 4) reduces to\nρi(˜ω2−ω2\nAi)\nR+ρe(˜ω2−ω2\nAe)\na\n−iπ\n|∆|/parenleftBigrA\na/parenrightBigm/parenleftbigggB\nB2\n0ρe(˜ω2−ω2\nAe)+2mfBBϕ\nµ0rAB0R/parenrightbigg\n×/parenleftbigggB\nm(˜ω2−ω2\nAi)−2fBBϕB0\nµ0ρir2\nA/parenrightbigg\n= 0.(29)\nNotice that we look for the frequencies in the range of ωAi< ω < ω Ae, in which the resonant\nabsorption occurs. Here ωAi=kzB0√µ0ρiandωAe=kzB0√µ0ρeare the interior and exterior Alfv´ en\nfrequencies, respectively. In the next section, we solve Eq . (29) analytically to obtain the\nfrequency and damping rate of the kink MHD modes.\n5 Analytical results\nHere, we are interested in studying the effect of the twist para meterαon the frequencies, ω, and\nthe damping rates, γ, of the kink ( m= 1) MHD modes in a resonantly damped coronal loop.\nTo this aim, we need to solve Eq. (29). First, we use the dimens ionless quantities ¯ r=r/R,\n¯L=L/R,¯B=B/B0and¯˜ω= ˜ω/(vAi/R). Thus, Eq. (29) can be recast in dimensionless form\nas\nζ(˜ω2−ω2\nAi)+1\nq(˜ω2−ω2\nAe)\n−iπ\n|∆|/parenleftBigrA\na/parenrightBigm/parenleftbigg\ngB(˜ω2−ω2\nAe)+2ζmfBBϕ\nrA/parenrightbigg\n×/parenleftbigggB\nm(˜ω2−ω2\nAi)−2fBBϕ\nr2\nA/parenrightbigg\n= 0,(30)\n11where we have dropped the bars for simplicity. Here q≡a/R. Equation (30) yields a quadratic\nequation for ˜ ω2as follows\nc1˜ω4+c2˜ω2+c3= 0, (31)\nwhere\nc1=−iπ\n|∆|/parenleftbiggrA\nq/parenrightbiggmg2\nB\nm,\nc2=/parenleftbigg\nζ+1\nq/parenrightbigg\n+iπ\n|∆|/parenleftbiggrA\nq/parenrightbiggm\n×/bracketleftbiggg2\nB\nm/parenleftbig\nω2\nAi+ω2\nAe/parenrightbig\n+2fBgBBϕ\nr2\nA(1−ζrA)/bracketrightbigg\n,\nc3=−ζω2\nAi/parenleftbigg\n1+1\nq/parenrightbigg\n−iπ\n|∆|/parenleftbiggrA\nq/parenrightbiggm\nζ\n×/bracketleftBigg\ng2\nB\nmω4\nAi−4mf2\nBB2\nϕ\nr3\nA+2fBgBBϕω2\nAi\nr2\nA(1−rA)/bracketrightBigg\n.(32)\nIt is straightforward to find the solutions of Eq. (31), ˜ ω=ω−iγ, as follows\nω±=1/radicalBig\nR2\n±+I2\n±/parenleftbigg/radicalBig\nχ±2+Θ2\n±+χ±/parenrightbigg1/2\n,\nγ±=1/radicalBig\nR2\n±+I2\n±/parenleftbigg/radicalBig\nχ±2+Θ2\n±−χ±/parenrightbigg1/2\n,(33)\nwhere\nχ±=c3RR±+c3II±,\nΘ±=c3IR±−c3RI±,\nR±=−c2R±/parenleftBigg/radicalbig\nc2\n4+c2\n5+c4\n2/parenrightBigg1/2\n,\nI±=−c2I±/parenleftBigg/radicalbig\nc2\n4+c2\n5−c4\n2/parenrightBigg1/2\n,(34)\nwith\nc4=c2\n2R−c2\n2I+4c1Ic3I,\nc5= 2c2Rc2I−4c1Ic3R.(35)\nHere, the subscripts RandIdenote the real and imaginary parts of a given complex quanti ty.\nFor a typical coronal loop, we take L= 105km,R/L= 0.005,0.01,ζ=ρi/ρe= 2,4,\nρi= 2×10−14g cm−3andB0= 100 G. So, one finds vAi=B0/√µ0ρi= 2000 km s−1in this\nloop. We assume that the magnetic twist takes place in a thin l ayer of thickness R−a= 0.05R.\nSo we have q= 0.95. In the thin boundary approximation, we can set the locati on of the\nresonance point at the tube surface, i.e. rA≃R.\n12Using Eq. (33) we obtain two roots of ω±which their values take place in the range of\nωAi< ω−< ωAeandω+> ωAe. Therefore, ω−is our physical root and ω+should be ruled\nout, because it does not give rise to the resonant absorption . As we have already mentioned,\nin the limit of q→1 (i.e.a→R), we have ∆ → ∞. In this case, from Eq. (32) we obtain\nc1=c2I=c3I= 0. Hence, in the absence of twisted annulus region, form Eqs . (33)-(35) we get\nω−=/radicalBigg\n2ζ\nζ+1ωAi=ωkink,\nγ−= 0,(36)\nwhereωkinkis the kink mode frequency of an untwisted loop in the thin tub e approximation\n(see e.g. Goossens et al. 2009).\nHere, we are interested to obtain a minimum value for the twis t parameter α=αminwhich\nis required to excite the resonance. The exact value of αminis determined as the root of the\nfollowing equation\nω−|α=αmin=ωA(R)|α=αmin. (37)\nWe solve this numerically and conclude that the result obtai ned forαminsatisfies the following\nrelation\nω−|α=αmin≃ωkink. (38)\nEquating right hand sides of Eqs. (37) and (38), one can get an approximate expression for αmin\nas\nαmin≃π/parenleftbiggR\nL/parenrightbigg/parenleftbiggl\nm/parenrightbigg/parenleftBigg/radicalBigg\n2ζ\nζ+1−1/parenrightBigg\n. (39)\nFor instance, taking ζ= 2 and R/L= 0.01, for the fundamental ( l= 1), first overtone ( l= 2)\nand second overtone ( l= 3) kink ( m= 1) modes, Eq. (39) gives αmin=0.0049, 0.0097 and\n0.0146, respectively. The relative error of the results of E q. (39) with respect to the exact\nsolutions of Eq. (37) is about 0 .6%.\nFigures 4, 5, and 6 illustrate the frequencies ( ω−=ωml), the damping rates ( γ−=γml)\nand the ratio of the frequency to the damping rate ( ωml/γml) of the fundamental ( l= 1) and\nfirst/second overtone ( l= 2,3) kink ( m= 1) modes versus the twist parameter α, respectively.\nFigures show that (i) the frequencies and damping rates incr ease when the twist parameter\nincreases. (ii) The ratio of the oscillation frequency to th e damping rate ω/γdecreases when the\ntwist parameter increases. The result, interestingly enou gh, is that for the fundamental ( l= 1)\nkind (m= 1) mode, for the twist parameter α= 0.0147 we obtain ω11/(2πγ11) = 3 which is in\ngood agreement with the observations reported by Nakariako v et al. (1999), Wang & Solanki\n(2004), and Verwichte et al. (2004). Here, the behaviour of ωandγversus the twist parameter\nαare the same as that obtained by Karami & Bahari (2010) but for the resonant absorption\ndue to the radial density structuring. (iii) For a given αandζ, whenR/Lincreases then ω\nincreases, γdecreases and ω/γincreases. (iv) For a given αandR/L, whenζincreases, the\nfrequency and the ratio ω/γincrease. However, this behavior for the damping rate holds only\nfor large values of the twist parameter. Moreover, increasi ngζdoes not affect the ratio ω/γ\nfor large values of α. (v) Note that based on Eq. (39), αminis a function of m,l,ζandR/L.\nTherefore, for each set of these parameters, the start point s of the diagrams in Figs. 4 to 6 have\n13Figure 4: Frequency of the fundamental ( l= 1) kink ( m= 1) mode (top), its damping rate\n(middle) and the ratio of the oscillation frequency to the da mpingrate (Bottom) versus the twist\nparameter α. Solid line: ζ= 2,R/L= 0.01; Dashed line: ζ= 4,R/L= 0.01; Dash-Dotted line:\nζ= 2,R/L= 0.005. Other auxiliary parameters as in Fig. 1. Both frequenci es and damping\nrates are in units of vAi/R= 2 rad s−1.\ndifferent αminvalues. Also to avoid the kink instability in our model, foll owing Hood & Priest\n(1979) we consider the twist value φtwist/lessorsimilarφc= 3.3πand consequently obtain an upper limit\nfor the twist parameter as αmax= (R/L)φc. Hence, the diagrams in Figs. 4 to 6 with different\nR/Lhave different αmaxcut-offs.\nFigure 7 presents variations of ω,γandω/γfor the fundamental ( l= 1) kink ( m= 1) mode\nversus the thickness of the twisted layer, d≡R−a. Figure shows that when dincreases both\nfrequencyanddampingrateincreasebuttheratioofthefreq uencytothedampingratedecreases.\nThis behaviour holds also for the first and second overtone ( l= 2,3) kink modes. Figure 7\nclarifies that in the limit of d→0 (i.e.a→R), we have ω11→ωkink=/radicalBig\n2ζ\nζ+1ωAi≃0.03628\n(in units of 2 rad s−1) andγ11→0. Note that the result of ω/γfor the fundamental mode of\nkink oscillations is in good agreement with that obtained fo r the resonant absorbtion due to the\nradial density structuring (see e.g. Ruderman & Roberts 200 2; Ruderman & Terradas 2013).\nIn Fig. 8, the periodratios of the fundamental to the firstove rtone,P1/P2, and to the second\novertone, P1/P3, of the kink ( m= 1) modes are plotted versus the twist parameter α. Figure\nshows that (i) when the twist parameter increases, the value s ofP1/P2andP1/P3decrease from\ntheir canonical values, 2 and 3, respectively. (ii) The dens ity ratio ζ=ρi/ρedoes not affect the\n14Figure 5: Same as Fig. 4 but for the first overtone ( l= 2) kink ( m= 1) mode.\n15Figure 6: Same as Fig. 4 but for the second overtone ( l= 3) kink ( m= 1) mode.\n16Figure 7: Frequency of the fundamental ( l= 1) kink ( m= 1) mode (top), its damping rate\n(middle) and the ratio of the oscillation frequency to the da mping rate (bottom) for α= 0.05\nversusthethicknessofthetwistedlayer, d(inunitsof R= 1000 km). Otherauxiliaryparameters\nas in Fig. 1.\n17Figure 8: Period ratios of the fundamental to the first overto ne,P1/P2, and to the second\novertone, P1/P3, of the kink ( m= 1) modes versus the twist parameter α. Solid line: ζ=\n2,R/L= 0.01. Dashed line (which overlaps the solid one): ζ= 4,R/L= 0.01. Dash-Dotted\nline:ζ= 2,R/L= 0.005. Other auxiliary parameters as in Fig. 1.\nFigure 9: Period ratios of the fundamental to the first overto ne,P1/P2, and to the second\novertone, P1/P3, of the kink ( m= 1) modes for α= 0.05 versus the thickness of the twisted\nlayer,d(in units of R= 1000 km). Other auxiliary parameters as in Fig. 1.\n18Table 1: Magnetic twist parameter α=Bϕ/Bzand corresponding twist value φtwist=/parenleftbigL\nR/parenrightbig\nα\npredictedbyourmodelforsomeobservational periodratios P1/P2andP1/P3ofthekink( m= 1)\nmodes. Other auxiliary parameters as in Fig. 1.\nReference P1/P2 αφtwist/π\nVan Doorsselaere et al. (2007) 1 .795±0.051 0.0718 2.28\nVan Doorsselaere et al. (2009) 1 .980±0.002 0.0295 0.94\nBallai et al. (2011) 1 .82±0.02 0.0678 2.16\nReference P1/P3 αφtwist/π\nVan Doorsselaere et al. (2009) 2 .89±0.14 0.0460 1.46\nperiod ratio. Notice that the solid curve overlaps the dashe d one. (iii) For a given αandζ, the\nperiod ratio decreases when R/Ldecreases. In Fig. 9, we plot variations of the period ratios\nP1/P2andP1/P3versus the thickness of the twisted layer d. Figure 9 shows that the period\nratios decrease with increasing d.\nNote that the deviations of the period ratios P1/P2andP1/P3from their canonical values\nprovide useful information about the magnetic twist struct uring within the loop. For instance,\ntheobservedvalue P1/P2= 1.795reportedbyVanDoorsselaereetal. (2007) canbejustifie dwith\nthe twist parameter α= 0.0718. Van Doorsselaere et al. (2007) also reported the ident ification\nof the second overtone of the kink mode by re-analyzing the tr ansverse oscillations of coronal\nloops observed by TRACE on 13 May 2001. The period ratio of fun damental to second overtone\nkink mode found to be P1/P3= 2.89 which can be justified with the twist parameter α= 0.0460.\nTable 1 summarizes the twist parameters and the correspondi ng twist values, φtwist=/parenleftbigL\nR/parenrightbig\nα,\npredicted by our model in order to justify some observed peri od ratios P1/P2andP1/P3of the\nkink (m= 1) modes. Note that the magnetic twist values φtwistpredicted by our model are in\nthe range of the observational values reported by Kwon & Chae (2008) and Wang et al. (2015).\n6 Conclusions\nHere, we investigated the resonant absorption of kink MHD mo des by magnetic twist in coronal\nloops. To this aim, we considered a thin straight cylindrica l flux tube with a twisted magnetic\nfield in a thin layer at the boundary of the loop and a straight m agnetic field everywhere else.\nThe magnetic twist causes a radial Alfv´ en frequency gradie nt and consequently gives rise to\nthe resonant absorption. We assumed the plasma density to be constant but different in the\ninterior and exterior regions of the loop. We obtained the so lutions of ideal MHD equations\nfor the interior and exterior regions of the tube. Then, we in vestigated a number of possible\ninstabilities that may arise in our model and concluded that these instabilities can be avoided in\nthe present model. We also derived the dispersion relation b y using the appropriate connection\nformula introduced by Sakurai et al. (1991a). In thin tube th in boundary approximation, we\nsolved analytically the dispersion relation and obtained t he frequencies and damping rates of\nthe fundamental ( l= 1) and first/second overtone ( l= 2,3) kink ( m= 1) modes. Our results\nshow the following.\n•The frequencies and damping rates increase when the twist pa rameter increases.\n•The ratio of the fundamental frequency to its corresponding damping rate of the kink\n(m= 1) modes can well justify the rapid damping of kink MHD waves (ω/(2πγ)≃3)\nreported by the observations. This confirms the high efficienc y of resonant absorbtion due\nto the magnetic twist.\n19•For a given twist parameter αand density ratio ζ=ρi/ρeby increasing R/L, the frequen-\ncies increase, the damping rates decrease and the ratio of th e frequency to the damping\nrate increases.\n•By increasing the thickness of the twisted layer, the freque ncy and the damping rate\nincrease but the ratio of the frequency to the damping rate an d the period ratio decrease.\n•For a given αandζ, the period ratio decreases with decreasing R/L. 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Press, Oxford\n23"    },    {        "title": "1904.11247v1.Low_damping_magnetic_properties_and_perpendicular_magnetic_anisotropy_with_strong_volume_contribution_in_the_Heusler_alloy_Fe1_5CoGe.pdf",        "content": "arXiv:1904.11247v1  [cond-mat.mtrl-sci]  25 Apr 2019Low damping magnetic properties and perpendicular magneti c anisotropy with strong\nvolume contribution in the Heusler alloy Fe 1.5CoGe\nAndres Conca,1,∗Alessia Niesen,2G¨ unter Reiss,2and Burkard Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Center for Spintronic Materials and Devices, Physics Depar tment, Bielefeld University, 100131 Bielefeld, Germany\n(Dated: April 26, 2019)\nWe present a study of the dynamic magnetic properties of TiN- buffered epitaxial thin films of\nthe Heusler alloy Fe 1.5CoGe. Thickness series annealed at different temperatures a re prepared and\nthe magnetic damping is measured, a lowest value of α= 2.18×10−3is obtained. The perpen-\ndicular magnetic anisotropy properties in Fe 1.5CoGe/MgO are also characterized. The evolution\nof the interfacial perpendicular anisotropy constant K⊥\nSwith the annealing temperature is shown\nand compared with the widely used CoFeB/MgO interface. A lar ge volume contribution to the\nperpendicular anisotropy of (4 .3±0.5)×105J/m3is also found, in contrast with vanishing bulk\ncontribution in common Co- and Fe-based Heusler alloys.\nTheneedforstrongperpendicularmagneticanisotropy\n(PMA) [1–5] and low damping properties [6–9] in next-\ngeneration spin-transfer-torque magnetic memory (STT-\nMRAM)generatesalargeinteresttowardsHeusleralloys.\nIn addition, large tunneling magnetoresistance (TMR)\nratios with MgO tunneling barriers have been reported\nforseveralofthem [10, 11]. For the application in devices\nbased on spin transfer torque switching, a low damp-\ning parameter αis important since the critical switching\ncurrent is proportional to αM2\nS[12] for in-plane magne-\ntized films. With perpendicular magnetization, the crit-\nical current is further reduced and it is proportional to\nαMS[13]. Therefore, a large effort is directed to study\nthe PMA properties of Heusler alloys with low damping.\nFor the PMA of thin Heusler films, the interface-\ninduced perpendicular anisotropy is essential and its\nstrength is given by the perpendicular interfacial\nanisotropy constant K⊥\nS. The interface properties, and\ntherefore the value of K⊥\nS, are strongly influenced by the\nconditions of the annealing, which is required to improve\nthe crystalline order of the Heusler and MgO layers and\nto achieve large TMR values.\nHere, we report on the evolution of the PMA proper-\nties with annealing of the PMA properties in Fe 1.5CoGe\nwith a MgO interface, by measuring different thickness\nseries and a comparison is made with the well-known\nCoFeB/MgO interface. The Gilbert damping parameter\nαchange with varying thickness and annealing tempera-\nture is also discussed.\nThe films were grown by sputtering, Rf-sputtering was\nused for the MgO deposition and dc-sputtering for the\nrest. For Fe 1.5CoGe, the layerstackis MgO(S) /TiN(30)\n/ Fe1.5CoGe(d) / MgO(7) / Si(2) with d= 80, 40, 20,\n15, 11 and 9 nm. Four series were deposited and three of\nthem annealed for one hour at 320◦C, 400◦C and 500◦C.\nFor CoFeB, the layer stack structure is MgO(s) / Ta(5)\n/ Ru(30) / Ta(10) / MgO(7) / CoFeB( d) / MgO(7) /\nTa(5) / Ru(2) d= 80, 40, 20, 15, 11, 9, 7 and 5 nm. The\nannealing was performed at 325◦C and 360◦C.FIG. 1. (Color online) (a) X-ray diffraction patterns of 40 nm\nthin Fe 1.5CoGe layers as-deposited, and annealed at 500◦C.\nThe (002) superlattice and the fundamental (004) peak of\nthe Fe 1.5CoGe are clearly visible, confirming the partial B2\ncrystalline order. (b) X-ray reflectometry data correspond ing\nto the samples in (a).\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). For\nthis, the samples were placed facing the strip-line and\nthe S12transmission parameter was recorded.\nCrystallographic properties of the CFA thin films were2\nFIG. 2. (Color online) Linewidth dependence on the fre-\nquency for Fe 1.5CoGe thin films with a thickness of 20 nm\nfor different annealing temperatures. The data sets have a\nvertical offset to improve visibility. The lines correspond to\na linear fit to extract the damping parameter α. The hollow\npoints are not considered for the fits (see main text).\ndetermined using x-ray diffraction (XRD) measurements\nin a Philips X’Pert Pro diffractometer equipped with a\nCu anode.\nThe XRD data corresponding to two 40 nm thick sam-\nples in the as-deposited state and annealed at 500◦C are\nshownin Fig.1(a). The (002)superlatticeandthe funda-\nmental (004) peak of Fe 1.5CoGe can be observed already\nfor the as-deposited state but they experience a strong\nintensity increase with the thermal treatment. The TiN\nlayeracts as a seed layer and its role in improving growth\nhas been reported also for other alloys [1, 14]. Due to\nthe similar lattice constant of TiN and MgO, the TiN\nfilm diffraction peaks are close to the substrate reflec-\ntions and therefore difficult to separate. The films are\nB2-ordered, the presence of the (111) is not proven and\ntherefore L2 1order cannot be confirmed.\nFigure 1(b) shows X-ray reflectometry (XRR) data for\nthe same films as in the (a) panel. The large number\nof oscillations prove the low roughness of the interfaces.\nThis is due to the low roughness below 1 nm of the TiN\nbuffer [14]. The similarity between both data sets also\nproves that the topology of the interfaces do not vary in\nthe studied temperature range.\nFigure 2 shows the dependence of the field linewidth\n∆Hof the FMR peak on the resonance frequency fFMR\nfor the sample series with no thermal treatment (as-\ndeposited) and for the series annealed at 320◦C and\n400◦C. In order to prevent poor visibility due to data\noverlapp, the sets are shifted in the vertical axis, except\nthe one corresponding to 320◦C. The actual linewidth at\n6 GHz is in the range 3 .25±0.15mT. The lines represent\nthe result to a linear fit to Eq. 1 to extract the dampingFIG. 3. (Color online) Dependence of the Gilbert damping\nparameter αon the thickness dfor three sample series: as-\ndeposited, annealed at 320◦C, and annealed at 400◦C.\nparameter α:\nµ0∆H=µ0∆H0+4παfFMR\nγ. (1)\nHere,∆H0istheinhomogenousbroadeningandisrelated\nto film quality, and γis the gyromagnetic ratio.\nA deviation from this simple linear behavior is ob-\nserved for the lower frequency range and these points\nhave not been considered for the fit (hollow circles). This\nfaster increase of linewidth with frequency is common\nin fully epitaxial Heusler layers [7, 15] and has been re-\nlated with an increased anisotropic two-magnon scatter-\ning in the thin films for low frequency values resulting\nin an anisotropic ∆ H. This anisotropy is not exclusive\nto Heusler alloys but it is expected in any epitaxial fer-\nromagnetic film [16–18]. The exact conditions for ob-\nservation, however, depend on the material parameters\nand the spin-wave dispersion. For instance, in epitaxial\nFe films, the low frequency ∆ Hbehavior deviates only\nfrom a linear behavior when magnetic dragging due to\ncrystalline anisotropy is dominant [19].\nAn additional sample set annealed at 500◦C showedno\nvisible FMR peak pointing to a degradation of the mag-\nnetic properties of Fe 1.5CoGe for high annealing temper-\nature. This is in constrast with Co-based Heusler alloys\nwhere large temperatures are typically required for opti-\nmal properties. For instance, for Co 2FeAl, lowest damp-\ning is achieved at 600◦C [7] and for Co 2MnSi, very low\ndamping is still present at 750◦C [20].\nThe results for the damping parameter αobtained\nfrom the linear fits are summarized in Fig. 3. A re-\nduction of damping is observed when comparing the as-\ndeposited samples to the annealed ones but the sam-\nples annealed at 400◦C show larger damping than the\nones annealed at 320◦C. Combined with the absence of3\nFIG. 4. (Color online) Dependence of Meffextracted from the Kittel fit on the inverse thickness 1 /dfor three Fe 1.5CoGe sample\nseries: as-deposited, annealed at 320◦C and annealed at 400◦C. The lines are a fit to Equation 2. The inset shows the evoluti on\nofK⊥\nSwith the annealing temperature.\nan FMR peak for 500◦C, this reinforces the conclusion\nthat the optimal annealing temperature for good dy-\nnamic properties of Fe 1.5CoGe is low. The lowest damp-\ning is 2.18±0.03×10−3for the 20 nm thick film an-\nnealed at 320◦C. For Co-based Heusler alloys, the lowest\nreported damping is achieved to be 7 ×10−4in Co2MnSi\n[20]. For Co 2FeAl, values around 1-3 ×10−3are reported\ndepending on the annealing conditions [7, 8]. Concern-\ning Fe-based alloys, values in the range of 1.2-1.9 ×10−3\nare reported for Fe 1+xCo2−xSi [9], for Fe 2Cr1−xCoxSiα\nvaries between 9 ×10−3with the lowest value of 8 ×10−4\nfor Fe2CoSi [21]. Therefore, the obtained value for the\ndamping parameter in our alloy is in the lower range of\npreviously reported ones and slighty reduced compared\nto those reported for the related alloy CoFeGe [22]. It\nis also smaller than the ones reported for widely used\npolycrystalline CoFeB [23, 24] and permalloy [25–28].\nThe thickness dependence of αshows a minimum\naround 20 nm and an increase for larger and smaller\nthicknesses. This behavior has been already observed\nfor Co 2FeAl [8], and the reasons are similar to that al-\nloy and different for the two thickness ranges. For soft\nmagnetic thin films, a strong damping increase with in-\ncreasing thickness is expected starting at a certain value.\nAn example can be found for NiFe in the literature [29].\nThe reason is a non-homogeneous magnetization state\nfor thicker films which opens new loss channels via in-\ncreased magnon scattering. For the thinner films, the\ndamping increase is due to two reasons. When the thick-\nness is reduced and the effect of the interface anisotropy\nis becoming larger the magnetization state is becoming\nmore inhomogeneous due to the counterplay between the\ndemagnetization field and the anisotropy field [30]. In\naddition, other effects related to an increased role of sur-\nface roughness with decreasing film thickness play also a\nrole.\nThe effective magnetization Meffis extracted using a\nfit to Kittel’s formula [31] to the dependence of the reso-nance field HFMRon the resonance frequency fFMR. For\na more detailed description of the FMR measurement\nand analysis procedure see Ref. [32]. Meffis related to\nthe saturation magnetization of the film by [34–36]\nMeff=Ms−H⊥\nK=Ms−1\nµ0Ms/parenleftbiggK⊥\nS\nd+K⊥\nV/parenrightbigg\n(2)\nwhereK⊥\nSandK⊥\nVare the perpendicular surface (or in-\nterfacial)andthe bulk anisotropyconstants, respectively.\nFigure 4 shows the dependence of M effon the inverse\nthickness 1 /dfor the three sample series: as-deposited,\nannealed at 320◦C and annealed at 400◦C. The slope\nprovides the value of K⊥\nS. The constant shows a positive\nvalue for the as-deposited series, 0 .41±0.12 mJ/m2, i.e.\nfavouring a perpendicular orientation of the magnetiza-\ntion. However, the value is small in absolute value and\nit grows only slighthy upto 0 .51±0.17 mJ/m2when the\nsamples are treated at 320◦C. The annealing at 400◦C\nchanges the situation drastically. The value of K⊥\nSis\nmuch larger, −1.36±0.14 mJ/m2, but it also suffers a\nchange of sign which implies that the interface induces\nan in-plane orientation of the magnetization. It is re-\nmarkable that this change of the magnetic properties of\nthe interface developes without a large modification of\nthe morphology, as proven by the XRR data shown in\nFig. 1(b). The inset in Fig. 4 summarizes the depen-\ndence of K⊥\nSon the annealing temperature.\nTaking into account the saturation magnetization Ms\nobtained by alternating gradient magnetometer (AGM),\n1100±120kA/m, it is possible to determine also the vol-\nume contribution to the perpendicular anisotropy to be\n(4.3±0.5)×105J\nm3. This large value ensures, that for a\n1.5 nm thin film and even for the 320◦C case, the PMA\nproperties are dominated by the bulk contribution. Re-\ncently, we reported on the evolution of the PMA prop-\nerties of the Co 2FeAl/MgO interface [8] and the situa-\ntion is very different for that alloy. First, no interface-4\ngenerated PMA is present in the as-deposited samples\nand it only appears after annealing. Second, K⊥\nSis al-\nways positive and larger than for Fe 1.5CoGe/MgO and\nthe absence of a remarkable volume contribution makes\nthe PMA there controlled only by the interface. Also in\nthe related alloy Co 20Fe50Ge30there is no bulk contri-\nbution to the perpendicular anisotropy and a interface\ncontribution (0 .9 mJ/m2) larger than that obtained here\n[33]. The most probable reason for the difference is the\nlower Fe content in our case. The presence of this strong\nbulk contribution to PMA is quite remarkable since it is\nabsent in common Co- and Fe-based Heusler alloys and\nonly observed in tetragonally distorted MnGa or MnGe\nrelated Heusler alloys [37–39].\nForcomparison,thePMApropertiesofthewidelyused\nCoFeB/MgO interface were also measured. The data\nis shown in Fig. 5 for annealing temperatures of 325◦C\nand 360◦C. An as-deposited series was not characterized\nsince CoFeB is amorphous in that state. The lines are\na fit to Eq. 2 with a prefactor 2 before K⊥\nSto account\nfor the presence of two interfaces since a trilayer system\nMgO/CoFeB/MgO was used.\nThe CoFeB/MgO interface shows a robust interface\nperpendicular anisotropy, three times larger than for\nFe1.5CoGe/MgO, and slighty decreases with tempera-\nture. The bulk contribution is zero or too small to be\nFIG. 5. (Color online) Dependence of Meffextracted\nfrom the Kittel fit on the inverse thickness 1 /dfor two\nMgO/CoFeB/MgO sample series: annealed at 325◦C and an-\nnealed at 360◦C. The lines are a fit to Equation 2 with a\nprefactor 2 (see text).detectable. These results are comparable to the liter-\nature [40–42]. From this we conclude that while the\nCo2FeAl [8] andCoFeB/MgOareverysimilarin absolute\nvalues, thermal evolution and relative weight of interface\nand bulk contribution to PMA, Fe 1.5CoGe/MgO differs\nstrongly.\nA comment must be made concerning the relation be-\ntween the values of K⊥\nSshown in Fig. 4 and the increase\nin damping for the thinnest films shown in Fig. 3. When\ndiscussing the increase of α, a possible connection is sug-\ngested with the interplay between the demagnetization\nfield and the perpendicular anisotropy field. An addi-\ntional hint supporting this interpretation is given by the\nfact that the for the 400◦C samples series, for which no\ninterface PMA is present, the relative increase in αfrom\ntheseriesminimum isthesmallest, aswellastheabsolute\nvalue for the thinnest film.\nIn summary, the damping properties of the Heusler\nalloy Fe 1.5CoGe and the perpendicular magnetic\nanisotropy of the Fe 1.5CoGe/MgO system have been\nstudied. From the thickness dependent magnetic prop-\nerties for as-deposited and annealed series we obtained\na minimum value for αof 2.18±0.03×10−3for a\n20 nm thick film. The evolution of the interface per-\npendicular anisotropy constant on the annealing temper-\nature is shown and compared with the standard inter-\nface CoFeB/MgO. We found a large and dominant vol-\nume contribution to the PMA, which differs from CoFeB\nor other well studied alloys as Co 2FeAl and the inter-\nface contribution suffers a sign change depending on the\nannealing temperature. We explained the increase on\ndamping with decreasing thickness in terms of a coun-\nterplay between demagnetizing field and interface PMA\nand correlate it with the obtained values for K⊥\nS.\nACKNOWLEDGEMENTS\nFinancial support by M-era.Net through the\nHEUMEM project is gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] A. Niesen, J. Ludwig, M. Glas, R. Silber, J.-M. Schmal-\nhorst, E. Arenholz, and G. Reiss, J. Appl. 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There isobservational evidence of propagating kink waves d rivenby photospheric motions. These disturbances, interp reted\nas kinkmagnetohydrodynamic (MHD)waves are attenuated as t heypropagate upwards inthe solar corona.\nAims.Toshow that resonant absorption provides a simple explanat ion tothe spatialdamping of these waves.\nMethods. KinkMHD wavesare studiedusingacylindricalmodel of solar magnetic fluxtubes whichincludes anon-uniform layerat\nthe tube boundary. Assuming that the frequency is real and th e longitudinal wavenumber complex, the damping length and d amping\nper wavelength produced by resonant absorption are analyti cally calculated in the thin tube (TT) approximation, valid for coronal\nwaves. Thisassumption is relaxedinthe case of chromospher ic tube waves and filament threadwaves.\nResults.Thedampinglengthofpropagatingkinkwavesdueresonantab sorptionisamonotonicallydecreasingfunctionoffrequen cy.\nFor kink waves with low frequencies the damping length is exa ctly inversely proportional to frequency and we denote this as the\nTGVrelation.Whenmoving tohighfrequencies the TGVrelati oncontinues tobe anexceptionally goodapproximation ofth e actual\ndependency of the damping length on frequency. This depende ncy means that resonant absorption is selective as it favour s low\nfrequency waves and can e fficiently remove high frequency waves from a broad band spectr um of kink waves. The e fficiency of the\ndampingduetoresonantabsorptiondepends onthepropertie softheequilibriummodel,inparticularonthewidthofthen on-uniform\nlayer and the steepness of the variation of the local Alfv´ en speed.\nConclusions. Resonant absorption is an e ffective mechanism for the spatial damping of propagating kin k waves. It is selective as\nthe damping length is inversely proportional to frequency s o that the damping becomes more severe with increasing frequ ency. This\nmeans that radial inhomogeneity can cause solar waveguides to be a natural low-pass filter for broadband disturbances. H ence kink\nwave trains travelling along, e.g., coronal loops, will hav e a greater proportion of the high frequency components diss ipated lower\ndown inthe atmosphere. This could have important consequen ces withrespect to the spatial distribution of wave heating inthe solar\natmosphere.\nKey words. Magnetohydrodynamics (MHD) —Waves —Magnetic fields— Sun: a tmosphere—Sun: oscillations\n1. Introduction\nThe first observations of post-flare transversal coronal loo p\noscillations by the Transition Region and Coronal Explorer\n(TRACE) (e.g., Aschwandenet al., 1999; Nakariakovet al.,\n1999; Aschwandenet al., 2002), inspired much development i n\nmagnetohydrodynamic(MHD) wave theory. This observationa l\nbreakthrough was important since estimated wave parameter s,\nsuch asfrequencyandamplitudeallowedus to implementmag-\nnetoseismological techniques to probe the plasma fine struc -\nture of the Sun’s atmosphere, an idea initially proposedby e .g.,\nUchida(1970)andRobertset al.(1984).Itisnowcommonlyac -\nceptedthatthesetransversalwavesarethekinkmodefromMH D\nwavetheory(seee.g.,Edwin& Roberts,1983),ahighlymagne t-\nicallydominated,i.e.,Alfv´ enicwave(seeGoossenset al. ,2009,\nfor a discussion on the nature of kink waves). The observed\npost-flare kink waves in coronal loops have two main defining\ncharacteristics, firstly they are standing modes and second ly,\nthey are strongly damped oscillations (in about 1-4 periods ,\nsee e.g., Aschwandenet al., 2003). Initially there were sev eral\nphysical mechanisms proposed to explain the observed damp-\nSend offprint requests to : J. Terradas, e-mail:\njaume.terradas@uib.esing, e.g., footpoint leakage (Berghmans&deBruyne, 1995;\nDe Pontieuet al., 2001), phase mixing (Heyvaerts& Priest,\n1983; Roberts, 2002; Ofman&Aschwanden, 2002) and res-\nonant absorption (Ruderman&Roberts, 2002; Goossenset al. ,\n2002)andmorerecentlyloopcooling(Morton& Erd´ elyi,200 9).\nThusfar, resonantabsorption,caused by plasma inhomogene ity\nin the direction transverse to the magnetic field, has proved the\nmost likely candidate in explaining the observed short damp -\ning times in coronal loops (see Goossens, 2008, for review).\nConsistent seismological studies based on resonant absorp tion\nusing the observed values of periods and damping times of\nstanding kink waves were carried out by Arreguiet al. (2007)\nandGoossensetal.(2008).Thesetwostudiesshowthat,atle ast\nfora collectionof11loops,resonantabsorptionprovidesa n ex-\nplanation of the observed periods and damping times. The ob-\nservational signatures of the alternative cooling loop dam ping\nmechanism proposed by Morton& Erd´ elyi (2009), di ffer from\nresonant absorption by the fact that frequency changes as a\nfunction of time (the resonant damping theory developed so f ar\nis restricted to static equilibria and therefore frequency is ex-\npected to be constant in time). Extensive MHD modelling has\nalso shown that resonant absorption is the most likely expla na-\ntion for the damping of transverse oscillations in prominen ce\n1Terradaset al.:Spatialdamping of propagating kinkwaves\nfine structures (see Arreguiet al., 2008; Soleret al., 2009a ,b;\nArregui&Ballester, 2010).\nIf indeed, coronal loop kink oscillations are being at-\ntenuated by the process of resonant absorption, one can\nexploit this to estimate the transverse plasma inhomogenei ty\nlength scales using observed frequencies and damping times\n(Goossensetal., 2006). The original equilibrium models by\ne.g., Ruderman&Roberts (2002), to study resonant absorpti on\nconsisted of monolithic loop structures. However, there ha s\nbeen some observational evidence by e.g., Aschwanden (2005 )\nthat coronal loops are composed of many di fferent strands,\npossibly at different temperatures and densities. To model\nthis loop multi-thread structure, there was an initial nume rical\nstudy by Terradaset al. (2008) and it was found that the\nprocess of resonant absorption was still an e fficient damping\nmechanism in more complex and realistically structured loo p\nmodels. Further study into the properties of standing kink\nwaves was also undertaken relating plasma inhomogeneity\nin the direction of the magnetic field caused by e.g., den-\nsity (D´ ıazet al., 2004; Andrieset al., 2005b; Arreguiet al .,\n2005; Dymova&Ruderman, 2006; Erd´ elyi&Verth, 2007;\nVerthetal., 2007) or magnetic (Verth&Erd´ elyi, 2008;\nRudermanetal., 2008) stratification. It was shown that\neigenfrequencies and eigenfunctions of coronal loops with\nlongitudinal stratification, were altered in such a way that\none could determine, e.g., the coronal density scale height\nby estimating the ratio of the fundamental mode to that of\nhigher overtones, a technique first proposed by Andrieset al .\n(2005a) and later developedby Verthet al. (2008) to correct for\nmagneticstratification.\nAll this theoretical development to describe kink waves in\ncoronal loops with realistic plasma stratification in the tr ans-\nverse and longitudinal direction was restricted to the stud y of\nstanding waves, since this was what was detected in TRACE\ndata. However, recently it has come to light that there are\nalso ubiquitous small amplitude propagating transversal M HD\nwaves in the solar atmosphere. These were first observed by\nthe novel Coronal Multi-Channel Polarimeter (CoMP) instru -\nment (Tomczyket al., 2007). Moreover, Tomczyk&McIntosh\n(2009) have been able to separate outward and inward propa-\ngating wave power. It was found that the outward power was\ngreater than the inward power by about a factor of two and this\ncan only be explained if the waves are damped in situ (see also\nPascoeet al., 2010). The reasonwe could notdetect these pro p-\nagatingwavespreviouslywithTRACEisthattheamplitudesa re\nof the order 50 km, while the TRACE resolution is only about\n800km.Tomczyket al.(2007)originallyinterpretedthesew ave\nasAlfv´ enwaves,however,VanDoorsselaereet al.(2008)su bse-\nquentlyarguedthattheobservedwaveswereactuallymoreco n-\nsistent with the propagating kink mode. Although both modes\nare dominatedby the restoringforce of magnetictension, in the\ngeometry of a solar flux tube, e.g., a magnetic cylinder, a pur e\nAlfv´ en wave is strictly torsional with no transverse compo nent\nand therefore completely incompressible. On the other hand , a\nkink wave propagating in a flux tube has a transverse perturba -\ntioncomponentandisweaklycompressible,atleastintheli near\nregime(seee.g.,Goossensetal., 2009).\nIn the present paper we shall restrict the study to inves-\ntigating the effect of transverse plasma inhomogeneity on the\npropagatingkink mode.Althoughthe two problemsof standin g\nand propagating kink waves are closely related, since a stan d-\ning wave is a superpositionof two propagatingwaves, there a re\nsomedifferencesthatneedtobeconsidered.Thestandingtrans-\nverse oscillation is the result of an initial value problem, i.e., aninitialdisturbanceinthesolar corona,e.g.,a CME oraflare ,in-\nducestheoscillationsoftheloopsat theirnaturalfrequen ciesor\neigenmodes. Alternately, the transverse travelling waves have a\nforcednaturesincethephotosphereactsasadriver.Thefre quen-\ncies of the kink waves observed by CoMP show a peak around\n5min,indicatinga p-modedrivenphotosphericorigin.Thespa-\ntial scale of the driver at the base of the tube is also importa nt\ninordertoexcitekinkoscillations.Fromthepropertiesof MHD\nwavesina fluxtubeweknowthata purelyincompressibleexci-\ntation excites purely incompressible Alfv´ en waves if the d river\nisstrictlylocalisedinsideoroutsidethetube(assuminga homo-\ngeneous loop model). On the other hand, an excitation locate d\nbothinsideandoutsidethetubeinvariablyexcitestransve rseos-\ncillations since an almost incompressiblesurface wave bet ween\nthetwo mediawill beestablished,i.e.,thekinkmode.\n2. Waveguidemodel\nTounderstandthee ffectofradialinhomogeneityonpropagating\nkinkwavesweconsiderthe relativelysimple equilibriummo del\nof a cylindrical axi-symmetric flux tube of radius Rwith a con-\nstantaxialmagneticfield Bzandwithadensitycontrastof ρi/ρe.\nThe subindex “i” and “e” refer to the internal and external pa rt\nof the tube, respectively. It is also assumed that the tube ha s a\nsmooth variationof density across the waveguideboundary( lo-\ncatedatr=R)withacharacteristicspatialscale l.Forsimplicity\na sinusoidal density profile connecting the internal and ext ernal\npart of the tube is implemented(see e.g., Ruderman&Roberts ,\n2002; Goossenset al., 2002; Terradaset al., 2006). The role of\nthe inhomogeneity at the loop boundary is crucial since this is\nwherethe processof resonantabsorptioninvariablytakesp lace.\nThis basically means that the transverse displacement of th e\nwhole tube is converted into azimuthal motions localised at the\ntubeboundary.Duetothisenergyconversion,thetransvers emo-\ntion is attenuated, while at the same time small scales are cr e-\nated in the nonuniform layer due to phase mixing. Phase mix-\ning causes a cascade of energy to smaller length scales, wher e\nthe dissipationbecomesmore e fficient.The readeris referredto\nGoossens (2008); Ruderman&Erd´ elyi (2009); Terradas(200 9)\nandreferencesthereinforfurtherdetailsaboutthisrobus tdamp-\ning mechanism. Recent studies by Soleret al. (2009a) in the\ncontext of modelling kink oscillations observed in solar pr omi-\nnences (complex coronal magnetic structures with relative ly\ncool and dense plasma), show that the process of resonant ab-\nsorptionstill survivesevenwhentheplasmaispartiallyio nised.\n3. Spatialdampingin theTT approximation\nBefore we embark on the analysis of MHD waves in the thin\ntube (TT) approximation, let us explain what this approxima -\ntion actuallyis. In what followswe shall considerbothstan ding\nand propagatingwaves. A standingwave is the superposition of\ntwo propagating waves travelling with the same frequency an d\nwavenumber but in opposite directions. For the standing wav e\nproblem the axial wavelength (or the axial wavenumber kz) is\nspecified and the correspondingfrequencyis determined. A T T\ninthiscasemeansthattheaxialwavelengthismuchlongerth an\nthe radius of the tube so that kzR≪1. Hence the TT approx-\nimation for standing waves is the long wavelength approxima -\ntion.Forpropagatingwavesthefrequency ωisspecifiedandthe\ncorrespondingwavelength is determined.In this situation , a TT\nmeansthatduringoneperiod,definedbythefrequency ω,asig-\nnal travelling at the Alfv´ en speed can cross the waveguide i n\n2Terradaset al.:Spatialdamping of propagating kinkwaves\nthe radial direction many times or ω/(vA/R)≪1. Hence for\npropagating waves, the TT approximation is the low frequenc y\napproximation. The benefit of the TT approximation is that it\nenablesusto obtainsimplemathematicalexpressionswhich are\nvery accurate, allowing us to gain a physical insight into th e\nproblem.\n3.1. Homogeneousmagneticcylinder\nFor a homogeneousmagnetic cylinder (i.e., no inhomogeneou s\nlayer,l=0) the dispersion relation for the kink mode ( m=1)\nand the fluting modes ( m>1) in the TT approximation is (see\nGoossenset al., 2009)\nρi(ω2−ω2\nAi)+ρe(ω2−ω2\nAe)=0, (1)\nwhere\nω2\nA=k2\nzv2\nA,v2\nA=B2\nz/(µρ). (2)\nThedispersionrelationEq.(1)specifiesafunctionaldepen dence\non frequency ωand wavenumber kz. This can be studied for the\ncasesofeitherstandingorpropagatingwaves.Forthepropa gat-\ning wave study, the wavenumber kzis specified and the disper-\nsionrelationissolvedforfrequency ω,leadingtothewellknown\nresult\nω2=ρiω2\nAi+ρeω2\nAe\nρi+ρe≡ω2\n∗, (3)\nwhereω∗is real. For equal magnetic field strength inside and\noutsidethecylinderthisexpressioncanbefurthersimplifi edto\nω2=2B2\nz\nµ(ρi+ρe)k2\nz=2ρi\nρi+ρev2\nAik2\nz≡ω2\n∗. (4)\nFor propagating waves we consider waves generated at a given\nlocation with a real frequency ω∗and solve the dispersion rela-\ntionforkz, resultingin\nk2\nz=µ(ρi+ρe)\n2B2zω2\n∗=ρi+ρe\n2ρiω2\n∗\nv2\nAi≡k2\n∗, (5)\nwherek∗is also real. The solution givenby Eq. (5) corresponds\ntothewellknownundampedkinkwave(andtheflutingmodes).\nTo derive Eq. (1) it is implicitly assumed that ωR/vA≪1 but\nsince not all frequencies will satisfy these conditions lat er we\npresent results for any frequency of driver without this res tric-\ntion.\n3.2. Inhomogeneousmagneticcylinder\nThe thin boundary (TB) approximation means that the non-\nuniformity is confined to [ R−l/2,R+l/2], withl/R≪1, so\nthat the non-uniform layer coincides with the dissipative l ayer.\nThis approximation results in the mathematical simplificat ion\nthat MHD waves can be described by solutions for uniform\nplasmas that are connected over the dissipative layer by jum p\nconditions (see e.g., Sakuraiet al., 1991; Goossensetal., 1995;\nTirry& Goossens, 1996). The use of the TB approximationhas\nbeen applied in e.g., Goossensetal. (2006), Goossens (2008 )\nand Goossenset al. (2009). The inclusion of the e ffect of an in-\nhomogeneouslayer is reasonably simple in the case of the thi n\nboundaryapproximationand results in the following disper sion\nrelation,\nρi(ω2−ω2\nAi)+ρe(ω2−ω2\nAe)=\niπm/rA\nρ(rA)|∆A|ρi/parenleftBig\nω2−ω2\nAi/parenrightBig\nρe/parenleftBig\nω2−ω2\nAe/parenrightBig\n.(6)HererAdenotes the position of the Alfv´ en resonance. In the\nTB approximation it is natural to adopt rA=Rsincel/R≪1\nandrA∈]R−l/2,R+l/2[. Use of the jump condition is not re-\nstricted to the thin non-uniform layer as can be seen from e.g .,\nTirry& Goossens (1996); Tirryetal. (1997, 1998). However,\nthis condition requires numerical integration of the ideal MHD\nequations in a non-uniform plasma up to the dissipative laye r.\nIn the present paper we do not intend to use numerical integra -\ntion of the ideal MHD equations relating to thick boundaries .\nHowever,weshallusetheresultsobtainedwiththeTBapprox i-\nmationforthickboundariesfora comparisonwiththe result sof\na fullnumericalcalculationin Section5.\nNote that the effect of resonance is contained in the right\nhand side of Eq. (6). Again we can view the dispersion relatio n\nEq.(6)asa relationforeitherstandingorpropagatingwave s.In\nthe case ofstandingwavesthe wavenumberisreal ( kz=k∗) and\nthe frequencyis complex. This case has been considered prev i-\nouslybye.g.,Goossenset al.(1992,2009).Letusnowfocuso n\npropagating waves with a given real frequency ( ω=ω∗) and\ncomplex wavenumber. The imaginary part of the wavenumber\nindicatesthat the wave, as it propagates,is dampeddue to re so-\nnantabsorption.Nowweassume that\nkz=kR+ikI. (7)\nThe purely imaginary term in Eq. (7) reflects the damping im-\nposed on the wave and since the damping is in the spatial do-\nmainthewavenumberisnowcomplex.Ifweapproximate k2\nzby\nk2\nR+2ikRkI(we assume weak damping, i.e., kI≪kR) we have\nthatkR≈k∗(givenbyEq.[5])andaftersomealgebrawe find\nkI\nk∗=π\n8m\nR1\nρ(rA)|∆A|(ρi−ρe)2\n(ρi+ρe)ω2\n∗. (8)\nBy thefactthat,\nρ(rA)|∆A|=ω2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ\ndr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglerA, (9)\nwe finallyobtainthefollowingexpression\nkI\nk∗=π\n8m\nR(ρi−ρe)2\nρi+ρe1\n|dρ/dr|rA. (10)\nBecausekzis complex we define the damping length as LD=\n1/kIwhilethewavelengthissimply λ=2π/k∗.Ausefulquantity\nisthethe dampingperwavelengthwhichis\nLD\nλ=4\nπ2R\nmρi+ρe\n(ρi−ρe)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ\ndr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglerA. (11)\nForasinusoidaldensityprofileit canbeshownthat\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ\ndr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglerA=π\n2ρi−ρe\nl, (12)\nandEq.(11)reducesto\nLD\nλ=2\nπ1\nmR\nlρi+ρe\nρi−ρe. (13)\nEquation(13) clearlyshowsthe dependenceofthe dampingpe r\nwavelength on the thickness of the layer. The wider the layer ,\nthe stronger the spatial attenuation of the wave. This is not sur-\nprising, since we can relate this result to the expression fo r the\ntemporaldamping,i.e., wavenumberassumedreal ( kz=k∗) and\n3Terradaset al.:Spatialdamping of propagating kinkwaves\nfrequency complex. The well known formula in the case of the\ntemporaldampingforastandingwave is\nτD\nP=2\nπ1\nmR\nlρi+ρe\nρi−ρe. (14)\nIf we compare this expression with Eq. (13) we note that the\ndampingperwavelengthforpropagatingwavesandthedampin g\nperperiodforstandingwavesareexactlythesame.Hence,in the\nTTapproximationspatial andtemporaldampingarecomplete ly\nequivalent.FromEqs.(13)and(14) weobtainthesimpleresu lt\nLD\nλ=τD\nP. (15)\nPropagating kink waves were recently studied by\nVasheghaniFarahaniet al. (2009) in the long wavelength\nlimit in X-ray jets in the solar atmosphere. These authors co m-\nputed the ratio of the damping time to the period for standing\nwaves and used this quantity to discuss the spatial damping o f\npropagating waves. The TT approximation result presented i n\nEq. (15), therefore validates the use of the damping express ion\nderived for kink standing waves by VasheghaniFarahaniet al .\n(2009) to interpret the attenuation of propagating kink wav es.\nThe TT damping relations given by Eqs. (13) and (14) have\nfurthersignificantconsequences.Forstandingwaveswerew rite\nEq.(14) as\nτD\nP=ξE\nm, (16)\nwhere\nξE=2\nπR\nlρi+ρe\nρi−ρe, (17)\nwhichonlydependsontheparametersoftheequilibriummode l,\nnot on the particular type of MHD wave mode defined by the\nvalue ofm(note that for the kink mode m=1). The period is\ndefinedby\nP=2π\nω, (18)\nandforstandingwaves ωisrelatedtothewavenumber kzbythe\ndispersionrelationgivenbyEq.(4),i.e.,\nω=nvAiπ\nL/radicalBigg\n2ρi\nρi+ρe, (19)\nwherevAiistheinternalAlfv´ enspeed, n=1,2,3, ...isthelon-\ngitudinal mode number and Lis the total length of the waveg-\nuide, e.g., a coronal loop (we have used that kz=nπ/L).\nEquation(16) canthenbewrittenas\nτD\nτAi=2ξE\nm/radicalbiggρi+ρe\n2ρi1\nn, (20)\nwhereτAi=L/vAiis the Alfv´ en transit time in the longitudinal\ndirection. Equation (20) has interesting consequences. Fi rstly,\nit indicates that fluting modes ( m>1) have shorter damp-\ning times than the kink mode ( m=1). Secondly, it shows\nthat the damping time for a standing wave is inversely pro-\nportional to the longitudinal mode number n, i.e., the damping\ntime is inversely proportional to the wavelength of the stan d-\ning wave, so that higher overtones (with shorter periods) ar e\ndamped faster than low order overtones, e.g, the fundamenta lmode. Fortunately, there have been some signatures of over-\ntones in coronal loop standing kink waves detected in TRACE\ndata(seee.g.Verwichteetal.,2004;De Moortel&Brady,200 7;\nVerthetal., 2008; VanDoorsselaereet al., 2009). For a part icu-\nlarlyclearexampleofthefirstovertonedampingbeforethef un-\ndamental mode, that could be explained by resonant absorpti on\nattenuating the higher harmonic faster, see the Morlet wave let\ntransforminFigure5 ofVerwichteet al.(2004).\nNow considering the spatial damping of propagating waves\nwe canalsowriteEq.(13)as\nLD\nλ=ξE\nm. (21)\nThe wavelength is defined as λ=2π/kzand is related to the\nfrequency by the dispersion relation in Eq. (5). Equation (2 1)\ncanthenberewrittenin termsofthefluxtuberadiusas\nLD\nR=2πvAi\nωRξE\nm/radicalBigg\n2ρi\nρi+ρe, (22)\nwhereωR/vAiis a dimensionless frequency (note that the TT\napproximationmeans ωR/vAi≪1).Denotingthisfrequencyas\nf=ωR\nvAi, (23)\nthen\nLD\nR=2πξE\nm/radicalBigg\n2ρi\nρi+ρe1\nf. (24)\nWe shall refer to the relation between the damping length and\nfrequency defined in Eq. (24) as the TGV relation for propa-\ngating kink waves. In what follows, it will become clear that\nthe TGV relation is actually a very good approximation of the\ndependencyofdampinglengthonfrequencyforall relevantf re-\nquenciesasillustratedinFig.2.TheexpressiongiveninEq .(24)\nhas important consequences, as it shows that LD/Ris inversely\nproportionalto fforpropagatingwaves,i.e.,thedampinglength\nis inversely proportional to frequency, so that high freque ncy\nwaves are damped in shorter spatial scales than their lower f re-\nquencycounterparts.Thismeansthatfordrivenwavespropa gat-\ningupwardsfromthephotosphereeachfrequencyhasadi fferent\npenetrationheightintothe solarcorona.Therefore,reson antab-\nsorptionprovidesa naturalfiltering mechanismfora broadb and\ndisturbances,e.g., like those observedby Tomczyk&McInto sh\n(2009), with lower frequency waves being least a ffected by the\ndamping process, propagating to higher heights in the solar\ncorona.\n4. SpatialdampingbeyondtheTTapproximation\nTheTTapproximationisusefulbecauseit makesthedispersi on\nrelation analytically solvable. In Table 1 we list observed esti-\nmates of kzRfor various MHD wave modes and it can be seen\nthatformanysolaratmosphericwaves,e.g.,standingkinkw aves\nin coronal loops observed by Aschwandenetal. (2002), the TT\napproximation is reasonably valid. However, for kink waves in\nfilament threads (Linet al., 2009) or torsional Alfv´ en wave s in\nchromospheric magnetic bright points (Jesset al., 2009) ma y\nhave values of kzR≈1. To address the kink waves observed\ninthisregime,itisrelativelystraightforwardtorelaxth eTTap-\nproximationandto take into accountthe e ffect of the finite tube\nradiusondamping.\n4Terradaset al.:Spatialdamping of propagating kinkwaves\nTable 1.Estimatedrangeof kzRinobservedpropagating(P)andstanding(S)MHDwavemodes.\nReference MHD wave mode Widthof Wavelength kzR\ninterpretationof authors waveguide (Mm)\n(Mm)\n1 Kink(S) 5 .9−11.5 328−552 0.03−0.11\n2 Alfv´ en(P) 0.2 >4.0<0.16\n3 Kinkor sausage waves (S) 0.4 2 .5−3.1 0.41−0.5\n4 Torsional Alfv´ en(P) 2.0 4 .5−6.0 1.04−1.39\n5 Kink(P) 0 .2−1.0 3.4−12.9 0.03−0.9\n6 Kink(P) 0 .43−0.66>250 <0.12\n7 Alfv´ en(P) 9 .0 >393 <0.07\n8 Kinkor Alfv´ enwaves (P) 9 .0 >180 <0.16\nReferences. (1)Aschwanden etal.(2002);(2)De Pontieuet al.(2007);(3 )Fujimura & Tsuneta(2009);(4)Jess et al.(2009);(5)Linet al.(2009);\n(6) Okamoto etal.(2007);(7) Tomczyk etal. (2007);(8) Tomc zyk & McIntosh (2009).\n4.1. Homogeneousmagneticcylinder\nFor a homogeneous magnetic cylinder the dispersion relatio n,\ntakingintoaccounta finiteradius,is\nρi(ω2−ω2\nAi)+ρe(ω2−ω2\nAe)F(ω,kz)=0, (25)\nwhere\nF(ω,kz)=−ki\nkeJ′\nm(kiR)Km(keR)\nJm(kiR)K′m(keR), (26)\nk2\ni=ω2−ω2\nAi\nv2\nAi,k2\ne=−ω2−ω2\nAe\nv2\nAe. (27)\nNotethatthefunction Fdependsbothonfrequency ωandwave\nnumberkz,takingintoaccountthatthetubehasafinitewidth.In\nthe limit of a TT, F≡1 (the Bessel functionsare approximated\nbytheirsmallargumentsexpressions)andwerecoverthedis per-\nsionrelationgivenbyEq.(1).Forastandingwave kzisfixedand\nEq. (25) is solved for ω. For a propagatingwave ωis fixed and\nEq. (25) is solved for kz. The solution to Eq. (25) corresponds\nagain to the undamped kink wave. To solve this equation ana-\nlyticallyfora finitewidthtubeonewouldhavetouseadditio nal\ntermsintheasymptoticexpansionsoftheBesselfunctions, com-\nplicating matters. To avoid these cumbersome calculations we\nsimply solve Eq. (25) numerically, with the solution di ffering\nfromthesolutiongivenbyEq.(5) outwiththeTT regime.\n4.2. Inhomogeneousmagneticcylinder\nWhen we add a thin non-uniform layer we obtain the complex\ndispersionrelation\nρi(ω2−ω2\nAi)+ρe(ω2−ω2\nAe)F(ω,kz)=\niπm/R\nρ(rA)|∆A|ρi/parenleftBig\nω2−ω2\nAi/parenrightBig\nρe/parenleftBig\nω2−ω2\nAe/parenrightBig\nG(ω,kz),(28)\nwhere\nG(ω,kz)=−m\nR1\nkeKm(keR)\nK′m(keR). (29)For the TT we can again use the asymptotic expansions of the\nBessel function so that G≡1, hence in the TT limit, Eq. (28)\nis simplified to Eq. (6). Equation (28) can be solved for a com-\nplex frequency ωwith a specified real wavenumber k∗or for a\ncomplex wavenumber kzfor a specified real frequency ω∗. By\nsolvingEq.(28) in thecase of a propagatingwavewith real fr e-\nquencyω∗we find that the real part of complex kzisk∗(we get\nagainEq.[25],note thatnow k∗is differentfromEq.[6]),while\ntheimaginarypartis givenby\nkI\nk∗=T\nN, (30)\nwhere\nT=π\n2m\nR(ρi−ρe)2\n/bracketleftbigρi+ρeF(ω∗,k∗)/bracketrightbig[1+F(ω∗,k∗)]2×\nF(ω∗,k∗)G(ω∗,k∗)\n|dρ/dr|rA, (31)\nand\nN=1+k∗\n2ρi−ρe/bracketleftbigρi+ρeF(ω∗,k∗)/bracketrightbig[1+F(ω∗,k∗)]∂F\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ω∗,k∗).(32)\nThe expressions are slightly more complicated than in the TT\napproximation(see Eq.[10]), but once k∗is knownthe different\nterms can be easily calculated. The corresponding damping p er\nwavelengthis\nLD\nλ=1\n2πN\nT. (33)\nIn the TT limit, i.e., when F(ω∗,k∗) andG(ω∗,k∗)→1 and\n∂F/∂k→0,werecoverEq.(11).Forasinusoidaldensityprofile\nwe simply have to make use of Eq. (12) in Eq. (31). Now let us\ndefine\n/tildewideN=1−ω∗\n2ρi−ρe/bracketleftbigρi+ρeF(ω∗,k∗)/bracketrightbig[1+F(ω∗,k∗)]∂F\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ω∗,k∗).(34)\n5Terradaset al.:Spatialdamping of propagating kinkwaves\nAs in the previous Section we compare the damping per wave-\nlength for propagating waves with the damping per period,\nwhichinthenon-TTapproximationisgivenby\nτD\nP=1\n2π/tildewideN\nT. (35)\nBy Eqs. (33) and (35), there is a clear correspondence betwee n\nthetemporalandthespatialdampingbeyondtheTTapproxima -\ntion.Furthermore,ageneralexpressionthatrelatesthete mporal\ndamping of standing waves and the spatial damping of propa-\ngatingwavescanbederivedfollowingthe analysisofAppend ix\nA in Taggeretal. (1995). We can write the complex dispersion\nrelationgivenbyEq.(28) as\nDR(ω,k)+iDI(ω,k)=0. (36)\nTomakeit clear,in Eq.(36), DRis equalto theLHS ofEq.(28)\nandDIequal to minus the RHS of the equation. In the case of\nspatial damping, if ω∗andk∗are the solutions of DR(ω,k)=0\nthen it is easy to see by making a Taylor expansion around the\nsolutionthat\nkI=−DI\n∂DR/∂k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ω∗,k∗), (37)\nwhilefortemporaldamping\nωI=−DI\n∂DR/∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ω∗,k∗). (38)\nCombiningthese twoexpressionswe findthat\nLD\nλ=k∗\nω∗∂DR/∂k\n∂DR/∂ωτD\nP, (39)\nwhichissimply(seealsoPascoe etal.,2010)\nLD\nλ=vgr\nvphτD\nP. (40)\nThisresult isvalidwhenthedampingis nottoostrong(see e. g.,\nTaggeret al., 1995), an assumption that we have already made\nin the derivation of the damping per wavelength ( kI≪kR). In\nthe TT approximationkink waves are weakly dispersive so tha t\nvgr≈vphhence,inagreementwiththeresultsfoundinSection3,\nthe dampingper wavelengthis exactlythe same as the damping\nper period but in the regime when the TT approximation is not\napplicablethegroupspeedcandi fferfromthephasespeed.\nFrom the previousresults we can also derivealternativefor -\nmulae for the damping per wavelength and damping per period\nintermsofthe imaginarypartofthedispersionrelation,\nLD\nλ=k∗\n2π1\nDI(ω∗,k∗)×\n/braceleftBigg\n−2k∗/bracketleftBig\nρiv2\nAi+ρev2\nAeF(ω∗,k∗)/bracketrightBig\n+ρe(ω2\n∗−k2\n∗v2\nAe)∂F\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ω∗,k∗)/bracerightBigg\n,\n(41)\nτD\nP=ω∗\n2π1\nDI(ω∗,k∗)×\n/braceleftBigg\n2ω∗/bracketleftbigρi+ρeF(ω∗,k∗)/bracketrightbig+ρe(ω2\n∗−k2\n∗v2\nAe)∂F\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ω∗,k∗)/bracerightBigg\n.\n(42)\nUsing Eq. (40) we can identify in these expressions the terms\nrelatedtothephaseandgroupspeed.Fig.1.Dampingperwavelengthasafunctionofthedimensionless fr e-\nquency (f=ωR/vAi) for three different widths of the inhomogeneous\nlayers. The solid line corresponds to the analytical result s, the dashed\nline represents the full numerical solution of the resistiv e eigenvalue\nproblem. The dotted line corresponds to the TT approximatio n, valid\nwhenf→0. Inthisplot ρi/ρe=3.\nNext we study, using the analytical results given in this\nSection,howthedampingperwavelengthdependsonthedimen -\nsionlessfrequencyofthedriver f,definedbyEq.(23),forpartic-\nularcases(seeFigure1).Weseethatthewiderthelayerthem ore\nefficient the attenuation (smaller damping per wavelengths), i n\nagreement with the analytical results in the TT approximati on.\nFor small fthe damping per wavelength tends to the TT value\n(see dotted line). The value of LD/λincreases monotonically\nwithω. The deviation with respect to the TT results is smaller\nforthicklayers.IntheTTapproximation,whichforpropaga ting\nwaves, is the low frequency approximation, LD/λis indepen-\ndent of frequency, since this approximation does not take in to\naccount the variation of frequency, i.e., the frequency is o nly\npresumed to be small. The analytical results for LD/λbeyond\nthe TT approximation,take into accountthe dependenceon fr e-\nquency. The value of LD/λnow undergoesa moderate increase\nwhen we move from low to high frequencies. However, the re-\nallyinterestingquantitytocalculateisthedampinglengt hitself.\nIn order to make the dependence of LDon frequency more ex-\nplicit we follow the same line of reasoning in Section 3.2 and\nrewriteEq.(13) as\nLD\nλ=ξEW, (43)\nwhere the quantity ξEWnow depends on the parameters of the\nequilibrium model and the characteristics of the wave itsel f.\nAgainλ=2π/kzandkzis related to the frequency by the dis-\npersion relation Eq. (25). Since we have abandoned the TT ap-\nproximation, a simple analytical formula that relates kztoωis\nnotreadilyavailablebutwe canalwayswrite\nkzR=fψ(f), (44)\nwherefis the dimensionless frequencydefined in Eq. (23) and\nψ(f) is a function that we can determine numerically. Hence,\nimplementingthefunction ψ(f),we havethat\nLD\nR=2πξEW\nψ(f)1\nf, (45)\n6Terradaset al.:Spatialdamping of propagating kinkwaves\nwhich in the low frequency limit is equivalent to the TGV rela -\ntiongivenbyEq.(24)for m=1,i.e.,\nξEW→ξEandψ→/radicalbiggρi+ρe\n2ρi(46)\nasf→0, whereξEis defined by Eq. (17). In Eq. (45), govern-\ning the damping length for propagating waves, since ψ(f) and\nξEWare slowly varying functions of f, the main dependence of\nLDonfis contained in the factor 1 /f, as in the low frequency\nlimit shown by Eq. (24). This is illustrated in Figure 2, wher e\nLD/Ris plotted as a functionof ffor several values of l/R. The\ndependence of the curves on 1 /fis very clear and the non-TT\nand TT solutions (compare the solid with the dotted lines) te nd\nto overlap in the limit of f→0, where both cases are accu-\nrately described by the TGV relation. Note that even for f→1\nthe TGV relation still describes quite well the behaviour of the\ndamping length with frequency. Again, the potential of reso -\nnant absorption as a frequency filter is clearly demonstrate d in\nFigure 2, with high frequency waves being damped in shorter\nspatialscalesthanlowfrequencywaves(forfixed l/R).\nFig.2.Damping length normalised to the loop radius as a function of\nthe dimensionless frequency ( f=ωR/vAi) for three different widths\nof the inhomogeneous layers. The solidlinecorresponds tot he analyti-\ncal results, the dashed line represents the full numerical s olution of the\nresistiveeigenvalue problem.Thedottedlinecorresponds totheTTap-\nproximation calculated using Eq. (24), valid when f→0. In this plot\nρi/ρe=3.\n5. Resistivecalculations\nThe results based on TB approximation,described in the prev i-\nous Sections, are compared with the full resistive calculat ions\n(seeTerradasetal.,2006,forthedetailsaboutthemethod) .This\ncomparison is useful since in the resistive eigenvalue prob lem\nthere are no implicit assumptions about the TT or TB approx-\nimation, i.e., the TT and TB approximation are not used. The\nresistivecalculationappliesto anyfrequencywhetherit i ssmall\ncompared to vAi/Ror not and also to equilibrium models that\nhaveathinnon-uniformlayerorarefullynon-uniform.Weha ve\nsolvedtheeigenvalueproblem(for ω) andhaveusedEq.(40)to\ntranslate from temporaldampingto spatial damping.Due to t he\ninclusionofresistivity,theeigenvalueproblemforthewa venum-\nber is more difficult to solve than the eigenvalue problem for\nthefrequency.Theresultsoftheresistivecalculationsar eshownin Figures 1 and 2 where the damping per wavelength and the\ndamping length are plotted (see dashed lines) as a function o f\nthe frequency of the driver. The agreement between the resis -\ntive computationsand the analytical or semi-analyticalme thods\nis very good. The small di fferences are due to the fact that in\nthe analytical approximations we have assumed that the reso -\nnance is always located at r=Rand we have calculated the\nderivativeof the densityat thisposition.Thisexplainsth e small\ndeviations from the resistive estimations. A more precise d eter-\nminationcouldbedonecalculatingtheexactlocationofthe res-\nonance and then using a slightly modified version of Eq. (28),\nbutsincetheanalyticalapproximationsthatwehavealread yde-\nrived are quite satisfactory there is no pressing need to exp lore\nthisissuefurther.\n6. Conclusionsand discussion\nThe spatial damping due to resonant absorption of driven kin k\nwaveshasbeeninvestigated.Themainconclusionofthework is\nthatthedampinglengthofpropagatingkinkwavesdueresona nt\nabsorptionis a monotonicallydecreasingfunctionof frequ ency.\nThe TGV relation for kink waves was derived, demonstrating\nthat for low frequenciesthe dampinglength is exactly inver sely\nproportionalto frequency.In the high frequencyrangethe T GV\nrelationcontinuestobeanexcellentapproximationofthea ctual\ndependency of the damping length on frequency. Certainly, f or\nall physically relevant frequencies the dependency of damp ing\nlengthonfrequencyisaccuratelydescribedbytheTGVrelat ion.\nThis dependency means that resonant absorption is selectiv e as\nit favours low frequency waves and can e fficiently remove high\nfrequency waves from a broad band spectrum of kink waves.\nThis has great significance for solar atmospheric kink waves ,\nsince high frequency waves will tend to lose more power than\ntheir low frequency counterparts before reaching high alti tudes\ninthesolarcorona,withtheexactpercentagepowerlossdep end-\ning on the properties of the equilibrium, in particular the w idth\nof the non-uniformlayer and steepness in variation of the lo cal\nAlfv´ en speed. With respect to mode conversion, the process of\nresonantabsorptionwill causethe higherfrequencywavest obe\nattenuatedmorebecausetheglobalkinkmodewillbeconvert ed\ninto localised Alfv´ enic modes at lower heights. If the ener gy of\nthese Alfv´ enic motions is eventually dissipated, then res onant\nabsorptionshouldproduceacharacteristicdistributiono ftheen-\nergy as a functionof height in the solar atmosphere. This cou ld\nhaveimportantconsequenceswithrespecttothespatialdis tribu-\ntionofwaveheatingin thesolaratmosphere.\nIt was also shown that spatial andtemporaldampingare ba-\nsicallyequivalent.IntheTTapproximationthedampingper pe-\nriod and the damping per wavelength are exactly the same. The\ndifferencesin these two quantitiesarise in the regime where the\nTT is not valid, but even in this situation it is easy to relate the\nspatial and the temporal damping rates through the group and\nphase speeds of the kink MHD waves. This allows us to trans-\nlate the results from the temporally dampedwaves ( ωcomplex,\nkreal) to spatially attenuated waves ( ωreal,kcomplex) due to\nresonant absorption. This mechanism requires the frequenc y of\nthedrivertobebetweentheinternalandtheexternalAlfv´ e nfre-\nquencyofthetube.Thismightseemaveryrestrictivecondit ion,\nbut in fact it is just the opposite. In the driven problem the f re-\nquency is fixed but the system chooses the proper wavelength\n(along the waveguide) to accommodate the kink mode in the\ntube. This kink mode generated at the base of the loop propa-\ngates upwardsalong the tube and at the same time is attenuate d\nduetotheinhomogeneityat thetubeboundary.\n7Terradaset al.:Spatialdamping of propagating kinkwaves\nAn interesting result is that both the damping length in the\nspatial problem and the damping time in the temporal problem\narealwayssmallerthanintheTTapproximation(when f→0),\nmeaning that waves with short wavelengths or large frequen-\ncies are always more e fficiently damped.Note that the observa-\ntions of standing kink waves observedwith TRACE and for the\npropagatingkink waves detectedwith the CoMP instrumentar e\nprecisely in the regime where the TT is applicable, i.e., whe re\nthe waves are less a ffected byresonant absorption.Thefact that\novertonesof standing kink waves and high frequencypropaga t-\ning waves have proveddi fficult to detect may be a direct conse-\nquenceofthefilteringduetoresonantabsorption.Fromadi ffer-\nentperspective,wehavealsoshownthat thedampingperwave -\nlength (and the damping per period) has a weak dependenceon\nthefrequency.\nIt is necessaryto remarkthat ourresults are basedon a sim-\nple magnetic flux tube model, i.e., a straight cylinder, with no\ngravity and pressure. The only inhomogeneity in this model i s\nplasma density in the radial direction, however this a chara cter-\nistic property of solar waveguides that is observed at all at mo-\nspheric heights, e.g., chromospheric magnetic bright poin ts or\ncoronal loops. Thus the theory of resonant damping of propa-\ngating kink waves due to radial plasma density inhomogeneit y\noffersa natural explanationfor the dissipation observedby e. g.,\nTomczyk&McIntosh (2009). However, a detailed comparison\nbetweentheobservationsandtheexpectedfrequencydepend ent\nresponse due to resonant absorption is needed to quantify th e\nprecise spatial distribution of wave heating due to this mec ha-\nnisminthesolarplasma,e.g.,kinkwavedissipationasafun ction\nof both frequency and height. This problem will be the subjec t\nofafuturework(Verthet al.2010,inpreparation).\nAcknowledgements. 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Deutsch\nDepartment of Physics, University of California, Santa Cru z, CA 95064.\nA. Berger\nCIC Nanogune, Mikeletegi Pasealekua 56, 301 E-20009 Donost ia Spain\nWe show for the Ising model that is possible construct a discr ete time stochastic model analogous\nto the Langevin equation that incorporates an arbitrary amo unt of damping. It is shown to give\nthe correct equilibrium statistics and is then used to inves tigate nonequilibrium phenomena, in\nparticular, magnetic avalanches. The value of damping can g reatly alter the shape of hysteresis\nloops, andfor small dampingandhighdisorder, themorpholo gyoflarge avalanchescanbedrastically\neffected. Small damping also alters the size distribution of avalanches at criticality.\nPACS numbers: 75.40.Mg, 75.60.Ej, 05.45.Jn,\nI. INTRODUCTION\nIn many situations, it is useful to discretize continu-\nous degrees of freedom to better understand them, both\nfrom a theoretical standpoint and for numerical effi-\nciency. Ising models are perhaps the best example of this\nand have been the subject of numerous theoretical and\nnumerical studies. Renormalization group arguments1\nhave explained the reason why this discretization gives\nequilibrium critical properties of many experimental sys-\ntems, and these kinds of arguments have been extended\nto understanding their equilibrium dynamics2. For non-\nequilibrium situations, such as the study of avalanches,\nsuch arguments probably do also apply to large enough\nlength and time scales as well. However there are many\nsituations where it would be desirable to understand\nsmaller length scales where other factors should become\nrelevant.\nThis is particularly true with dynamics of magnetic\nsystems, where damping is often weak in comparison to\nprecessional effects. For studies of smaller scales, it has\nbeen necessary to use more time consuming micromag-\nnetic simulations utilizing continuous degrees of freedom,\nsuch as the Landau Lifshitz Gilbert equations3which is a\nkind of Langevin equation that gives the stochastic evo-\nlution of Heisenberg spins.\nds\ndt=−s×(B−γs×B), (1)\nwheresis a microscopic magnetic moment, Bis the local\neffective field, and γis a damping factor, measuring the\nrelative importance of damping to precession. In real\nmaterials it ranges4from small damping γ=.01, to 1.\nIn contrast, the dynamical rules implemented for Ising\nmodels are most often “relaxational” so that energy is\ninstantaneously dissipated when a spin flips, as with the\nMetropolis algorithm.\nHowever there is a class of “microcanonical” Ising dy-\nnamics5reviewed in section II where auxiliary degrees of\nfreedom are introduced and all moves conserve the total\nenergy. The other degrees of freedom can be taken to bevariables associated with each spin, and allowed moves\ncan change both the state of the spins and the auxiliary\nvariables. Thiscanbe thoughtofcrudely, asadiscretized\nanalogyto moleculardynamics, and is also similar to dis-\ncrete lattice gas models of fluids6,7. These models give\nthe correct equilibrium Ising statistics of large systems\nand can also be used to understand dynamics in a differ-\nent limit than the relaxational case.\nReal spin systems are intermediate between these two\nkindsofdynamicsand asmentioned above, arebetter de-\nscribed by Langevin dynamics. In the context of spins,\nthe question posed and answered here is: how does one\nformulate a discrete time version of stochastic dynamics\nthat includes damping and gives the correct equilibrium\nstatistics? In section IIA we are able to show that there\nis a fairly simple method for doing this using a combina-\ntion of microcanonical dynamics, and an elegant proce-\ndure that incorporates damping and thermal noise. This\nprocedure differs from that of the Langevin equation in\nthat it requires non-Gaussian noise. Despite this, the\nnoise has surprisingly simple but unusual statistics.\nWe will then showthat this proceduregivesthe correct\nequilibrium statistics and verify this numerical in section\nIIB with a simulation ofthe two dimensional Ising model\nwith different amounts of damping.\nBecause the value of damping is an important phys-\nical parameter in many situations it is important that\nthere is a straightforward way of incorporating its effects\nin Ising simulations. This is particularly noteworthy as\nIsing kinetics are a frequently used means of understand-\ning dynamics in many condensed matter systems.\nAfter this in section III we will turn to nonequilibrium\nproblems where, using this approach, we can study the\neffects of damping on a number of interesting properties\nof systems displaying avalanches and Barkhausen noise8.\nWe firstshowhowto modify the kinetics forthis caseand\nthen study systems in two and three dimensions. With\nmodest amounts of computer time, we can analyze prob-\nlems that are out of the reach of micromagnetic simula-\ntions and allow us to probe the effects of damping on the\nproperties of avalanches. This is related to recent work92\nby the present authors using both the Landau Lifshitz\nGilbert equation, Eqn. 1, and theoretical approaches, to\nunderstand how relaxational dynamics of avalanches10,\nare modified at small to intermediate scales by this more\nrealistic approach. With the present approach we find\nnew features and modifications of avalanche dynamics.\nWe find that the shape ofhysteresisloopscanbe strongly\ninfluenced by the amount of damping. One of the most\nstriking findings is that there exists a parameter regime\nof high disorder and small damping where single system-\nsize avalanches occur that are made up of a large number\nof disconnected pieces. We can also analyze the criti-\ncal properties of avalanches when damping is small and\ngive evidence that there is a crossoverlength scale, below\nwhich avalanches have different critical properties.\nII. NON-RELAXATIONAL DYNAMICS\nWestartbyconsideringamodelforamagnetwithcon-\ntinuous degrees of freedom, such as a Heisenberg model\nwith anisotropy. The Ising approximation simplifies the\nstate of each spin to either up or down, that is si=±1,\ni= 1,...,N. One important effect that is ignored by\nthis approximation is that of spin waves that allow the\ntransfer of energy between neighbors, and for small os-\ncillations, give an energy contribution per spin equal to\nthe temperature T(here we set kB= 1). This moti-\nvates the idea that there are extra degrees of freedom\nassociated with every spin that can carry (a positive)\nenergyei. Creutz introduced such degrees of freedom5\nand posited that they could take any number of dis-\ncrete values. He used these auxiliary variables eito\nconstruct a cellular automota to give the correct equi-\nlibrium statistics for the Ising model, in a very efficient\nway that did not require the generation of random num-\nbers. Thus we have a Hamiltonian Htotthat is the sum\nof both spin Hspinand auxiliary degrees of freedom He:\nHtot=Hspin+He.Hspincan be a general Ising spin\nHamiltonian and He=/summationtext\niei. In our model there is a\nsingle auxiliary variable eiassociated with each lattice\nsitei, that can take on any real value ≥0.\nHowever for the purposes of trying to model dynamics\nof spins, it also makes sense to allow the ei’s to interact\nand exchange energy between neighbors. For example,\none precessing spin should excite motion in its neighbors.\nThis exchange was formulated in the context of solidifi-\ncation using a Potts model instead of an Ising model by\nContiet al.11, but can equally well be used here.\nNow we can formulate a microcanonical algorithm for\nthe Ising model using a procedure very similar to their\nprescription. In each step:\n1. We choose a site iat random.\n2. We randomly pick with equal probability either a\nspin or an auxiliary degree of freedom, siofei:(a)si’s: We attempt to move spins (such as the\nflipping of a single spin). If the energy cost in\ndoing this is ≤eiwe perform the move and\ndecrease eiaccordingly. Otherwise we reject\nthe move.\n(b)ei’s: We pick a nearest neighbor j, and repar-\ntition the total energy with uniform probabil-\nity between these two variables. That is, af-\nter repartitioning, e′\ni= (ei+ej)rande′\nj=\n(ei+ej)(1−r), where 0 < r <1 is uniform\nrandom variable.\nNote that these rules preserve the total energy and the\ntransitions between any two states have the same proba-\nbility. Therefore this will give the correct microcanonical\ndistribution. For large N, this is, for most purposes12,\nequivalent to the canonical distribution ∝exp(−βHtot).\nNote that the probability distribution for each variable\nei,P(ei) =βexp(−βei), so that the ∝angbracketleftei∝angbracketright=T. That is,\nmeasurement of averageof ei’s directly gives the effective\ntemperature of the system.\nA. Extension To Damping\nThe question we asked, is how to extend this equilib-\nrium simulation method to include damping. In this case\nthe system is no longer closed and energy is exchanged\nwith an outside heat bath through interaction with the\nauxiliaryvariables. As with the Langevinequation, there\nare two effects. The first is that the energy is damped.\nCall the dissipation parameter for each step α, which will\nlie between 0 and 1. Then at each time step we lower\nthe energy with ei→αeifor all sites i. By itself, this\nclearly will not give a system at finite temperature and\nwe must also include the second effect of a heat bath,\nwhich adds energy randomly to the system. In the case\nof the Langevin equation, a Gaussian noise term n(t) is\nadded to keep the system at finite temperature. A dis-\ncretized version of this, that evolves the energy e(t) at\ntime step tis\ne(t+1) =αe(t)+n(t). (2)\nThis equation will not work if the noise n(t) is Gaussian\nas this does not give the Gibbs distribution Peq(e) =\nβexp(−βe). Therefore we need to modify the statistics\nofn(t). It is possible to do so if we choose n(t) at each\ntimetfrom a distribution\np(n) =αδ(n)+(1−α)βe−βnθ(n) (3)\nwhereθis the Heaviside step function. To show this, we\nwrite down the corresponding equation for the evolution\nof the probability distribution for e:3\nP(e′,t+1) =∝angbracketleftδ(e′−(αe+n))∝angbracketright=/integraldisplay /integraldisplay\nP(e,t)p(n)δ(e′−(αe+n))de dn (4)\nWe require that the Past→ ∞obeysP(e,t+ 1) =\nP(e,t) =Peq(e) =βexp(−βe), fore >0. It is easily ver-\nified thatby choosingthisform of P(e,t)and bychoosing\nP(n) as in Eq. 3, we satisfy Eq. 4.\nTherefore to add damping to this model, we add the\nfollowing procedure to the steps stated above:\n3. Chooseauniformrandomnumber0 < r <1. Ifr <\nα, thenei→αei. Otherwise ei→αei−Tln(r′),\nwherer′is another uniform random number be-\ntween 0 and 1.\nIf we assume that the probability distribution for the\ntotal system is of the form PGibbs∝exp(−βHtot) =\nexp(−βHspin)exp(−βHe), we will now show that the\nsteps 1, 2, and 3, of this algorithm preserve this distri-\nbution. Following the same reasoning as above for the\nmicrocanonical simulation, moves implementing steps 1\nand 2 do not change the total energy, and they preserve\nthe form of PGibbsbecause PGibbsdepends only on the\ntotal energy ( Htot), and 1 and 2 explore each state in\nan energy shell with uniform probability. Because of\nthe form of PGibbs, its dependence on the variable eiis\n∝exp(−βei). According to the above argument, after\nstep 3, it will remain unchanged. Therefore all steps in\nthis algorithm leave PGibbsunchanged. The algorithm is\nalsoergodic,and thereforethis willconvergetothe Gibbs\ndistribution13ast→ ∞.\nBecausethe stepseachpreservethe Gibbsdistribution,\nthe ordering of them is not important in preserving equi-\nlibrium statistics. For example, we could sweep through\nthe lattice sequentially instead of picking iat random.\nWe could perform step 3 after steps 1 and 2 were per-\nformedNtimes.\nB. Equilibrium Tests\nWe performed tests on this algorithm and verified that\nit did indeed work as expected. We simulated the two\ndimensional Ising model on a 1282lattice with differ-\nent values of the damping parameter, and compared it\nwith the exact results. The average magnetization per\nspinmis plotted in Fig. 1 as a function of the tempera-\ntureTand compared with the exact result14for large N\n(dashedcurve). The ×’sarethecase α= 1, whichisthen\njust an implementation of the microcanonical method11\ndescribed above. In this case, the temperature was ob-\ntained by measuring ∝angbracketleftei∝angbracketrightbecause the energy was fixed\nat the start of the simulation. The only point which is\nslightly off the exact solution is in the critical region, as\nis to be expected. The case α= 0.5 is shown with the\n+’s and lie on the same curve. Results were obtained forα= 0.9 but are so close as to be indistinguishable and\nare therefore not shown. We also checked that the distri-\nbution of auxiliary variables had the correct form. The\nprobability distribution for the energy eis shown in Fig.\n2. Fig. 2 plots the distribution P(ei) versus energy ei,\naveraged over all sites ion a linear-log scale for α= 0.5\nandT= 0.8, and 1.1. The curves are straight lines over\nfour decades and show the correct slopes, for T= 0.8,\n∝angbracketleftei∝angbracketright= 0.8002 and for T= 1.1,∝angbracketleftei∝angbracketright= 1.1003.\nFIG. 1: Plot of results obtained for the two dimensional Isin g\nmodel on a1282lattice for twodifferent values of the damping\nparameter. This is a plot of the average magnetization per\nspinmvs.T. The×’s are for no dissipation, α= 1, which is\na purely microcanonical simulation. The +’s are for α= 0.5.\nThe dashed curve is the exact solution to this model in the\nthermodynamic limit.\nIII. AVALANCHE DYNAMICS\nAvalanche dynamics of spin systems have been mainly\nstudied using models that are purely relaxational. There\nis a whole rangeof interesting phenomena that have been\nelucidated by such studies and have yielded very inter-\nesting properties. The simplest model that can be used\nin this context is the random field Ising model (RFIM)\nwith a Hamiltonian\nH=−/summationdisplay\n<ij>Jsisj−/summationdisplay\nihisi−h/summationdisplay\nsi(5)\nwhereJis the strength of the nearest neighbor coupling,\nhiis a random field, with zero mean, and his an exter-\nnally applied field. A magnet is placed in a high field h4\nFIG. 2: Plot of results obtained for the two dimensional\nIsing model on a 1282lattice for the probability distribution\nfor the auxiliary variables ei, at two different temperatures\nwith a damping parameter α= 0.5. The upper curve is for\nT= 1.1 and the lower for T= 0.8.\nandthen this isveryslowlylowered. As thishappens, the\nspins will adjust to the new field byflipping to lowertheir\nenergy. In the usual situation, the system is taken to be\natT= 0, so that only moves that lower the energy are\naccepted. The flipping of one spin can cause a cascade of\nadditional spins to flip, causing the total magnetization\nMto further decrease. The occurrence of these cascades\nis called an “avalanche”. At zero temperature there is\none parameter jthat characterizes the system, the ratio\nof nearest neighbor coupling to the distribution width of\nthe random field. One considers the behavior of a sys-\ntem when its starts in a high field and is slowly lowered.\nWhenjis small the system is strongly pinned and the\nsystem will have a number of small avalanches generat-\ning a smooth hysteresis loop. For large j, the system\nwill have a system-size avalanche involving most of the\nspins in the system, leading to a precipitous drop in the\nhysteresis loop. There is a critical value of jwhere the\ndistribution of avalanche sizes is a power law and self-\nsimilar scaling behavior is observed.\nHere we investigate how this is modified by adding\ndamping to these zero temperature dynamics according\nto the following rules:\n1. The field is slowly lowered by finding the next field\nwhere a spin can flip.\n2. The spins then flip, exchanging energy with auxil-\niaryvariables eias describedabove. The numberof\ntimes this is attempted is nmtimes the total num-\nber of spins in the system. Here we set nm= 16.\nIn more detail:\n(i)Spin moves: An attempt to move each spin\non the lattice is performed by attempting to\nflip sequentially every third spin, in order tominimize artifacts in the dynamics due to up-\ndating contiguous spins. (The lattice sites are\nlinearly ordered using “skew” boundary con-\nditions). Then all three sublattices are cycled\nover.\n(ii)Energy moves: Exchange of energy\nwith nearest neighbors is performed cycling\nthrough all directions of nearest neighbors.\nUsing the same sequence of updates, the ei’s\nexchange energy with their nearest neighbors\nin one particular direction.\n(iii)Dissipation: The energyofeach eiis lowered\ntoαei.\n3. We check for when the spins have settled down as\nfollows: if the ei’s are not all below some energy\nthreshold ethresh, set below to be 10−4, or the spin\nconfiguration has changed, step 2 is repeated until\nthese conditions are both met.\n4. When the spins have settled down, we go to step 1.\nThe parameters nmandethreshwere varied to check that\nthe correct dynamics were obtained. The larger α, the\nsmaller the dissipation and the larger the number of iter-\nations necessary to achieve the final static configuration.\nFIG. 3: The major branch of the descending hysteresis loop\nfor 642systems using different values of the damping parame-\nter and the spin coupling. Strong damping, α= 0.5 is shown\nin the left most curve (as judged from the top of the plot)\nfor coupling j= 0.3 which starts decreasing from M= 1 at\nh=−0.2, and does not have large abrupt changes. All the\nother curves are for weak damping, α= 0.99. In this case but\nalso forj= 0.3, we see that although Mstarts to decrease at\nthe same location as for strong damping, it drops abruptly as\nthe field is lowered. As the coupling jis decreased, smooth\ncurvesare eventuallyseenagain. Goinglefttoright, as jud ged\nfrom the top, are j= 0.3, 0.25, 0.2, and 0.15.5\n(a)\n (b)\nFIG. 4: (a) The spin configuration for a 2562system with j= 0.35,α= 0.9 during a system size avalanche at the field\nh=−0.400007. (b) A gray-scale plot of the auxiliary variables at t he same time.\n(a)\n (b)\n (c)\nFIG. 5: Spin configurations for a 2562system with j= 0.25,α= 0.99 during an avalanche at the field h=−7×10−5. (a)\nThe beginning of the avalanche. (b) When the avalanche is of o rder of half the system size. (c) The final configuration of the\navalanche.\nA. Two Dimensional Patterns\nWe first investigate the case of two dimensions where\nit is simpler to visualize the avalanches in various condi-\ntions than in three dimensions. Much experimental work\nand theoretical work on avalanches has been done on two\ndimensionalmagneticfilmsandthiscaseshouldbehighly\nrelevant10.\nWe first examine how the hysteresis loops change as\na function of the coupling jand the damping parame-\nterαfor a 642system. The major downwards hystere-\nsis loops are shown in Fig. 3 for a variety of param-\neters described below. We first examine strong damp-\ningα= 0.5. Forj= 0.3 the hysteresis curve is quite\nsmooth with all avalanches much less than the systemsize (left most curve). Now consider the same value of j\nbut with with small damping, α= 0.99. The curve now\nis a single downwards step with a small tail at negative\nh. The lower damping has allowed that system to form\na system size avalanche. The difference is due to the\nfact that with small damping, the energy of avalanched\nspinsis not immediately dissipatedand asaconsequence,\nheats up neighboring spins, allowing them to more easily\navalanche as well. Therefore a system size avalanche is\nseenin the smalldampingcase, leadingtothe precipitous\ndrop in the hysteresis loop.\nWhen the value of the coupling jis lowered to 0 .15 for\nα= 0.99, smooth loops are obtained. The Fig. 3 shows\nintermediate values of the coupling parameter as well.\nTo better understand the reason why the energy of6\nthe auxiliary variables can trigger further spins to flip, in\nFig. 4 we show the state of a system during a system size\navalanche for j= 0.35 and a moderately small damping\nvalue,α= 0.9,withh=−0.400007. Fig. 4(a)showsthat\nthe flipped spins form a fairly compact cluster and Fig.\n4(b) shows the corresponding values of the ei’s in a gray\nscale plot, suitably normalized. It has the appearance of\na halo around the growth front of the avalanche. The\nspins in the growth front have just flipped and so energy\nthere has not had a chance to diffuse or dissipate and\nso has a higher spin temperature. The interior is cold\nbecause damping has removed energy from the auxiliary\ndegreesoffreedom. This highertemperature diffuses into\nthe the unflipped region allowing spins to flip by thermal\nactivation.\nBecause large avalanches are possible for small damp-\ning in a parameter range where the relative effect of the\nrandom field is much larger, it is of interest to see if\navalancheshaveadifferentmorphologythan typicallarge\navalanches for high damping systems. Fig. 5 shows such\nspin configurationsfirst at the beginning of the avalanche\nandfurtheralongduringpropagationwhenithasreached\nroughly half the system size, and finally when it has\nreached its final configuration and the maximum aux-\niliary variable value is <4×10−4. The morphology of\nthisisverydifferentthanwhatisseenforlargeavalanches\nwith strongercoupling, for exampleFig. 4. At verysmall\nfields, in this figure h=−7×10−5, surface tension pre-\ncludes the formation of minority domains, but because\ndisorder is large, there will be many small regions where\nthe local field is much stronger and these will want to\nform downward oriented (black) domains. There is a fi-\nnite activation barrier to forming these that can only be\novercome at finite temperature. However the majority\nof the spins still strongly disfavor flipping. But because\ndamping is small, heat has a chance to diffuse through\nthese regions into the favorable regions, allowing discon-\nnected regions to change orientation by thermal activa-\ntion. Note that we have checked numerically that small\ndamping with strong coupling also leads to compact con-\nfigurations, so disorder is an essential ingredient in this\nnew morphology.\nB. Three Dimensions\nWe first check that as with two dimensions, the value\nof the damping parameter can have a large effect on the\nshape of a hysteresis loop. Fig. 6 shows the downward\nbranches of the major hysteresis loop when the only pa-\nrameter that is changed is the damping, α. The system\nis a 323lattice with j= 0.19. A value for high damping,\nα= 0.5, is the upper line. The lower line is for small\ndamping with α= 0.99.\nA more subtle effect, is that of damping on what hap-\npens near criticality. In this case the value of the criti-\ncaljwill depend on the value of αas is apparent from\nthe results of Fig. 6. At this point, the distribution ofFIG. 6: The major branch of the descending hysteresis loops\nin two 323systems with j= 0.19, for two different values of\nthe dissipation, upper curve: α= 0.5, lower curve: α= 0.99.\navalanchesizesisexpected tofollowapowerlawdistribu-\ntion for large sizes. We located this point and examined\nsystem properties in this vicinity. Fig. 7 shows examples\nof such runs for 323systems. Fig. 7(a) shows a plot of\nthe magnetization per spin M, versus the applied field h\nforj= 0.165 and j= 0.167. For larger values of j, the\navalanches rapidly become much larger as is seen in Fig.\n6, and for smaller values, avalanches all become small.\nFig. 7(b) shows a plot of the same quantity with relax-\national dynamics near criticality. The avalanches take\nplace over a much smaller range in applied field.\nTo quantify this difference, we studied the avalanche\nsize distribution exponent that is obtained by calculating\nthe distribution of avalanche sizes over the entire hys-\nteresis loop. This was studied by averaging avalanches of\nmanyruns, (200for α= 0.99)for 323systemsand fordif-\nferentvaluesofparameters. Weshowacomparisonofthe\navalanche size distribution for α= 0.99, shown with +’s\nand forα= 0.9, shown with ×’s in Fig. 8. For α= 0.99\nthe curve fits quite well to a power law with an exponent\nof−1.4±.1 as shown in the figure. For purely relax-\national dynamics, the same exponent has been carefully\nmeasured15to be 2.03±.03 (which is consistent with our\nresults for relaxational dynamics on much smaller sys-\ntems than theirs). With smaller damping we expect to\nhave a crossover length corresponding to the length scale\nassociated with the damping time, above which the dy-\nnamics should appear relaxational. α= 0.9 appears to\nshowsuchacrossoverfromaslope ofapproximately −1.4\nfor small avalanches, to a higher slope for large ones. A\nlinewithslopeof −2isshownforcomparisonandappears\nto be consistent with this interpretation.7\n(a) (b)\nFIG. 7: (a) Magnetization versus field for the Ising model wit h damping described in the text. The system size is 323and\nthe two lines represent two runs close to criticality, one wi th a coupling of j= 0.165 and the other of 0 .167. (b) The plot for\nrelaxational dynamics (large damping) with couplings of .21 and.212.\nFIG. 8: The avalanche size distribution, measured of the\nentirehysteresisloopfor α= 0.99(+symbols)and α= 0.9(×\nsymbols). The x-axis is the number of avalanches normalized\nby it’s mean size. The y-axis is the normalized distribution\nof sizes. The less negative sloped straight line is a fit of the\nα= 0.99 curve and has a slope of −1.4. The more strongly\nsloped one has a slope of −2.\nIV. DISCUSSION\nThis paper has introduced a new set of dynamics for\nIsing models that incorporates damping in a way that\nhas not before been achieved. The dynamics that have\nbeen devised have a lot in common with Langevin dy-\nnamics, except they are for discrete rather than contin-\nuous systems. In Langevin equations, a continuous set\nof stochastic differential equations are used to model a\nsystem. It differs from molecular dynamics in that ther-\nmal noise and damping are both added so that the sys-\ntem obeys the correct equilibrium statistics. In the case\nstudied here, we start by considering microcanonical dy-namics5,11whichintroduces auxiliarydegreesoffreedom.\nWe then add damping and thermal noise. Whereas the\nthermal noise is typically Gaussian in the case of the\nLangevin equation, here it must be taken to be of a spe-\ncial exponential form, Eq. 3, in order for it to satisfy the\ncorrect equilibrium statistics.\nThe form of this noise, although quite unusual, can\nbe understood, to some extent qualitatively. For large\ndamping, or small α, the strength of the δfunction be-\ncomes small, and the effect is dominated by the second\ntermwhichis ∝exp(−βn)(forpositive n). Althoughthis\nisnon-Gaussian, ncanbethoughtofasarandomamount\nof positive energy. In the Langevin equation, noise is of-\nten added to a velocity degree of freedom. In terms of\na velocity, the exponential form that we have obtained\nwould correspond to a Gaussian if this was expressed in\nterms of a velocity instead. In the limit of small damp-\ning, where αis close to 1, the effect of the noise becomes\nsmallbecausethefirstterm, whichisto addnonoise, will\ndominate the distribution. This is in accord with what\nhappens in the Langevin equation where if dissipation is\nsmall, little thermal noise is needed to keep the system\nat a given temperature.\nThe fact that it is possible to model damped systems\nin this discrete manner should have many useful applica-\ntions, and is easily extended to other kinds of systems,\naside from Ising models, especially in applications where\ncomputational efficiency is an important criterion.\nThe case of avalanches in magnetic systems is an in-\nteresting nonequilibrium use of these dynamics. Al-\nthoughonemightexpectthatinmostsituations,forlarge\nenough distance and time scales, finite damping will be\nunimportant, physics at smaller scales is still of great in-\nterest and effects at those scales can propagate to larger\nscales. Because damping in real materials can be quite\nsmall, their effects are readily observable experimentally.\nThis work is expected to be important at intermediate\nscales. We haveinvestigatedthephenomenonseenin this\nmodel with varying degrees of damping and found that8\nit makes a qualitative difference to many of the features\nseen on small and intermediate scales. This workis by no\nmeans exhaustive and there are many other effects that\ncan be investigated by straightforward extensions. The\neffect of dipolar interactions in conjunction with damp-\ning could alsobe explored. We havechosento update the\nspinandauxiliaryvariablesatequalfrequencies. Varying\nthis should lead to a different value for the heat diffusion\ncoefficient which should change the quantitative values\nfor length and time scales.\nThe phenomena we have found was in qualitative\nagreement with earlier work using the Landau Lifshitz\nGibbs equations9. As avalanches progress, the effective\ntemperature, which we have seen can be quantified by\n∝angbracketleftei∝angbracketrightat sitei, will increase as energy is released. Thisenergy then diffuses to the surrounding regions, giving\nthosespins the opportunity tolowertheir energyby ther-\nmal activation. This allows avalanches to more easily\nprogress when the damping is small in contrast to relax-\national dynamics, which has effectively infinite damping,\nα= 0. This can lead to some substantial differences in\navalanche morphology, particularly as for small damp-\ning, highly disordered systems can avalanche. At low\nfields this leads to a single avalanche being composed of\nmany disconnected pieces. Experiments have been de-\nvised16that are close experimental realization of the two\ndimensional random field Ising model, and it would be\ninteresting to determine if systems such as this one, or\nsimilar to it, show avalanches with this morphology.\n1S.K.Ma, “Modern Theory of Critical Phenomena”, Fron-\ntiers in Physics, No. 46, Perseus Books (1976).\n2B.I. Halperin, P.C. Hohenberg, and S.K. Ma, Phys. Rev.\nB10, 139 (1974).\n3F.H. de Leeuw, R. van den Doel and U. Enz, Rep. Prog.\nPhys.43, 689 (1980).\n4Q.PengandH.N.Bertram, J. Appl.Phys. 81, 4384 (1997);\nA. Lyberatos, G. Ju, R.J.M. van de Veerdonk, and D.\nWeller, J. Appl. Phys. 91, 2236 (2002).\n5M. Creutz, Ann. Phys 16762 (1986).\n6U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett.\n56, 1505 (1986).\n7G. Zanetti, Phys. Rev. A 40, 1539 (1989).\n8H.Barkhausen, Z. Phys. 20, 401 (1919); P.J. Cote and\nL.V. Meisel, Phys. Rev. Lett. 671334 (1991); J.P. Sethna,\nK.A. Dahmen and C.R. Myers Nature 410242(2001); B.\nAlessandro, C. Beatrice, G. Bertotti, and A. Montorsi, J.\nAppl. Phys. 682901 (1990); ibid.682908 (1990); J.S. Ur-\nbach, R.C.Madison, andJ.T. Markert, Phys.Rev.Lett. 75\n276 (1995); O. Narayan, Phys. Rev. Lett. 773855 (1996);S. Zapperi, P. Cizeau, G. Durin, and H.E. Stanley, Phys.\nRev. B586353 (1998);\n9J.M. Deutsch and A. Berger, Phys. Rev. Lett. 99, 027207\n(2007)\n10J.P. Sethna, K.A. Dahmen and O. Perkovic, “The Science\nof Hysteresis II” , edited by G. Bertotti and I. Mayergoyz,\nAcademic Press, Amsterdam, 2006, p. 107-179.\n11M. Conti, U. M. B. Marconi and A. Crisanti, Europhys.\nLett.47338 (1999).\n12M. Lax, Phys. Rev. 971419, (1955).\n13J.P. Sethna “Statistical Mechanics Entropy, Order Param-\neters and Complexity” Page 170. Oxford University Press\n(2006),\n14B.M. McCoy and T.T. Wu, “The Two-dimensional Ising\nModel” Harvard University Press, Cambridge, MA, 1973.\n15O.Perkovi´ c., K.Dahmen, J. Sethna.Phys.Rev.B 59, 6106\n(1999).\n16A. Berger, A. Inomata, J. S. Jiang, J. E. Pearson, and S.\nD. Bader, Phys. Rev. Lett. 854176 (2000);"    },    {        "title": "0801.3437v2.Damped_Bloch_Oscillations_of_Bose_Einstein_Condensates_in_Disordered_Potential_Gradients.pdf",        "content": "arXiv:0801.3437v2  [cond-mat.other]  7 Apr 2008Damped Bloch Oscillations of Bose-Einstein\nCondensates in Disordered Potential Gradients\nS. Drenkelforth1, G. Kleine B¨ uning1, J. Will1, T. Schulte1,\nN. Murray1, W. Ertmer1, L. Santos2, and J.J. Arlt1\n1Institut f¨ ur Quantenoptik, Leibniz Universit¨ at Hannover, Welfe ngarten 1,\nD-30167 Hannover, Germany\n2Institut f¨ ur Theoretische Physik, Leibniz Universit¨ at Hannover , Appelstraße 2,\nD-30167 Hannover, Germany\nE-mail:drenkelforth@iqo.uni-hannover.de\nAbstract. We investigate both experimentally and theoretically disorder induce d\ndamping of Bloch oscillations of Bose-Einstein condensates in optical lattices. The\nspatially inhomogeneous force responsible for the damping is realised by a combination\nof a disordered optical and a magnetic gradient potential. We show t hat the\ninhomogeneity of this force results in a broadening of the quasimome ntum spectrum,\nwhich in turn causes damping of the centre-of-mass oscillation. We q uantitatively\ncompare the obtained damping rates to the simulations using the Gro ss-Pitaevskii\nequation. Our results are relevant for high precision experiments o n very small forces,\nwhich require the observation of a large number of oscillation cycles.Damped Bloch Oscillations 2\n1. Introduction\nThe ability to realise ultracold quantum gases in periodic and disordere d potentials has\nenabled detailed studies of fascinating effects originating in solid stat e physics. Ongoing\ninvestigationsofsingleparticlephenomenasuchasAndersonlocalisa tionaswell asmany\nparticle effects like the Bose-Glass phase and the Mott insulator [1, 2 , 3, 4, 5, 6] show\nthe variety of possibilities these ensembles offer. Especially the non- intuitive dynamics\nof quantum gases in periodic potentials is of interest for theoretica l and experimental\ninvestigations, since quantum gases have enabled the first direct o bservation of Bloch\noscillations [8] in tilted periodic potentials [9, 10, 11]. In these system s the periodicity\nleads to an oscillatory motion instead of a linear acceleration of the pa rticles subjected\nto an external force.\nIn solid state systems scattering at imperfections of the crystal structure leads to\ndamping of Bloch oscillations on timescales much shorter than the osc illation period\nitself. Therefore Bloch oscillations of electrons are only observable in semiconductor\nsuper lattices [12], where the large spatial period leads to high oscillat ion frequencies,\nwhich are faster than the damping. Optical lattices on the other ha nd constitute perfect\noptical crystals and allow for the observation of long lived Bloch oscilla tions [13, 14, 15,\n16]. The experimental control of lattice parameters such as lattic e depth and spacing,\nthe possibility to detect the atomic cloud with absorption imaging and t he very small\nmomentum spread of Bose-Einstein condensates (BEC) have enab led detailed studies\nof this quantum effect.\nAcomparisonof thesesystems gives riseto thequestion howtheco ntrolled addition\nof disorder to an optical lattice will affect the dynamics of particles in such a periodic\npotential. Disorder can be realised with additional optical potentials [17, 18, 19, 2, 20,\n21, 22, 5, 7], impurity atoms [23, 24] or the roughness of the trapp ing potential close to\nthe surface of atomchips [25]. The simultaneous applicationof a homo geneous forceand\nthe optical disorder potential constitutes a spatially inhomogeneo us acceleration. This\ninhomogeneity can have important consequences for theapplicatio n of Bloch oscillations\nas a sensitive tool for high precision measurements of small forces [15, 26, 27, 14, 13],\nsince it leads to a dephasing of the quasimomentum andthus to a damp ing ofthe centre-\nof-mass oscillation [28]. Therefore a detailed quantitative understa nding of the effect of\nthe disorder and the underlying mechanism is indispensable for futur e applications.\nWe investigate the effect of a small disordered potential on Bloch os cillations of\nBose-Einstein condensates in a 1D optical lattice potential. Our res ults indeed show\nthat the inhomogeneity leads to significant damping of the centre-o f-mass motion. The\ndamping rate increases with disorder depth and is quantitatively com pared to numerical\nsimulations using the Gross-Pitaevskii equation (GPE). Furthermo re we show that the\ndisorderinducedbroadeningofthequasimomentumdistribution, wh ichistheunderlying\nmechanism for the damping of Bloch oscillations, reduces the fractio n of atoms in the\nBEC.Damped Bloch Oscillations 3\n2. Bloch oscillations in periodic potentials\nThe acceleration of particles in periodic potentials leads to an oscillato ry motion instead\nof a linear increase in velocity. Since this is a pure single particle effect, it is possible\nto describe the underlying physics in a 1D model, while a quantitative an alysis of the\neffects of disorder and interactions require a 3D description, as re ported in a previous\nwork [28]. We briefly discuss the main features necessary for the un derstanding of Bloch\noscillations while a comprehensive review can be found in [29].\nThe periodicity of the potential implies that the eigenfunctions obey the same\ntranslational symmetry as the potential\nφ(z+d) =eiqdφ(z), (1)\nwhere the phase difference qfrom site-to-site is called quasimomentum and dis the\nlattice constant. Since the eigenfunctions are periodic, it is possible to restrict the\ndescription of the dynamics to the first Brillouin zone [-π\nd,π\nd]. According to the\nacceleration theorem q(t) =q(0) +F\n¯htan additional potential gradient will cause\nthe quasimomentum to evolve linearly in time, because the energy offs et from site-\nto-site leads to a linear increase of the phase difference from site-t o-site during the time\nevolution. In combination with the periodicity of the band structure this causes an\noscillatory motion, since the group velocity is proportional to the de rivative of the band\nstructure. The oscillation period T= 2π¯h/F ddepends on the applied force Fand the\nlattice constant. The resulting amplitude zBO= ∆/2|F|is given by the width of the\nfirst band ∆ and the force. Since this amplitude is only a few micrometr es for typical\nexperimental parameters, theBlochoscillations areconveniently a nalysed inmomentum\nspace via time-of-flight (TOF) absorption imaging.\nWhile the oscillation can be described in a simple 1D model, the complex dyn amics\nin a disordered potential gradient and the role of interactions have to be analysed with\na full 3D model. All numerical simulations presented in this work are ob tained using\nthe 3D Gross-Pitaevskii equation\n/bracketleftBigg\n−¯h2\n2m∇2+VL(z)+VMF(r)+Vgrad(z)+g|Ψ(r)|2/bracketrightBigg\nΨ(r) =µΨ(r).(2)\nThe cylindrically symmetric magnetic trapping potential is VMF(r) =1\n2mω2\nρρ2+\n1\n2mω2\nzz2with trapping frequencies ωρandωz.The optical potential is VL(z) =\nsErcos2(kz), where sis the lattice depth in units of the recoil energy Er= ¯h2k2/2m\nandk= 2π/λis the wave vector of the optical lattice. The additional homogeneo us\ngradient potential is given by Vgrad(z) =F z.\nDue to interactions between the atoms, Bloch oscillations of Bose-E instein\ncondensates suffer, contrarytothecaseoffermions[13], from dynamical instabilities [31,\n32]. Intheouter halfof theBrillouin zone thenonlinear coupling leads t o anexponential\ngrowth of small perturbations. Independent of inhomogeneities in the potential they\nare responsible for a damping of the Bloch oscillations. Therefore it is necessary to\nreduce the dynamical instability in order to investigate the effect of the disorder on theDamped Bloch Oscillations 4\n0\n10\n18Bloch oscillation time [ms]\nx[ȝm]0 -400 400\nFigure 1. Absorptionimages after aTOF of 30ms, for varyingBloch oscillationt imes.\nThe lattice depth was 2 Erand the acceleration 2 .4m/s2.\nBloch oscillations. This can be accomplished with a combination of a high p otential\ngradient [35] and a reduction of the nonlinear interactions. In rece nt work long lived\nBloch oscillations of Bose-Einstein condensates were realised [16, 14 ] by decreasing\nthe s-wave scattering length with a Feshbach resonance, essent ially turning off the\natomic interaction. In the experiments reported here the interac tions were reduced\nby decreasing the density of the BEC, to enable the observations o f disorder induced\ndamping of Bloch oscillations.Damped Bloch Oscillations 5\n/s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s45/s49/s50/s48/s45/s49/s48/s48/s45/s56/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s99/s101/s110/s116/s114/s101/s45/s111/s102/s45/s109/s97/s115/s115/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s91/s181/s109/s93\n/s66/s108/s111/s99/s104/s32/s111/s115/s99/s105/s108/s108/s97/s116/s105/s111/s110/s32/s116/s105/s109/s101/s32/s91/s109/s115/s93\nFigure 2. Centre-of-mass position of Bose-Einstein condensates after a T OF of 30ms,\nas a function of the Bloch oscillation time. The parameters were ident ical to Fig. 1.\n3. Experimental realisation\nThe experiments were performed with87Rb Bose-Einstein condensates in the F =\n2,mF= 2 state. A detailed description of our apparatus is given in [33]. Near ly\npure BEC of up to N= 3×105Atoms are produced, however all experiments described\nhere were carried out with N= 5×104atoms to reduce the interaction energy, and to\nensure that no significant thermal background is present. After production of the BEC\nthe magnetic offset field was adiabatically increased to 91G within 440m s, thus lowering\nthe radial trapping frequency to ω⊥= 2π·29Hz. This reduces the interaction energy\nby a factor of 6.25, which in combination with low atom numbers sufficien tly inhibits\nthe dynamical instability, such that up to four undamped Bloch oscilla tion periods\ncan be observed. The following procedure was used to observe the Bloch oscillations.\nAfter decreasing the radial trapping frequency the intensity of t he optical lattice was\nadiabatically increased to its final value within 60ms. Subsequently th e atoms were\nsubjected to either a homogeneous potential gradient or the spa tially inhomogeneous\none for a variable time. Finally all potentials were turned off at the sam e time and the\natomic cloud was detected after a time-of-flight of 30ms by absorp tion imaging.\nThe 1D optical lattice is provided by a standing light field at a wavelengt h of\n825nm, which is superimposed on the axial direction of the magnetic t rap with a waist\nofω0= 140µm at the position of the atoms. The investigations were performed a t a\nlattice depth of 2 Er, since low lattice depths lead to a large width of the energy band\nand therefore to a high maximal group velocity. This results in an osc illation amplitudeDamped Bloch Oscillations 6\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s45/s56/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s32/s32/s32/s99/s101/s110/s116/s114/s101/s45/s111/s102/s45/s109/s97/s115/s115/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s91/s181/s109/s93\n/s66/s108/s111/s99/s104/s32/s111/s115/s99/s105/s108/s108/s97/s116/s105/s111/s110/s32/s116/s105/s109/s101/s32/s91/s109/s115/s93\nFigure 3. Centre-of-mass oscillation position of a BEC for a lattice depth of 2 Er, an\nacceleration of 2 .4m/s2and disorder depths of 0 Er(black, solid line), 70 ×10−3Er\n(red, dotted line) and 130 ×10−3Er(blue, dashed line), obtained from a numerical\nsolution of the GPE.\nof thecentre-of-mass motionof 80 µmafter a TOF of30ms, which caneasily bedetected,\nwhile the Bloch oscillations are not affected by Landau-Zener tunnellin g [34].\nThehomogeneouspotentialgradientisprovidedbymagneticcoilsina nti-Helmholtz\nconfiguration, which produce gradients of up to 3 .7G/cm. This corresponds to an\nacceleration of 2 .4m/s2.\nThe inhomogeneity is realised by a disordered optical dipole potential, generated by\nimaging a randomly structured chrome substrate radially onto the B EC, as described\nin a previous publication [2]. The correlation length of the disorder is 8 µm and its\ndepth was varied between 0 and 135 ×10−3Er, where the depth is defined as twice the\nstandard deviation analogue to [18].\nFigure 1 shows absorption images of Bose-Einstein condensates un dergoing Bloch\noscillations after 30ms TOF for a lattice depth of 2 Erwithout disorder. One clearly\nrecognises the oscillation as a motion of the central peak and the pe riodic appearance\nof a second peak replacing the main one during an oscillation cycle. Figu re 2 shows the\ncentre-of-mass oscillation as well as a theoretical prediction base d on a band structure\ncalculation without free parameters. The position of the Bose-Eins tein condensate zBO\nwas calculated by numerically summing the weighted axial positions zBO=/summationtext\nzN(z)\nNtotalz.\nThe measured Bloch period was 4 .5ms and the oscillation amplitude was 73 µm which\nis in good agreement with the theoretical calculations of 4 .66ms and 80 µm for an\nacceleration of 2 .4m/s2and a lattice depth of 2 Er.Damped Bloch Oscillations 7\n4. Damped Bloch oscillations\nFigure 4. Centre-of-mass position of Bose-Einstein condensates perform ing damped\nBloch oscillations. The disorder depths were 35, 105 and 135 ×10−3Er, from top\ndown. A fit to the data (solid red line) and the Gaussian envelope (das hed black line)\ndue to the damping are also shown. Note that the recorded oscillatio n time is reduced\nfor increased disorder depth since the broadening of the quasimom entum spectrum\ncauses a strong reduction of the contrast (see Fig. 8). This signifi cantly reduces the\nsignal-to-noise ratio of the absorption images.\nThe theory of disorder induced damping of Bloch oscillations was inves tigated in\ndetail in a previous publication [28] and we only briefly review the import ant features\nfor the interpretation of the experimental results.\nFor the analysis of Bloch oscillations in an inhomogeneous potential gr adient theDamped Bloch Oscillations 8\nFigure 5. Damping coefficient 1 /σof the centre-of-mass oscillation in the disordered\nlattice potential. The damping coefficients were obtained by applying t he same\nfit procedure to the experimental data of Fig. 4 and the results of the numerical\nsimulations. The red dots represent the experimental data and th e black dots the\nsimulations. Similar to the experimental case, the number of Bloch os cillation periods\nused in the fits to the simulations was reduced for increasing disorde r depth. The\nshadedareacorrespondstoatomnumbersusedinthesimulationsr angingfrom3 .5×104\nto 6.5×104.\nGPE (equation 2) has to be modified by expanding the acceleration te rm to\nVgrad=F z+Vdis(z), (3)\nwhereVdis(z) denotes the additional optical potential which constitutes the d isorder.\nFigure 3 shows the centre-of-mass position obtained from numeric al simulations of\ndampedBlochoscillationsforatypical disorderedpotential usedint heexperiment. One\nclearly recognises that the damping of the oscillations strongly depe nds on the disorder\ndepth. The damping can be understood qualitatively in terms of the e volution of the\nphase difference from site-to-site. For an undisturbed homogene ous potential gradient\nit develops in time according to\n∆φ(t) =δE\n¯ht, (4)\nwhereδEis the energy offset and ∆ φthe time dependent phase difference between\nneighbouring sites.\nThe disorder gives rise to a spatially varying energy difference from s ite-to-site\nδE(z). Therefore the phase evolution and the quasimomentum vary acr oss the lattice.\nThis broadening of the quasimomentum spectrum causes the dampin g of the Bloch\noscillations, since each qcorresponds to one group velocity.Damped Bloch Oscillations 9\n0\n1\n2\n3\n4\n5\n-2 ħk 0 2 ħk Bloch oscillation time [ms]\nmomentum\nFigure 6. Grey scale plot of the evolution of the momentum distribution during a\nBloch oscillation for a disorder depth of 105 ×10−3Er. The lattice depth and the\nacceleration were identical to the case of Fig. 3.\n0\nx[µm]-400 400\n0\np[ k]ħ-2 2 0\np[ k]ħ-2 20\nx[µm]-400 400\nFigure 7. Absorption images of Bose-Einstein condensates for an oscillation t ime of\n0.5 Bloch periods at an acceleration of 2 .4m/s2and a lattice depth of 2 Erafter a\nTOF of 30ms. The left column shows the case without added disorder , whereas the\nright column corresponds to a disorder depth of 105 ×10−3Er. The top row shows the\nabsorption images, the middle row corresponds to the associated a xial density profiles\nand the bottom row contains the momentum spectrum obtained fro m our simulation.\nFigure 4 shows the experimental observation of disorder induced d amping of Bloch\noscillations for various disorder depths. Note that the lattice dept h and the acceleration\nare identical to the undamped oscillation in Fig. 2. The graph clearly sh ows the distinct\nreduction of the oscillation amplitude for increased disorder. The so lid lines are a fit\nto the data, with the damping coefficient and the periodicity as free p arameters. We\ngenerate the fit function for the damped Bloch oscillation zDBO(t) by multiplying theDamped Bloch Oscillations 10\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s32/s32/s78\n/s102/s114/s97/s99\n/s66/s108/s111/s99/s104/s32/s111/s115/s99/s105/s108/s108/s97/s116/s105/s111/s110/s32/s116/s105/s109/s101/s32/s91/s109/s115/s93\nFigure 8. Fraction of atoms in the BEC after a time-of-flight of 30ms as a func tion\nof the Bloch oscillation time. Shown are four depths of the disorder p otential of 0 Er\n(black circles), 35 ×10−3Er(red squares), 70 ×10−3Er(blue dots), and 135 ×10−3Er\n(green triangles). The curves correspond to a fit with an exponen tial decay. Residual\nimaging effects (see figure 7) result in an overestimate of the total number of atoms.\nTherefore Nfracis 0.5 even for pure BEC without any discernible thermal fraction.\ntime evolution of the undamped oscillation zBO(t), obtained from the band structure\ncalculation, with a Gaussian envelope\nzDBO(t) =A0e−t2/σ2zBO(t). (5)\nTheshapeoftheenvelopeisaconsequenceofthebroadeningofth equasimomentum\ndistributionintime. Thewidthofthisdistributiondetermines theoscilla tionamplitude,\nwhich is reduced compared to the single particle picture, because se parate parts of\nthe ensemble simultaneously undergo different phases of the oscillat ion. Based on the\nassumption that the width of this distribution increases linearly in time , a Gaussian\nenvelope of the damping amplitude is expected [36, 30].\nTo quantitatively compare the experimental data to the numerical solutions of the\nGPE, we show the resulting damping coefficients as a function of the d isorder depth\nin Fig. 5. The parameters of the simulations correspond to the expe rimental ones\nfor a typical realisation of the disordered potential. Since the damp ing rate strongly\ndepends on the depth of the disorder potential, the horizontal er ror bars represent\nan experimental uncertainty in this depth of 25%. This was estimate d by evaluating\nthe depth of the used disorder potential at different positions, wh ile small deviations\nbetween theexact shape ofthedisorder potential intheexperime nt andinthe numerical\nsimulation were not accounted for. The shaded area corresponds to an uncertainty inDamped Bloch Oscillations 11\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s32/s32/s91/s109/s115/s93\n/s68/s105/s115/s111/s114/s100/s101/s114/s32/s100/s101/s112/s116/s104/s32/s91/s49/s48/s45/s51\n/s32/s69\n/s114/s93\nFigure 9. The decay times τof Fig. 8 are shown as a function of the disorder depth.\nStrong reduction is observed even for moderate disorder depths .\nthe atom number of 30%, accounted for in the simulations. Within this uncertainty, we\nobserve good agreement between experimental and theoretical values of the damping\ncoefficients.\nTo show, that the broadening of the quasimomentum spectrum is th e underlying\nmechanism of the damping, we analyse the experimental data with a s econd approach.\nSince the dephasing of the quasimomenta increases the number of a toms with varying\nmomenta, the number of atoms in the BEC peaks is reduced [35]. This picture is\nconfirmed by Fig. 6, which shows numerical results for the time evolu tion of the\nmomentum spectrum in the presence of a disordered potential gra dient. The disorder\ninduced dephasing leads to an increased blurring of the sharp momen tum distribution,\nas the quasimomenta undergo a non-uniform evolution.\nThis behaviour is consistent with the absorption images shown in Fig. 7 . In the\nundisturbed case sharp BEC peaks are visible whereas the disorder ed case exhibits\na clearly discernible background and a broadening of the peaks due t o the dephased\nquasimomenta.\nTo analyse this effect quantitatively, we estimate the number of ato ms in the BEC\nNBECat different times of the Bloch oscillation by fitting Thomas-Fermi pro files to\nthe characteristic peaks of the BEC. The fraction of atoms in the B EC is calculated\nby comparing the number of atoms in these peaks with the total ato m number\nNfrac=NBEC/Ntotal.\nFigure 8 shows this fraction as a function of the Bloch oscillation time f or variousDamped Bloch Oscillations 12\ndepths of thedisorder potential. The lines arefitsto thedata witha nexponential decay.\nA clear reduction in the occupation of the BEC peaks during the evolu tion of the Bloch\noscillation is visible. In all cases the dephasing of the quasimomenta pr ecedes the onset\nof the damping shown in Fig. 4. This is in agreement with a Gaussian shap e of the\ndamping envelope given in Eqn. 5 and confirms that the dephasing of t he quasimomenta\nis the underlying mechanism for the damping of the centre-of-mass motion. Hence only\na significant broadening of the quasimomentum leads to a relevant da mping.\nNote that even at a disorder depth of 0 Era reduction and therefore a dephasing\nis observed due to the interparticle interactions, while the centre- of-mass oscillation in\nFig. 4 is still unaffected.\nThedecaytimes τfromFig.8areshowninFig.9asafunctionofthedisorderdepth.\nThe decreasing fractionofatomsintheBEC peaksagainconfirmsth at stronger disorder\nleads to faster dephasing of the momentum distribution and that th is broadening of the\nquasimomentum is the underlying mechanism for the damping of the Blo ch oscillations.\n5. Conclusion\nWe have presented the first experimental investigation on disorde r induced damping\nof Bloch oscillations of Bose-Einstein condensates. The application o f an additional\ndisorder potential during the oscillation leads to a strong damping of the centre-of-mass\nmotion and to a significant reduction of the fraction of atoms in the B EC. The observed\ndamping rates are in good agreement with predictions based on nume rical solutions of\nthe full Gross-Pitaevskii equation and show that the underlying ph ysical mechanism for\nthe damping is the broadening of the quasimomentum spectrum due t o the spatially\nvarying phase evolution of the condensate.\nWe show that even very small disorder results in fast dephasing of t he\nquasimomentum and therefore damping of the Bloch oscillation. Since the disorder\npresented here is equivalent to a spatially inhomogeneous force, th e results are of special\ninterest for the application of Bloch oscillations for high precision spe ctroscopy of very\nsmall forces. To reach high precision in such experiments it is essent ial to follow a large\nnumber of Bloch oscillations. This number may be reduced if the obser ved force is\nspatially inhomogeneous on length scales comparable to the extend o f the condensate.\nThe good agreement between theory and experiment shows the ap plicability of our\nmethod to analyse the effects of spatially varying forces, and allows for estimates of the\neffect of small inhomogeneities for future experiments.\n6. Acknowledgments\nWethankM.Lewenstein forfruitful discussions. This workwassup ported bytheCentre\nfor Quantum Engineering and Space-Time Research and by the Deut sche Forschungs-\ngemeinschaft within the SFB 407, the SPP1116 and within the Europe an Graduate\nCollege Interference and Quantum Applications.Damped Bloch Oscillations 13\n[1] B. Damski et al., Phys. Rev. Lett. 91, 080403 (2003).\n[2] T. Schulte et al., Phys. Rev. Lett. 95, 170411 (2005).\n[3] L. Sanchez-Palencia et al., Phys. Rev. Lett. 98, 210401 (2007).\n[4] P. Lugan et al.,Phys. Rev. Lett. 99, 180402 (2007).\n[5] L. Fallani et al., Phys. Rev. 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Phys. 5, 112 (2003).\n[32] M. Modugno, C. Tozzo, and F. Dalfovo, Phys. Rev. A 70, 043625 (2004).\n[33] L. Cacciapuoti, et al.Phys. Rev. A 68, 053612 (2003).\n[34] G. Zener, Proc. R. Soc. London A 137, 696 (1932);\n[35] M. Cristiani et al., Optics Express 12, 4-10 (2004).\n[36] T. Hartmann, F. Keck, H. J. Korsch, and S. Mossmann, New J. Phys.6, 2 (2004);"    },    {        "title": "1908.10753v2.Spin_functional_renormalization_group_for_quantum_Heisenberg_ferromagnets__Magnetization_and_magnon_damping_in_two_dimensions.pdf",        "content": "Spin functional renormalization group for quantum Heisenberg ferromagnets:\nMagnetization and magnon damping in two dimensions\nRaphael Goll, Dmytro Tarasevych, Jan Krieg, and Peter Kopietz\nInstitut f ur Theoretische Physik, Universit at Frankfurt,\nMax-von-Laue Strasse 1, 60438 Frankfurt, Germany\n(Dated: November 21, 2019)\nWe use the spin functional renormalization group recently developed by two of us [J. Krieg and\nP. Kopietz, Phys. Rev. B 99, 060403(R) (2019)] to calculate the magnetization M(H;T) and the\ndamping of magnons due to classical longitudinal \ructuations of quantum Heisenberg ferromagnets.\nIn order to guarantee that for vanishing magnetic \feld H!0, the magnon spectrum is gapless\nwhen the spin rotational invariance is spontaneously broken, we use a Ward identity to express\nthe magnon self-energy in terms of the magnetization. In two dimensions our approach correctly\npredicts the absence of long-range magnetic order for H= 0 at \fnite temperature T. The magnon\nspectrum then exhibits a gap from which we obtain the transverse correlation length. We also\ncalculate the wave-function renormalization factor of the magnons. As a mathematical by-product,\nwe derive a recursive form of the generalized Wick theorem for spin operators in frequency space\nwhich facilitates the calculation of arbitrary time-ordered connected correlation functions of an\nisolated spin in a magnetic \feld.\nCONTENTS\nI. Introduction 1\nII. Exact \row equations for quantum spin systems 3\nIII. Vertex expansion 5\nA. Exact \row equations 5\nB. Deformation scheme: switching o\u000b the\ntransverse interaction 7\nC. Initial conditions 7\nIV. Magnetization of two-dimensional ferromagnets 8\nA. One-loop approximation with sharp\nmomentum cuto\u000b 9\nB. Self-energy and vertex corrections 10\n1. Magnon self-energy from the Ward\nidentity 10\n2. Vertex correction from the Katanin\nsubstitution 11\n3. Vertex correction from \row equations 12\n4. Wave-function renormalization 13\nC. Magnetization curves 14\nV. Magnon damping due to classical longitudinal\n\ructuations 15\nVI. Summary and outlook 16\nAcknowledgments 17\nAPPENDIX A: Relations between irreducible\nvertices and connected correlation functions 17\nAPPENDIX B: Generalized blocks and Wick\ntheorem for spin operators 19\nAPPENDIX C: Ward identity and hierarchy of\nequations of motion 21APPENDIX D: Tree approximation for\ncorrelation functions and vertices 23\nReferences 24\nI. INTRODUCTION\nRecently two of us have proposed a new approach to\nquantum spin systems based on a formally exact renor-\nmalization group equation for the generating functional\nof the connected spin correlation functions.1Our method\nworks directly with the physical spin operators, thus\navoiding any representation of the spin operators in terms\nof canonical bosons or fermions acting on a projected\nHilbert space. A similar strategy was adopted half a\ncentury ago by Vaks, Larkin, and Pikin (VLP),2,3who\ndeveloped an unconventional diagrammatic approach to\nquantum spin systems based on a generalized Wick the-\norem for spin operators. A detailed description of this\napproach can be found in the textbook by Izyumov and\nSkryabin.4It turns out, however, that the diagrammatic\nstructure of this approach is rather complicated, which is\nperhaps the reason why this method has not gained wide\nacceptance. As pointed out in Ref. [1], by combining the\nVLP approach2,3with modern functional renormaliza-\ntion group (FRG) methods,5{8we can reduce the prob-\nlem of calculating the spin correlation functions to the\nproblem of solving the bosonic version of the Wetterich\nequation9with a special initial condition determined by\ntheSU(2)-spin algebra. Our spin functional renormaliza-\ntion group (SFRG) approach is an extension of the lattice\nnon-perturbative renormalization group approach devel-\noped by Machado and Dupuis10for classical spin mod-\nels, and by Ran\u0018 con and Dupuis11{15for bosonic quan-\ntum lattice models. Another FRG approach to quantum\nspin systems is the so-called pseudofermion FRG,16{19\nwhich uses the representation of spin-1 =2 operators inarXiv:1908.10753v2  [cond-mat.stat-mech]  21 Nov 20192\nterms of Abrikosov pseudofermions20to approximate the\nrenormalization group \row of spin-1 =2 quantum spin sys-\ntems by a truncated fermionic FRG \row.8,21The pseud-\nofermion FRG has been quite successful to map out the\nphase diagram of various types of frustrated magnets\nwithout long-range magnetic order.16{19In contrast to\nthe pseudofermion FRG, our SFRG can be used for ar-\nbitrary spin Sbecause it does not rely on the represen-\ntation of the spin operators in terms of auxiliary degrees\nof freedom.\nAs a \frst application of our SFRG approach, in\nRef. [22] we have shown how Anderson's poor man's scal-\ning equations for the anisotropic Kondo model emerge\nfrom a simple weak coupling truncation of the SFRG \row\nequations for the irreducible vertices of this model. Our\nultimate goal is to develop the SFRG method into a quan-\ntitative tool for studying quantum spin systems which is\nvalid both in the magnetically ordered and the disordered\nphases. To realize this program, it is important to test\nthe method for model systems where the physics is well\nunderstood and quantitatively accurate results can be\nobtained by means of other methods. Here we will use\nthe SFRG to study the spin- Sferromagnetic quantum\nHeisenberg model in two dimensions. At any \fnite tem-\nperatureT > 0 the magnetization M(H;T > 0) of this\nmodel is rigorously known23to vanish when the external\nmagnetic \feld Happroaches zero, so that conventional\nspin-wave theory breaks down for H!0. If we never-\ntheless try to calculate the magnetization for H= 0 per-\nturbatively, we encounter infrared divergencies signalling\nthe breakdown of perturbation theory.24\nTheoretical investigations of quantum Heisenberg fer-\nromagnets in two dimensions have a long history. In the\n1980s several approximate analytical methods have been\ndeveloped to calculate the thermodynamics and the spec-\ntrum of renormalized magnons at \fnite temperature. For\nexample, in Takahashi's modi\fed spin-wave theory25,26\nthe vanishing of the magnetization for H= 0 is en-\nforced by adding an e\u000bective chemical potential to the\nmagnon energies which regularizes the infrared diver-\ngence encountered in perturbation theory. Note, how-\never, that for \fnite magnetic \feld Hthis strategy is not\napplicable because then the magnetization does not van-\nish. One possibility to generalize Takahashi's modi\fed\nspin-wave theory for \fnite magnetic \feld is to consider\nthe spin-wave expansion of the Gibbs potential for \fxed\nmagnetization.27Another useful method is based on the\nrepresentation of the spin operators in terms of Schwinger\nbosons.28This representation maps the exchange inter-\naction between spins onto a quartic boson Hamiltonian\nwith an additional constraint on the physical Hilbert\nspace. Often a simple mean-\feld treatment of the result-\ning e\u000bective boson problem gives already sensible results,\nso that Schwinger boson mean-\feld theory continues to\nbe a popular analytical method for studying spin sys-\ntems without long-range magnetic order. However, going\nbeyond the mean-\feld approximation is rather di\u000ecult\nwithin the Schwinger boson approach.29{32Another ana-lytical approach to quantum ferromagnets is based on the\ndecoupling of the equations of motion for the Green func-\ntions of the spins.33{35Of course, the physical properties\nof quantum ferromagnets can be obtained with high ac-\ncuracy numerically using Monte Carlo simulations.36{38\nIn Ref. [37] the correlation length and the susceptibility\nof a two-dimensional quantum Heisenberg ferromagnet\nhave been calculated via a special implementation of the\nmomentum-shell Wilsonian renormalization group (RG)\ntechnique using the Holstein-Primako\u000b transformation to\nexpress the spin operators in terms of bosons. At the level\nof a one-loop approximation the results for the correla-\ntion length and the susceptibility agree with modi\fed\nspin-wave theory25,26and Schwinger boson mean-\feld\ntheory,28but the two-loop corrections have been found\nto modify the one-loop results.37In the present work,\nwe will show that the one-loop \row equations derived in\nRef. [37] can be obtained in a straightforward way within\nour SFRG formalism from a truncated \row equation for\nthe magnetization. We then use the SFRG to derive an\nimproved \row equation for the magnetization which takes\ninto account self-energy and vertex corrections neglected\nin Ref. [37]. Moreover, we will also calculate the damping\nof spin-waves due to the coupling to classical longitudi-\nnal spin \ructuations. This decay channel of magnons\nis neglected in conventional spin-wave theory where the\nlongitudinal \ructuations are not treated as independent\ndegrees of freedom.\nThe rest of this work is organized as follows. In Sec. II,\nwe de\fne a new hybrid generating functional \u0000 \u0003[m;\u001e]\ndepending on the transverse magnetization mas well\nas on a longitudinal exchange \feld \u001eand show that this\nfunctional satis\fes the Wetterich equation.9In Sec. III we\ngive the general structure of the expansion of \u0000 \u0003[m;\u001e] in\npowers of the \felds and explicitly write down the exact\n\row equations for the magnetization and the two-point\nvertices. Section IV is devoted to the explicit calcula-\ntion of the magnetization M(H;T) of a two-dimensional\nHeisenberg ferromagnet. We establish the relation of our\nSFRG approach to the momentum shell RG of Ref. [37]\nand go beyond this work by including self-energy and\nvertex corrections. In Sec. V, we calculate the damp-\ning of spin waves in two-dimensional ferromagnets due\nto the coupling to classical longitudinal \ructuations. In\nSec. VI, we summarize our main results and point out\npossible extensions of our method.\nIn four appendices, we give additional technical details.\nIn Appendix A, we derive the relation between the con-\nnected spin correlation functions and the irreducible ver-\ntices generated by our hybrid functional \u0000 \u0003[m;\u001e]. In Ap-\npendix B, we give the time-ordered connected spin corre-\nlation functions of a single spin in an external magnetic\n\feld with up to four spins and derive the corresponding\nirreducible vertices; we also present a simple recursive\nform of the generalized Wick theorem for spin opera-\ntors in frequency space. In Appendix C, we write down\nequations of motion for the time-ordered connected spin\ncorrelation functions and derive a Ward identity which3\nwe use in the main text to close the \row equation for\nthe magnetization. Finally, in Appendix D, we derive\ninitial conditions for the irreducible vertices in tree ap-\nproximation where all terms involving loop integrations\nover momenta are neglected.\nII. EXACT FLOW EQUATIONS FOR\nQUANTUM SPIN SYSTEMS\nIn this section and in the following Sec. III, we con-\nsider a general anisotropic Heisenberg Hamiltonian of the\nform\nH=\u0000HX\niSz\ni+1\n2X\nij\u0002\nJ?\nijS?\ni\u0001S?\nj+Jz\nijSz\niSz\nj\u0003\n;(2.1)\nwhere the external magnetic \feld His measured in units\nof energy, S?\ni= (Sx\ni;Sy\ni) is the transverse part of the\nspin operator Si, andJ?\nijandJz\nijare transverse and lon-\ngitudinal exchange couplings. In Sec. IV, we will consider\nisotropic ferromagnets by specifying J?\nij=Jz\nij=\u0000Vij<\n0, but at this point, we work with general anisotropic ex-\nchange couplings. Following Ref. [1], we now modify the\nHamiltonian (2.1) by replacing J?\nijandJz\nijby deformed\nexchange couplings depending on a continuous parame-\nter \u0003,\nJ?\n\u0003;ij=J?\nij+R?\n\u0003;ij; Jz\n\u0003;ij=Jz\nij+Rz\n\u0003;ij;(2.2)\nwhere the regulators R?\n\u0003;ijandRz\n\u0003;ijshould be chosen\nsuch that for some initial \u0003 = \u0003 0the deformed model\ncan be solved in a controlled way, and for some \fnal\nvalue of \u0003 the regulators vanish so that we recover our\noriginal model. For example, \u0003 can be a momentum\nscale acting as a cuto\u000b for long-wavelength \ructuations,\nor simply a dimensionless parameter in the interval [0 ;1]\nwhich multiplies the bare interaction. At this point, it\nis not necessary to specify the deformation scheme. Let\nus write the deformed Hamiltonian in the form H\u0003=\nH0+V\u0003, where\nH0=\u0000HX\niSz\ni (2.3)\nis the Hamiltonian of isolated spins in a constant mag-\nnetic \feld and\nV\u0003=1\n2X\nij\u0002\nJ?\n\u0003;ijS?\ni\u0001S?\nj+Jz\n\u0003;ijSz\niSz\nj\u0003\n(2.4)\nrepresents the coupling between the spins. The gener-\nating functional of the deformed Euclidean time-ordered\nspin correlation functions can then be written as1\nG\u0003[h] = ln Trh\ne\u0000\fH0TeR\f\n0d\u001c[P\nihi(\u001c)\u0001Si(\u001c)\u0000V\u0003(\u001c)]i\n:\n(2.5)\nHere\fis the inverse temperature, Tdenotes time-\nordering in imaginary time, hi(\u001c) are \ructuating source\n\felds, and the time dependence of all operators is in theinteraction picture with respect to H0. By simply taking\na derivative of both sides of Eq. (2.5) with respect to \u0003 we\ncan derive an exact functional \row equation for the gen-\nerating functional G\u0003[h]. Moreover, the Legendre trans-\nform ofG\u0003[h] satis\fes an exact functional \row equation\nwhich is formally identical to the bosonic version of the\nWetterich equation.1,9A technical complication of this\nmethod is that in a scheme where initially the exchange\ncouplings are completely switched o\u000b the Legendre trans-\nform ofG\u0003=0[h] does not exist1,15because in this limit\nthe longitudinal spin \ructuations do not have any dy-\nnamics. In Ref. [1] we have already pointed out that\nthis problem can be avoided by working with the gener-\nating functional of the amputated connected correlation\nfunctions. Here we show how this idea is implemented\nin practice. Actually, because only the longitudinal \ruc-\ntuations are initially static, it is convenient to introduce\na hybrid functional F\u0003[h?;s] which generates connected\ncorrelation functions for transverse \ructuations, but am-\nputates the external legs associated with the longitudinal\n\ructuations. Formally, this functional can be de\fned by\nF\u0003[h?;s] =G\u0003h\nh?\ni;hz\ni=\u0000X\njJz\n\u0003;ijsji\n\u00001\n2Z\f\n0d\u001cX\nijJz\n\u0003;ijsi(\u001c)sj(\u001c); (2.6)\nwhere h?\ni= (hx\ni;hy\ni) is a transverse magnetic source \feld,\nandsi(\u001c) is a longitudinal source \feld which can be in-\nterpreted as a \ructuating magnetic moment in the di-\nrection of the external \feld. Note that a similar hybrid\nfunctional of \\partially amputated connected\" correla-\ntion functions has been introduced earlier in Ref. [39] to\nderive partially bosonized FRG \row equations for inter-\nacting fermions. By expanding F\u0003[h?;s] in powers of\nthe source \felds h?\ni(\u001c) andsi(\u001c), we obtain connected\nspin correlation functions with the additional property\nthat external legs associated with longitudinal propaga-\ntors are amputated.1As a consequence, correlation func-\ntions generated by F\u0003[h?;s] with longitudinal external\nlegs involve powers of the longitudinal interaction Jz\n\u0003;ij\nand therefore vanish for Jz\n\u0003;ij!0. For example, the\nlongitudinal two-point function is given by\nF\u0003;ij(\u001c;\u001c0) =\u000e2F\u0003[h?\ni= 0;s]\n\u000esi(\u001c)\u000esj(\u001c0)\f\f\f\f\ns=0\n=\u0000\u000e(\u001c\u0000\u001c0)Jz\n\u0003;ij+X\nklJz\n\u0003;ikJz\n\u0003;jlGzz\n\u0003;kl(\u001c;\u001c0);(2.7)\nwhere\nGzz\n\u0003;ij(\u001c;\u001c0)\u000e2G\u0003[h]\n\u000ehz\ni(\u001c)\u000ehz\nj(\u001c0)\f\f\f\f\f\nh=0(2.8)\nis the longitudinal part of the time-ordered two-spin cor-\nrelation function. The relations between the higher order4\nlongitudinal correlation functions is for n\u00153,\nFz:::z\n\u0003;i1:::in(\u001c1;:::;\u001cn) =\u000enF\u0003[h?= 0;s]\n\u000esi1(\u001c1):::\u000esin(\u001cn)\f\f\f\ns=0\n= (\u00001)nX\nj1:::jnJz\n\u0003;i1j1:::Jz\n\u0003;injnGz:::z\n\u0003;j1:::jn(\u001c1;:::;\u001cn);(2.9)\nwhere\nGz:::z\n\u0003;i1:::in(\u001c1;:::;\u001cn) =\u000enG\u0003[h]\n\u000ehz\ni1(\u001c1):::\u000ehz\nin(\u001cn)\f\f\f\nh=0:(2.10)\nIf we work with a deformation scheme where initially\nall exchange couplings vanish, all correlation functions\ngenerated byF\u0003[h?;s] involving longitudinal legs also\nvanish at the initial scale, so that at the \frst sight this\nfunctional does not give rise to a convenient initial condi-\ntion for this cuto\u000b scheme. However, the Legendre trans-\nform of the functional F\u0003[h?;s] has a well-de\fned limit\nfor vanishing exchange couplings because the externals\nlegs associated with the longitudinal propagators are re-\nmoved in the Legendre transform. As usual, we subtract\nthe regulator terms from the Legendre transform and de-\n\fne the generating functional of the irreducible vertices\nas follows,\n\u0000\u0003[m;\u001e] =Z\f\n0d\u001cX\ni(mi\u0001h?\ni+\u001eisi)\u0000F \u0003[h?;s]\n\u00001\n2Z\f\n0d\u001cX\nij\u0010\nR?\n\u0003;ijmi\u0001mj+R\u001e\n\u0003;ij\u001ei\u001ej\u0011\n;(2.11)\nwhere the transverse and longitudinal regulators are\ngiven by\nR?\n\u0003;ij=J?\n\u0003;ij\u0000J?\nij; (2.12)\nR\u001e\n\u0003;ij=\u0000[ Jz\n\u0003]\u00001\nij+ [ Jz]\u00001\nij: (2.13)Here Jz\n\u0003is a matrix on the spatial labels with matrix\nelements [ Jz\n\u0003]ij=Jz\n\u0003;ij. Note that on the right-hand\nside of Eq. (2.11) the sources h?\ni(\u001c) andsi(\u001c) should be\nexpressed in terms of the transverse magnetization mi(\u001c)\nand the longitudinal exchange \feld \u001ei(\u001c) by inverting the\nrelations\nmi(\u001c) =\u000eF\u0003[h?;s]\n\u000eh?\ni(\u001c)=hTS?\nj(\u001c)i; (2.14)\n\u001ei(\u001c) =\u000eF\u0003[h?;s]\n\u000esi(\u001c)\n=\u0000X\njJz\n\u0003;ij\u0002\nsj(\u001c) +hTSz\nj(\u001c)i\u0003\n;(2.15)\nwhere the expectation values hTS?\nj(\u001c)iandhTSz\nj(\u001c)i\nshould be evaluated for \fnite sources h?\ni(\u001c) andsi(\u001c).\nFrom the last line in Eq. (2.15), we see that for si(\u001c) = 0\nthe \feld\u001ei(\u001c) can be identi\fed with the exchange cor-\nrection to the external magnetic \feld. Obviously, if we\nuse Eq. (2.11) to express si(\u001c) as a functional of \u001ei(\u001c)\nwe obtain a factor of [ Jz\n\u0003]\u00001\nijwhich cancels the factors\nofJz\n\u0003;ijin the expansion of the functional F\u0003[h?;s] in\npowers of the longitudinal sources si(\u001c). The irreducible\nvertices generated by \u0000 \u0003[m;\u001e] are therefore well-de\fned\neven if we use a deformation scheme where initially all\nexchange couplings vanish. Another way to see this is\nto explicitly express the vertices generated by \u0000 \u0003[m;\u001e]\nin terms of the connected spin correlation functions, see\nAppendix A.\nIt is now straightforward to derive formally exact\nFRG \row equations of the two functionals F\u0003[h?;s] and\n\u0000\u0003[m;\u001e]. Following the procedure outlined in Ref. [1],\nwe \fnd that the functional F\u0003[h?;s] satis\fes the \row\nequation\n@\u0003F\u0003[h?;s] =\u00001\n2Z\f\n0d\u001cX\nij(@\u0003J?\n\u0003;ij)X\n\u000b=x;y\"\n\u000e2F\u0003[h?;s]\n\u000eh\u000b\ni(\u001c)\u000eh\u000b\nj(\u001c)+\u000eF\u0003[h?;s]\n\u000eh\u000b\ni(\u001c)\u000eF\u0003[h?;s]\n\u000eh\u000b\nj(\u001c)#\n+1\n2Z\f\n0d\u001cX\nij\u0000\n@\u0003[ Jz\n\u0003]\u00001\nij\u0001\"\n\u000e2F\u0003[h?;s]\n\u000esi(\u001c)\u000esj(\u001c)+\u000eF\u0003[h?;s]\n\u000esi(\u001c)\u000eF\u0003[h?;s]\n\u000esj(\u001c)#\n+1\n2Tr\u0002\nJz\n\u0003@\u0003(Jz\n\u0003)\u00001\u0003\n: (2.16)\nHereJz\n\u0003is a matrix in all labels (spin component, lattice site, imaginary time) with matrix elements de\fned by\n[Jz\n\u0003]\u000b\u000b0\ni\u001c;j\u001c0=\u000e(\u001c\u0000\u001c0)\u000e\u000bz\u000e\u000b0z[ Jz\n\u0003]ij=\u000e(\u001c\u0000\u001c0)\u000e\u000bz\u000e\u000b0zJz\n\u0003;ij: (2.17)\nWith this notation the trace in the last term of Eq. (2.16) is over all labels; the formally divergent \u000e-function\u000e(\u001c= 0)\nhidden in the last term of Eq. (2.16) cancels when we combine this term with a similar contribution from the second\nterm. Di\u000berentiating both sides of Eq. (2.11) with respect to the deformation parameter \u0003 and using Eq. (2.16), we\nobtain\n@\u0003\u0000\u0003[m;\u001e] =1\n2Z\f\n0d\u001cX\nij\u0014X\n\u000b=x;y\u000e2F\u0003[h?;s]\n\u000eh\u000b\ni(\u001c)\u000eh\u000b\nj(\u001c)@\u0003R?\n\u0003;ij+\u000e2F\u0003[h?;s]\n\u000esi(\u001c)\u000esj(\u001c)@\u0003R\u001e\n\u0003;ij\u0015\n\u00001\n2Tr\u0002\nJz\n\u0003@\u0003(Jz\n\u0003)\u00001\u0003\n:(2.18)\nNote that the quadratic subtractions in Eq. (2.11) elim- inate the terms on the right-hand side of the \row equa-5\ntion (2.16) corresponding to diagrams which are reducible\nwith respect to cutting either a transverse propagator line\nor a longitudinal interaction line. Finally, we express the\nsecond derivatives on the right-hand side of Eq. (2.18) in\nterms of the matrix \u000000\n\u0003[m;\u001e] of second functional deriva-\ntives of \u0000 \u0003[m;\u001e] and obtain an exact functional \row\nequation for our quantum spin system which formally\nresembles the Wetterich equation,9\n@\u0003\u0000\u0003[m;\u001e]\n=1\n2Trnh\n(\u000000\n\u0003[m;\u001e] +R\u0003)\u00001+Jz\n\u0003i\n@\u0003R\u0003o\n:(2.19)\nHere the matrix elements of \u000000\n\u0003[m;\u001e] are given by\n(\u000000\n\u0003[m;\u001e])i\u001c\u000b;j\u001c0\u000b0=\u000e2\u0000\u0003[m;\u001e]\n\u000e\b\u000b\ni(\u001c)\u000e\b\u000b0\nj(\u001c0); (2.20)\nwhere we have combined the two components of mi(\u001c)\ntogether with \u001ei(\u001c) into a three-component \feld\n0\n@\bx\ni\n\by\ni\n\bz\ni1\nA=0\n@mx\ni\nmy\ni\n\u001ei1\nA; (2.21)\nandR\u0003is diagonal in the \feld labels with matrix ele-\nments\n[R\u0003]xx\ni\u001c;j\u001c0= [R\u0003]yy\ni\u001c;j\u001c0=\u000e(\u001c\u0000\u001c0)R?\n\u0003;ij;(2.22)\n[R\u0003]zz\ni\u001c;j\u001c0=\u000e(\u001c\u0000\u001c0)R\u001e\n\u0003;ij: (2.23)\nFor \fnite external \feld Hor in the presence of a \fnite\nspontaneous magnetization the expectation value hSz\ni(\u001c)i\nis \fnite even for vanishing sources si(\u001c). Then the func-\ntional \u0000 \u0003[m= 0;\u001e] is extremal for a \fnite value \u001e\u0003of\nthe exchange \feld,\n\u000e\u0000\u0003[m= 0;\u001e]\n\u000e\u001ei(\u001c)\f\f\f\f\n\u001e=\u001e\u0003= 0; (2.24)\nwhere\u001e\u0003can be identi\fed with the scale-dependent\nrenormalization of the external magnetic \feld due to the\nexchange interaction. It is then convenient to shift the\n\feld\u001ei(\u001c) =\u001e\u0003+'i(\u001c) and consider the \row of\n~\u0000\u0003[m;'] = \u0000 \u0003[m;\u001e\u0003+']; (2.25)which is given by\n@\u0003~\u0000\u0003[m;'] =@\u0003\u0000\u0003[m;\u001e]j\u001e!\u001e\u0003+'\n+Z\f\n0d\u001cX\ni\u000e~\u0000\u0003[m;']\n\u000e'i(\u001c)@\u0003\u001e\u0003\n=1\n2Tr\u001a\u0014\u0010\n~\u000000\n\u0003[m;'] +R\u0003\u0011\u00001\n+Jz\n\u0003\u0015\n@\u0003R\u0003\u001b\n+Z\f\n0d\u001cX\ni\u000e~\u0000\u0003[m;']\n\u000e'i(\u001c)@\u0003\u001e\u0003:(2.26)\nIII. VERTEX EXPANSION\nA. Exact \row equations\nTo construct an approximate solution of the exact \row\nequation (2.26), we expand the functional ~\u0000\u0003[m;'] in\npowers of the transverse magnetization mi(\u001c) and the\n\ructuation 'i(\u001c) of the longitudinal exchange \feld. Then\nwe obtain an in\fnite hierarchy of \row equations for the\nirreducible vertices generated by ~\u0000\u0003[m;']. It is conve-\nnient to formulate the expansion in momentum-frequency\nspace. Introducing a collective label K= (k;i!) for mo-\nmentum kand Matsubara frequency i!, the Fourier ex-\npansion of the \felds can be written as\nmi(\u001c) =Z\nKei(k\u0001ri\u0000!\u001c)mK; (3.1a)\n'i(\u001c) =Z\nKei(k\u0001ri\u0000!\u001c)'K; (3.1b)\nwhereR\nK= (\fN)\u00001P\nk;!. In terms of the spherical\ncomponents of the transverse magnetization,\nm+\nK=mx\nK+imy\nKp\n2; m\u0000\nK=mx\nK\u0000imy\nKp\n2= (m+\n\u0000K)\u0003;\n(3.2)\nthe vertex expansion of ~\u0000\u0003[m;'] up to fourth order in\nthe \felds is of the form6\n~\u0000\u0003[m;'] =\fNf \u0003+Z\nK\u0000+\u0000\n\u0003(K)m\u0000\n\u0000Km+\nK+1\n2!Z\nK\u0000zz\n\u0003(K)'\u0000K'K\n+Z\nK1Z\nK2Z\nK3\u000e(K1+K2+K3)\u0000+\u0000z\n\u0003(K1;K2;K3)m\u0000\nK1m+\nK2'K3\n+1\n3!Z\nK1Z\nK2Z\nK3\u000e(K1+K2+K3)\u0000zzz\n\u0003(K1;K2;K3)'K1'K2'K3\n+Z\nK1Z\nK2Z\nK3Z\nK4\u000e(K1+K2+K3+K4)\u001a1\n(2!)2\u0000++\u0000\u0000\n\u0003 (K1;K2;K3;K4)m\u0000\nK1m\u0000\nK2m+\nK3m+\nK4\n+1\n2!\u0000+\u0000zz\n\u0003 (K1;K2;K3;K4)m\u0000\nK1m+\nK2'K3'K4+1\n4!\u0000zzzz\n\u0003(K1;K2;K3;K4)'K1'K2'K3'K4\u001b\n+::: ; (3.3)\nwhere\u000e(K) =\fN\u000e k;0\u000e!;0and we have omitted vertices\nwith \fve and more external legs. The superscripts at-\ntached to the vertices refer to the \feld types of the as-\nsociated external legs. The fact that in the expansion\n(3.3) the \feld m\u0000\nKis associated with the superscript+\nin \u0000\u0001\u0001\u0001+\u0001\u0001\u0001\n\u0003 (\u0001\u0001\u0001K\u0001\u0001\u0001) is related to the fact that m\u0000\nKis gen-\nerated by di\u000berentiating G\u0003[h] with respect to h+\nK, and\nm+\nKcan be obtained by di\u000berentiating G\u0003[h] with respect\ntoh\u0000\nK(note the alternating superscripts). The relation\nbetween the time-ordered spin correlation functions and\nthe irreducible vertices de\fned via the expansion (3.3) is\nexplicitly constructed in Appendix A for vertices with up\nto four external legs.\nThe exact \row equations for the vertices can be\nobtained from the functional \row equation (2.26) by\nexpanding both sides in powers of the \felds and\ncomparing coe\u000ecients of a given power after proper\nsymmetrization.7To write down the \row equations, we\nneed the regularized transverse and longitudinal propa-\ngators\nG\u0003(K) =1\n\u0000+\u0000\n\u0003(K) +R?\n\u0003(k); (3.4)\nF\u0003(K) =1\n\u0000zz\n\u0003(K) +R\u001e\n\u0003(k); (3.5)\nand the corresponding single-scale propagators\n_G\u0003(K) =\u0000[G\u0003(K)]2@\u0003R?\n\u0003(k); (3.6)\n_F\u0003(K) =\u0000[F\u0003(K)]2@\u0003R\u001e\n\u0003(k): (3.7)\nHere the regulator in momentum space are given by\nR?\n\u0003(k) =J?\n\u0003(k)\u0000J?(k); (3.8)\nR\u001e\n\u0003(k) =1\nJz(k)\u00001\nJz\n\u0003(k); (3.9)\nwhere we have used the discrete translational invariance\nof the lattice to expand the exchange couplings in mo-\nmentum space,\nJ?\n\u0003;ij=1\nNX\nkeik\u0001(ri\u0000rj)J?\n\u0003(k); (3.10)\nJz\n\u0003;ij=1\nNX\nkeik\u0001(ri\u0000rj)Jz\n\u0003(k): (3.11)With this notation the \row equation for the constant f\u0003\nin Eq. (3.3), which can be identi\fed with the free energy\nper lattice site, can be written as\n@\u0003f\u0003=Z\nKh\nG\u0003(K)@\u0003R?\n\u0003(k)\n+1\n2[F\u0003(K) +Jz\n\u0003(k)]@\u0003R\u001e\n\u0003(k)i\n: (3.12)\nRecall that according to Eq. (2.7) the longitudinal prop-\nagatorF\u0003(K) is related to the longitudinal two-spin cor-\nrelation function Gzz\n\u0003(K) via\nF\u0003(K) =\u0000Jz\n\u0003(k) + [Jz\n\u0003(k)]2Gzz\n\u0003(K); (3.13)\nso that the \row equation (3.12) can alternatively be writ-\nten as\n@\u0003f\u0003=Z\nK\u0014\n_J?\n\u0003(k)G\u0003(K) +1\n2_Jz\n\u0003(k)Gzz\n\u0003(K)\u0015\n;(3.14)\nwhere _J?\n\u0003(k) =@\u0003J?\n\u0003(k) and _Jz\n\u0003(k) =@\u0003Jz\n\u0003(k). Next,\nconsider the \row equation for the exchange \feld \u001e\u0003,\nwhich can be obtained from the condition that the ex-\npansion of the generating functional ~\u0000\u0003[m;'] does not\nhave a term linear in the \ructuation '(see Refs. [7 and\n39]). This is guaranteed if \u001e\u0003\rows according to\n\u0000zz\n\u0003(0)@\u0003\u001e\u0003=\u0000Z\nK_G\u0003(K)\u0000+\u0000z\n\u0003(\u0000K;K; 0)\n\u00001\n2Z\nK_F\u0003(K)\u0000zzz\n\u0003(\u0000K;K; 0):(3.15)\nFinally, let us also write down the exact \row equations\nfor the transverse and longitudinal two-point vertices,7\n@\u0003\u0000+\u0000\n\u0003(K) = \u0000+\u0000z\n\u0003(\u0000K;K; 0)@\u0003\u001e\u0003+Z\nQ_G\u0003(Q)\u0000++\u0000\u0000\n\u0003 (\u0000K;\u0000Q;Q;K ) +1\n2Z\nQ_F\u0003(Q)\u0000+\u0000zz\n\u0003 (\u0000K;K;\u0000Q;Q)\n\u0000Z\nQ[_G\u0003(Q)F\u0003(Q\u0000K) +G\u0003(Q)_F\u0003(Q\u0000K)]\u0000+\u0000z\n\u0003(\u0000K;Q;K\u0000Q)\u0000+\u0000z\n\u0003(\u0000Q;K;Q\u0000K); (3.16)\n@\u0003\u0000zz\n\u0003(K) = \u0000zzz\n\u0003(\u0000K;K; 0)@\u0003\u001e\u0003+Z\nQ_G\u0003(Q)\u0000+\u0000zz\n\u0003 (\u0000Q;Q;\u0000K;K ) +1\n2Z\nQ_F\u0003(Q)\u0000zzzz\n\u0003(\u0000Q;Q;\u0000K;K )\n\u0000Z\nQ[_G\u0003(Q)G\u0003(Q\u0000K) +G\u0003(Q)_G\u0003(Q\u0000K)]\u0000+\u0000z\n\u0003(\u0000Q;Q\u0000K;K )\u0000+\u0000z\n\u0003(K\u0000Q;Q;\u0000K)\n\u00001\n2Z\nQ[_F\u0003(Q)F\u0003(Q\u0000K) +F\u0003(Q)_F\u0003(Q\u0000K)]\u0000zzz\n\u0003(\u0000Q;Q\u0000K;K )\u0000zzz\n\u0003(K\u0000Q;Q;\u0000K): (3.17)\nGraphical representations of the exact \row equations\n(3.15){(3.17) are shown in Fig. 1. Following VLP,2it\nis convenient to parametrize the longitudinal spin cor-\nrelation function Gzz\n\u0003(K) in terms of the interaction-\nirreducible polarization \u0005 \u0003(K) by setting\nGzz\n\u0003(K) =\u0005\u0003(K)\n1 +Jz\n\u0003(k)\u0005\u0003(K): (3.18)\nUsing the relation (3.13) this implies that the longitu-\ndinal two-point function F\u0003(K) generated by our func-\ntionalF\u0003[h?;s] can be written as\nF\u0003(K) =\u0000Jz\n\u0003(k)\n1 +Jz\n\u0003(k)\u0005\u0003(K): (3.19)\nHence, the longitudinal propagator F\u0003(K) can be iden-\nti\fed with an e\u000bective screened interaction between lon-\ngitudinal spin \ructuations. Finally, using Eq. (3.5) to\nexpressF\u0003(K) in terms of the longitudinal two-point ver-\ntex \u0000zz\n\u0003(K) which is generated by our hybrid functional\n~\u0000\u0003[m;'] de\fned via Eqs. (2.11) and (2.25), we obtain\n\u0000zz\n\u0003(K) =\u00001\nJz(k)\u0000\u0005\u0003(K): (3.20)\nWe conclude that, up to a minus sign, the right-hand side\nof the \row equation (3.17) for \u0000zz\n\u0003(K) can be identi\fed\nwith the \row equation for the irreducible longitudinal\npolarization \u0005 \u0003(K) de\fned via Eq. (3.18).\nB. Deformation scheme: switching o\u000b the\ntransverse interaction\nLet us now specify our deformation scheme. In general,\nthe deformed exchange interactions J?\n\u00030;ijandJz\n\u00030;ijat\nthe initial value \u0003 = \u0003 0of the deformation parameter\nshould be chosen such that the correlation functions of\nthe deformed model can be calculated in a controlled\nway.1The simplest possibility is to completely switch o\u000b\nthe deformed exchange couplings, J?\n\u00030;ij=Jz\n\u00030;ij= 0, so\nthat the initial system consists of non-interacting spins\nsubject to an external magnetic \feld. Note, however,that even in this case the time-ordered spin correlation\nfunctions and the corresponding irreducible vertex func-\ntions are rather complicated and re\rect the non-trivial\non-site spin correlations between di\u000berent spin compo-\nnents implied by the SU(2)-spin algebra. In Appendix B\nwe explicitly derive the corresponding initial values for\nthe correlation functions and the vertices with up to four\nexternal legs.\nIn order to establish the relation of our SFRG ap-\nproach and the previous momentum-shell RG calcula-\ntion for quantum Heisenberg ferromagnets,37we choose\nin this work a di\u000berent deformation scheme, where only\nthe transverse interaction is initially switched o\u000b while\nthe longitudinal interaction is not deformed at all,\nJz\n\u0003(k) =Jz(k): (3.21)\nFor our purpose it is su\u000ecient to regularize the long-\nwavelength modes of the transverse interaction via a\nsharp cuto\u000b,\nJ?\n\u0003(k) = \u0002(k\u0000\u0003)J?(k); (3.22)\nwhich amounts to the following choice of the transverse\nregulator:\nR?\n\u0003(k) =J?\n\u0003(k)\u0000J?(k) =\u0000\u0002(\u0003\u0000k)J?(k):(3.23)\nThe initial value \u0003 0of the deformation parameter \u0003\n(which has units of momentum) should be chosen of the\norder of the inverse lattice spacing. Since @\u0003Jz\n\u0003(k) = 0\nin this cuto\u000b scheme, all terms involving the longitudi-\nnal single-scale propagator _F\u0003(K) in the \row equations\n(3.15{3.17) shown graphically in Fig. 1 can be omitted.\nOur deformed model at the initial scale \u0003 = \u0003 0is then\nan Ising model, which is still nontrivial.\nC. Initial conditions\nFor the deformation scheme discussed above, the initial\nconditions for the SFRG \row are determined by the mag-\nnetization and the correlation functions of a spin S Ising\nmodel, which cannot be calculated exactly. However, in\nthis work we are not interested in the critical regime of8\nFIG. 1. Graphical representation of the exact FRG \row\nequations (3.15), (3.16) and (3.17) for the exchange \feld and\nthe two-point vertices. (a) Flow equation for the exchange\n\feld\u001e\u0003; the dotted cross represents the scale derivative @\u0003\u001e\u0003.\nThe dot above the circle represents the derivative @\u0003with re-\nspect to the deformation parameter. (b) Flow equation for\nthe transverse two-point vertex \u0000+\u0000\n\u0003(K). (c) Flow equation\nfor the longitudinal two-point vertex \u0000zz\n\u0003(K). We use the\nsame color coding as in Ref. [7]: two-point vertices are red,\nthree-point vertices are green, and four-point vertices are blue.\nOutgoing arrows represent the spherical component m\u0000\n\u0000Kof\nthe transverse magnetization \feld, while incoming arrows rep-\nresent the conjugate \feld m+\nK. Wavy lines are associated with\nthe \ructuating part 'of the longitudinal exchange \feld. Solid\narrows and wavy lines represent transverse and longitudinal\npropagators, while lines with an extra slash represent the cor-\nresponding single-scale propagators. A diagram with a cross\ninside a loop represents the sum of two similar diagrams where\neach of the propagators forming the loop is successively re-\nplaced by the corresponding single-scale propagator (product\nrule).\nthe Ising model, but we focus on the low-temperature\nregimeT\u001cjJz(0)jwhere a perturbative calculation of\nthe correlation functions is possible. In order to calcu-\nlate the initial conditions in a controlled way, we follow\nVLP and assume that the range r0of the exchange in-\nteraction is large compared with the lattice spacing. In\nthis case, momentum integrals are controlled by the small\nparameter 1 =r0, and the initial values of the correlation\nfunctions can be calculated by solving the corresponding\nhierarchy of equations of motion perturbatively in powers\nof loop integrations, see Appendix C. To leading order,\ni.e., in the so called tree approximation, one may simply\nneglect all loops. As discussed in Appendix D, then the\nin\fnite hierarchy of equations of motion decouples. The\nresulting initial magnetization M0=hSz\niiis given by theself-consistent mean-\feld approximation, obtained from\nthe solution of\nM0=b(\f(H+\u001e0)); (3.24)\nwhere\n\u001e0=\u0000Jz(0)M0 (3.25)\nis the initial value of the exchange \feld introduced in\nEq. (2.24) and\nb(y) =\u0000\nS+1\n2\u0001\ncoth\u0014\u0000\nS+1\n2\u0001\ny\u0015\n\u00001\n2cothhy\n2i\n(3.26)\nis the spin-SBrillouin function. The transverse two-spin\ncorrelation function is then simply given by\nG0(K) =M0\nH+\u001e0\u0000i!; (3.27)\nwhile the longitudinal two-spin correlation function is\nGzz\n0(K) =\f\u000e!;0b0(\f(H+\u001e0))\n1 +Jz(k)\f\u000e!;0b0(\f(H+\u001e0)):(3.28)\nHereb0(y) is the derivative of the Brillouin function. Us-\ning Eq. (3.4) we \fnd that in this case the initial value of\nthe transverse two-point vertex is\n\u0000+\u0000\n0(K) =J?(k) +H+\u001e0\u0000i!\nM0\n=H+M0[J?(k)\u0000Jz(0)]\u0000i!\nM0;(3.29)\nwhile from Eq. (3.18) we conclude that the intial value\nfor the polarization is\n\u00050(K) =\f\u000e!;0b0(\f(H+\u001e0)); (3.30)\nwhich according to Eq. (3.20) implies for the initial value\nof the longitudinal two-point vertex,\n\u0000zz\n0(K) =\u00001\nJz(k)\u0000\f\u000e!;0b0(\f(H+\u001e0));(3.31)\nThe initial values of the higher-order spin correlation\nfunctions and corresponding irreducible vertices are de-\nrived in Appendix D.\nIV. MAGNETIZATION OF\nTWO-DIMENSIONAL FERROMAGNETS\nIn this section we calculate the magnetic equation of\nstateM(H;T) of an isotropic Heisenberg ferromagnet in\ntwo dimensions at \fnite temperature. For H= 0 the\nmagnetization vanishes at any \fnite temperature,23and\na naive spin-wave expansion gives a divergent result for\nM(H= 0;T). Non-perturbative methods are thus neces-\nsary to obtain reliable results for M(H;T) for su\u000eciently9\nsmall magnetic \felds.24To simplify our notation, we set\nfrom now on\nJ?(k) =Jz(k) =\u0000Vk; (4.1)\nwhere for a ferromagnet the Fourier transform Vkof the\nexchange couplings satis\fes V0\u0011Vk=0>0. At low\ntemperatures only long-wavelength modes are thermally\nexcited, so that we may expand40\nVk=V0\u0000\u001a0\nM0k2+O(k4); (4.2)\nwhere the value \u001a0of the bare spin-sti\u000bness depends on\nthe range of the interaction. For nearest-neighbor ferro-\nmagnetic exchange J > 0 on aD-dimensional hypercu-\nbic lattice with lattice spacing a, we haveV0= 2DJand\n\u001a0=M0Ja2. However, the initial conditions discussed\nin Sec. III C are only valid if the exchange interaction is\nlong-range; for short-range interaction, the tree approxi-\nmaton for the vertices at the initial scale is uncontrolled.\nNevertheless, the comparision of our FRG results with\nMonte Carlo simulations for two-dimensional ferromag-\nnets with nearest-neighbor interaction shows that even in\nthis case our truncation works rather well, see Sec. IV C\nbelow.\nA. One-loop approximation with sharp momentum\ncuto\u000b\nAs already mentioned in Sec. III C, to establish the\nprecise relation between our SFRG approach and an old\nmomentum-shell renormalization group calculation for\nquantum ferromagnets,37we use the sharp momentum\ncuto\u000b (3.22) for the transverse exchange couplings and no\ncuto\u000b at all for the longitudinal couplings. In this case,\nall terms involving the longitudinal single-scale propaga-\ntor_F\u0003(K) in our \row equations should be simply omit-\nted, so that the \row equation (3.15) for the exchange\n\feld\u001e\u0003=\u0000Jz(0)M\u0003=V0M\u0003reduces to\n\u0000zz\n\u0003(0)@\u0003\u001e\u0003=\u0000Z\nK_G\u0003(K)\u0000+\u0000z\n\u0003(K;K; 0);(4.3)\nwhere the transverse single-scale propagator de\fned in\nEq. (4.20) is with our cuto\u000b scheme given by\n_G\u0003(K) =\u0000@\u0003R?\n\u0003(k)\n[\u0000+\u0000\n\u0003(K) +R?\n\u0003(k)]2\n=\u0000\u000e(k\u0000\u0003)Vk\n[\u0000+\u0000\n\u0003(K) + \u0002(\u0003\u0000k)Vk]2: (4.4)\nA technical complication of the sharp momentum cuto\u000b\nscheme is that it leads to expressions involving both \u000e\nand \u0002 functions, which should be carefully de\fned using\nthe identity7,41\n\u000e(x)f(\u0002(x)) =\u000e(x)Z1\n0dtf(t): (4.5)Taking this into account the single-scale propagator reads\n_G\u0003(K) =\u000e(k\u0000\u0003)\u00141\n\u0000+\u0000\n\u0003(K) +Vk\u00001\n\u0000+\u0000\n\u0003(K)\u0015\n:(4.6)\nIn the simplest approximation, the \row of the vertices\n\u0000zz\n\u0003(0) and \u0000+\u0000z\n\u0003(K;K; 0) is neglected, so that these ver-\ntices are approximated by their initial values at scale\n\u0003 = \u0003 0where the deformed transverse exchange interac-\ntion vanishes. Moreover, at low temperatures, the deriva-\ntives of the Brillouin function are exponentially small\nand can be neglected. In this case, we may approximate\n\u0000zz\n0(0)\u00191=V0and \u0000+\u0000z\n0(K;K; 0)\u00191=M0[see Eq. (D 7)\nin Appendix D], so that Eq. (4.3) reduces to the following\n\row equation for the scale-dependent magnetization:\n@\u0003M\u0003=\u00001\nM0Z\nK_G\u0003(K): (4.7)\nNeglecting self-energy corrections to the transverse prop-\nagator, we may approximate\n_G\u0003(K)\u0019\u0000\u000e(k\u0000\u0003)\n\u0000+\u0000\n0(K)=\u0000\u000e(k\u0000\u0003)M0\nH+Ek\u0000i!;\n(4.8)\nwhere we have used the initial value (3.29) for the trans-\nverse two-point function and introduced the magnon dis-\npersion\nEk=M0(V0\u0000Vk)\u0019\u001a0k2: (4.9)\nRecall that the bare spin sti\u000bness \u001a0is de\fned via the\nexpansion (4.2) of the Fourier transform Vkof the ex-\nchange coupling. Substituting Eq. (4.8) into Eq. (4.7)\nand performing the integrations and the Matsubara sum,\nwe obtain in Ddimensions\n@\u0003M\u0003=KDaD\u0003D\u00001\ne\f(H+\u001a0\u00032)\u00001; (4.10)\nwhereKDis the surface area of the D-dimensional unit\nsphere divided by (2 \u0019)D. In Fig. 2, the result of the nu-\nmerical integration of the \row equation (4.10) for D= 2\nis represented by dashed lines. Obviously, for su\u000eciently\nsmall magnetic \feld and high temperatures the magne-\ntization \rows to unphysical negative values within this\napproximation. This is clearly an artefact of the simple\napproximations used to derive Eq. (4.10). In Sec. IV B,\nwe show how this problem can be cured by taking into ac-\ncount a Ward identity relating the magnon self-energy to\nthe magnetization. From Eq. (4.10), it is furthermore\nstraightforward to recover the RG \row equations for\nquantum ferromagnets derived 30 years ago in Ref. [37]\n(see also Ref. [38]) using the momentum-shell RG tech-\nnique. In this approach, the low-temperature behavior\nof quantum ferromagnets in Ddimensions is encoded in\nthe RG \row of the following three dimensionless rescaled10\nFIG. 2. RG \row of the magnetization M\u0003as a function of the\nlogarithmic \row parameter l= ln(\u0003 0=\u0003) forH=J = 0:125 and\nT=J = 1;0:5;0:2 (red, gray, blue; the curves are labeled by the\ncorresponding values of T=J) . The dashed lines represent the\nsolution of the simplest truncation (4.10) of the \row equation\nfor the magnetization ( M\u0003;simp) which completely neglects\nself-energy and vertex corrections. The solid lines show the\nsolution of the improved \row equation (4.22) ( M\u0003;WI) which\ntakes into account the RG \row of the magnon self-energy via\nthe Ward identity (4.19). Note that M\u0003;WI, in contrast to\nM\u0003;simp, does not \row to unphysical negative values for high\ntemperatures.\ncoupling constants:\nt=T(a\u0003)D\u00002\nJSM \u0003; (4.11a)\ng=(a\u0003)D\nM\u0003; (4.11b)\nh=H\nJS(a\u0003)\u00002; (4.11c)\nwhere \u0003 = \u0003 0e\u0000lis the running momentum cuto\u000b and\nthe energy Jis de\fned by40\nJ=\u001a0=(M0a2): (4.12)\nUsing the \row equation (4.10) for the scale-dependent\nmagnetization M\u0003, we \fnd that the above rescaled cou-\nplings satisfy the system of \row equations:\n@lt= (2\u0000D)t+KDgt\neg(1+h)=t\u00001;(4.13a)\n@lg=\u0000Dg+KDg2\neg(1+h)=t\u00001; (4.13b)\n@lh= 2h; (4.13c)\nin agreement with the \row equations given in Refs. [37\nand 38]. By integrating these \row equations from l= 0\nup to some \fnite l=l\u0003where the renormalized dimen-\nsionless temperature t\u0003is of the order of unity, we can\nestimate the correlation length \u0018in units of the lattice\nspacinga. ForH= 0, the result is\n\u0018\na/r\nJS\nTe2\u0019JS2=T; (4.14)where the precise prefactor cannot be determined with\nthis method. Note also that \row equations similar to\nEqs. (4.13) for quantum antiferromagnets have been ob-\ntained by Chakravarty, Halperin, and Nelson42,43by ap-\nplying the conventional momentum-shell RG technique\nto the quantum nonlinear sigma model which is believed\nto describe the low-energy and long-wavelength physics\nof quantum Heisenberg antiferromagnets in the renormal-\nized classical regime (see Ref. [44] for a derivation within\nthe FRG).\nB. Self-energy and vertex corrections\nFrom Fig. 2, we see that in two dimensions the \row\nequation (4.10) implies that at su\u000eciently small mag-\nnetic \feld and large temperature the \rowing magneti-\nzationM\u0003(H;T) becomes negative at some \fnite scale\n\u0003c=e\u0000lc, so that we cannot integrate the \row all the\nway down to \u0003 = 0. This is of course an unphysical\nfeature of our truncation. Within our SFRG approach\nwe can construct a better truncation by including the\n\row of the magnon self-energy \u0006 \u0003(K) as well as vertex\ncorrections. However, in the presence of a \fnite spon-\ntaneous magnetization, the magnon spectrum must be\ngapless, so that the self-energy \u0006 \u0003(K= 0) must vanish\nforH!0. To implement this condition without \fne-\ntuning the initial condition, it is crucial to truncate the\nin\fnite hierarchy of \row equations such that the Ward\nidentity\n\u001f?\u0011G(k= 0;i!= 0) =M\nH(4.15)\nis satis\fed at least at the end of the \row. This identity\nrelating the exact transverse uniform susceptibility \u001f?to\nthe exact magnetization Mhas been discussed long time\nago by Patashinskii and Prokrovskii.45In Appendix C,\nwe give a rigorous derivation of this identity using the\nHeisenberg equations of motion.\n1. Magnon self-energy from the Ward identity\nWe de\fne the \rowing magnon self-energy \u0006 \u0003(K) by\nwriting the transverse two-point vertex \u0000+\u0000\n\u0003(K) in the\nform\n\u0000+\u0000\n\u0003(K) =H+Ek\u0000i!\nM0+ \u0006 \u0003(K); (4.16)\nso that the regularized transverse propagator is\nG\u0003(K) =M0\nH+Ek\u0000i!+M0R?\n\u0003(k) +M0\u0006\u0003(K):\n(4.17)\nObviously, the Ward identity (4.15) can be implemented\nby demanding that the \rowing transverse two-point ver-\ntex at vanishing momentum and frequency is related to11\nthe \rowing magnetization via\n\u0000+\u0000\n\u0003(0) =H\nM\u0003; (4.18)\nwhich implies that the magnon self-energy \u0006 \u0003(0) at van-\nishing momentum and frequency is related to the mag-\nnetization via\n\u0006\u0003(0) =H\nM\u0003\u0000H\nM0: (4.19)\nNeglecting the momentum- and frequency-dependence of\nthe self-energy, we then obtain for the transverse single-\nscale propagator instead of Eq. (4.8),\n_G\u0003(K) =\u0000\u000e(k\u0000\u0003)\nH+Ek\u0000i!\nM0+ \u0006 \u0003(0)\n=\u0000\u000e(k\u0000\u0003)M0\n\u0001\u0003+Ek\u0000i!(4.20)\nwhere\n\u0001\u0003=HM 0\nM\u0003(4.21)\ncan be interpreted as a scale-dependent gap of the\nmagnon dispersion. The resulting modi\fed \row equa-\ntion for the magnetization is\n@\u0003M\u0003=KDaD\u0003D\u00001\ne\f(\u0001\u0003+\u001a0\u00032)\u00001: (4.22)\nA numerical solution of this modi\fed \row equation is\nshown by the solid lines in Fig. 2. Note that now the\nmagnetization never \rows to unphysical negative values.\nThe de\fnition (4.11c) of the rescaled dimensionless mag-\nnetic \feld should then be replaced by\nh=H\nJM\u0003(a\u0003)\u00002; (4.23)\nand the modi\fed system of \row equations for the rescaled\ncouplingst,g, andhreads\n@lt= (2\u0000D)t+KDgt\neg(1+h)=t\u00001;(4.24a)\n@lg=\u0000Dg+KDg2\neg(1+h)=t\u00001; (4.24b)\n@lh= 2h+KDgh\neg(1+h)=t\u00001: (4.24c)\nIn contrast to our earlier \row equation (4.13c), the renor-\nmalized magnetic \feld now has a nontrivial \ructuation\ncorrection.\nIn order to quantify the e\u000bects of additional higher\norder vertex corrections on the magnetization, we inves-\ntigate in the following two more advanced truncations.2. Vertex correction from the Katanin substitution\nA simple approximate method to take vertex correc-\ntions into account is the so-called Katanin substitution,\nwhich amounts to replacing the single-scale propagator\nin the \row equations by a total derivative,46\n_G\u0003(K)!@\u0003G\u0003(K): (4.25)\nThis substitution, known from the FRG formulation of\ninteracting Fermi systems,8,46has been found to be es-\nsential to obtain meaningful results in the pseudofermion\nFRG approach to quantum spin systems.16{19The e\u000bect\nof higher-order vertices is thereby partially taken into\naccount in a weak coupling truncation. In this way, the\nviolation of Ward identities is shifted to a higher order\nin the vertex expansion. Within our simple truncation\nthe Katanin substitution amounts to replacing the single-\nscale propagator in Eq. (4.20) by\n_G\u0003(K)!@\u0003\u0002(k\u0000\u0003)\nH+Ek\u0000i!\nM0+ \u0006 \u0003(0)\n=\u0000\u000e(k\u0000\u0003)\nH+Ek\u0000i!\nM0+ \u0006 \u0003(0)\u0000\u0002(k\u0000\u0003)@\u0003\u0006\u0003(0)\n\u0002H+Ek\u0000i!\nM0+ \u0006 \u0003(0)\u00032:\n(4.26)\nWith this replacement the right-hand side of our trun-\ncated \row equation (4.7) for the magnetization becomes\na total derivative. Using again the Ward identity (4.19)\nto express \u0006(0) at the end of the \row in terms of the\nmagnetization M, we obtain\nM=M0\u00001\nNX\nk1\ne\f(HM0=M+Ek)\u00001: (4.27)\nIn one and two dimensions, the zero-\feld susceptibility\n\u001f= limH!0M=H is expected to be \fnite for any non-\nzero temperature, so that the gap \u0001 \u0003approaches a \fnite\nlimit forH!0. In this limit Eq. (4.27) reduces to the\nfollowing equation for the zero-\feld susceptibility:\nM0=1\nNX\nk1\ne\f(M0=\u001f+Ek)\u00001: (4.28)\nAt low temperatures ( T\u001cV0), we can solve this equa-\ntion by approximating\n1\ne\f(M0=\u001f+Ek)\u00001\u0019T\nM0=\u001f+\u001a0k2(4.29)\nand imposing an ultraviolet cuto\u000b \u0003 0on the momentum-\nintegration such that \f\u001a0\u00032\n0= 1. De\fning the energy\nscale40Jin terms of \u001a0as in Eq. (4.12) and using the\nfact that at low temperatures M0\u0019S, we obtain for the\nsusceptibility at vanishing magnetic \feld,\n\u001f=M0\nTe4\u0019JS2=T: (4.30)12\nThe transverse correlation length \u0018(H;T) can be de\fned\nby writing the magnon dispersion for small momenta as\nHM 0=M+Ek\u0019\u001a0(\u0018\u00002+k2): (4.31)\nForH!0, this leads to the identi\fcation\n\u0018=p\n\u001a0\u001f=M 0: (4.32)\nWith\u001fgiven by Eq. (4.30), this yields\n\u0018\na=r\nJS\nTe2\u0019JS2=T; (4.33)\nwhich agrees (up to a numerical prefactor of the or-\nder of unity) with previous one-loop calculations based\non modi\fed spin-wave theory,25Schwinger-Boson mean-\n\feld theory,28and a one-loop momentum shell renor-\nmalization group calculation.37As shown in Ref. [37],\nhowever, a more accurate two-loop calculation gives an\naddition factor of T=(JS2) in front of the exponential,\ni.e.,\u0018/(T=JS )1=2e2\u0019JS2=T. Similarly, the temperature\ndependence of the prefactor in the expression (4.30) for\nthe susceptibility is known to be modi\fed by two-loop\ncorrections.37Moreover, the one-loop approximation for\nthe zero-\feld susceptibility given in Refs. [25, 28, and 37]\nis a factor of T=(JS) smaller than the result (4.30). A\npossible reason for this discrepancy is that in our trun-\ncation we have neglected the frequency and momentum\ndependence of the magnon self-energy.\nWithin our approach, it is possible to quantify the ver-\ntex correction which is implicitly taken into account via\nthe Katanin substitution. Therefore we use the Ward\nidentity (4.19) to express the scale-derivative of the self-\nenergy on the right-hand side of Eq. (4.26) in terms of\nthe derivative of the magnetization,\n@\u0003\u0006\u0003(0) =\u0000H\nM2\n\u0003@\u0003M\u0003: (4.34)\nOur truncated \row equation (4.7) can then be written as\n@\u0003M\u0003=M0\n\fNX\nk;!\u0000+\u0000z\n\u0003;Kat\u000e(k\u0000\u0003)\n\u0001\u0003+Ek\u0000i!; (4.35)\nwhere the scale-dependent mixed three-point vertex is\ngiven by\nM0\u0000+\u0000z\n\u0003;Kat=1\n1 +\u0001\u0003\nM\u0003R\nK\u0002(\u00030\u0000k)\u0002(k\u0000\u0003)\n[\u0001\u0003+Ek\u0000i!]2: (4.36)\nWe conclude that the Katanin substitution amounts\nto the assumption that the mixed three-legged vertex\n\u0000+\u0000z\n\u0003(\u0000K;K; 0) in the exact FRG \row equation (3.15)\nfor the exchange \feld \u001e\u0003can be approximated by a\nmomentum- and frequency-independent constant \u0000+\u0000z\n\u0003;Kat\nwhich is linked to the \rowing magnetization M\u0003via\nEq. (4.36). In Fig. 3 we show the \row of \u0000+\u0000z\n\u0003;Katin two\ndimensions for T >0 and \fnite magnetic \feld. Although\nthe vertex correction becomes important when the mag-\nnetization is signi\fcantly reduced from its initial value,\nthe \row of the magnetization is very similar to the \row\nobtained without vertex correction.\nFIG. 3. (a) RG \row of the vertex \u0000+\u0000z\n\u0003as a function of the\nlogarithmic \row parameter l= ln(\u0003 0=\u0003) forT=J = 1;0:5;0:2\n(red, gray, blue) and H=J = 0:125, calculated either us-\ning the Katanin substitution (\u0000+\u0000z\n\u0003;Kat, see Eqs. (4.35) and\n(4.36)), or the approximate solution of the SFRG \row equa-\ntions (\u0000+\u0000z\n\u0003;vert, see Eqs. (4.45) and (4.43)). The qualita-\ntive behavior of \u0000+\u0000z\n\u0003;vertand \u0000+\u0000z\n\u0003;Katis similar throughout the\ntemperature range considered. (b) Resulting magnetization\nM(H;T) =M\u0003!0(H;T) as a function of temperature. The\ndotted line represents the solution of the \row equation (4.22)\nwithout vertex correction where M0\u0000+\u0000z\n\u0003= 1. The di\u000berent\nmagnetization curves coincide for high and low temperatures\nand show only small deviations in the intermediate range.\n3. Vertex correction from \row equations\nWe have shown in the previous subsection that the\nKatanin substitution (4.26) amounts to replacing the\nthree-legged vertex \u0000+\u0000z\n\u0003(\u0000K;K; 0) in the \row equa-\ntion (4.3) for the expectation value \u001e\u0003of the exchange\n\feld by a momentum- and frequency-independent cou-\npling \u0000+\u0000z\n\u0003;Katgiven by Eq. (4.36). To check the validity of\nthis substitution, we now give an independent calculation\nof the vertex \u0000+\u0000z\n\u0003. In principle, we could write down\nthe corresponding exact \row equation, which depends on\nvarious higher-order vertices. Alternatively, we can use\nthe Ward identity (4.19) to determine \u0000+\u0000z\n\u0003from the re-13\nquirement that the \row equations for @\u0003M\u0003and@\u0003\u0006\u0003(0)\ngive consistent results. Within our cuto\u000b scheme where\nonly the transverse part of the exchange interaction is de-\nformed, the exact \row equation (3.16) for the transverse\ntwo-point vertex reduces to\n@\u0003\u0006\u0003(0) = \u0000+\u0000z\n\u0003(0;0;0)V0@\u0003M\u0003\n+Z\nQ_G\u0003(Q)\u0000++\u0000\u0000\n\u0003 (0;\u0000Q;Q; 0)\n\u0000Z\nQ_G\u0003(Q)F\u0003(Q)\u0000+\u0000z\n\u0003(0;Q;\u0000Q)\u0000+\u0000z\n\u0003(\u0000Q;0;Q):\n(4.37)\nTo simplify the calculation, we replace the vertices in\nthe second and third lines of Eq. (4.37) by their initial\nvalues given by the tree approximation discussed in Ap-\npendix D, see Eq. (D 10). This amounts to neglecting\nthe contribution involving the three-legged vertices in the\nlast line of Eq. (4.37), and replacing in the second line\nZ\nQ_G\u0003(Q)\u0000++\u0000\u0000\n\u0003 (0;\u0000Q;Q; 0)!U0Z\nQ_G\u0003(Q);(4.38)\nwhere the coupling constant U0is given by\nU0\u0019H+V0M0\nM3\n0: (4.39)\nUsing the Ward identity in the form (4.34) to express\n@\u0003\u0006\u0003(0) in terms of @\u0003M\u0003, we \fnally obtain\n@\u0003M\u0003=\u0000U0\nH=M2\n\u0003+V0\u0000+\u0000z\n\u0003Z\nK_G\u0003(K):(4.40)\nOn the other hand, for a momentum-independent three-\nlegged vertex, the \row equation (4.3) for M\u0003is of the\nform\n@\u0003M\u0003=\u0000\u0000+\u0000z\n\u0003Z\nK_G\u0003(K); (4.41)\nso that in this approximation the vertex \u0000+\u0000z\n\u0003 satis\fes\nthe compatibility condition\n\u0000+\u0000z\n\u0003=U0\nH=M2\n\u0003+V0\u0000+\u0000z\n\u0003: (4.42)\nSolving for \u0000+\u0000z\n\u0003we obtain\n\u0000+\u0000z\n\u0003=\"\u0012H\n2V0M2\n\u0003\u00132\n+U0\nV0#1=2\n\u0000H\n2V0M2\n\u0003:(4.43)\nKeeping in mind that U0=V0= 1=M2\n0+H=(V0M3\n0), we\nsee that at the initial scale \u0003 = \u0003 0whereM\u00030=M0the\nright-hand side of Eq. (4.43) reduces to 1 =M0, while for\nsmallM\u0003the vertex vanishes as\n\u0000+\u0000z\n\u0003\u0018H+V0M0\nHM2\n\u0003\nM2\n0: (4.44)With a sharp momentum cuto\u000b the \row of the magneti-\nzation is therefore given by\n@\u0003M\u0003=M0\n\fNX\nk;!\u0000+\u0000z\n\u0003\u000e(k\u0000\u0003)\n\u0001\u0003+Ek\u0000i!: (4.45)\nIn Fig. 3 we compare the \row of the vertex (4.43) with\nthe corresponding vertex \row implied by the Katanin\nsubstitution, see Eq. (4.36). The qualitative behavior is\nsimilar, so that the Katanin substitution at least quali-\ntatively takes the vertex correction due to the \row of the\nmixed three-point vertex into account. Figure 3(b) re-\nveals furthermore that the resulting magnetization curves\nof the three di\u000berent approximations used in this section\ndo not di\u000ber signi\fcantly, indicating that the ful\flment\nof the Ward identity (4.15) is the essential feature of any\nof these truncations.\n4. Wave-function renormalization\nFinally, let us include the wave-function renormaliza-\ntion factor Z\u0003of the magnons which is related to the\nfrequency-dependence of the magnon self-energy via\n\u0006\u0003(k;i!) = \u0006 \u0003(0) +M\u00001\n0\u0000\n1\u0000Z\u00001\n\u0003\u0001\ni!+O(!2;k2):\n(4.46)\nFor our purpose it is su\u000ecient to approximate the vertices\nin the exact \row equation (3.16) for the transverse two-\npoint vertex by their initial values given in Appendix D.\nThen we obtain\n@\u0003\u0006\u0003(K) =@\u0003\u0006a\n\u0003(K) +@\u0003\u0006b\n\u0003(K); (4.47)\nwith\n@\u0003\u0006a\n\u0003(k;i!) =1\n\fNM2\n0X\nk0;!0_G\u0003(k0;i!0)\n\u0002h\nG\u00001\n1(!) +G\u00001\n1(!0)\u0000Vk\u0000k0\u0000V0i\n;\n(4.48)\n@\u0003\u0006b\n\u0003(k;i!) =\u00001\n\fNM2\n0X\nk0;!0_G\u0003(k0;i!0)\f\u000e!;!0b0\n1\u0000\fb0Vk\u0000k0\n\u0002[G\u00001\n1(!)\u0000Vk\u0000k0][G\u00001\n1(!0)\u0000Vk\u0000k0];\n(4.49)\nwhere with our cuto\u000b scheme\nG\u00001\n1(!) =H+V0M0\u0000i!\nM0; (4.50)\nsee Eq. (3.27). At low temperatures, the derivatives of\nthe Brillouin function are exponentially small, so that the\nsecond contribution \u0006b\n\u0003(k;i!) can be neglected. From\nthe frequency-dependence of \u0006a\n\u0003(k;i!), we obtain for the\n\rowing wave-function renormalization\n@\u0003Z\u0003=\u0000Z2\n\u0003\nM2\n0Z\nK_G\u0003(K)\n=KD(a\u0003)D\u00001\nM0Z3\n\u0003\ne\fZ\u0003(\u0001\u0003+\u001a0\u00032)\u00001:(4.51)14\nFIG. 4. (a) Magnetization curve M(T;H) inD= 2 as a\nfunction of temperature obtained from the numerical integra-\ntion of Eqs. (4.51) and (4.52) for H=J = 1:0;0:6;0:2;0:05\n(top to bottom; the values of H=J are written next to the\ncurves). Here J=\u001a0=(M0a2) is de\fned in terms of the\nbare spin-sti\u000bness \u001a0, see Eqs. (4.2) and (4.12). The dots\nare the Monte Carlo results of Ref. [36] for a spin S= 1=2\nHeisenberg ferromagnet with nearest neighbor exchange J.\n(b)M(H;T) as a function of the magnetic \feld for tempera-\nturesT=J = 1;0:5;0:2 (red, gray, blue; the values of T=J are\nwritten next to the curves).\nIn the following section, we discuss the resulting magne-\ntization and wave-function renormalization as functions\nof temperature and magnetic \feld strength.\nC. Magnetization curves\nTaking into account the wave-function renormalization\nfactorZ\u0003and vertex correction given in Eq. (4.43), our\nmodi\fed \row equation for the magnetization becomes\n\u0003@\u0003M\u0003=M0KDZ\u0003\u0000+\u0000z\n\u0003(a\u0003)D\ne\fZ\u0003(\u0001\u0003+\u001a0\u00032)\u00001; (4.52)\nwhich should be solved simultaneously with the \row\nequation (4.51) for the wave-function renormalization\nFIG. 5. (a) Wave-function renormalization factor Z(H;T)\ninD= 2 as a function of temperature obtained from the\nnumerical integration of Eqs. (4.51) and (4.52) for H=J =\n1:0;0:6;0:2;0:05 (top to bottom). (b) Z(H;T) as a function\nof magnetic \feld for temperatures T=J = 1;0:5;0:2 (red, gray,\nblue).\nfactor. Note that the vertex \u0000+\u0000z\n\u0003 and the gap \u0001 \u0003=\nHM 0=M\u0003are functions of the \rowing magnetization\nM\u0003, see Eqs. (4.43) and (4.21). By integrating the\n\row equations (4.52) and (4.51) from some large ini-\ntial scale \u0003 0of the order of the inverse lattice spacing\ndown to \u0003 = 0, we obtain nonperturbative expressions\nfor the magnetic equation of state M(H;T) and the corre-\nsponding wave-function renormalization factor Z(H;T),\nwhich remain well-de\fned in low dimensions even in the\nlimit of vanishing magnetic \feld, see Figs. 4 and 5. Re-\ncall that in Eq. (4.12) we have de\fned the energy scale\nJ=\u001a0=(M0a2) in terms of the bare spin sti\u000bness.40For\nlow temperatures and \fnite magnetic \feld, both M=S\nandZremain close to unity, indicating that magnons\nare well-de\fned quasiparticles. When T=J is not small\nthe magnetization and the wave-function renormalization\nfactor both decrease. The identity (4.31) implies that\nthen also the correlation length decreases. For not too\nsmall magnetic \feld, the quantitative agreement of our\nSFRG calculation with Monte Carlo simulations36ex-15\ntends to temperatures of order J, while forH\u001cJ, our\nSFRG result for M(H;T) agrees with the Monte Carlo\nresults only for temperatures T.0:2J. This is due to the\nfact that in our truncation we have neglected all terms\ninvolving the derivatives of the Brillouin function, which\nforH\u001cJis only justi\fed for temperatures T\u001cJ.\nFrom Fig. 5 (b), we also see that for any \fnite mag-\nnetic \feldHthe wave-function renormalization factor Z\nremains \fnite. On the other hand, Z(H;T) vanishes if\nwe take the limit H!0 forT > 0; the correspond-\ning steepening of the slope @M(H;T)=@HjH=0re\rects\nthe exponential behavior of the susceptibility calculated\nin Eq. (4.30). The fact that the Monte Carlo calcula-\ntions have been performed for a nearest neighbor Heisen-\nberg model whereas our truncation of the hierarchy of the\nSFRG \row equations is controlled only for long-range in-\nteractions40does not a\u000bect the quantitative accuracy of\nour SFRG calculation at low temperatures because in this\nregime the relevant energy scale is set by the spin sti\u000b-\nness\u001a0de\fned in terms of the small-momentum expan-\nsion (4.9) of the magnon dispersion. We conclude that at\nlow temperatures the magnetization curves predicted by\nour SFRG approach agree with controlled Monte Carlo\nsimulations for two-dimensional quantum Heisenberg fer-\nromagnets.36Similar results have been obtained with a\n1=N-expansion,31with Green function methods,34and\nwith an exact diagonalization calculation.35\nV. MAGNON DAMPING DUE TO CLASSICAL\nLONGITUDINAL FLUCTUATIONS\nLet us now calculate the damping of magnons due\nto the coupling to classical longitudinal spin \ructua-\ntions at intermediate temperatures. This decay chan-\nnel is not properly taken into account in the usual spin-\nwave expansion where \ructuations of the length of the\nmagnetic moments are not treated as independent de-\ngrees of freedom.24For a three-dimensional ferromagnet,\nthe leading perturbative contribution to this decay pro-\ncess has already been studied by VLP3using their spin-\ndiagrammatic approach. To obtain their result for the\ndecay rate within our SFRG approach, we can neglect\nself-energy corrections to the single-scale propagator on\nthe right-hand side of the \row equation for the self-energy\n\u0006b\n\u0003(k;i!) in Eq. (4.49). The right-hand side of the \row\nequation is then a total derivative which can easily be in-\ntegrated over the \row parameter. After analytic continu-\nation to real frequencies we then obtain the perturbative\nresult for the damping given by VLP,3\n\r(k;!) =\u0000M0Im\u0006b\n\u0003(k;!+i0)\n=\u0019b0\nNX\nk0\u000e(H+Ek0\u0000!)(Vk0\u0000Vk\u0000k0)2\n1\u0000\fb0Vk\u0000k0:(5.1)\nIn two dimensions, this expression is not valid for small\nmagnetic \feld, because we know that self-energy correc-\ntions generate a gap in the magnon spectrum and there-\nfore cannot be neglected. Unfortunately, the inclusion ofself-energy corrections requires also a consistent renor-\nmalization of the higher-order interaction vertices which\nis beyond the scope of this work. A simple phenomeno-\nlogical way to take self-energy e\u000bects into account is to\nreplace the bare magnetic \feld Hin Eq. (5.1) by the gap\n\u0001 =HM 0\nM(H;T)(5.2)\nand divide the external frequency by the wave-function\nrenormalization factor Z, both of which are calculated by\nmeans of our SFRG approach. This leads to the following\nexpression for the magnon damping,\n\r(k;!) =\u0019b0\nNX\nk0\u000e(\u0001 +Ek0\u0000!=Z)(Vk0\u0000Vk\u0000k0)2\n1\u0000\fb0Vk\u0000k0:\n(5.3)\nFormally, the renormalizations described by \u0001 and Zcan\nbe generated by replacing the mean-\feld inverse propa-\ngatorsG\u00001\n1(i!) in the second line of Eq. (4.49) by ap-\npropriate renormalized propagators, which amounts to\nassuming that the structure of renormalized vertex re-\nsembles the bare one. Evaluating \r(k;!) within a small-\nmomentum expansion, we plot in Fig. 6 the resulting\nspectral density function\nS(k;!) =1\n\u0019ImG(k;i!!!+i0)\n=1\n\u0019M0\r(k;!)\n[\u0001 +Ek\u0000!=Z]2+\r(k;!)2(5.4)\nas a function of temperature and magnetic \feld for a\ngeneric momentum k0. Note that the threshold for\nthe spectral weight is controlled by the \u000efunction in\nEq. (5.3), which sets S(k;!) = 0 for! < Z (\u0001 +Ek0).\nIn the limit H!0, this implies that the spectral weight\nS(k;!) vanishes for ! < ZM 0=\u001f(T) +Ek0, where\u001f(T)\nis the susceptibility for vanishing \feld. With increas-\ning magnetic \feld and decreasing temperature the re-\nsulting quasiparticle-peaks grow and sharpen, so that\nthe magnons are stabilized in this regime. The same\nholds true when the magnitude of the magnon momen-\ntumk0is lowered (not shown in Fig. 6). The momen-\ntum and frequency dependence of \r(k;!) furthermore\nleads to a non-Lorentzian asymmetry of the spectral\nlineshape, which increases when the quasiparticle-peaks\nbroaden. It should be emphasized that at low temper-\natures (T\u001cJ) the derivative b0of the Brillouin func-\ntion and hence also the magnon damping in Eq. (5.3) are\nexponentially small. In this regime we expect that the\nmagnon damping is dominated by two-body scattering\nof renormalized magnons. The proper treatment of these\ntype of processes, taking into account that at \fnite tem-\nperature magnons cannot propagate over distances larger\nthan the correlation length, is beyond the scope of this\nwork.16\nFIG. 6. Spectral density function S(k0;!) as a func-\ntion of frequency and temperature for k0a= 0:75 and dif-\nferent magnetic \felds H=J = 0:001;0:01;0:1 (from left to\nright). The quasiparticle peaks sharpen with increasing\nmagnetic \feld strength and decreasing temperature. Note\nthat the lineshapes of the peaks are not symmetrical around\n!=Z\u0001 +ZEk0but show a slight asymmetry due to the\n!-dependence of \r(k0;!).\nVI. SUMMARY AND OUTLOOK\nIn this work we have used the spin functional renor-\nmalization group approach (SFRG) recently proposed by\ntwo of us1to study the thermodynamics and dynamics\nof two-dimensional quantum Heisenberg ferromagnets at\nlow temperatures. We have established the precise re-\nlation between the SFRG and an old momentum-shell\nRG-calculation,37and have derived the temperature de-\npendence of the susceptibility and transverse correlation\nlength, which agree up to a prefactor in the suscepti-\nbility with Takahashi's modi\fed spin-wave theory25and\nSchwinger-Boson calculations,28respectively. Our SFRG\nresults for the magnetic equation of state in two dimen-\nsions agrees quite well with numerical simulations. More-\nover, we have also shown how the damping of transverse\nspin-waves due to the coupling to longitudinal spin \ruc-\ntuations can be obtained within our SFRG approach. At\nintermediate temperatures, this damping can be substan-\ntial in two dimensions and generates an asymmetry in the\nspectral line shape.\nThis work also contains several technical advances\nwhich will be helpful for future applications of the SFRG\nto quantum spin systems. Of particular importance are\nthe following three points.\n1. The construction of the hybrid functional \u0000 \u0003[m;\u001e]\nin Eq. (2.11) which generates vertices which are ir-\nreducible in the transverse propagator line and in\nthe longitudinal interaction line. This functional\nsatis\fes the Wetterich equation (2.19) and has a\nwell-de\fned initial condition even if we start from\na deformed Hamiltonian where the exchange in-\nteractions are completely switched o\u000b. The two-point functions generated by this functional can\nbe identi\fed physically with the self-energy \u0006 \u0003(K)\nof the transverse magnons and the irreducible po-\nlarization \u0005 \u0003(K) of longitudinal spin \ructuations.\nThis establishes the precise relation between our\nSFRG approach and the diagrammatic method for\nquantum spin systems developed by VLP2,3and by\nothers.4,47\n2. The use of the Ward identity (4.15) to obtain a\nclosed \row equation for the magnetization. This\nis crucial in the regime where the spin-rotational\nsymmetry is spontaneously broken and the magnon\nspectrum is gapless. The Ward identity guaran-\ntees in this case that the gap in the magnon spec-\ntrum vanishes in the entire symmetry broken phase\nwithout \fne-tuning the initial conditions. Note also\nthat, in contrast to Takahashi's modi\fed spin-wave\ntheory,25,26in our approach, we do not assume a\npriori that for H= 0 the magnetization vanishes in\ntwo dimensions.\n3. In a deformation scheme where initially the ex-\nchange interactions are completely switched o\u000b, the\ninitial condition for the renormalization group \row\nis determined by the connected imaginary-time or-\ndered correlation functions of a single spin in a mag-\nnetic \feld. The diagrammatic calculation of these\ncorrelation function using the generalized Wick the-\norem for spin operators derived by VLP2,3are\nrather cumbersome. We have derived an explicit\nrecursive form of the generalized Wick theorem for\nspin operators in frequency space, see Eq. (B 14),\nwhich provides us with an e\u000ecient method to cal-\nculate the higher-order connected spin correlation\nfunctions.\nAnother challenging problem where our SFRG ap-\nproach promises to be useful is the calculation of the\nlongitudinal part of the dynamic structure factor of an or-\ndered ferromagnet. VLP3suggested that for su\u000eciently\nsmall wave-vectors the longitudinal structure factor of\nan ordered ferromagnet should exhibit a di\u000busive peak\nat vanishing frequency and two inelastic peaks at fre-\nquencies corresponding to the magnon energies. How-\never, the diagrammatic resummation within the frame-\nwork of the spin-diagrammatic approach to con\frm this\nscenario has not been found.3,48Recently, it has been\nshown by means of a kinetic equation approach that in\nthe absence of momentum-relaxing scattering processes\n(such as umklapp processes or scattering by disorder) the\nlongitudinal structure factor of a two-dimensional ferro-\nmagnet in a magnetic \feld exhibits a linearly dispersing\nhydrodynamic sound mode at low frequencies, which is\ninduced by \ructuations of the magnon density.49In three\ndimensions, such a mode was also found by Izyumov et\nal.47by means of the spin-diagram technique.2{4It would\nbe interesting to see whether such a sound-like magnon\nmode can also be obtained within our SFRG approach.17\nFinally, let us point out that our SFRG approach can\nbe extended in many directions. Our goal is to develop\nour method to become a useful tool for studying vari-\nous types of frustrated spin systems in the regime with-\nout long-range magnetic order. Note that in the past\nfew years these types of systems have been studied us-\ning the so-called pseudofermion FRG,16{19which relies\non the represention of the spin-1 =2 operators in terms of\nfermionic operators.20The unphysical states which are\nintroduced by this representation have to be projected\nout which in practice can only be achieved approximately.\nThis problem does not arise within our SFRG approach\nwhere we directly work with the physical spin operators.\nACKNOWLEDGMENTS\nThis work was \fnancially supported by the Deutsche\nForschungsgemeinschaft (DFG) through project KO\n1442/10-1.\nAPPENDIX A: RELATIONS BETWEEN\nIRREDUCIBLE VERTICES AND CONNECTED\nCORRELATION FUNCTIONS\nIn this appendix we work out the precise relation\nbetween the irreducible vertices generated by our hy-\nbrid functional ~\u0000\u0003[m;'] = \u0000\u0003[m;\u001e\u0003+'] de\fned via\nEqs. (2.25) and (3.3) on the one hand, and the connected\nspin correlation functions generated by G\u0003[h] de\fned in\nEq. (2.5). With a slight variation of the notation intro-\nduced in Eq. (2.21), we de\fne the three-component \feld\n\bKwith spherical transverse components,\n\bK=0\n@\b+\nK\n\b\u0000\nK\n\bz\nK1\nA=0\n@m+\nK\nm\u0000\nK\n\u001eK1\nA; (A 1)\nwhere the spherical Fourier components m\u0006\nKare de\fned\nas in Eq. (3.1) implying ( m+\nK)\u0003=m\u0000\n\u0000K, see Eq. (3.2).\nNote that in this appendix the superscript \u000bassumes thevalues +;\u0000;z. We also introduce the conjugate three-\n\ravor source \feld\njK=0\n@j+\nK\nj\u0000\nK\njz\nK1\nA=0\n@h+\nK\nh\u0000\nK\nsK1\nA: (A 2)\nThe generating functional \u0000 \u0003[\b] = \u0000 \u0003[m;\u001e] de\fned in\nEq. (2.11) can then be written as\n\u0000\u0003[\b] =Z\nK\by\nKjK\u0000F\u0003[j]\u00001\n2Z\nK\by\nKR\u0003(k)\bK;(A 3)\nwhere the hybrid functional F\u0003[j] =F\u0003[h+;h\u0000;s] is de-\n\fned in Eq. (2.6), i.e.,\nF\u0003[j] =G\u0003h\nh+\nK;h\u0000\nK;hz\nK=\u0000Jz\n\u0003(k)sKi\n\u00001\n2Z\nKJz\n\u0003(k)s\u0000KsK: (A 4)\nThe regulator matrix R\u0003(k) is diagonal in \ravor space,\nR\u0003(k) =0\n@R?\n\u0003(k) 0 0\n0R?\n\u0003(k) 0\n0 0 R\u001e\n\u0003(k)1\nA; (A 5)\nwhere the transverse and longitudinal regulators R?\n\u0003(k)\nandR\u001e\n\u0003(k) in momentum space are given in Eqs. (3.8)\nand (3.9). To take into account that the third component\n\u001eKof\bKcan have a \fnite expectation value, we set\n\u001eK=\u000e(K)\u001e\u0003+'Kand de\fne [see Eq. (2.25]\n~\u0000\u0003[~\b] = \u0000 \u0003[m+\nK;m\u0000\nK;\u001eK=\u000e(K)\u001e\u0003+'K];(A 6)\nwhere the third component of\n~\bK=0\n@m+\nK\nm\u0000\nK\n'K1\nA (A 7)\ncontains the \ructuating part 'Kof the longitudinal ex-\nchange \feld \u001eK. The expansion of ~\u0000\u0003[~\b] in powers of\n~\bde\fnes the irreducible hybrid vertices, see Eq. (3.3).\nSimilarly, we may expand the functional F\u0003[j] in powers\nof the sources j,\nF\u0003[j] =F\u0003[0] +Z\nKG+\u0000\n\u0003(K)h\u0000\n\u0000Kh+\nK+1\n2!Z\nKF\u0003(K)s\u0000KsK\n+Z\nK1Z\nK2Z\nK3\u000e(K1+K2+K3)F+\u0000z\n\u0003(K1;K2;K3)h\u0000\nK1h+\nK2sK3\n+1\n3!Z\nK1Z\nK2Z\nK3\u000e(K1+K2+K3)Fzzz\n\u0003(K1;K2;K3)sK1sK2sK3+::: : (A 8)\nThe relations between the correlation functions generated\nbyF\u0003[j] and the vertices generated by its (subtracted)Legendre transform ~\u0000\u0003[~\b] can be obtained by successive\ndi\u000berentiation of the \feld-dependent relation of the Hes-18\nsian matrices,\n\u0010\n~\u000000\n\u0003[~\b] +R\u0003\u0011\naa0=\u0010\nF00\n\u0003[j]\u0011\u00001\naa0; (A 9)\nwhere the collective labels a= (\u000b;K) anda0= (\u000b0;K0)\nrepresent the \ravour index \u000b= +;\u0000;zin combination\nwith the momentum-frequency label K= (k;i!), and the\ndouble-primes represent the second functional derivativeswith respect to the corresponding \felds,\n\u0010\n~\u000000\n\u0003[~\b]\u0011\naa0=\u000e2~\u0000\u0003[~\b]\n\u000e~\b\u000b\nK\u000e~\b\u000b0\nK0; (A 10)\n\u0010\nF00\n\u0003[j]\u0011\naa0=\u000e2F\u0003[j]\n\u000ej\u000b\nK\u000ej\u000b0\nK0: (A 11)\nBy taking additional derivatives of Eq. (A 9) with respect\nto~\b\u000b\nKwe \fnd for the third- and fourth-order derivative\ntensors,\n\u0010\n~\u0000(3)\n\u0003[~\b]\u0011\na1a2a3=\u0000X\na0\n1a0\n2a0\n3\"3Y\ni=1(F00\n\u0003[j])\u00001\naia0\ni#\u0010\nF(3)\n\u0003[j]\u0011\na0\n1a0\n2a0\n3; (A 12)\n\u0010\n~\u0000(4)\n\u0003[~\b]\u0011\na1a2a3a4=\u0000X\na0\n1a0\n2a0\n3a0\n4\"4Y\ni=1(F00\n\u0003[j])\u00001\naia0\ni#\u0010\nF(4)\n\u0003[j]\u0011\na0\n1a0\n2a0\n3a0\n4\n+Sa1;a2;a3;a41\n2X\na0\n1a0\n2\u0010\n~\u0000(3)\n\u0003[~\b]\u0011\na1a2a0\n1(F00\n\u0003[j])\u00001\na0\n1a0\n2\u0010\n~\u0000(3)\n\u0003[~\b]\u0011\na0\n2a3a4: (A 13)\nHere, the operator Sa1;a2;a3;a4symmetrizes the expression to its right with respect to the exchange of all labels,7and\nthe summation over the internal labels is de\fned byP\na=R\nKP\n\u000b=+;\u0000;z. By setting ~\b= 0 and j= 0 in Eqs. (A\n12) and (A 13) we obtain the desired expansion of the three-point and four-point vertices generated by our hybrid\nfunctional ~\u0000\u0003[~\b] in powers of the correlation functions generated by F\u0003[j]. Using the fact that for vanishing sources\n(F00\n\u0003[0])+\u0000\nKK0=\u000e(K+K0)G\u0000+\n\u0003(K) =\u000e(K+K0)G\u0003(\u0000K); (A 14a)\n(F00\n\u0003[0])\u0000+\nKK0=\u000e(K+K0)G+\u0000\n\u0003(K) =\u000e(K+K0)G\u0003(K); (A 14b)\n(F00\n\u0003[0])zz\nKK0=\u000e(K+K0)F\u0003(K); (A 14c)\nwhere the scale-dependent transverse propagator interaction G\u0003(K) is given in Eq. (3.4) and the longitudinal e\u000bective\ninteraction F\u0003(K) is given in Eq. (3.5), we obtain from Eq. (A 12) for the three-point vertices de\fned via the vertex\nexpansion (3.3) in momentum-frequency space,\n\u0000+\u0000z\n\u0003(K1;K2;K3) =\u0000G\u00001\n\u0003(\u0000K1)G\u00001\n\u0003(K2)F\u00001\n\u0003(K3)F+\u0000z\n\u0003(\u0000K1;\u0000K2;\u0000K3); (A 15)\n\u0000zzz\n\u0003(K1;K2;K3) =\u0000\"3Y\ni=1F\u00001\n\u0003(Ki)#\nFzzz\n\u0003(\u0000K1;\u0000K2;\u0000K3): (A 16)\nUsing the relation (2.6) between the hybrid functional F\u0003[j] and the generating functional G\u0003[h] of the connected spin\ncorrelation functions, we can express the three-point functions F+\u0000z\n\u0003(\u0000K1;\u0000K2;\u0000K3) andFzzz\n\u0003(\u0000K1;\u0000K2;\u0000K3)\non the right-hand side of Eqs. (A 15) and (A 16) in terms of the connected spin correlation functions. In momentum-\nfrequency space we obtain\nF\u00001\n\u0003(K3)F+\u0000z\n\u0003(\u0000K1;\u0000K2;\u0000K3) =F\u00001\n\u0003(K3)\u0000\n\u0000Jz\n\u0003;k3\u0001\nG+\u0000z\n\u0003(\u0000K1;\u0000K2;\u0000K3)\n=\u0002\n1 +Jz\n\u0003;k3\u0005\u0003(K3)\u0003\nG+\u0000z\n\u0003(\u0000K1;\u0000K2;\u0000K3); (A 17)\n\"3Y\ni=1F\u00001\n\u0003(Ki)#\nFzzz\n\u0003(\u0000K1;\u0000K2;\u0000K3) =\"3Y\ni=1\u0000\n1 +Jz\n\u0003;ki\u0005\u0003(Ki)\u0001#\nGzzz\n\u0003(\u0000K1;\u0000K2;\u0000K3): (A 18)\nNote that the relation (A 18) between longitudinal three-point functions follows also directly from Eq. (2.9). Sub-\nstituting Eqs. (A 17) and (A 18) into Eqs.(A 15) and (A 16), we obtain the expansion of the three-point vertices in\nterms of the connected spin correlation functions,\n\u0000+\u0000z\n\u0003(K1;K2;K3) =\u0000G\u00001\n\u0003(\u0000K1)G\u00001\n\u0003(K2)\u0002\n1 +Jz\n\u0003;k3\u0005\u0003(K3)\u0003\nG+\u0000z\n\u0003(\u0000K1;\u0000K2;\u0000K3); (A 19)\n\u0000zzz\n\u0003(K1;K2;K3) =\u0000\"3Y\ni=1\u0000\n1 +Jz\n\u0003;ki\u0005\u0003(Ki)\u0001#\nGzzz\n\u0003(\u0000K1;\u0000K2;\u0000K3): (A 20)19\nAnalogously, using Eq.(A 13) we \fnd for the four-point vertices,\n\u0000++\u0000\u0000\n\u0003 (K1;K2;K3;K4) =\u0000G\u00001\n\u0003(\u0000K1)G\u00001\n\u0003(\u0000K2)G\u00001\n\u0003(K3)G\u00001\n\u0003(K4)G++\u0000\u0000\n\u0003 (\u0000K1;\u0000K2;\u0000K3;\u0000K4)\n+n\n\u0000+\u0000z\n\u0003(K1;K3;\u0000K1\u0000K3)F(\u0000K1\u0000K3)\u0000+\u0000z\n\u0003(K2;K4;\u0000K2\u0000K4) + (K3$K4)o\n;\n(A 21)\n\u0000+\u0000zz\n\u0003 (K1;K2;K3;K4) =\u0000G\u00001\n\u0003(\u0000K1)G\u00001\n\u0003(K2)[1 +Jz\n\u0003;k3\u0005(K3)][1 +Jz\n\u0003;k4\u0005(K4)]G+\u0000zz\n\u0003 (\u0000K1;\u0000K2;\u0000K3;\u0000K4)\n+n\n\u0000+\u0000z\n\u0003(K1;\u0000K1\u0000K3;K3)G\u0003(\u0000K1\u0000K3)\u0000+\u0000z\n\u0003(\u0000K2\u0000K4;K2;K4) + (K3$K4)o\n+ \u0000zzz\n\u0003(K3;K4;\u0000K3\u0000K4)F\u0003(\u0000K3\u0000K4)\u0000+\u0000z\n\u0003(K1;K2;\u0000K1\u0000K2): (A 22)\n\u0000zzzz\n\u0003(K1;K2;K3;K4) =\u0000\"4Y\ni=1\u0000\n1 +Jz\n\u0003;ki\u0005(Ki)\u0001#\nGzzzz\n\u0003(\u0000K1;\u0000K2;\u0000K3;\u0000K4)\n+n\n\u0000zzz\n\u0003(K1;K3;\u0000K1\u0000K3)[1 +Jz\n\u0003;\u0000k1\u0000k3\u0005(\u0000K1\u0000K3)]\u0000zzz\n\u0003(K2;K4;\u0000K2\u0000K4)o\n;\n(A 23)\nwhere we have introduced the notation ff(::;K 3;::) + (K3$K4)g=ff(::;K 3;::) +f(::;K 4;::)gwith an arbitrary\nfunctionf.\nAPPENDIX B: GENERALIZED BLOCKS AND\nWICK THEOREM FOR SPIN OPERATORS\nFor vanishing exchange interaction all spins are com-\npletely decoupled so that all spin correlation functions are\ndiagonal in the site index. The on-site time-ordered con-\nnected spin correlation functions can then be expanded\nin frequency space as follows,\nG\u000b1:::\u000bn\n0 (\u001c1;:::;\u001cn)\u0011hT [S\u000b1(\u001c1):::S\u000bn(\u001cn)]iconnected\n=1\n\fnX\n!1:::!ne\u0000i(!1\u001c1+:::+!n\u001cn)~G\u000b1:::\u000bn\n0 (!1;:::;!n):\n(B 1)\nTranslational invariance in imaginary time implies that\nwe can factor out a frequency-conserving \u000e-function,\n~G\u000b1:::\u000bn\n0 (!1;:::;!n)\n=\u000e(!1+\u0001\u0001\u0001+!n)G\u000b1:::\u000bn\n0 (!1;:::;!n);(B 2)\nwhere we have introduced the discretized \u000e-function in\nfrequency space,\n\u000e(!) =\f\u000e!;0: (B 3)\nIn the book by Izyumov and Skryabin4time-ordered con-\nnected spin correlation functions G\u000b1:::\u000bn\n0 (!1;:::;!n) are\ncalled generalized blocks . In the local limit the longitu-\ndinal correlations are purely static and can be expressed\nin terms of the derivatives of the Brillouin function,\n~Gzz\n0(!1;!2) =\u000e(!1)\u000e(!2)b0; (B 4)\n~Gzzz\n0(!1;!2;!3) =\u000e(!1)\u000e(!2)\u000e(!3)b00; (B 5)\n~Gzzzz\n0(!1;!2;!3;!4) =\u000e(!1)\u000e(!2)\u000e(!3)\u000e(!4)b000:(B 6)\nThe transverse two-spin correlation functions are\n~G+\u0000\n0(!;!0) =\u000e(!+!0)G0(!); (B 7)\n~G\u0000+\n0(!;!0) =\u000e(!+!0)G0(\u0000!); (B 8)where\nG0(!) =b\nH\u0000i!; (B 9)\nand the spin- SBrillouin function b=b(\fH) is given in\nEq. (3.26). The mixed three-spin correlation function is\nbG+\u0000z\n0(!1;!2;!3) =\u0000G0(!1)G0(\u0000!2) +G0(!1)\u000e(!3)b0;\n(B 10)\nand the purely transverse connected four-spin correlation\nfunction is\nb2G++\u0000\u0000\n0 (!1;!2;!3;!4) =\n\u0000G0(!1)G0(!2)[G0(\u0000!3) +G0(\u0000!4)]\n+G0(!1)G0(!2)[\u000e(!1+!3) +\u000e(!1+!4)]b0:(B 11)\nThere is also a mixed four-spin correlation function in-\nvolving two transverse and two longitudinal spin compo-\nnents,\nb2G+\u0000zz\n0 (!1;!2;!3;!4) =\nG0(!1)G0(\u0000!2)[G0(!1+!3) +G0(!1+!4)]\n\u0000G0(!1)G0(\u0000!2)[\u000e(!3) +\u000e(!4)]b0\n+G0(!1)\u000e(!3)\u000e(!4)bb00: (B 12)\nThe above expressions for the connected spin correlation\nfunctions up to fourth order have have \frst been derived\nby VLP,2see also Ref.[4]. The higher-order connected\nspin correlation functions can in principle be calculated\ndiagrammatically using the generalized Wick theorem for\nspin operators derived by VLP.2Since the diagrammatic\napproach is rather tedious, we have developed an alterna-\ntive method to generate the higher-order connected spin\ncorrelation functions in frequency space based on the hi-\nerarchy of equations of motion given in Eq. (C10). Not-\ning that in the limit of vanishing exchange couplings only20\nthe \frst two terms on the right-hand side of Eq. (C10) survive, and denoting the resulting local limit of the con-\nnected correlation function by\nG(n;n;m )\n0 (!1:::!n;!0\n1:::!0\nn;!00\n1:::!00\nm) =Gnz}|{\n+\u0001\u0001\u0001+nz}|{\n\u0000\u0001\u0001\u0001\u0000mz}|{z\u0001\u0001\u0001z\n0 (!1:::!n;!0\n1:::!0\nn;!00\n1:::!00\nm) (B 13)\nwe obtain from Eq. (C10) the following expression relating the correlation function involving 2 ntransverse and m\nlongitudinal spin components to a linear combination of connected correlation functions involving at most 2 n+m\u00001\nspins,\nbG(n;n;m )\n0 (!1:::!n;!0\n1:::!0\nn;!00\n1:::!00\nm) =\u0000G0(!1)mX\n\u0017=1G(n;n;m\u00001)\n0 (!1+!00\n\u0017;!2:::!n;!0\n1:::!0\nn;!00\n1:::=!00\n\u0017:::!00\nm)\n+G0(!1)nX\n\u0017=1G(n\u00001;n\u00001;m+1)\n0 (!2:::!n;!0\n1:::=!0\n\u0017:::!0\nn;!00\n1:::!00\nm;!1+!0\n\u0017); (B 14)\nwhere the slashed symbol =!00\n\u0017in the list!00\n1:::=!00\n\u0017:::!00\nm\nmeans that !00\n\u0017should be omitted. The recur-\nsion relation (B 14) can be viewed as an algebraic\nform of the generalized Wick theorem for spin op-\nerators. By iterating this relation, we can express\nG(n;n;m )\n0 (!1:::!n;!0\n1:::!0\nn;!00\n1:::!00\nm) as a linear com-\nbination of terms involving products of the transverse\npropagators G0(!) and purely longitudinal blocks\nGkz}|{z\u0001\u0001\u0001z\n0 (!1:::!k) = k\u00001Y\ni=1\u000e(!i)!\nb(k\u00001)(B 15)\nup tok=n+m, whereb(k)denotes the k-th derivative\nof the Brillouin function.\nThe corresponding irreducible vertices can be obtained\nusing the relations between irreducible vertices and con-\nnected spin correlation functions derived in Appendix A,\nsee Eqs. (A 19{A 23). We \fnd that the longitudinal\nvertices are simply given by the negative of the corre-sponding generalized blocks,\n\u0000zz\n0(!) =\u0000\u000e(!)b0; (B 16)\n\u0000zzz\n0(!1;!2;!3) =\u0000\u000e(!1)\u000e(!2)b00; (B 17)\n\u0000zzzz\n0(!1;!2;!3;!4) =\u0000\u000e(!1)\u000e(!2)\u000e(!3)b000:(B 18)\nThe transverse two-point vertex is\n\u0000+\u0000\n0(!) =G\u00001\n0(!) =H\u0000i!\nb; (B 19)\nand the mixed three-point vertex is related to the mixed\nthree-spin correlation function via\n\u0000+\u0000z\n0(!1;!2;!3) =\n\u0000G\u00001\n0(\u0000!1)G\u00001\n0(!2)G+\u0000z\n0(\u0000!1;\u0000!2;\u0000!3);(B 20)\nwhich gives\nb\u0000+\u0000z\n0(!1;!2;!3) = 1\u0000G\u00001\n0(!2)\u000e(!3)b0: (B 21)\nThe transverse and the mixed four-point vertices are\ngiven by\n\u0000++\u0000\u0000\n0 (!1;!2;!3;!4) =\u0000G\u00001\n0(\u0000!1)G\u00001\n0(\u0000!2)G\u00001\n0(!3)G\u00001\n0(!4)G++\u0000\u0000\n0 (\u0000!1;\u0000!2;\u0000!3;\u0000!4); (B 22)\n\u0000+\u0000zz\n0 (!1;!2;!3;!4) =\u0000G\u00001\n0(\u0000!1)G\u00001\n0(\u0000!2)G+\u0000zz\n0 (\u0000!1;\u0000!2;\u0000!3;\u0000!4)\n+\b\n\u0000+\u0000z\n0(!1;\u0000!1\u0000!3;!3)G0(\u0000!1\u0000!3)\u0000+\u0000z\n0(\u0000!2\u0000!4;!2;!4) + (!3$!4)\t\n;(B 23)\nwhich gives\nb2\u0000++\u0000\u0000\n0 (!1;!2;!3;!4) =G\u00001\n0(!3) +G\u00001\n0(!4)\n\u0000[\u000e(!1+!3) +\u000e(!1+!4)]b0G\u00001\n0(!3)G\u00001\n0(!4);\n(B 24)and\nb2\u0000+\u0000zz\n0 (!1;!2;!3;!4) =\u0000[\u000e(!3) +\u000e(!4)]b0\n+\u000e(!3)\u000e(!4)G\u00001\n0(!2)[2(b0)2\u0000bb00]:(B 25)\nIn a cuto\u000b scheme where initially the exchange interac-\ntion is completely switched o\u000b, the above expressions for21\nthe vertices de\fne the initial condition for the SFRG \row\nequations.\nAPPENDIX C: WARD IDENTITY AND\nHIERARCHY OF EQUATIONS OF MOTION\nIn the regime where the spin-rotational invariance is\nspontaneously broken, Goldstone's theorem guarantees\nthat the energy dispersion of the spin-waves is gapless\nfor vanishing external \feld H!0. To construct a trun-\ncation of the SFRG \row equations which does not violate\nthis, it is crucial to take into account an exact Ward iden-\ntity forcing the vanishing of the spin-wave gap when M\nremains \fnite for H!0. To derive this Ward iden-\ntity, consider the imaginary time Heisenberg equation ofmotion for the spin operator Si(\u001c) with deformed Hamil-\ntonianH\u0003=H0+V\u0003given by Eqs. (2.3) and (2.4) un-\nder the in\ruence of an additional \ructuating source \feld\nhi(\u001c). Writing for simplicity J?\nij=J?\n\u0003;ijandJz\nij=Jz\n\u0003;ij,\nand choosing our coordinate system such that the uni-\nform magnetic \feld H=Hezpoints inz-direction ez,\nthe Heisenberg equation of motion for the spin operators\nSi=Sz\niez+S?\ni=Sk\ni+S?\niin imaginary time can be\nwritten as\ni@\u001cSi(\u001c) =Si(\u001c)\u0002\u0014\nH+hi(\u001c)\n\u0000X\nj\u0010\nJz\nijSk\nj+J?\nijS?\nj(\u001c)\u0011\u0015\n:(C1)\nTaking the source-dependent average of both sides of\nEq. (C1) we obtain\ni@\u001chSi(\u001c)ih=hSi(\u001c)ih\u0002\u0014\nH+hi(\u001c)\u0000X\nj\u0010\nJz\nijhSk\nj(\u001c)ih+J?\nijhS?\nj(\u001c)ih\u0011\u0015\n\u0000X\nj\u0010\nJz\nijh\u000eSi(\u001c)\u0002\u000eSk\nj(\u001c)ih+J?\nijh\u000eSi(\u001c)\u0002\u000eS?\nj(\u001c)ih\u0011\n; (C2)\nwhere\u000eSi(\u001c) =Si(\u001c)\u0000hSi(\u001c)ihand the average symbol is de\fned as follows,\nh:::ih=Trh\ne\u0000\fH0TeR\f\n0d\u001c[P\nihi(\u001c)\u0001Si(\u001c)\u0000V\u0003(\u001c)]:::i\nTrh\ne\u0000\fH0TeR\f\n0d\u001c[P\nihi(\u001c)\u0001Si(\u001c)\u0000V\u0003(\u001c)]i: (C3)\nBy de\fnition, the averages in Eq. (C2) can be expressed in terms of the derivatives of the generating functional G\u0003[h]\nof the connected imaginary-time spin correlation functions de\fned in Eq. (2.5). It is convenient to express the part\nof the source \feld hi(\u001c) which is perpendicular to the external magnetic \feld in terms of the spherical components\nh\u0006\ni(\u001c) =1p\n2[hx\ni(\u001c)\u0006ihy\ni(\u001c)]: (C4)\nThen Eq. (C2) reduces to the following three identities:\n@\u001c\u0014\u000eG\u0003\n\u000ehz\ni(\u001c)\u0015\n=\u000eG\u0003\n\u000eh\u0000\ni(\u001c)h\u0000\ni(\u001c)\u0000\u000eG\u0003\n\u000eh+\ni(\u001c)h+\ni(\u001c)\n\u0000X\njJ?\nij\"\n\u000eG\u0003\n\u000eh\u0000\ni(\u001c)\u000eG\u0003\n\u000eh+\nj(\u001c)\u0000\u000eG\u0003\n\u000eh+\ni(\u001c)\u000eG\u0003\n\u000eh\u0000\nj(\u001c)+\u000e2G\u0003\n\u000eh\u0000\ni(\u001c)\u000eh+\nj(\u001c)\u0000\u000e2G\u0003\n\u000eh+\ni(\u001c)\u000eh\u0000\nj(\u001c)#\n; (C5a)\n@\u001c\u0014\u000eG\u0003\n\u000eh+\ni(\u001c)\u0015\n=\u000eG\u0003\n\u000eh+\ni(\u001c)[H+hz\ni(\u001c)]\u0000\u000eG\u0003\n\u000ehz\ni(\u001c)h\u0000\ni(\u001c)\n\u0000X\nj\"\nJz\nij\u000eG\u0003\n\u000eh+\ni(\u001c)\u000eG\u0003\n\u000ehz\nj(\u001c)\u0000J?\nij\u000eG\u0003\n\u000ehz\ni(\u001c)\u000eG\u0003\n\u000eh+\nj(\u001c)+Jz\nij\u000e2G\u0003\n\u000eh+\ni(\u001c)\u000ehz\nj(\u001c)\u0000J?\nij\u000e2G\u0003\n\u000ehz\ni(\u001c)\u000eh+\nj(\u001c)#\n;(C5b)\n@\u001c\u0014\u000eG\u0003\n\u000eh\u0000\ni(\u001c)\u0015\n=\u0000\u000eG\u0003\n\u000eh\u0000\ni(\u001c)[H+hz\ni(\u001c)] +\u000eG\u0003\n\u000ehz\ni(\u001c)h+\ni(\u001c)\n+X\nj\"\nJz\nij\u000eG\u0003\n\u000eh\u0000\ni(\u001c)\u000eG\u0003\n\u000ehz\nj(\u001c)\u0000J?\nij\u000eG\u0003\n\u000ehz\ni(\u001c)\u000eG\u0003\n\u000eh\u0000\nj(\u001c)+Jz\nij\u000e2G\u0003\n\u000eh\u0000\ni(\u001c)\u000ehz\nj(\u001c)\u0000J?\nij\u000e2G\u0003\n\u000ehz\ni(\u001c)\u000eh\u0000\nj(\u001c)#\n:(C5c)\nTo derive the Ward identity (4.15), we take another func-\ntional derivative\u000e\n\u000eh+\nn(\u001c0)of Eq. (C5c) and then set allsources equal to zero. For an isotropic ferromagnet with22\nJz\nij=J?\nij=\u0000Vijthe resulting equation of motion for\nthe transverse two-spin correlation function in real space\nand imaginary time can be written as\n@\u001cG+\u0000\nin(\u001c\u0000\u001c0) =\u0000HG+\u0000\nin(\u001c\u0000\u001c0) +\u000ein\u000e(\u001c\u0000\u001c0)M\n+X\njVijh\nG+\u0000\nin(\u001c\u0000\u001c0)M\u0000G+\u0000\njn(\u001c\u0000\u001c0)M\n+G+\u0000z\ninj(\u001c;\u001c0;\u001c)\u0000G+\u0000z\njni(\u001c;\u001c0;\u001c)i\n; (C6)\nwhereM=hSz\ni(\u001c)iis the local moment for the system\nwith deformed Hamiltonian H\u0003in the absence of sources,\ni.e. for hi(\u001c) = 0. Summing both sides of this equation\nover the site label iand integratingR\f\n0d\u001cwe see that the\nlast term on the right-hand side vanishes due to the anti-\nsymmetry of the terms in the square braces with respect\ntoi$j, while the left-hand side vanishes due to theperiodicity of the imaginary time spin correlation func-\ntions. Noting that the uniform transverse susceptibility\nis given by\n\u001f?=G+\u0000(K= 0) =Z\f\n0d\u001cX\niG+\u0000\nin(\u001c\u0000\u001c0);(C7)\nwe \fnally arrive at the Ward identity45\u001f?=M=H , see\nEq. (4.15) of the main text.\nBy successively taking higher-order derivatives of the\nfunctional relations (C5a){(C5c), we can derive equations\nof motion for all higher-order connected spin correlation\nfunctions. For example, starting from the third relation\n(C5c) we can obtain the equations of motion for the con-\nnected spin correlation functions\nG(n;n;m )(X1;\u0001\u0001\u0001;Xn;X0\n1;:::;X0\nn;X00\n1;:::;X00\nm) =Gnz}|{\n+\u0001\u0001\u0001+nz}|{\n\u0000\u0001\u0001\u0001\u0000mz}|{z\u0001\u0001\u0001z(X1;\u0001\u0001\u0001;Xn;X0\n1;:::;X0\nn;X00\n1;:::;X00\nm)\n(C8)\ninvolving 2n\u00152 transverse and mlongitudinal spin components by taking nderivatives with respect to the h+-sources,\nn\u00001 derivatives with respect to the h\u0000-sources, and mderivatives with respect to the hz-sources. In Eq. (C8) the\nsymbolsXi= (ri;\u001ci) are collective labels representing the lattice sites riand the imaginary time \u001ci. Transforming\nall objects to momentum-frequency space,\nG(n;n;m )(X1;\u0001\u0001\u0001;Xn;X0\n1;:::;X0\nn;X00\n1;:::;X00\nm) =Z\nK1:::Z\nKnZ\nK0\n1:::Z\nK0nZ\nK00\n1:::Z\nK00meiPn\ni=1(KiXi+K0\niX0\ni)+iPm\ni=1K00\niX00\ni\n\u0002\u000e\u0010Xn\ni=1(Ki+K0\ni) +Xm\ni=1K00\ni\u0011\nG(n;n;m )(K1;\u0001\u0001\u0001;Kn;K0\n1;:::;K0\nn;K00\n1;:::;K00\nm); (C9)\nwhereKiXi=ki\u0001ri\u0000!i\u001ci, we obtain the following in\fnite hierarchy of equations,\n(H\u0000i!1)G(n;n;m )(K1;K2:::Kn;K0\n1:::K0\nn;K00\n1:::K00\nm) =\n\u0000mX\n\u0017=1G(n;n;m\u00001)(K1+K00\n\u0017;K2:::Kn;K0\n1:::K0\nn;K00\n1:::=K00\n\u0017:::K00\nm)\n+nX\n\u0017=1G(n\u00001;n\u00001;m+1)(K2:::Kn;K0\n1:::=K0\n\u0017:::K0\nn;K00\n1:::K00\nm;K1+K0\n\u0017)\n+n\u00001X\n\u0017=0mX\n\u0016=0S(+)\nK2:::K\u0017+1;K\u0017+2:::KnS(\u0000)\nK0\n1:::K0\u0017;K0\n\u0017+1:::K0nS(z)\nK00\n1:::K00\u0016;K00\n\u0016+1:::K00m(\n\u0014\nJz\u0010X\u0017\ni=1(ki+1+k0\ni) +X\u0016\ni=1k00\ni\u0011\n\u0000J?\u0010\nk1+X\u0017\ni=1(ki+1+k0\ni) +X\u0016\ni=1k00\ni\u0011\u0015\n\u0002G(\u0017;\u0017;\u0016 +1)\u0010\nK2:::K\u0017+1;K0\n1:::K0\n\u0017;K00\n1:::K00\n\u0016;\u0000X\u0017\ni=1(Ki+1+K0\ni)\u0000X\u0016\ni=1K00\ni\u0011\n\u0002G(n\u0000\u0017;n\u0000\u0017;m\u0000\u0016)\u0010\nK1+X\u0017\ni=1(Ki+1+K0\ni) +X\u0016\ni=1K00\ni;K\u0017+2:::Kn;K0\n\u0017+1:::K0\nn;K00\n\u0016+1:::K00\nm\u0011)\n+Z\nQ[Jz(q)\u0000J?(k1+q)]G(n;n;m +1)\u0000\nK1+Q;K 2:::Kn;K0\n1:::K0\nn;K00\n1:::K00\nm;\u0000Q\u0001\n; (C10)\nwhere the slashed symbol =K00\n\u0017in the listK00\n1:::=K00\n\u0017:::K00\nm\nmeans that K00\n\u0017should be deleted from the list, and theoperatorsS(\u000b)(where\u000b= +;\u0000;z) symmetrize all ex-\npressions with respect to permutations of the K-labels23\nbelonging to the same spin component, see Ref. [7] for an\nexplicit de\fnition of these operators. To obtain the equa-\ntions of motion of purely longitudinal correlation func-\ntions, we take functional derivatives of the \frst equation\n(C5a) with respect to the longitudinal sources and then\nset all sources equal to zero. Note that only the terms\ninvolving the second functional derivative in the second\nline of Eq. (C5a) survive in this limit, which in Fourier\nspace involve a loop integral. If we deform our model\nsuch that the loop integrations are small (this is the case\nin the presence of a strong external magnetic \feld, or if\nthe exchange interaction is long-ranged), then all terms\ninvolving loops can be dropped. We call this the tree\napproximation. In this limit we can solve the simpli-\n\fed hierarchy of \row equations recursively, because the\nright-hand side of the hierarchy (C10) involves correla-\ntion functions of the same order as the left-hand side with\nthe same arguments or of lower order. In Appendix D\nwe explicitly give the tree approximation for the vertices\nwith up to four external legs.\nAPPENDIX D: TREE APPROXIMATION FOR\nCORRELATION FUNCTIONS AND VERTICES\nFor the approximate solution of the SFRG \row equa-\ntions it is convenient to include the exchange couplings\nbetween di\u000berent spins at the initial scale within the tree\napproximation, which graphically amounts to neglect-\ning all diagrams involving closed loops. In the spirit ofVLP,2we therefore assume that the exchange interac-\ntion is long-ranged so that its Fourier transform is domi-\nnated by momenta jkj.k0\u001c1=a, whereais the lattice\nspacing. Loop integrations over momenta are then sup-\npressed by powers of k0a, so that perturbation theory in\npowers of loops is controlled by the small parameter k0a.\nTo leading order, we may simply neglect all loops (tree\napproximation). As already pointed out at the end of\nAppendix C, in this limit the in\fnite hierarchy of equa-\ntions of motion decouples. To simplify our notation, let\nus rename here the Fourier transform of the exchange\ncouplings as follows,\nJz(k) =\u0000Vz\nk; J?(k) =\u0000V?\nk: (D 1)\nThe tree approximation for the transverse propagator is\nG0(K) =G0(k;i!) =M0\nH+Ek\u0000i!; (D 2)\nwhere the magnetization M0in self-consistent mean-\n\feld approximation is de\fned in Eq. (3.24), and Ek=\nM0(Vz\n0\u0000V?\nk) is the magnon dispersion. The longitu-\ndinal e\u000bective interaction is in tree approximation given\nby\nF0(K) =F0(k;i!) =Vz\nk\n1\u0000\u000e(!)b0Vz\nk; (D 3)\nwhile the tree approximation for the mixed three-spin\ncorrelation function can be written as\nM0G+\u0000z\ntree(K1;K2;K3) =\u0000G0(K1)G0(\u0000K2)(\n1\u0000\u0002\nG\u00001\n0(\u0000K2) +V?\nk2\u0000Vz\nk3\u0003\u000e(!3)b0\n1\u0000\fb0Vz\nk3)\n: (D 4)\nMoreover, from the equation of motion (C10) we obtain for transverse four-spin correlation function in tree approxi-\nmation\nM2\n0G++\u0000\u0000\ntree (K1;K2;K3;K4) =\u0000G0(K1)G0(K2)G0(\u0000K3)G0(\u0000K4)(\nG\u00001\n0(\u0000K3) +V?\nk3\u0000Vz\nk1+k3\n\u0000\u000e(!1+!3)b0\n1\u0000\fVz\nk1+k3b0\u0002\nG\u00001\n0(\u0000K3) +V?\nk3\u0000Vz\nk1+k3\u0003\u0002\nG\u00001\n0(\u0000K4) +V?\nk4\u0000Vz\nk1+k3\u0003\n+ (K3$K4))\n: (D 5)\nFinally, the tree approximation for the mixed four-spin correlation function can be written as\nM2\n0G+\u0000zz\ntree (K1;K2;K3;K4) =G0(K1)G0(\u0000K2)\n[1\u0000\u000e(!3)b0Vz\nk3][1\u0000\u000e(!4)b0Vz\nk4](\nG\u00001\n0(\u0000K2) +V?\nk2\u0000Vz\nk3+k4\n1\u0000\fb0Vz\nk3+k4\u000e(!3)\u000e(!4)M0b00\n+\u001a\nG0(K1+K3)h\n1 +\u000e(!3)b0V?\nk1+k3ih\n1\u0000\u000e(!4)b0\u0000\nG\u00001\n0(\u0000K2) +V?\nk2\u0001i\n+ (K3$K4)\u001b)\n: (D 6)\nNote that for vanishing exchange couplings Eqs. (D 4){(D 6) reduce to the corresponding generalized blocks given in\nEqs. (B 10){(B 12).\nGiven the connected spin correlation functions, we can construct the corresponding irreducible vertices generated\nby our hybrid functional ~\u0000\u0003[m;'] de\fned via Eqs. (2.11) and (2.25) using the general relations (A 19){(A 23) derived\nin Appendix A. The two-point vertices \u0000+\u0000\ntree(K) and \u0000zz\ntree(K) in tree approximation are given in Eqs. (3.29) and (3.31)24\nof the main text. Substituting the tree approximation G+\u0000z\ntree(K1;K2;K3) given in Eq. (D 4) for the mixed three-spin\ncorrelation function on the right-hand side of Eq. (A 19), we obtain\nM0\u0000+\u0000z\ntree(K1;K2;K3) = 1\u0000\u0002\nG\u00001\n0(K2) +V?\nk2\u0003\n\u000e(!3)b0= 1\u0000G\u00001\n1(!2)\u000e(!3)b0; (D 7)\nwhereG1(!) is de\fned by\nG\u00001\n0(K) +V?\nk=H+M0(Vz\n0\u0000V?\nk)\u0000i!\nM0+V?\nk=H+M0Vz\n0\u0000i!\nM0\u0011G\u00001\n1(!): (D 8)\nNote that with the substitutions b!M0andG\u00001\n0(!2)!G\u00001\n1(!2) the mixed three-legged vertex in tree approximation\ncan be obtained from the corresponding irreducible vertex for vanishing exchange interaction given in Eq. (B 21).\nNext, consider the transverse four-point vertex in tree approximation, which according to Eqs. (A 21) and (D 5) is\ngiven by\nM2\n0\u0000++\u0000\u0000\ntree (K1;K2;K3;K4) =G\u00001\n0(K3) +G\u00001\n0(K4) +V?\nk3+V?\nk4\u0000Vz\nk1+k3\u0000Vz\nk1+k4\n\u0000(\n\u000e(!1+!3)b0\n1\u0000\fVz\nk1+k3b0\u0000\nG\u00001\n0(K3) +V?\nk3\u0000Vz\nk1+k3\u0001\u0000\nG\u00001\n0(K4) +V?\nk4\u0000Vz\nk1+k3\u0001\n+ (K3$K4))\n+n\nM0\u0000+\u0000z\ntree(K1;K3;\u0000K1\u0000K3)F0(\u0000K1\u0000K3)M0\u0000+\u0000z\ntree(K2;K4;\u0000K2\u0000K4) + (K3$K4)o\n:(D 9)\nIn the last line we substitute again the tree approximation (D 7) for the three-point vertices and obtain after some\nre-arrangements\nM2\n0\u0000++\u0000\u0000\ntree (K1;K2;K3;K4) =G\u00001\n1(!3) +G\u00001\n1(!4)\u0000[\u000e(!1+!3) +\u000e(!1+!4)]b0G\u00001\n1(!3)G\u00001\n1(!4);(D 10)\nwhich again can be obtained from the corresponding expression (B 24) for vanishing exchange couplings by replacing\nG0(!)!G1(!). Finally, for the mixed four-point vertex, we obtain from Eqs. (A 22) and (D 6),\nM2\n0\u0000+\u0000zz\ntree (K1;K2;K3;K4) =\u0000G\u00001\n0(\u0000K2)\u0000Vz\nk3+k4\n1\u0000\fb0Vz\nk3+k4\u000e(!3)\u000e(!4)M0b00\n\u0000G0(K1+K3)\u0000G0(K1+K4) +G\u00001\n0(\u0000K2)\u0002\nG0(K1+K3)\u000e(!4) +G0(K1+K4)\u000e(!3)\u0003\nb0\n+n\nM0\u0000+\u0000z\ntree(K1;\u0000K1\u0000K3;K3)G0(K2+K4)M0\u0000+\u0000z\ntree(\u0000K2\u0000K4;K2;K4) + (K3$K4)o\n+M0\u0000zzz\ntree(K3;K4;\u0000K3\u0000K4)F0(\u0000K3\u0000K4)M0\u0000+\u0000z\ntree(K1;K2;\u0000K1\u0000K2): (D 11)\nSubstituting Eqs (D 7) and (B 17) for the three-point vertices, we \fnally obtain\nM2\n0\u0000+\u0000zz\ntree (K1;K2;K3;K4) =\u0000\u0002\n\u000e(!3)[1\u0000V?\nk2+k4G0(K2+K4)] +\u000e(!4)[1\u0000V?\nk2+k3G0(K2+K3)]\u0003\nb0\n+\u000e(!3)\u000e(!4)G\u00001\n1(!2)\u0002\n[2\u0000V?\nk2+k4G0(K2+K4)\u0000V?\nk2+k3G0(K2+K3)](b0)2\u0000M0b00\u0003\n: (D 12)\nIn the limit of vanishing exchange interaction, we re-\ncover again the corresponding single-site irreducible ver-\ntex given in Eq. (B 25). 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Their intrinsic damping is largely determined by electron-magnon scat-\ntering induced by spin-orbit interactions. In the low scattering limit, damping is dominated by\nintra-band electronic transitions, which has been theoretically shown to be proportional to the elec-\ntronic density of states at the Fermi level. In this work, we focus on body-centered-cubic iron as a\nparadigmatic ferromagnetic material. We comprehensively study its electronic structure using first-\nprinciples density functional theory simulations and account for finite lattice temperature, boron\n(B) doping, and structure amorphization. Our results indicate that temperature induced atomic\ndisorder and amorphous atomic geometries only have a minor influence. Instead, boron doping no-\nticeably decreases the density of states near the Fermi level with an optimal doping level of 6.25 %.\nIn addition, we show that this reduction varies significantly for different atomic geometries and\nreport that the highest reduction correlates with a large magnetization of the material. This may\nsuggest materials growth under external magnetic fields as a route to explore in experiment.\nI. INTRODUCTION\nHybrid magnonics is gaining growing interest due to its\npotential for coherent quantum information processing\n[1–4]. This was triggered by the experimental demon-\nstration of coherent coupling between a magnon and a\nsuperconducting qubit, mediated by a microwave cavity,\nby Tabuchi et al. [5] As one key ingredient in hybrid\nmagnonic systems, magnons possess unique advantages\nsuch as easily tunable resonance frequencies through ex-\nternal magnetic fields or materials anisotropy, microwave\nbandwidths that match state-of-the-art superconduct-\ning quantum devices, and intrinsic non-reciprocity that\nis promising for noise-resilient quantum state transduc-\ntion [4, 6]. In recent years, more research has established\ncoupling between magnons and microwave photons in a\ncavity or a coplanar circuit structure [7–10].\nHowever, one essential challenge is the damping of\nmagnon excitations that limits the coherence time of hy-\nbrid quantum states [4, 11]. Therefore, exploring mate-\nrials with low magnetic damping is crucial for achieving\nhybrid magnonic quantum devices with long coherence\ntime. One of the best-known materials in this context\nis the ferrimagnetic insulator yttrium iron garnet (YIG),\nY3Fe5O12, whose magnetic damping parameter can be\nas low as 10−5in bulk crystals [12]. However, insulators\nsuch as YIG are not desired for many spintronics appli-\ncations that require a charge current through the mate-\nrial [13]. In addition, YIG is not well suited to be inte-\n∗schleife@illinois.edugrated into on-chip devices for circuit quantum electrody-\nnamics due to experimental constraints. One of the rea-\nsons, is that in order to have sufficient crystalline quality\nfor low damping, YIG films need to be grown on specific\nsubstrates for achieving epitaxy. The standard substrate\nchoice for this task is gadolinium gallium garnet (GGG,\nGd3Ga5O12), but at very low temperatures the increased\nmagnetic susceptibility due to ordering of the Gd mag-\nnetic moments can increase the magnetic damping of YIG\nfilms considerably [14–18]. Ferromagnetic metals and al-\nloys are an alternative category of promising materials\nthat are significantly easier to integrate on-chip. It is\ntherefore desirable to explore and optimize metallic mag-\nnets towards low magnetic damping for hybrid magnonic\nquantum devices [19].\nA major contribution to magnetic damping in metal-\nlicmagnets arises from conduction electrons that dis-\nsipate magnons through the spin-orbit (SO) interaction\n[20–22]. This mechanism is described by two early the-\nories, the breathing Fermi surface (BFS) model and the\nlater torque correlation (TC) model that were developed\nby Kambersk´ y [21, 23]. The TC model is more general\nand describes how the SO torque ⟨n,k|[σ−,ˆHSO]|m,k⟩\ninduces intra-band ( m=n) and inter-band ( m̸=n) elec-\ntronic transitions which are interpreted as conductivity-\nlike and resistivity-like damping pathways for magnons,\nrespectively [22, 24–26]. In its low scattering limit, when\nthe spectral overlap between different bands is small,\ndamping is largely dominated by the intra-band part,\nin agreement with the BFS model. Damping is shown\nto be approximately proportional to the electronic den-\nsity of states (EDOS) at the Fermi level [21, 23, 27].\nMany experimental works confirm that the lowest damp-arXiv:2401.08076v1  [cond-mat.mtrl-sci]  16 Jan 20242\ning coincides with the lowest EDOS at the Fermi level\n(EDOS-FL) for ferromagnetic alloys such as cobalt-iron\n(Co-Fe) and iron-vanadium (Fe-V) [28–30]. Therefore,\none practical way to reduce magnetic damping is to min-\nimize EDOS-FL by electronic-structure engineering.\nIn this work, we focus on the paradigmatic ferromag-\nnetic material, body-centered-cubic (BCC) iron (Fe). We\nuse first-principles electronic-structure theory to compre-\nhensively study its EDOS-FL under the conditions of (i)\nthermal atomic disorder, (ii) boron (B) doping, and (iii)\nstructure amorphization. We account for the effect of\nthermal disorder on EDOS-FL since real devices always\nwork at non-zero temperature, although for supercon-\nducting quantum devices the operating temperature can\nbe extremely low. Typical superconducting quantum cir-\ncuits, such as qubits, are operated at temperatures of a\nfew 10s mK. Meanwhile, recent experimental work has\nshown that doping carbon (C) or B into ferromagnetic\nalloys can make the structure amorphous and reduce\nmagnetic damping [31, 32]. Hence, it is interesting to\ninvestigate the origin of the reduced damping in doped\namorphous alloys and the extent to which it can be at-\ntributed to reduced EDOS-FL. To explore this, we study\nthe effects of B doping and structure amorphization indi-\nvidually and compare EDOS-FL for crystalline Fe doped\nwith B and amorphous Fe, as well as pure crystalline\nFe. Finally, we consider amorphous Fe with B doping to\ninvestigate the combined effect.\nThe remainder of this paper is organized as follows:\nWe first introduce the methods that are used to con-\nstruct our simulation cells including BCC Fe supercells\nwith phonon excitations, BCC Fe supercells doped with\nB, amorphous Fe supercells, and amorphous Fe supercells\ndoped with B, in Sec. II. The details of our DFT simula-\ntions are also explained in Sec. II. In Sec. III, we discuss\nour main results for the EDOS of the different structures.\nIn general, we find that phonon excitations do not signif-\nicantly change EDOS-FL up to at least 500 K. Doping B\ninto BCC Fe decreases EDOS-FL, while making the Fe\namorphous increases EDOS-FL. Overall, amorphous Fe\ndoped with B can have lower EDOS-FL than pure BCC\nFe. Finally, we give our conclusions in Sec. IV.\nII. METHODOLOGY\nThermal disorder of the atomic positions in the BCC\nFe lattice is modeled by considering phonons that dis-\nplace nuclei from their ideal 0 K lattice positions of the\nBCC crystal structure [see Fig. 1(a)]. Simulation cells of\nthe thermally disordered lattice are constructed by super-\nimposing harmonic phonon modes with random phases\nand amplitudes according to classical statistics at differ-\nent temperatures [34]. In this work, we consider three\ndifferent temperatures of 10 K, 300 K, and 500 K. For\neach temperature, 50 random snapshots of the disordered\n2×2×2 BCC Fe supercell are generated for simulations.\nIn Fig. 1(b), we show the disordered supercell superim-\n(a)\n(b) (c)\n(d) (e)\n1234\n5678\n9101112\n13141516\n5.665 Å 5.665 Å \n5.665 Å 8.267 Å \n8.267 Å FIG. 1. Simulation supercells used in this work. (a) The\n2×2×2 supercell of ideal BCC Fe at the temperature of\nT= 0 K. The supercell includes 16 Fe atoms (brown spheres)\nthat are labeled by the numbers on them. (b) BCC Fe su-\npercell at T= 300 K, with 50 instantaneous snapshots of the\nvibrating lattice superimposed. (c) BCC Fe supercell doped\nwith 6.25% B (green spheres), corresponding to nB= 1 in\nTable I. (d) Snapshot of amorphous Fe (a-Fe) with a volume\nof a cubic 3 ×3×3 supercell. This supercell is cut from a\nlarger one constructed by molecular dynamics (MD) simula-\ntions (see text). (e) Snapshot of amorphous Fe doped with\n7.5% B (a-FeB). All these supercell images are produced by\nVESTA [33].\nposing 50 snapshots for T= 300 K as an example. The\nsupercell snapshots for T= 10 K and T= 500 K are\nsimilar but with different amplitudes of the atomic dis-\nplacements. For analyzing the EDOS, we average over\nall 50 snapshots for each temperature.\nLattice geometries for B doped BCC Fe are constructed\nusing a cluster-expansion method which was used before\nfor studying binary alloys [35–40]. This method starts\nwith a 2 ×2×2 BCC Fe supercell with 16 atoms and\nreplaces nBFe atoms with B atoms. Generally, there\nareCnB\nnFe+nBways of distributing nFeFe atoms and nBB\natoms, however, symmetries of the lattice structure re-\nduce the total number of non-equivalent configurations\nto a few classes with different folds of degeneracy (see\nTable I for all different nBconsidered in this paper).\nElectronic-structure simulations are carried out for only\none representative of each class, which significantly re-3\nnBclass degeneracy representation\n11 16 1\n22 64 1, 2\n3 24 1, 3\n4 24 1, 7\n5 8 1, 15\n36 192 1, 2, 3\n7 192 1, 2, 7\n8 64 1, 2, 15\n9 48 1, 3, 5\n10 48 1, 3, 13\n11 16 1, 7, 11\n412 48 1, 2, 3, 4\n13 384 1, 2, 3, 5\n14 96 1, 2, 3, 6\n15 192 1, 2, 3, 8\n16 384 1, 2, 3, 13\n17 96 1, 2, 3, 14\n18 96 1, 2, 3, 16\n19 48 1, 2, 7, 8\n20 128 1, 2, 7, 11\n21 96 1, 2, 7, 12\n22 96 1, 2, 7, 16\n23 16 1, 2, 15, 16\n24 12 1, 3, 5, 7\n25 16 1, 3, 5, 9\n26 48 1, 3, 5, 11\n27 48 1, 3, 5, 15\n28 12 1, 3, 13, 15\n29 4 1, 7, 11, 13\nTABLE I. All non-equivalent classes (second column) of ar-\nranging nBB atoms (first column) on 16 Fe BCC lattice sites,\nFe16−nBBnB. Only these representative structures are simu-\nlated in our work. The third column provides the degeneracy\nfor each class determined by crystal structure symmetry and\nthe fourth column contains one representative atomic geome-\ntry that was used in this work (see atom labels in Fig. 1).\nduces the computational effort and allows us to study an\nalloy with nFe+nB=16 atoms. In this work, we consider\nnB=1 (see Fig. 1(c) for Fe 15B1), 2, 3, and 4 that corre-\nspondingly have 1, 4, 6, and 18 non-equivalent classes (see\nTable I). For each doping level nB, we compute the EDOS\nby averaging over the non-equivalent classes weighted by\ntheir degeneracies (third column in Table I). There are\ndifferent methods to determine the weights used for aver-\naging, introduced in Ref. [35], corresponding to different\nthermodynamic conditions of the alloy. The weights we\nuse here correspond to a simplification of the strict reg-\nular solution, with the macroscopic alloy composition or\ndoping level consisting exclusively of microscopic struc-\ntures with exactly the same concentration of elements.\nWe construct amorphous Fe (a-Fe) atomic geometries\nby simulating heating and quenching processes using\nmolecular dynamics (MD) simulations as implemented\nin the LAMMPS code [41]. The inter-atomic interac-\ntions between Fe atoms are described via a ‘magnetic’ in-\nteratomic potential that was developed by Dudarev and\nDerlet [42–44], based on the embedded atom method.The inter-atomic potential for our simulation is obtained\nfrom the OpenKim website [45–48]. The same potential\nhas been used previously [49] to generate a-Fe geome-\ntries and was shown to produce characteristic features\nobserved in the experimentally measured radial distri-\nbution function (RDF) g(r) [50]. In our NVT canonical\nensemble MD simulations, we use a 10 ×10×10 supercell\nof BCC Fe with 2000 atoms. The system is then heated\nto 10,000 K using a Nos´ e-Hoover thermostat, leading to\na completely liquid state. We then implement a cool-\ning process where first the temperature Tis decreased to\n4,000 K in 50 picoseconds (ps) and maintained at 4,000 K\nfor 250 ps. Subsequently, Tis decreased to 3,000 K in\n250 ps. Finally, Tis decreased to 300 K (room tempera-\nture) in 50 ps and maintained for another 50 ps. We com-\npute the RDF g(r) for this final structure and find good\nagreement with the result from Ma et al. [49] and the\nexperimental observation from Ichikawa [50] (see Fig. 1\nin the supplemental information).\nWe then extract smaller cubic cells from this result,\ncorresponding to volumes of a 3 ×3×3 BCC Fe cell [see\nFig. 1(d)], and use these for first-principles calculations of\nthe electronic structure. We cut 10 such a-Fe snapshots\nfrom random positions of the large a-Fe structure, sim-\nulate all of them, and average the results. To generate\na-Fe doped with B (a-FeB), we use a single a-Fe snap-\nshot and randomly replace Fe atoms by B atoms accord-\ning to the doping level [see Fig. 1(e)]. For each doping\nlevel, 10 different configurations of random replacement\nare constructed for our simulations in order to approach\na statistical average.\nFor all these different atomic geometries, we compute\nelectronic densities of states (EDOS) using density func-\ntional theory (DFT) as implemented in the Vienna Ab-\ninitio Simulation Package (VASP) [51, 52]. A plane-wave\nbasis with a cutoff energy of 500 eV is applied to expand\nKohn-Sham states. The generalized-gradient approxima-\ntion (GGA) parametrized by Perdew, Burke, and Ernz-\nerhof (PBE) is used for the exchange-correlation func-\ntional [53]. All simulations are carried out with spin-\npolarized DFT which does not include the effect of spin-\norbit coupling (SOC). We use different k-point grids to\nsample the Brillouin zone (BZ) of the different simulation\ncells: These parameters are determined based on the con-\nvergence of the BCC Fe unit-cell for which a 12 ×12×12\nMPk-grid is enough to achieve convergence of the total\nenergy per atom to within 2 meV. For larger supercells,\nwe maintain a similar k-grid density by scaling the sam-\npling points inversely with the supercell size. For the su-\npercell of ideal BCC Fe and its thermally disordered lat-\ntices, we use a 8 ×8×8 Monkhorst-Pack (MP) grid [54].\nFor the B doped crystalline Fe structures, Fe nFeBnB, a\n6×6×6 MP grid is applied, and we tested that this\nconverges the total energy to within 1 meV/atom. For\namorphous structures, we use a 3 ×3×3 MP grid. Due\nto the reduced symmetry of a-Fe, a 4 ×4×4 MP k-grid\nis computationally too expensive and we use 3 ×3×3\npoints.4\n02.55\nT=10 KT=0 K\n02.55Density of states (1/eV)T=300 KT=0 K\n-3 -2 -1 0 1 2 3\nEEf (eV)\n02.55\nT=500 KT=0 K\nFIG. 2. Electronic density of states (EDOS) of pure Fe,\nnormalized per atom, at different lattice temperatures of T=\n10 K, 300 K, and 500 K. For each temperature, the EDOS\nof 50 snapshots is plotted. The EDOS of perfect BCC Fe at\nT= 0 K is included in every subplot for comparison.\nStructural relaxations for ideal BCC Fe and B doped\nBCC Fe are performed with a force tolerance of 5 meV/ ˚A.\nWe first relax undoped a-Fe structures within classical\nMD to a force tolerance of 0.01 eV/ ˚A and refer to these\nas “MD relaxed” a-Fe. They are then further relaxed\nby DFT, labeled as “MD+DFT relaxed” a-Fe, with a\nforce tolerance of 0.15 eV/ ˚A, which is the level that we\ncan achieve within a reasonable computational time. We\ngenerate B doped a-Fe from the “MD relaxed” geometries\nand relax these within DFT to the same force tolerance of\n0.15 eV/ ˚A. Thermally disordered lattices are not relaxed\nto maintain the frozen snapshots of the vibrating lattice.\nIII. RESULTS AND DISCUSSION\nA. Thermally induced atomic disorder\nFirst, we investigate the effect of finite lattice temper-\nature and the resulting atomic disorder on the EDOS\nof Fe for temperatures of T= 10 K, 300 K, and 500 K.\nPhonon excitations displace nuclei from their equilibrium\npositions, which increasingly disorders atomic geometries\nwith increasing temperature. The EDOS results for all\n50 calculated snapshots are plotted in Fig. 2 for each\ntemperature and compared to the EDOS of ideal BCC\nFe at T= 0 K.\nThese results show that increasing the temperature\ngenerally broadens the linewidth of the EDOS relative\nto the zero-temperature counterpart due to the random-\nness in the atomic positions for the disordered lattice\nstructure. For T= 300 and 500 K, noticeable changes\nto the EDOS are observed especially near peaks at −2.6\neV,−0.8 eV, and 1.8 eV as well as valleys near −2.0 eV\nand 0.6 eV, see Fig. 2. While the density of states at\nthe Fermi level n(Ef) (EDOS-FL) is much less affected,\nwe provide a quantitative analysis of these changes sincethey may directly influence electron-magnon damping.\nTowards this, we compute the average and standard de-\nviation of EDOS-FL for each temperature and find for\nT= 10 K that n(Ef) = 0 .991±0.004 eV−1, for T=\n300 K n(Ef) = 1 .038±0.012 eV−1, and for T= 500 K\nn(Ef) = 1 .058±0.019 eV−1. These three values are all\nlarger than n(Ef) = 0 .974 eV−1for ideal BCC Fe at\nzero temperature, showing that increasing temperature\nincreases EDOS-FL for Fe. However, even the increase\nof about 8 .6 % at T= 500 K is not significant, compared\nto the influence of doping and amorphization that we\ndiscuss later.\nWhile our results show that EDOS-FL of Fe does\nnot significantly change at low temperatures, this does\nnot imply that magnetic damping is temperature in-\ndependent, since mechanisms other than conductivity\nlike damping also contribute. Conductivity-like electron-\nmagnon damping in the intra-band scattering limit can\nbe approximately described by an empirical formula\nαintra∼n(Ef)|Γ−|τ[22, 24, 55, 56], where |Γ−|is the\nstrength of the spin–orbit interaction near the Fermi level\nandτis the electron relaxation time from the Drude\nmodel. The proportionality of αintra to the electron\nrelaxation time τimplies that conductivity-like damp-\ning [22, 24] is more pronounced at lower temperatures\nor in cleaner crystals where τis large. This somewhat\ncounter-intuitive increase of magnetic damping with in-\ncreasing τhas been experimentally observed in a recent\nwork on Fe [57] and an earlier work on cobalt (Co) and\nNickel (Ni) [58]. We note that in this work, we only\nstudy the dependence of αintra onn(Ef), however, the\nrelaxation time also can dependent on the phonon tem-\nperature or lattice disorder, including defects. Our simu-\nlation results clearly indicate that the EDOS-FL, one of\nthe factors determining the intrinsic magnetic damping\nin metallic magnets, is not significantly affected by low\ntemperatures. In addition to conductivity like damping,\nresistivity like damping may also be affected by temper-\nature, but is not discussed in this study.\nB. Boron doping\nNext, we compare the EDOS of BCC Fe doped with\nB to that of pure BCC Fe. Four different doping lev-\nels are considered and modeled by the structures Fe 15B1\n(6.25% B), Fe 14B2(12.5% B), Fe 13B3(18.75% B), and\nFe12B4(25% B), constructed using the cluster expansion\nmethod (see Sec. II). Except for the case nB= 1 which is\nrepresented by only one class, we calculate weighted av-\nerage and standard deviation (error bar) over all classes\nin Table I for the other doping levels.\nWe find that doping B into BCC Fe leads to a re-\nduced EDOS-FL compared to pure BCC Fe for all dop-\ning levels considered here [see Fig. 3(a)]. This is because\ndoping with B shifts the entire EDOS to lower energies,\nsuch that the Fermi level shifts to the dip in the orig-\ninal non-doped EDOS, as indicated by the black arrow5\n-2 -1 0 1 2\nEEf (eV)\n01234Density of states (1/eV)(a)\n-0.2-0.1 00.10.20.60.81.01.2BCC Fe\nFe15B1\nFe14B2\nFe13B3\nFe12B4\nFeFe15B1Fe14B2Fe13B3Fe12B4\nCompositions0.40.81.2n(Ef) (1/eV)(b)\nFIG. 3. (a) Electronic density of states (EDOS), normalized\nper atom, for Fe and Fe doped with B, simulated using a 2 ×2×\n2 supercell with different numbers of B atoms, Fe 16−nBBnB,\nfornB=1, 2, 3, and 4. Cluster expansion averages are shown\nfornB>1. The inset magnifies EDOS-FL and clearly shows a\nreduction by B doping. (b) Average and error bar of EDOS-\nFL for different doping concentrations Fe 16−nBBnB. For pure\nFe and Fe 15B1only one non-equivalent class (see Table. I) is\nused and, thus, we compute no standard deviation.\nin Fig. 3(a). Main peaks in the EDOS, i.e. at −0.784\neV and 1 .853 eV, are also shifted to lower energies with\nincreasing B doping. Meanwhile, peak intensities are de-\ncreased and peaks widths are broadened with increasing\nB doping.\nOur data also points to a non-monotonic dependence\nofn(Ef) on the B doping concentration and, as a result,\nthe possibility of optimizing the doping level to minimize\nmagnon damping. Figure 3(b) shows the average value\nand the error bar for different doped systems and pure\nBCC Fe. Our results indicate an optimal B doping level\nof 6.25%, which leads to the lowest n(Ef) = 0 .662 eV−1.\nAt the same time, with increasing number of B atoms in\nthe doped cell, there are more non-equivalent classes as\nseen from Table I, and the error bar increases from nB=2\ntonB=4. In particular, EDOS-FL for Fe 12B4has a large\nerror bar, which implies that the value of n(Ef) for the 18\nnon-equivalent classes (see Table I) spreads over a large\nrange. These classes, although having the same composi-\ntion, are microscopically quite different regarding the dis-\ntribution of B on the Fe lattice. In Fig. 4, we analyze the\nground state total energy E0, the magnetic moment m,\n-8-7.5E0 (eV)(a)\n1.51.9m (B)\n(b)\n12 14 16 18 20 22 24 26 28\nClass index0.00.51.0n(Ef) (1/eV)(c)\nEDOS-FL for BCC FeFIG. 4. (a) Ground-state total energy E0in eV, (b) mag-\nnetic moment minµB, and (c) electronic density of states at\nthe Fermi level n(Ef), normalized per atom, for all 18 non-\nequivalent classes corresponding to the structure Fe 12B4(see\nTable I). The horizontal dashed line in (c) indicates n(Ef) for\npure BCC Fe.\nand EDOS-FL n(Ef) for the 18 non-equivalent classes.\nWhile n(Ef) spans over a big range from 0.327 eV−1to\n1.030 eV−1, all values remain smaller than the value of\nthe pure BCC Fe, except for the class with the index 25\n(see Table I). This confirms that doping B still decreases\nEDOS-FL in general. The difference between maximum\nand minimum value of E0,m, and n(Ef) amounts to\n4.09 %, 9.08 %, and 98.74 %, respectively, relative to their\naverage values. We do not identify obvious correlation\nbetween the total energy and EDOS-FL in Fig. 4, with\na correlation coefficient of −0.110. For the lowest-energy\nclass with index of n= 12, n(Ef) = 0 .772 eV−1, which\nis larger than the average, while the class with index 28\nhas a relatively large energy, but the lowest EDOS-FL.\nHowever, we do find noticeable negative correlation be-\ntween the magnetic moment and EDOS-FL with a cor-\nrelation coefficient of −0.683. This observation is ben-\nefitial for real magnetic devices, where lower magnetic\ndamping and higher magnetization is often desirable for\napplications. It could also suggest that growing B doped\nFe under an external magnetic field, and thus making\nhigh magnetic moment atomic geometries more likely,\ncould lead to samples with lower magnetic damping. It\nis already well established that synthesis or annealing of\namorphous ferromagnets in applied magnetic fields may\nlead to induced magnetic anisotropies due to modifica-\ntions of the short-range order [59]. Similar, it is con-\nceivable that applying fields during growth or during a\npost-annealing may stabilize larger saturation magneti-\nzations [60].6\n(a)\n(b)\n(c)\n(d)\nFIG. 5. Amorphous Fe (a-Fe) structure and the radial dis-\ntribution function (RDF) g(r). (a) Illustration of the large\ncubic a-Fe supercell constructed by MD. The yellow dashed\nbox shows a small cubic a-Fe cell cut out for DFT simula-\ntions. (b) One representative small a-Fe cell for which DFT\nrelaxation does not change the RDF away from the amor-\nphous state. (c) One representative small a-Fe cell for which\nDFT relaxation leads to re-crystallization, i.e., the RDF ap-\nproaches the RDF of ideal BCC Fe. (d) RDF for the large\na-Fe cell created from MD, compared to the average g(r) of\neight small a-Fe cells that are randomly cut from it.\nC. Structure amorphization\nNext, we study the influence of structure amorphiza-\ntion by characterizing a-Fe via the radial distribution\nfunction (RDF) g(r). As explained in Sec. II, we use\nmolecular dynamics (MD) to compute atomic positions\nfor a large a-Fe supercell, from which we randomly cut\nten small cubic cells [see yellow box in Fig. 5(a)] and sub-\nsequently relax using MD (“MD-relaxed”) or MD and\nDFT (“MD+DFT relaxed”). For eight of these cells,\nMD and DFT relaxation does not change the RDF and\nmaintains amorphousness, while the other two relax and\nrecrystallize, as evidenced by their RDF. This is shown\nin Fig. 5(b) for one representative a-Fe cell for which re-\nlaxation does not change the amorphousness but simply\nsmoothens g(r). In contrast, Fig. 5(c) shows one example\n-2 -1 0 1 2\nEEf (eV)\n01234Density of states (1/eV)BCC Fe\na-Fe (MD relaxed)\na-Fe (MD+DFT relaxed)FIG. 6. Electronic density of states (EDOS), normalized per\natom, averaged over 8 simulation cells for amorphous Fe (a-\nFe). EDOS-FL increases for both the MD relaxed structures\nand “MD+DFT” relaxed structures, relative to ideal Fe. The\nerror bar is computed from the standard deviation.\nfor which DFT relaxation recrystallizes the amorphous\nstructure and discrete sharp peaks appear that coincide\nwith the RDF of ideal BCC Fe. For our purpose of study-\ning EDOS-FL for a-Fe, we exclude the two recrystallized\nsamples. The averaged RDF of the remaining eight a-\nFe cells is calculated and compared in Fig. 5(d) to the\nRDF of the large initial a-Fe supercell, depicted in Fig.\n5(a). We can see that for large rthey almost overlap.\nFor small raround 2 .5˚A, the two curves slightly differ,\nbut still show similar peak structure. Based on this, we\nconclude that the average of the small a-Fe cells approxi-\nmates the large a-Fe cell well enough to study EDOS-FL\nfor a-Fe.\nFigure 6 shows that a-Fe has a broadened distribution\nof the average EDOS and an increased value of EDOS-\nFL, compared to ideal BCC Fe. The peak structure of\nthe EDOS of BCC Fe is lost for a-Fe and the entire EDOS\nis smoother and broadened. We note that performing a\nDFT relaxation on top of the MD relaxed structure can\ndecrease n(Ef) even further, see Fig. 6. However, the\nincrease of EDOS-FL for a-Fe is larger than the error\nbar for both structure relaxation approaches. Since DFT\nrelaxations with tight force convergence criteria are com-\nputationally expensive, we do not explore this in more\ndetail in this work. Instead, we conclude that it is not\ndesirable to reduce n(Ef) of Fe by structure amorphiza-\ntion.\nFinally, our g(r) results illustrate that the a-Fe struc-\ntures generated from our MD simulations are fundamen-\ntally different from the Fe structure up to a temperature\nofT=500 K, even though in both cases the original per-\nfect BCC Fe lattice structure is appreciably disordered\n(see details in Fig. S2 of SI). Comparing to the g(r) of the\nperfect BCC Fe at zero lattice temperature, the temper-\nature induced disorder just broadens the original peaks.\nSome individual peaks are merged if they are too close to\neach other along the r-axis. The a-Fe structure, however,7\n-2 -1 0 1 2\nEEf (eV)\n01234Density of states (1/eV)(a)\n00.71.01.3\nBCC Fe\na-Fe\na-FeB (3.7% B)\na-FeB (7.5% B)\na-FeB (11.3% B)\na-FeB (15.1% B)\na-FeB (18.9% B)\n0 5 10 15 20\nB percentage (%)0.70.91.11.31.5n(Ef) (1/eV)(b)before DFT relaxation\nafter DFT relaxation\nFIG. 7. (a) EDOS per atom for ideal BCC Fe (same data\nas in Fig. 3), a-Fe (the single snapshot used to build doped a-\nFe through random replacement), and a-FeB (B doped a-Fe)\nwith different doping levels of 3.7 %, 7.5 %, 11.3 %, 15.1 %, and\n18.9 %. For each doping level of a-FeB the plotted result is the\naverage over 10 different snapshots with random replacements\nof Fe atoms by B atoms, starting from a single original a-Fe\nsupercell. The inset magnifies the EDOS near the Fermi level.\n(b) Average and standard deviation of EDOS-FL for different\nlevels of B doping. Results before DFT relaxation are shown\nfor comparison.\nhas a fundamentally different g(r). A new distinct peak\nappears below the lowest inter-atomic distance of BCC\nFe, which is even lying outside the broadening observed\nforT=500 K. Meanwhile, g(r) for larger rhas a nearly\ncontinuous distribution instead of a discrete distribution\nsharply peaked at specific distances. This is consistent\nwith our result of only a minor influence of lattice tem-\nperature on EDOS-FL and a more noticeable change for\na-Fe.\nD. Amorphous iron with boron doping\nFinally, after showing in Sec. III B that B doping can\ndecrease the electronic density of states at the Fermi level\n(EDOS-FL) for crystalline Fe, we now consider B doping\nof a-Fe. For each doping level, the random distribution\nof B sites is considered by calculating ten possible con-\nfigurations. They are constructed as described in Sec. II\nand subsequently relaxed using DFT until forces on all\natoms are smaller than 0 .15 eV/ ˚A, i.e., the same force\ntolerance used for “MD+DFT relaxed” a-Fe. For each of\nthese the EDOS is computed and averaged for analysis.\nOur results in Fig. 7(a) show that B doping decreases\nthe averaged EDOS at the Fermi level n(Ef) (EDOS-Structures EDOS-FL (1 /eV)Error\nBCC FeT=0 K 0.974 —\nT=10 K 0.991 0.004\nT=300 K 1.038 0.012\nT=500 K 1.058 0.019\nB-doped BCC Fe6.25 % B 0.662 —\n12.50 % B 0.739 0.086\n18.75 % B 0.786 0.096\n25.00 % B 0.712 0.149\na-Fe 1.199 0.065\na-FeB3.7 % B 0.965 0.049\n7.5 % B 0.866 0.025\n11.3 % B 0.842 0.024\n15.1 % B 0.817 0.027\n18.9 % B 0.803 0.029\nTABLE II. Electron density of states of BCC Fe at the Fermi\nlevel (EDOS-FL) for all different structures studied in this\nwork, i.e., non-zero lattice temperature, B doping, amorphous\nlattice (a-Fe), and B doped amorphous Fe (a-FeB). Values are\nnormalized per atom and we report averages and standard de-\nviations for the cases where multiple configurations contribute\n(see text).\nFL) for a-FeB, similar to what we observed for crystalline\nBCC Fe. With increasing B doping, the EDOS-FL of a-\nFeB monotonically decreases in the range from 0 % to\n18.9 % B, and starts to saturate around 15 %. The inset\nin Fig. 7(a) focuses on this trend near the Fermi energy\n(see black arrow). Interestingly, doping B into a-Fe can\nlead to an EDOS-FL that is even smaller than that of\npure BCC Fe, overcoming the increase we discussed for\na-Fe (see Fig. 6). This implies that a-FeB is potentially\nadvantageous for reducing intrinsic intra-band magnon-\nelectron damping, even though undoped a-Fe is not.\nThis is illustrated more clearly in Fig. 7(b) via aver-\nage and standard deviation. The decrease of n(Ef) with\nincreasing B doping from 0 to 18.9 % is outside the sta-\ntistical error bars and gradually saturates when the dop-\ning percentage is ≳15 %. Therefore, we do not consider\nlarger doping percentages in this work. It is possible\nthat further increasing the doping percentage may even\nincrease n(Ef) similar to what we have observed for B\ndoped BCC Fe in Fig. 3(b). Exploring this is outside the\nscope of this work, since it is undesired for reducing mag-\nnetic damping. Our simulations indicate that 18 .9 % of B\ndoping is near the optimal doping level that corresponds\nto the lowest n(Ef) for a-FeB.\nFinally, for all simulated a-FeB structures, DFT relax-\nation leads to a reduction of EDOS-FL, similar to what\nwe observed for a-Fe in Fig. 6. Figure 7(b) shows for\ndifferent B doping percentages that this reduction is be-\nyond the range of the statistical error bar. The trend in\nthis figure indicates that while relaxing the amorphous\nstructure to even lower force tolerance is computationally\ncostly, it could lead to slightly lower values of EDOS-FL.\nWe summarize our numerical results for averages and\nstandard deviations of EDOS-FL for all simulated struc-\ntures in Table II. This shows that crystalline BCC Fe8\ndoped with 6.25 % B leads to the lowest EDOS-FL among\nall structures considered. Previous experimental work\ndemonstrated that doping carbon into Co-Fe alloys can\nnaturally make the structure amorphous and that amor-\nphous Co-Fe-C alloys can have smaller magnetic damp-\ning than polycrystalline Co-Fe alloys [31]. Recent ex-\nperiments confirmed that similar effects can be achieved\nwith B doping [32]. While this is explained by smaller\ngrain sizes in the amorphous material and, therefore, re-\nduced sample inhomogeneity leading to lower damping,\nour results indicate that B doping of amorphous Fe is an\nadditional factor that can reduce the intrinsic magnetic\ndamping by decreasing the EDOS-FL.\nIV. CONCLUSIONS\nWe used first-principles simulations to quantitatively\nstudy the influence of lattice temperature, boron doping,\nand structure amorphization on the electronic structure\nof iron. Using our results, we discussed how these fac-\ntors affect the electronic density of states at the Fermi\nlevel, as one important quantity determining the intrin-\nsic magnetic damping in metallic magnets within Kam-\nbersk´ y’s breathing Fermi surface model. Generally, we\nshowed that both structural effects modify the density\nof states near the Fermi level to a lesser extent. In-stead, doping with B is advantageous in crystalline and\namorphous Fe to reduce damping and an optimal doping\nlevel of 6.25 % B reduces EDOS-FL by 32 % in crystalline\nFe. We also showed that different arrangements of the B\natoms on Fe sites affect the resulting magnetization and\nEDOS-FL values differently. This possibly suggests ma-\nterials growth under an external magnetic field to favor\narrangements with large magnetization and small damp-\ning. While this work points to a reduction of EDOS-FL\nas one reason for reduced magnetic damping in B doped\nFe, we expect similar mechanisms to govern C doped ma-\nterial and Co-Fe alloys.\nACKNOWLEDGMENTS\nWe thank Brian R. Robinson for insightful discussions\nand valuable suggestion during the development of this\nmanuscript. This work was supported by the U.S. DOE,\nOffice of Science, Basic Energy Sciences, Materials Sci-\nences and Engineering Division under contract No. DE-\nSC0022060. 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Then, we\ninvestigate the properties of global attractors and the exi stence of exponential attractors. Finally,\nthe upper semicontinuity of global attractors has been inve stigated.\nMathematics Subject Classification(2010): 35B40, 35B41, 35L20, 74K1 0\nKeywords. Piezoelectric system; the well posedness; global attracto r; upper semicontinuity of\nattractors.\n1 Introduction\nIn this paper, we investigate the piezoelectric beam system with magnetic effect:\n\n\nρvtt−αvxx+γβpxx+f1(v,p)+g1(vt) =εh1,(0,L)×(0,T),\nµptt−βpxx+γβvxx+f2(v,p)+g2(pt) =εh2,(0,L)×(0,T),(1.1)\nwhereε∈[0,1],α=α1+γ2β, andα,ρ,γ,β,µ> 0 mean elastic stiffness, the mass density per unit\nvolume, piezoelectric coefficient, water resistance coeffici ent, magnetic permeability, respectively,\nwhich is supplemented by the following initial boundary con ditions:\n\n\nv(0,t) =αvx(L,t)−γβpx(L,t) = 0, t> 0,\np(0,t) =px(L,t)−γvx(L,t) = 0, t> 0,\nv(x,0) =v0, vt(x,0) =v1, 0<x<L,\np(x,0) =p0, pt(x,0) =p1, 0<x<L.(1.2)\nIt is well known that the ability to convert mechanical energ y and electrical energy into each\nother is a characteristic of piezoelectric materials. In ad dition, it is smaller, less expensive, and more\nefficient [31]. Therefore, it has been applied in automotive [ 23, 42], medical [12, 13], space structures\n[4, 26] and many other fields [43]. In the previous modeling st age, the magnetic effect was basically\nignored due to its slight [36], but the latest results show th at the magnetic effect may influence the\n∗Email addresses: gongweiliu@haut.edu.cn (G. Liu), wangme ngru97@163.com (M. Wang), dpymath@haut.edu.cn\n(P. Ding)\n1Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 2\nperformance of the system [38], so it is crucial to study piez oelectric beam models with magnetic\neffects. Nowadays, a growing number of scholars are beginning to do related research in this field.\nThe establishment of mathematical models of piezoelectric beams has a long history. In the early\ndays, Tebou [41] and Haraux [19] established the model of a si ngle beam due to Maxwell equation\n[24, 25] and the dynamic interaction between electromagnet ism is ignored. Therefore, they obtained\nthe following equation:\n\n\nρvtt−α1vxx= 0, (x,t)∈(0,L)×R+,\nv(0,t) =α1vx(L,t)+δvt(L,t) = 0, t∈R+.\nBased on the magnetic effects and variational method, the equa tions of a single beam were derived\nby Morris and ¨Ozer [28], as follows:\n\n\nρvtt−αvxx+γβpxx= 0, (x,t)∈(0,L)×R+,\nµptt−βpxx+γβvxx= 0, (x,t)∈(0,L)×R+,\nv(0,t) =p(0,t) =αvx(L,t)−γβpx(L,t) = 0, t∈R+,\nβpx(L,t)+V(t)\nh= 0, t ∈R+.(1.3)\nwhereV(t) =pt(L,t) denotes electrical feedback controller. Then, they estab lished the further\nresults: the well-posedness of solution, strong stability of piezoelectric beam and so on [29, 30].\nIn [34], Ramos et al. inserted a dissipative term δvtin the first equation of (1.3) and considered\nthe following boundary condition\n\n\nv(0,t) =αvx(L,t)−γβpx(L,t) = 0,0<t<T,\np(0,t) =px(L,t)−γvx(L,t) = 0, 0<t<T.\nThe authors applied energy method to prove the energy of the s ystem and estimated that the energy\nis exponentially stable. After that, Ramos et al. changed th e boundary condition in [35]:\n\n\nv(0,t) =αvx(L,t)−γβpx(L,t)+ξ1vt(L,t)\nh= 0,0<t<T,\np(0,t) =px(L,t)−γvx(L,t)+ξ2pt(L,t)\nh= 0, 0<t<T,\nand proved the exponential stability regardless of any rela tionship between system coefficients and\nthere is equivalent to exact observability at the boundary.\nMoreover, we know there are many ways to control piezoelectr ic vibration, among which time-\ndelayed feedback control is a common way to improve stabilit y of the system. Therefore, more and\nmore researchers studied the influence of time-delayed effect on the stability of hyperbolic systems,\nand also proved the so-called destabilizing effect.\nDakto et al. [8, 9, 10, 11] has obtained a series of research re sults in this direction. Among them,\nDakto studied the equation as follows,\nutt−uxx+2ut(x,t−τ) = 0,(x,t)∈(0,1)×(0,+∞),Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 3\nwhereτ >0, and proved that the time delay in the dissipative term char catered by velocity term\ncould make the beams lose stability.\nTherefore, it is necessary to add some controlled term to sta bilize the hyperbolic systems.\nIn 2021, Freitas and Ramos [15] considered the longitudinal vibration of the piezoelectric beams\nwith thermal effect, magnetic effect and fraction damping:\n\n\nρvtt−αvxx+γβpxx+δθx+f1(v,p)+vt=h1,(0,L)×(0,T),\nµptt−βpxx+γβvxx+Aνpt+f2(v,p) =h2, (0,L)×(0,T),\ncθt−κθxx+δvtx= 0, (0,L)×(0,T),\nthe authors applied the semigroup theory to prove the well-p osedness of solutions, then they showed\nthat the existence of global attractor, exponential attrac tor and the upper semi-continuity of global\nattractor when ν→0+.\nFreitas et al. [14] studied nonlinear piezoelectric beam sy stem with delay term:\n\n\nρvtt−αvxx+γβpxx+f1(v,p)+vt=h1,\nµptt−βpxx+γβvxx+f2(v,p)+µ1pt+µ2pt(x,t−τ) =h2,\nand proved the system is asymptotically smooth gradient, an d then they used stability estimation to\nobtain that the system is quasi-stable. Finally, they also p roved the global attractor has finite fractal\ndimension.\nMa et al. [27] studied long-time dynamics of semilinear wave equation:\n∂2\ntu−∆u+a(x)g(∂tu)+f(u) =εh(x) in Ω ×R+,\nwhere Ω ⊂R3is a bounded domain with a smooth boundary ∂Ω andε∈[0,1]. In this paper, they\nmainly proved the existence and properties of global attrac tors and attractors are continuous under\nautonomous perturbations.\nFreitas et al. [16] investigated the following wave equatio ns:\n/braceleftigg\nutt−∆u+(−∆)α1ut+g1(ut) =f1(u,v)+εh1,in Ω×R+,\nvtt−∆v+(−∆)α2vt+g2(vt) =f2(u,v)+εh2,in Ω×R+,\nwhere Ω ⊂R2is a bounded domain with smooth boundary ∂Ω,α1,α2∈(0,1) andε∈[0,1]. The\nauthors showed that the dynamical system is quasi-stable an d that the family of the global attractors\nis the continuous with respect to the parameter.\nRecently, Freitas and Ramos [17] also considered a system mo delinga mixtureof three interacting\ncontinuawithexternalforceswithaparameter ε∈[0,1], andgavethesmoothnessofglobalattractors,\nthe continuity of global attractors.\nThe main contributions of this paper are:\n(1) We consider the piezoelectric system with nonlinear dam ping and external forces with a\nparameter, we prove the existence of solutions by maximal mo notone operator theory.\n(2) We establish a quasi-stability estimate to prove the sys tem is quasi-stable and then we obtain\nthe existence of exponential attractor and some properties of the global attractor.\n(3) We consider the upper semicontinuity of global attracto rs with respect to the parameter\nε∈[0,1].Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 4\nThe present paper is organized as following. In section 2, we mainly establish the well-posedness\nof the solution by nonlinear operator theory. In section 3, w e apply the infinite dynamical system\ntheory to prove the existence of global attractors and expon ential attractors. In section 4, we mainly\nshow the upper semi-continuity of the global attractors.\n2 Existence of global solutions\nIn this section, we first give some assumptions and notations . Then, we are concerned with well-\nposedness of the solution of the system (1.1)-(1.2) by the si milar argument as in [7, 32, 33] and the\nreferences therein.\n2.1 Assumption\nNow, we give some assumptions which will be used hereinafter . And the following C,Ciall denote\npositive generic constants, which may be different in various lines.\n(1) The function F∈C2(R2) satisfies\n∇F= (f1,f2), (2.1)\nand there exists β0,mF≥0 such that\nF(v,p)≥ −β0(|v|2+|p|2)−mF, (2.2)\nwhere\n0≤β0<1\n2β1, (2.3)\nandβ1is the embedding constant defined in (3.6). Moreover, there e xist constant r≥1,Cf>0 such\nthat\n|fi(v)−fi(p)| ≤Cf(1+|v|r−1+|p|r−1)|v−p|. (2.4)\nFurthermore, for arbitrary v,p∈R, we get\n∇F(v,p)·(v,p)−F(v,p)≥ −β0(|v|2+|p|2)−mF. (2.5)\n(2) The external forces h1,h2∈L2(0,L),ε∈[0,1].\n(3) For the nonlinear damping gi(i= 1,2), we have\ngi∈C1(R), gi(0) = 0, (2.6)\nand there exists m,M1,q≥1,\nm≤g′\ni(v)≤M1(1+|v|q−1),∀v∈R. (2.7)\nIfq≥3, there exist l>q−1,M2>0 such that\ngi(v)v≥M2|v|l,|v| ≥1. (2.8)\nFurthermore, from (2.7), we obtain\n(gi(p)−gi(v))(p−v)≥m|p−v|2, p,v ∈R. (2.9)Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 5\n2.2 The Cauchy problem\nIn the rest of this paper, we denote\n/ba∇dblv/ba∇dblr=/ba∇dblv/ba∇dblLr(0,L), r≥1,(v,p)2= (v,p)L2(0,L).\nWe shall define the Sobolev Space\nHi\n∗(0,L) ={v∈Hi(0,L) :v(0) = 0}, i= 1,2.\nBecause of v(0) = 0, we obtain the Poincar´ e’s inequality\nλ1/ba∇dblv/ba∇dbl2\n2≤ /ba∇dblvx/ba∇dbl2\n2,∀v∈H1\n∗(0,L), (2.10)\nwhereλ1>0. Hence, we deduce /ba∇dblv/ba∇dblH1∗(0,L)=/ba∇dblvx/ba∇dbl2.\nThe space His defined by\nH=V×H= (H1\n∗(0,L))2×(L2(0,L))2.\nThe inner product on His\n(z,˜z)H=α1/integraldisplayL\n0vx˜vxdx+β/integraldisplayL\n0(γvx−px)(γ˜vx−˜px)dx+ρ/integraldisplayL\n0vt˜vtdx+µ/integraldisplayL\n0pt˜ptdx,\nwherez= (v,p,vt,pt)T,˜z= (˜v,˜p,˜vt,˜pt)T.\nFrom the inner product, we can define the norm as\n/ba∇dblz(t)/ba∇dbl2\nH=/ba∇dbl(v,p,˜v,˜p)T/ba∇dbl2\nH=α1/ba∇dblvx/ba∇dbl2+β/ba∇dblγvx−px/ba∇dbl2+ρ/ba∇dbl˜v/ba∇dbl2+µ/ba∇dbl˜p/ba∇dbl2, (2.11)\nMoreover, there exists a constant κ>0 such that\n/ba∇dblv/ba∇dbl2\n2+/ba∇dblp/ba∇dbl2\n2≤1\nλ1(/ba∇dblvx/ba∇dbl2\n2+/ba∇dblpx/ba∇dbl2\n2)≤κ(α1/ba∇dblvx/ba∇dbl2\n2+β/ba∇dblγvx−px/ba∇dbl2\n2), (2.12)\nIn fact, observing that\n/ba∇dblpx/ba∇dbl2\n2=/ba∇dblγvx−px−γvx/ba∇dbl2\n2≤2/ba∇dblγvx−px/ba∇dbl2\n2+2γ2/ba∇dblvx/ba∇dbl2\n2,\nwe have\n/ba∇dblvx/ba∇dbl2\n2+/ba∇dblpx/ba∇dbl2\n2≤(1+2γ2)/ba∇dblvx/ba∇dbl2\n2+2/ba∇dblγvx−px/ba∇dbl2\n2.\nwhereκ= max{(2γ2+1)α−1\n1,2β−1}, theinequality (2.12) holds. CombiningthePoincar´ e’s in equality\n(2.10) and the above formula, there exists β1=κ\nλ1>0 such that\n/ba∇dblv/ba∇dbl2\n2+/ba∇dblp/ba∇dbl2\n2≤β1(α1/ba∇dblvx/ba∇dbl2\n2+β/ba∇dblγvx−px/ba∇dbl2\n2). (2.13)\nLet us write the system (1.1)-(1.2) as an equivalent Cauchy p roblem\nd\ndtz(t)+Az(t) =Fz(t), z(0) =z0, (2.14)Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 6\nwherez0= (v0,p0,v1,p1)T∈ H,andA:D(A) ={(v,p,˜v,˜p)T∈ H:v,p∈H2(0,L),˜v,˜p∈\nH1\n∗(0,L),vx(L) =px(L) = 0} ⊂ H֒→ His defined by\nAz(t) =\n−˜v\n−˜p\n1\nρ(−αvxx+γβpxx+g1(vt))\n1\nµ(−βpxx+γβvxx+g2(pt))\n.\nThe function F:H → His defined by\nF(z) =\n0\n0\n1\nρ(εh1−f1(v,p)\n1\nµ(εh2−f2(v,p)\n. (2.15)\nBy a simple calculation, we have\n(Az,z)H=/integraldisplayL\n0(g1(˜v)˜v+g2(˜p)˜p)dx≥0, (2.16)\n2.3 Energy Identities\nDefinition 2.1. Letz(t) = (v,p,vt,pt)T∈C([0,∞),H)be a weak solution of the system (1.1)-(1.2),\nif for anyϕ,ψ∈H1\n∗(0,L)it satisfies\nρd\ndt(vt,ϕ)2+µd\ndt(pt,ψ)2+α1(vx,px)2+β(γvx−px)2(γϕx−ψx)2+(g1(vt),ϕ)2\n+(g2(pt),ψ)2+/integraldisplayL\n0f1(v,p)ϕdx+/integraldisplayL\n0f2(v,p)ψdx= (εh1,ϕ)2+(εh2,ψ)2.\nMoreover, if\nz∈C([0,∞);D(A))∩C1([0,∞);H),\nthenzis called the strong solution.\nThe total energy of solutions of (1.1)-(1.2) is defined by\nE(t) =E(t)+/integraldisplayL\n0F(v(t),p(t))dx−ε/integraldisplayL\n0(h1v(t)+h2p(t))dx, t ≥0, (2.17)\nwhereE(t) =1\n2/ba∇dblz(t)/ba∇dbl2\nH.\nLemma 2.1. Ifz= (v,p,vt,pt)Tis a strong solution of (1.1)-(1.2), the following conclusions will\nhold,\n(1)\nd\ndtE(t) =−/integraldisplay\n(g1(vt)vt+g2(pt)pt)dx\n≤ −m(/ba∇dblvt(t)/ba∇dbl2\n2+/ba∇dblpt(t)/ba∇dbl2\n2).(2.18)\n(2) There exist constants β2,CF>0such that\nβ2/ba∇dblz/ba∇dbl2\nH−CF≤ E ≤CF(1+/ba∇dblz/ba∇dblr+1\nH), t≥0. (2.19)Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 7\nProof.Multiplying the equations in (1.1) by vt,pt, respectively, and using integration by parts, and\napplying the inequality (2.9), we can obtain (2.18).\nApplying (2.2) and (2.14), we have\n/integraldisplayL\n0F(v,p)dx≥ −β0(/ba∇dblv/ba∇dbl2\n2+/ba∇dblp/ba∇dbl2\n2)−LmF\n≥ −β0β1(α1/ba∇dblvx/ba∇dbl2\n2+β/ba∇dblγvx−px/ba∇dbl2\n2)−LmF\n≥ −β0β1/ba∇dbl(v,p,vt,pt)T/ba∇dbl2\nH−LmF.(2.20)\nUsing (2.3) and (2.20),\nE(t) =/integraldisplayL\n0F(v,p)dx−ε/integraldisplayL\n0(h1v+h2p)dx+E(t)\n≥(1\n2−β0β1)/ba∇dblz/ba∇dbl2\nH−LmF−ε/integraldisplayL\n0(h1v+h2p)dx.\nLet\nβ2=1\n4(1−2β0β1)>0, (2.21)\nsinceε∈[0,1], we have\nε/integraldisplayL\n0(h1v+h2p)dx≤β2\nβ1(/ba∇dblv/ba∇dbl2\n2+/ba∇dblp/ba∇dbl2\n2)+β1\n4β2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2), (2.22)\nthe first part of (2.19) is obtained with\nCF=LmF+β1\n4β2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2).\nMoreover, by (2.4) we have\n/integraldisplayL\n0F(v,p)dx≤C(/ba∇dblvx/ba∇dblr+1\n2+/ba∇dblpx/ba∇dblr+1\n2+1). (2.23)\nBy (2.13) and (2.25), we can deduce the second inequality in ( 2.19).\n2.4 Well-Posedness\nIn this part, we will use the nonlinear operator theory to pro ve the well-posedness of solutions.\nDefinition 2.2. LetXbe a reflexive Banach space, the operator A:X→X′is called monotone if\nit satisfies\n/an}b∇acketle{tAz1−Az2,z1−z2/an}b∇acket∇i}ht ≥0,∀z1,z2∈D(A),\nFurthermore, if (A+I) :X→X′is onto, thenAis maximal.\nDefinition 2.3. LetXbe a reflexive Banach space, the operator B:X→X′is called hemicontinu-\nous, if\nlim\nλ→0/an}b∇acketle{tB(u+λv),w/an}b∇acket∇i}ht=/an}b∇acketle{tBu,w/an}b∇acket∇i}ht,∀u,v,w∈X.\nLemma 2.2. The operator F:H → Hdefined in (2.15)is locally Lipschitz continuous.Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 8\nProof.Let the solutions z1,z2∈ Hsuch that /ba∇dblz1/ba∇dblH,/ba∇dblz2/ba∇dblH≤ R, whereR>0.\nThen, we can deduce\n/ba∇dblF(z1)−F(z2)/ba∇dbl2\nH=1\nρ/integraldisplayL\n0|f1(v1,p1)−f1(v2,p2)|2dx+1\nµ/integraldisplayL\n0|f2(v1,p1)−f2(v2,p2)|2dx.\nBy (2.4),\n|fi(v1,p1)−fj(v2,p2)|2≤Cf(|v1|r−1+|p1|r−1+|v2|r−1+|p2|r−1+1)2(|v1−v2|2+|p1−p2|2).(2.24)\nIt follows from (2.24) that there exists some constant CR>0 such that\n/integraldisplayL\n0|fi(v1,p1)−fj(v2,p2)|2dx≤CR/ba∇dblz1−z2/ba∇dbl2\nH, i= 1,2.\nwhich implies that\n/ba∇dblF(z1)−F(z2)/ba∇dbl2\nH≤CR/ba∇dblz1−z2/ba∇dblH.\nNow, we are in the position to give the existence of the soluti ons.\nTheorem 2.1. Suppose the assumptions (2.1)-(2.7)hold. Ifz0∈ H, the system (1.1)-(1.2)has a\nunique weak solution zt()satisfiesz∈C([0,∞);H), and it depends continuously on the initial data\nz0. In addition, if z0∈D(A), the weak solution is strong solution.\nProof.Similar to (2.15) and (2.16), using the monotonicity of func tiongi,i= 1,2, we know Ais\nmonotone. For purpose of obtain Ais a maximal monotone operator, we need to show that there\nexistsz∈D(A), for arbitrary w∈ Hsuch that\nAz+z=w.\nIn fact, in the following, we decompose the operator Aas\nA=/parenleftbigg0−I\nB G/parenrightbigg\n,\nwhereB:D(B) ={(v,p)∈H2\n∗(0,L)×H2\n∗(0,L)} ⊂V ֒→H,\nB(v,p) =/parenleftigg1\nρ(−αvxx+γβpxx)\n1\nµ(−βpxx+γβvxx)/parenrightigg\n,\nG:H→Hand\nG(vt,pt) =/parenleftigg1\nρg1(vt)\n1\nµg2(pt)/parenrightigg\n.\nWritingz= (v,p,vt,pt)T= (τ,η)T∈V×H,w= (w1,w2)T∈V×H, so\nτ−η=w1, B (τ)+G(η)+η=w2,\nthen, we can analyze that η∈V. Hence, we need to prove K(η) = (B+I)η+G(η) :V→V′is onto.\nBy Corollary 2.2 of [1], we need only to prove Kis maximal monotone and coercive. From (2.11)\nand the embedding V ֒→H=H′֒→V′such that\n/an}b∇acketle{tBτ,η/an}b∇acket∇i}ht= (τ,η)V, τ,η∈V. /an}b∇acketle{tτ,η/an}b∇acket∇i}ht= (τ,η)H, τ∈H,η∈V.Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 9\nFirstly, let η1,η2∈V, then\n/an}b∇acketle{t(B+I)(η1−η2),η1−η2/an}b∇acket∇i}ht=/an}b∇acketle{tB(η1−η2),η1−η2/an}b∇acket∇i}ht+/an}b∇acketle{tη1−η2,η1−η2/an}b∇acket∇i}ht\n=/ba∇dblη1−η2/ba∇dbl2\nV+/ba∇dblη1−η2/ba∇dbl2\nH≥0.(2.25)\nBy (2.9), we have\n/an}b∇acketle{tG(η1)−G(η2),η1−η2/an}b∇acket∇i}ht= (G(η1)−G(η2),η1−η2)H≥0. (2.26)\nSo we obtain that B+IandGare monotone.\nSecondly, let τ,η,φ∈Vwhereτ= (τ1,τ2,τ3),η= (η1,η2,η3),φ= (φ1,φ2,φ3), andλ∈R,λn→\n0, we have\n|/an}b∇acketle{t(B+I)(τ+λη),φ/an}b∇acket∇i}ht−/an}b∇acketle{t(B+I)τ,φ/an}b∇acket∇i}ht|=|λ/an}b∇acketle{t(B+I)η,φ/an}b∇acket∇i}ht| ≤ |λ|(/an}b∇acketle{tBy,φ/an}b∇acket∇i}ht+/an}b∇acketle{tη,φ/an}b∇acket∇i}ht),\nwhich implies that the continuity of /an}b∇acketle{t(B+I)(τ+λη),φ/an}b∇acket∇i}htatλ= 0. Then\n/an}b∇acketle{tG(τ+λnη),φ/an}b∇acket∇i}ht=3/summationdisplay\ni=1/integraldisplayL\n0gi(τi+λnηi)φidx.\nClearly\ngi(τi+λnηi)φi→gi(τi)φia.e in(0,L).\nBy (2.7) and Dominated Convergence Theorem, we deduce\nlim\nn→∞/an}b∇acketle{tG(τ+λnη),φ/an}b∇acket∇i}ht= (G(τ),φ)H.\nSo we obtain that B+IandGare hemicontinuous.\nIn addition, we can deduce B+IandGare coercive from (2.25) and (2.26).\nTherefore, according to Theorem 2.6 of [1], we know that Kis coercive and maximal monotone,\nwhich implies that Kis onto. That is, Ais maximal monotone in H.\nInconclusion, since Aismaximalmonotoneand Fislocally Lipschitz, byapplyingTheorem7.2of\n[5], wecanobtain: when z0∈ H, theproblem(2.14)hasauniqueweak solution z(t)∈C([0,tmax);H).\nMoreover, if tmax<∞, then limsupt→t−\nmax/ba∇dblz/ba∇dblH=∞. Whenz0∈D(A), the problem (2.14) has a\nunique strong solution z(t)∈C([0,tmax),D(A)),(tmax≤ ∞).\nNow, we need to show the existence of global solutions, that i stmax=∞.In fact, let z(t) be a\nstrong solution defined in [0 ,tmax). From (2.18), we infer\nE(t)≤ E(0). (2.27)\nIt follows from (2.19) and (2.17) that\n/ba∇dblz/ba∇dbl2\nH≤1\nβ2(E(0)+CF).\nBy density argument, the conclusion for weak solution also h olds. Therefore tmax=∞.\nFinally, let z1,z2be two weak solutions, by the standard arguments, for any T >0, there exists\nconstantC >0 such that\n/ba∇dblz1(t)−z2(t)/ba∇dbl2\nH≤eCT/ba∇dblz1(0)−z2(0)/ba∇dbl2\nH, t∈[0,T]. (2.28)\nThe proof is complete.Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 10\n3 The existence of global attractor\nIn this section, for the sake of completeness, we collect som e known results in the theory of nonlinear\ndynamical systems (see [3, 6, 37, 40]).\nLet (v,p,vt,pt)Tbe the unique solution for the system (1.1)-(1.2). We can defi ne the operator\nS(t) :H → Hby\nS(t)(v0,p0,v1,p1)T= (v(t),p(t),vt(t),pt(t))T, t≥0.\nHence, (H,S(t)) constitutes a dynamical system.\nDefinition 3.1. LetB⊂ Hbe a positively invariant set of a dynamical system (H,S(t)).\n1. A function Φ(y)is said to be a Lyapunov function, if t→Φ(S(t)y)is a non-increasing function\nfor anyy∈B.\n2. A Lyapunov function is called strict, if there exist t0>0,y∈Bsuch that Φ(S(t0)y) = Φ(y),\ntheny=S(t)y, t>0.\nDefinition 3.2. The dynamical system (H,St)is quasi-stable on B⊂ H, if there exist a compact\nseminormχX(·)on the space Xand nonnegative scalar function a(t),b(t),c(t)∈R+satisfy:\n(1)a(t),c(t)are locally bounded on [0,∞);\n(2)b(t)∈L1(R+), andlimt→∞b(t) = 0;\n(3) for any y1,y2∈Bandt≥0, the estimates\n/ba∇dblS(t)y1−S(t)y2/ba∇dbl2\nH≤a(t)/ba∇dbly1−y2/ba∇dbl2\nH,\nand\n/ba∇dblS(t)y1−S(t)y2/ba∇dbl2\nH≤b(t)/ba∇dbly1−y2/ba∇dbl2\nH+c(t) sup\n0≤s≤t[χX(u1(s)−u2(s))]2,\nhold, where S(t) = (ui(t),ui\nt(t)),i= 1,2.\nLemma 3.1. The dynamical system (H,S(t))is gradient, that is, there exists a strict Lyapunov\nfunction Φ∈ H. What’s more,\nΦ(z)→ ∞ ⇐⇒ /ba∇dbl z/ba∇dblH→ ∞.\nProof.LetE(t) defined in (2.17) be a Lyapunov function Φ and z0= (v0,p0,v1,p1)T∈ H, we can\ninfer thatt→Φ(S(t)z0) is a non-increasing function from (2.18).\nSupposed that Φ( S(t)z0) = Φ(z0), fort≥0. Then\n/ba∇dblvt(t)/ba∇dbl2\n2=/ba∇dblpt(t)/ba∇dbl2\n2= 0, t≥0.\nWe obtain\nvt=pt= 0, a.e in (0,L), t≥0.\nConsequently, vt(t) =v0,pt(t) =p0, which implies S(t)z0=z(t) = (v0,p0,0,0)Tis a stationary\nsolution of ( H,S(t)).\nFrom (2.19), we obtain\nΦ(z)≤CF(1+/ba∇dblz/ba∇dblr+1\nH), t≥0.\nLet Φ(z)→ ∞, we have /ba∇dblz/ba∇dblH→ ∞. Then, it follows from (2.19) that\n/ba∇dblz/ba∇dbl2\nH≤Φ(z)+CF\nβ2.\nAs a result, we infer from /ba∇dblz/ba∇dblH→ ∞that Φ(z)→ ∞.Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 11\nLemma 3.2. The set of stationary points NofS(t)is bounded in H.\nProof.Letz= (v,p,0,0)T∈ Nbe the stationary solution of problem (1.1)-(1.2). Then, we have the\nfollowing elliptic equations/braceleftigg\n−αvxx+γβpxx+f1(v,p) =εh1,\n−βpxx+γβvxx+f2(v,p) =εh2.(3.1)\nMultiplying the equations in (3.1) by v,p, respectively, and integrating the result over (0 ,L), we\nhave\nα1/ba∇dblvx/ba∇dbl2\n2+β/ba∇dblγvx−px/ba∇dbl2\n2=−/integraldisplayL\n0(f1(v,p)v+f2(v,p)p)dx+ε/integraldisplayL\n0(h1v+h2p)dx.\nHence, using (2.1), (2.2), and (2.5), we obtain\n−/integraldisplayL\n0(f1(v,p)v+f2(v,p)p)dx≤2β0β1(α1/ba∇dblvx/ba∇dbl2\n2+β/ba∇dblγvx−px/ba∇dbl2\n2)+2LmF.\nBy (2.21), we have\n4β2(α1/ba∇dblvx/ba∇dbl2\n2+β/ba∇dblγvx−px/ba∇dbl2\n2)≤2LmF+ε/integraldisplayL\n0(h1v+h2p)dx.\nBy Young’s inequalities and (2.13), we infer\n/integraldisplayL\n0(h1v+h2p)dx≤β2\nβ1(/ba∇dblv/ba∇dbl2\n2+/ba∇dblp/ba∇dbl2\n2)+β1\n4β2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2)\n≤β2(α1/ba∇dblvx/ba∇dbl2\n2+β/ba∇dblγvx−px/ba∇dbl2\n2)+β1\n4β2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2).\nTherefore, we conclude\n3β2/ba∇dblz/ba∇dbl2\nH≤2LmF+β1\n4β2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2). (3.2)\nThe proof is complete.\nLemma 3.3. Suppose the Assumption 2.1 holds. Let Bbe a bounded forward invariant set in H\nandS(t)zi= (vi,pi,vi\nt,pi\nt)Tbe a weak solution of the problem (1.1)-(1.2)withzi\n0∈B,i= 1,2. Then\nthere exist constant σ,ς,CB>0independent of εsuch that\nE(t)≤ςE(0)e−σt+CBsup\ns∈[0,t](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ),∀t≥0.\nwhereE(t) =1\n2/ba∇dblz/ba∇dbl2\nH,v=v1−v2,p=p1−p2,θ≥2.\nProof.LetFi(v,p) =fi(v1,p1)−fi(v2,p2),i= 1,2,G1(vt) =g1(v1\nt)−g1(v2\nt),G2(pt) =g2(p1\nt)−g2(p2\nt).\nThenv=v1−v2,p=p1−p2satisfy\n\n\nρvtt−αvxx+γβpxx+G1(vt) =−F1(v,p),\nµptt−βpxx+γβvxx+G2(vt) =−F2(v,p),\n(v(0),p(0),vt(0),pt(0)) =z1−z2,\nv(0) =p(0) =vx(L) =px(L) = 0.(3.3)Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 12\nMultiplying the first equation of (3.3) by v, the second one by p, and integrating over [0 ,L]×[0,T],\nwe have\n/integraldisplayT\n0E(t)dt=/integraldisplayL\n0(ρvtv+µptp)dx/vextendsingle/vextendsingle/vextendsingleT\n0+/integraldisplayT\n0(ρ/ba∇dblvt/ba∇dbl2\n2+µ/ba∇dblpt/ba∇dbl2\n2)dt\n−1\n2/integraldisplayT\n0/integraldisplayL\n0(G1(vt)v)+G2(pt)p)dxdt−1\n2/integraldisplayT\n0/integraldisplayL\n0(F1(v,p)v+F2(v,p)p)dxdt.\nStep 1. By Poinc´ are’s and H¨ older’s inequalities, we have\n/integraldisplayL\n0(ρvtv+µptp)dx≤CE(t).\nThen /integraldisplayL\n0(ρvtv+µptp)dx/vextendsingle/vextendsingle/vextendsingleT\n0≤C(E(0)+E(T)),\nwhereCis a constant and C >0. From (2.9), we conclude\n/integraldisplayT\n0(ρ/ba∇dblvt/ba∇dbl2\n2+µ/ba∇dblpt/ba∇dbl2\n2)dt≤1\nm/integraldisplayT\n0/integraldisplayL\n0(G1(vt)v)+G2(pt)p)dxdt.\nStep 2. According to Young’s inequality and (2.9), we can deduce\n/integraldisplayT\n0/integraldisplayL\n0G1(vt)vdxdt≤1\n2/integraldisplayT\n0/integraldisplayL\n0G1(vt)vtdxdt+1\n2/integraldisplayT\n0/integraldisplayL\n0G1(vt)|v|2\nvtdxdt.\nApplying (2.7), we infer\n/integraldisplayL\n0G1(vt)vdx≤1\n2/integraldisplayL\n0G1(vt)vtdx+M1\n2/integraldisplayL\n0(1+|v1\nt|q−1+|v2\nt|q−1)|v|2dx.\nFor further estimation, we divide it into three cases.\nCase a:q= 1.It is easy to get\n/integraldisplayL\n0(1+|v1\nt|q−1+|v2\nt|q−1)|v|2dx≤3/ba∇dblv/ba∇dbl2\n2(1+/integraldisplayL\n0(g1(v1\nt)v1\nt+g1(v2\nt)v2\nt)dx).\nCase b: 1<q<3.Applying the assumption (2.9) and H¨ older’s inequality, we have\n/integraldisplayL\n0(1+|v1\nt|q−1+|v2\nt|q−1)|v|2dx≤C/ba∇dblv/ba∇dbl2\n2d1(/integraldisplayL\n0(1+|v1\nt|d(q−1)+|v2\nt|d(q−1))dx)1\nd\n≤C/ba∇dblv/ba∇dbl2\n2d1(L+/integraldisplayL\n0(g1(v1\nt)v1\nt+g1(v2\nt)v2\nt)dx)1\nd\n≤C/ba∇dblv/ba∇dbl2\n2d1(L+/integraldisplayL\n0(g1(v1\nt)v1\nt+g1(v2\nt)v2\nt)dx),\nwhered=2\nq−1, and1\nd+1\nd1= 1.\nCase c:q≥3.By the similar argument as in Case b, let d=l\nq−1>1, we can have the result.\nTo sum up, for some C >0,θ≥2, we can infer\n/integraldisplayL\n0G1(vt)vdx≤1\n2/integraldisplayL\n0G1(vt)vtdx+C/ba∇dblv/ba∇dbl2\nθ(L+/integraldisplayL\n0(g1(v1\nt)v1\nt+g1(v2\nt)v2\nt)dx).(3.4)Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 13\nSimilarly,\n/integraldisplayL\n0G2(pt)pdx≤1\n2/integraldisplayL\n0G2(pt)ptdx+C/ba∇dblp/ba∇dbl2\nθ(L+/integraldisplayL\n0(g2(p1\nt)p1\nt+g2(p2\nt)p2\nt)dx).(3.5)\nFurthermore, due to z1,z2∈B, and by (2.18), (2.19), we conclude there exists CB>0 such that\n/integraldisplayL\n0(g1(v1\nt)v1\nt+g1(v2\nt)v2\nt)dx≤CB,\n/integraldisplayL\n0(g2(p1\nt)p1\nt+g2(p2\nt)p2\nt)dx≤CB.\nCombining with (2.8), (3.4), (3.5), and applying L2θ(0,L)֒→Lθ(0,L), we deduce\n−1\n2/integraldisplayT\n0/integraldisplayL\n0(G1(vt)v+G2(pt)p)dxdt≤1\n2/integraldisplayT\n0/integraldisplayL\n0(G1(vt)vt+G2(pt)pt)dxdt\n+CTsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ).\nStep 3. Applying (2.4), H1\n∗(Ω)֒→Lr(Ω),r∈(1,∞) and H¨ older’s inequality, we can infer\n/integraldisplayL\n0F1(v,p)vdxdt≤Cf(/ba∇dblv1/ba∇dblr−1\n2θ+/ba∇dblv2/ba∇dblr−1\n2θ+/ba∇dblp1/ba∇dblr−1\n2θ+/ba∇dblp2/ba∇dblr−1\n2θ)(/ba∇dblv/ba∇dbl2θ+/ba∇dblp/ba∇dbl2θ)/ba∇dblv/ba∇dbl2\n≤CB((/ba∇dblv/ba∇dbl2θ+/ba∇dblp/ba∇dbl2θ)/ba∇dblv/ba∇dbl2\n≤CB((/ba∇dblv/ba∇dbl2\n2θ+/ba∇dblp/ba∇dbl2\n2θ).(3.6)\nSimilarly, we obtain/integraldisplayL\n0F2(v,p)pdxdt≤CB((/ba∇dblv/ba∇dbl2\n2θ+/ba∇dblp/ba∇dbl2\n2θ). (3.7)\nCombining (3.7) with (3.6), there exists CT>0, we have\n−1\n2/integraldisplayT\n0/integraldisplayL\n0(F1(v,p)v+F2(v,p)p)dxdt≤CTsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ).\nTherefore, combining the above estimates, we have\n/integraldisplayT\n0E(t)dt≤CB/integraldisplayT\n0/integraldisplayL\n0(G1(vt)vt+G2(pt)pt)dxdt\n+CTsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ)+C(E(0)+E(T)).(3.8)\nfor some constants CB,CT>0.\nStep 4. Multiplying the equations of (3.3) by vt,pt, respectively, and integrating over [0 ,L]×[s,T],\nwe obtain\nE(t) =E(s)−/integraldisplayT\ns/integraldisplayL\n0(G1(vt)vt+G2(pt)pt)dxdt−/integraldisplayT\ns/integraldisplayL\n0(F1(v,p)vt+F2(v,p)pt)dxdt. (3.9)\nDue to\n−/integraldisplayT\ns/integraldisplayL\n0(G1(vt)vt+G2(pt)pt)dxdt≤0,Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 14\nwe obtain\nE(t)≤E(s)−/integraldisplayT\ns/integraldisplayL\n0(F1(v,p)vt+F2(v,p)pt)dxdt.\nBy the similar argument, we have\n−/integraldisplayL\n0F1(v,p)vtdxdt≤CB((/ba∇dblv/ba∇dbl2θ+/ba∇dblp/ba∇dbl2θ)/ba∇dblvt/ba∇dbl2\n≤ζ/ba∇dblvt/ba∇dbl2\n2+CB\n4ζ(/ba∇dblv/ba∇dbl2\n2θ+/ba∇dblp/ba∇dbl2\n2θ).\nAnalogously,\n−/integraldisplayL\n0F2(v,p)ptdxdt≤ζ/ba∇dblpt/ba∇dbl2\n2+CB\n4ζ(/ba∇dblv/ba∇dbl2\n2θ+/ba∇dblp/ba∇dbl2\n2θ).\nConsequently,\n−/integraldisplayL\n0(F1(v,p)vt+F2(v,p)pt)dxdt≤ζE(t)+CB\n4ζ(/ba∇dblv/ba∇dbl2\n2θ+/ba∇dblp/ba∇dbl2\n2θ). (3.10)\nLetζ=1\nT, we have\nE(T)≤1\nT/integraldisplayT\n0E(t)dt+TCB/integraldisplayT\n0(/ba∇dblv/ba∇dbl2\n2θ+/ba∇dblp/ba∇dbl2\n2θ)dt+E(s).\nThen, integrating in [0 ,T], there exists a constant CT>0 such that\nTE(T)≤2/integraldisplayT\n0E(t)dt+CTsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ). (3.11)\nStep 5. Let (3.9) with s= 0 and (3.10) with ζ= 1, we have\n/integraldisplayT\n0/integraldisplayL\n0(G1(vt)vt+G2(pt)pt)dxdt≤E(0)−E(T)+/integraldisplayT\n0E(t)dt+TCBsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ).\nCombining the above estimates with (3.8), we can deduce\n/integraldisplayT\n0E(t)dt≤CTsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ)+CB(E(0) +E(T)).\nThen, from (3.11)\nTE(T)≤CTsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ)+CB(E(0)+E(T)).\nLetT >2CB, formT=CB\nT−CB<1, we can write\nE(T)≤mTE(0)+CTsup\ns∈[0,T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ).Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 15\nLetMn= sups∈[nT,(n+1)T](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ),n∈N. Then repeat the above argument progress on\nany [nT,(n+1)T], we have\nE(nT) =mn\nTE(0)+CTn/summationdisplay\nk=1mn+1−l\nTMj−1\n≤mn\nTE(0)+CT\n1−mTsup\ns∈[0,mT](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ).\nFor anyt≥0, andn∈N,k∈[0,T) so it hast=nT+k, then\nE(t)≤E(nT)≤m−1\nTmt\nT\nTE(0)+ sup\ns∈[0,t](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ).\nConsequently, for σ=−ln(mT)\nT,ς=m−1\nT,CB=CT\n1−mT,\nE(t)≤ςE(0)e−σt+CBsup\ns∈[0,t](/ba∇dblv(s)/ba∇dbl2\n2θ+/ba∇dblp(s)/ba∇dbl2\n2θ),∀t≥0.\nThe proof is complete.\nLemma 3.4. LetB⊂ Hbe a bounded forward invariant set, then dynamical system (H,S(t))is\nquasi-stable.\nProof.DefiningS(t)zi= (vi(t),pi(t),vi\nt(t),pi\nt(t))Tforzi∈B,i= 1,2, andv=v1−v2,p=p1−p2.\nThen, it follows from (2.28) that\n/ba∇dblS(t)z1−S(t)z2/ba∇dbl2\nH≤a(t)/ba∇dblz1−z2/ba∇dbl2\nH,\nwherea(t) =eCT.\nNow letX=H1\n∗(Ω)×H1\n∗(Ω) and the seminorm be\nχX(v,p) = (/ba∇dblv/ba∇dbl2\n2θ+/ba∇dblp/ba∇dbl2\n2θ)1\n2.\nSinceH1\n∗(Ω)֒→֒→L2θ(Ω), we can obtain that χXis compact on X.\nAccording to Lemma 3.3, we have\n/ba∇dblS(t)z1−S(t)z2/ba∇dbl2\nH≤b(t)/ba∇dblz1−z2/ba∇dbl2\nH+c(t) sup\ns∈[0,t][χX(v(s),p(s))]2,\nwithb(t) =ςe−σt,c(t) =CB.\nIt is easy to verify that b(t)∈L1(R+) and lim t→∞b(t) = 0.\nThen, since Bis a bounded subset of H, we have c(t) is locally bounded on [0 ,∞). By the Definition\n3.2, we have the dynamical system is quasistable on B⊂ H.\nSince dynamical system ( H,S(t)) is quasi-stable, we can give our main results as following .\nTheorem 3.1. Under the assumptions of Theorem 2.1, we obtain\n(1) The dynamical system (H,S(t))has a global attractor A ⊂ Hwhich is compact and connected.\nMoreover, the attractor can be characterized by the unstabl e manifold\nA=M+(N),Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 16\nemanating from the set of stationary solutions established in Lemma 3.2.\n(2) The attractor Ahas finite fractal dimension dimHA.\n(3) Every trajectory stabilizes to the set N, that is,\nlim\nt→+∞distH(Stz,N) = 0for anyz∈ H.\nIn particular, there exists a global minimal attractor Aminto the dynamical system, which is precisely\ncharacterized by the set of the stationary points N, that is Amin=N.\n(4) The attractor is bounded in H1= (H2(0,L)∩H1\n∗(0,L))2×(H1\n∗(0,L))2, and every trajectory\nz= (v,p,vt,pt)TinAhas the property\n/ba∇dbl(v,p)/ba∇dbl2\n(H2(0,L)∩H1∗(0,L))2+/ba∇dbl(vt,pt)/ba∇dbl2\n(H1∗(0,L))2+/ba∇dbl(vtt,ptt)/ba∇dbl2\n(L2(0,L))2≤R2, (3.12)\nfor some constant R>0independent of ε∈[0,1].\nProof.(1) It follows from Lemma 3.4 that ( H,S(t)) is quasi-stable. Hence, we have that ( H,S(t))\nis asymptotically smooth by Proposition 7.9.4 of [6]. Then, applying Lemma 3.1, Lemma 3.2 and\nCorollary 7.5.7 of [6], we can conclude ( H,S(t)) possesses a compact global attractor A. In addition,\nit can be characterized by A=M+(N).\n(2) Since the system ( H,S(t)) is quasi-stable, applying Theorem 7.9.6 of [6], we conclu de that\nattractor Ahas finite fractal dimension dimHA.\n(3) Combining Theorem 3.1-(1) and Theorem 7.5.10 of [6], we c an get the desired result immedi-\nately.\n(4) Since ( H,S(t)) is quasi-stable on A, the arbitrary trajectory z= (v,p,vt,pt)TinAhas the\nfollowing regularity properties\nvt,pt∈L∞(R;H1\n∗(0,L))∩C(R;L2(0,L)),\nvtt,ptt∈L∞(R;L2(0,L)).\nIt follows from (1.1) that\n\n\nαvxx=ρvtt+γβpxx+f1(v,p)+g1(vt)−εh1,\nβpxx=µptt+γβvxx+f2(v,p)+g2(pt)−εh2.(3.13)\nThen, we can deduce\nα1vxx=ρvtt+γµptt+γf2(v,p)+γg2(pt)−γεh2+f1(v,p)+g1(vt)−εh1.\nUsing the fact α1/ne}ationslash= 0,vt,pt∈L∞(R;H1\n∗(0,L))֒→L∞(R;L2(0,L)) andfi(v,p) is locally Lipschitz\ncontinuous, we have\nvxx∈L∞(R;L2(0,L)),\nv∈L∞(R;H2(0,L)∩H1\n∗(0,L)).\nIt follows from (3.13) that\nβpxx∈L∞(R;L2(0,L)),\np∈L∞(R;H2(0,L)∩H1\n∗(0,L)).\nThe proof is complete.Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 17\nTheorem 3.2. The system (H,S(t))has a generalized fractal exponential attractor. More prec isely,\nfor any given ξ∈(0,1], there exists a generalized exponential attractor Aexp,ξin the extended space\nH−ξwhich is defined as the interpolation of\nH0:=H,H−1:= (L2(0,L))2×(H−1\n∗(0,L))2.\nProof.Let us take B={z∈ H|Φ(z)≤R}where Φ is the strict Lyapunov functional given in\nLemma 3.1. Then we can derive that for sufficiently large RthatBis a positively invariant bounded\nabsorbing set, which shows that the system is quasi-stable o n the set B.\nThen for solution z(t) =S(t)z0with initial data z(0)∈ B, we can derive that, for any T >0,\n/integraldisplayT\n0/ba∇dblzt(s)/ba∇dbl2\nH−1ds≤C2\nBT,\nwhich shows that\n/ba∇dblS(t1)z−S(t2)z/ba∇dblH−1≤/integraldisplayt2\nt1/ba∇dblzt(s)/ba∇dblH−1ds≤CBT|t1−t2|1\n2\nwhereCBTis a positive constant and t1,t2∈[0,T]. Hence we obtain that for any initial data\nz0∈ Hthe mapt/mapsto→S(t)z0is1\n2-H¨ older continuous in the extended phase space H−1. Therefore, it\nfollows from Theorem 7.9.9 in [5] that the dynamical system ( H,S(t)) possesses a generalized fractal\nexponential attractor with finite fractal dimension in the e xtended space H−1.\nFurthermore, by the standard interpolation theorem, we can obtain the existence of exponential\nattractors in the extended space H−ξwithξ∈(0,1). The proof is complete.\n4 Upper semicontinuity of global attractor\nIn this section, we denote the attractor obtained in Theorem 3.1 as the Aε. Then, we investigate the\nupper semicontinuity of the attractors Aεasε→ε0.\nDefinition 4.1. [27] LetΛbe a complete metric space and Sσ(t)a family of semigroups on H, where\nσ∈Λ. The global attractors Aσis called upper semicontinuous on σ0∈Λif\nlim\nσ→σ0distH(Aσ,Aσ0) = 0,\nwheredistH{A,B}= supa∈Ainfb∈Bd(A,B)expresses the Hausdorff semi-distance in X. Similarly,\nAσis lower semicontinuous on σ0∈Λif\nlim\nσ→σ0distH(Aσ0,Aσ) = 0.\nThenAσis continuous on σ0∈Λif\nlim\nσ→σ0dH(Aσ,Aσ0) = 0,\nwheredH(A,B) = max{distH(A,B),distH(B,A)}expresses the Hausdorff metric in X.\nProposition 4.1. [22] Assume that\n(H1)(H,Sσ(t))has a global attractor Aσfor anyσ∈Λ,\n(H2) There exists a bounded set B⊂ Hsuch that Aσ∈Bfor everyσ∈Λ,\n(H3)Sσ(t)zis continuous in σfort>0and uniformly for zin bounded subsets of H.\nThen the global attractor is continuous on all I, whereIis a ”residual” set dense in Λ.Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 18\nLemma 4.1. There exists a set Idense in [0,1]such that the global attractor Aεobtained in Theorem\n3.1 is continuous at ε0∈I, that is\nlim\nε→ε0dH(Aε,Aε0) = 0. (4.1)\nProof.The argument is inspired by [2, 21]. We apply Proposition 4.1 with Λ = [0 ,1]. Then Theorem\n3.1 implies that the assumption (H1) holds.\nIt follows from (2.19) and the fact supz∈AεΦε(z)≤supz∈NεΦε(z) that\nsup\nz∈Aε/ba∇dblz/ba∇dbl2\nH≤supz∈AεΦε(z)+CF\nβ2\n≤CFsupz∈Nε/ba∇dblz/ba∇dblr+1\nH+2CF\nβ2.\nHence, we can derive from (3.2) that there exists a positive c onstantCindependent of εsuch that\nsup\nz∈Aε/ba∇dblz/ba∇dbl2\nH≤C,∀ε∈[0,1].\nThen we have that B={z∈ H:/ba∇dblz/ba∇dbl2\nH≤C}is a bounded set which is independent of εandAε⊂ B\nfor anyε∈[0,1]. Therefore, the assumption (H2) holds.\nLetBbe a bounded set of H. Then for any given ε1,ε2∈[0,1],z0∈B, we define\nSεi(t)z0= (vi(t),pi(t),vi\nt(t),pi\nt(t))T, i= 1,2,\nand\nv=v1−v2,p=p1−p2.\nThen (v,p,vt,pt) satisfies the following equations\n\n\nρvtt−αvxx+γβpxx= (ε1−ε2)h1−F1(v,p)−G1(vt),\nµptt−βpxx+γβvxx= (ε1−ε2)h2−F2(v,p)−G2(pt),(4.2)\nwhereFi(v,p) (i= 1,2)G1(vt) andG2(pt) are constructed by the same as in Lemma 3.3. Multiplying\nthe equations (4.2) by vt,pt, respectively. and integrating over [0 ,L] by parts, we have\n1\n2d\ndt/ba∇dblz/ba∇dbl2\nH=−/integraldisplayL\n0(F1(v,p)+F2(v,p))dx−/integraldisplayL\n0(G1(vt)vt+G2(pt))ptdx\n+(ε1−ε2)/integraldisplayL\n0(h1vt+h2pt)dx.(4.3)\nUsing (2.4), H¨ older’s inequality and H1\n∗(0,L)֒→Lr(0,L),r∈[0,∞), we can derive that\n/integraldisplayL\n0F1(v,p)vtdx≤Cf(1+/ba∇dblSε1(t)z0/ba∇dblr−1\nH+/ba∇dblSε2(t)z0/ba∇dblr−1\nH)(/ba∇dblv/ba∇dbl2r+/ba∇dblp/ba∇dbl2r)/ba∇dblvt/ba∇dbl2\n≤Cf(1+/ba∇dblSε1(t)z0/ba∇dblr−1\nH+/ba∇dblSε2(t)z0/ba∇dblr−1\nH)(/ba∇dbl∇v/ba∇dbl2+/ba∇dbl∇p/ba∇dbl2)/ba∇dblvt/ba∇dbl2.(4.4)\nSinceEε(t) is a non-increasing function, then for any z0∈B, we have\n/ba∇dblSεi(t)z0/ba∇dblr−1\nH≤Eεi(0)+CF\nβ2≤CB, i= 1,2.Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 19\nCombining the above estimate with (4.4), we have\n/integraldisplayL\n0F1(v,p)vtdx≤CB(/ba∇dbl∇v/ba∇dbl2+/ba∇dbl∇p/ba∇dbl2)/ba∇dblvt/ba∇dbl2≤CB(/ba∇dbl∇v/ba∇dbl2\n2+/ba∇dbl∇p/ba∇dbl2\n2)+/ba∇dblvt/ba∇dbl2\n2.\nSimilarly,/integraldisplayL\n0F2(v,p)ptdx≤CB(/ba∇dbl∇v/ba∇dbl2\n2+/ba∇dbl∇p/ba∇dbl2\n2)+/ba∇dblpt/ba∇dbl2\n2.\nTherefore,/integraldisplayL\n0(F1(v,p)vt+F2(v,p)pt)dx≤CB/ba∇dblz/ba∇dbl2\nH. (4.5)\nIt follows from the monotonicity property (2.9) that\n−/integraldisplayL\n0(G1(vt)vt+G2(pt)pt)dx≤0. (4.6)\nMoreover, it is easy to verify that\n(ε1−ε2)/integraldisplayL\n0(h1vt+h2p2)dx≤1\n2(/ba∇dblvt/ba∇dbl2\n2+/ba∇dblpt/ba∇dbl2\n2)+1\n2|ε1−ε2|2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2)\n≤1\n2/ba∇dblz/ba∇dbl2\nH+1\n2|ε1−ε2|2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2).(4.7)\nCombining (4.5), (4.6), (4.7) with (4.3), we have\nd\ndt/ba∇dblz/ba∇dbl2\nH≤CB/ba∇dblz/ba∇dbl2\nH+|ε1−ε2|2(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2). (4.8)\nUsing the Gronwall’s inequality and the fact /ba∇dblz(0)/ba∇dbl2\nH= 0, we can derive from (4.8) that\n/ba∇dblz(t)/ba∇dbl2\nH≤1\nCB(eCBt−1)(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2)|ε1−ε2|, t> 0.\nHence we have\n/ba∇dblSε1(t)z0−Sε2(t)z0/ba∇dblH≤(1\nCB(eCBt−1)(/ba∇dblh1/ba∇dbl2\n2+/ba∇dblh2/ba∇dbl2\n2))1\n2|ε1−ε2|.\nSo the assumption (H3) holds. As a conclusion, the equality ( 4.1) holds by Proposition 4.1.\nTheorem 4.1. Suppose the assumptions of Theorem 3.1 hold. Then the family of global attractors\nAεis upper semicontinuous at ε0, namely,\nlim\nε→ε0distH(Aε,Aε0) = 0. (4.9)\nProof.The argument is inspired by [18, 20]. Firstly, we suppose tha t (4.9) does not hold. Then there\nexistλ>0, the sequence εn→ε0andzn\n0∈ Aεnsuch that\ndistH(zn\n0,Aε0)≥λ>0, n∈N. (4.10)\nLetzn(t) = (vn(t),pn(t),vn\nt(t),pn\nt(t))Tbe a bounded full trajectory from the attracator Aεnwith\nzn(0) =zn\n0. It follows from (3.12) that znis bounded in L∞(R;H1).Long-time dynamical behavior for a piezoelectric system wi th magnetic effect 20\nBecause of H1֒→֒→ H, applying Simon’s Compactness Theorem (see [39]), we can ge tz∈\nC([−T,T];H) and a subsequence {zn}such that,\nzn→zinC([−T,T];H).\nThen, we can conclude that\nsup\nt∈R/ba∇dblz(t)/ba∇dblH<∞.\nLetz(t) = (v(t),p(t),vt(t),pt(t))Tbe a bounded full trajectory of the limiting semi-flow. Then,\nwe can infer that zsolves the limiting equation ( ε=ε0), namely,\n\n\nρvtt−αvxx+γβpxx+f1(v,p)+g1(vt) =ε0h1,\nµptt−βpxx+γβvxx+f2(v,p)+g2(pt) =ε0h2.(4.11)\nIn fact, from (1.1), we can get znsatisfies\n\n\nρvn\ntt−αvn\nxx+γβpn\nxx+f1(vn,pn)+g1(vn\nt) =εnh1,\nµpn\ntt−βpn\nxx+γβvn\nxx+f2(vn,pn)+g2(pn\nt) =εnh2.(4.12)\nwe can use the same argument as in the proof of (H3) in Lemma 4.1 , so we can infer that (4.11) is\nthe limit of (4.12) as n→ ∞.\nAs a result, we have\nzn\n0→z(0)∈ Aε0,\nwhich contradicts (4.10). The proof is complete.\nAcknowledgments\nThe authors would like to thank the referees for the careful r eading of this paper. 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Mendesb,†and Wilson Oliveirab‡\naGrupo de F´ ısica Te´ orica e Matem´ atica F´ ısica,\nDepartamento de F´ ısica,\nUniversidade Federal Rural do Rio de Janeiro\nBR 465-07, 23890-971,\nSerop´ edica, RJ, Brazil\nbDepartamento de F´ ısica, ICE,\nUniversidade Federal de Juiz de Fora,\n36036-330, Juiz de Fora, MG, Brazil\nJune 4, 2018\nDedicated to the memory of Prof. Emerson da Silva Guerra\nAbstract\nIt is well known that a direct Lagrangian description of radi ation damping is still missing. In this\npaperwewilluseaspecificapproachofthisproblemwhichist hestandardwaytotreattheradiation\ndamping problem. The objectives here are to construct: a N= 2 supersymmetric extension for the\nmodel describing the radiation damping on the noncommutati ve plane with electric and magnetic\ninteractions; a new dual equivalent action to the original o ne; the supercharge algebra and the total\nHamiltonian for the system.\nPACS numbers: 11.15.-q; 11.10.Ef; 11.10.-z; 41.60.-m\nKeywords: noncommutativity, supersymmetry, radiation damping , Noether symmetries\n∗Electronic address: evertonabreu@ufrrj.br\n†Electronic address: albert@fisica.ufjf.br\n‡Electronic address: wilson@fisica.ufjf.br\n1Symmetry considerations on radiation damping 2\nI. INTRODUCTION\nAn underlying feature of all charged particles is that an accelerate d charged particle\nradiate electromagnetic energy. During this process, the recoil m omentum of the emitted\nphotons is equivalent to a reaction force equivalent to the self-inte raction of the particle with\nits own electromagnetic field which creates the radiation damping [1– 3].\nThe analysis of dissipative systems in quantum theory is of strong int erest and relevance\neither because of fundamental reasons [4] or because of its prac tical applications [5–9]. The\nexplicit time dependence of the Lagrangian and Hamiltonian operator s introduces a major\ndifficulty in this study since the canonical commutation relations are n ot preserved by time\nevolution. Different approaches have been used in order to apply th e canonical quantization\nscheme to dissipative systems [10–15].\nAnother way to handle with the problem of quantum dissipative syste ms is to double\nthe target phase-space, so as we have to deal with an effective iso lated system composed by\nthe original system plus its time-reversed copy [16–18]. The new deg rees of freedom thus\nintroduced may be represented by a single equivalent (collective) de gree of freedom for the\nbath, which absorbs the energy dissipated by the system. In orde r to implement a canonical\nquantization formalism, we must first double the dimension of the tar get phase-space. The\nobjective of this doubling procedure is to comply with the canonical q uantization scheme,\nwhich requires an effective isolated system.\nTo study the quantum dynamics of an accelerated charge, it is prop er to use indirect\nrepresentations since it loses the energy, the linear momentum and the angular momentum\ncarried by the radiation field. The effect of these losses to the motio n of charge is known as\nradiation damping (RD) [1].\nThe reaction of a classical point charge to its own radiation was first discussed by Lorentz\nand Abraham more than a hundred years ago [2, 3]. There are two int eresting aspects of the\nAbraham-Lorentz theory: the self-acceleration and pre-accele ration.\nSelf-acceleration refers to the classical solutions where the char ge is under acceleration\neven in the absence of an external field. Pre-acceleration means t hat the charge begins to\naccelerate before the force begins to act.\nSo, a complete description of radiation damping is still missing. In this w ay, in this paper\nwe discussed some aspects of the RD framework concerning gauge symmetries, algebraic\nnoncommutativity and supersymmetry, as well as the correspond ing resulting physics, of\ncourse. Notice that to talk about these issues in a RD system is very difficult because, since\nwe have to deal with, in fact, two systems, the particle and the res ervoir, both mathematical\nand physical features are not so trivial, as we will see.\nWe investigated the existence of a dual equivalent model to the RD o ne through the\nNoether method. We introduced a N= 2 supersymmetric extension for the RD model\ncompleting the N= 1 supersymmetric version introduced in [19, 20].\nWe will describe a N= 2 supersymmetric extension of the nonrelativistic (2 + 1)-\ndimensional pseudo-Euclidean space model describing the RD (repr esented by the equation\n(3) below) on the noncommutative (NC) plane introducing an interac tion to the free model\nby theN= 2 superfield technique.\nHowever, it is important to notice that in fact there are two phase- spaces considered\nhere. The first one is where the radiation damping occurs and the se cond one is the double\nphase-space where details and relevance will be described in section II and in the referencesSymmetry considerations on radiation damping 3\nquoted there. In this doubled phase-space, which we believe that t he physics is not entirely\nunderstood, weperformedtheconsiderations described inthiswo rk, i. e., noncommutativity,\nduality and supersymmetry. For example, concerning only noncomm utativity, it can be\nshown easily that the original space is commutative whereas in this do uble phase-space we\nwill show precisely in this paper, that it is NC. We hope that our work ca n bring some light\nin the understanding of this extended space.\nThe organization of this paper is: in section 2 we will carry out a brief r eview of the\nmechanical model with a Chern-Simons term developed in [21] and its G alilean-symmetric\nversion, i.e., the LSZ model. We will obtain a dual equivalent model thro ugh the Noether\ndualization procedure in section 3. In section 4 we will present a symp lectic structure\nfor the model in order to introduce the noncommutativity through the variables used in\n[22, 23]. In section 5 we show the supersymmetric extension of the m odel, the supercharges\nand a supersymmetric version through the Hamiltonian formalism. Th e conclusions and\nperspectives are described in the last section.\nII. THE MODEL\nTheLSZ model. In[21]theauthorshaveintroducedanonrelativisticclassicalmech anics\nfree particle model with a Chern-Simons-like term as\nLLSZ=1\n2m˙x2\ni+λεijxi˙xj, i,j= 1,2, (1)\nwhereλhas dimension of mass/time and the second term can be seen as a par ticular electro-\nmagnetic coupling of an electromagnetic potential. This Lagrangian is neither invariant nor\ninvariant under a total derivative under the Galilean boosts transf ormations. A NC version\nof (1) was studied in [24]. To make (1) quasi-invariant under D= 2 Galilei symmetry the\nsecond term in (1) was modified and we have that\nLLSZ=1\n2m˙x2\ni−κεij˙xi¨xj, i,j= 1,2, (2)\nwhereκhas dimensions of mass ×time. It can be shown [25] that this Lagrangian is quasi-\ninvariant. This model (2) possess not a usual Galilei symmetry. We c an describe it by the\nexotic, two-fold centrally extended Galilei symmetry with non-comm utating boosts. It was\nanalyzed carefully in [21] and later in [26]. The authors in [21] demonstr ated that the model\ndescribes the superposition of a free motion in NC D= 2 spaces. A N= 2 supersymmetric\nextension of (2) was accomplished in [27] describing particles in the N C space with electric\nand magnetic interactions. A supersymmetrization of (2) was first ly obtained in [28]. Both\nmodels above are depicted here to help the reader to understand t he physical alternatives\nfor actions like (1) and (2). Other considerations can be found in [29 ].\nThe radiation damping model. In [20, 30] a new point of view concerning the study\nof RD [1, 31, 32] was presented, introducing a Lagrangian formalism to the model in two\ndimensions given by\nLRD=1\n2mgij˙xi˙xj−γ\n2εij˙xi¨xj, i,j= 1,2, (3)Symmetry considerations on radiation damping 4\nwhereεijis the Levi-Civita antisymmetric tensor, gijis the metric for the pseudo-Euclidean\nplane [33] which is given by\ngij=gij=diag(1,−1). (4)\nWe are using the Einstein sum convention for repeated indices. The m odel (3) was shown\nto have (1+1)-Galilean symmetry and the dynamical group structu re associated with that\nsystem is SU(1,1) [30]. The supersymmetrization N= 1 of (3) was studied in [34].\nThe Lagrangian (3) describes, in this pseudo-Euclidean space, a dis sipative system of a\ncharge interacting with its own radiation, where the 2-system repr esents the reservoir or heat\nbath coupled to the 1-system [19, 20]. It shows that the dissipativ e term, as a matter of fact,\nacts as a coupling term between the 1-system and the 2-system in t his space. In particular,\nwe have a system including the charge and its time-reversed image, t hat globally behaves\nlike a closed system described by equation (3).\nNote that the Lagrangian (3) is similar to the one discussed in [21] (ac tion (2)), which\nis a special nonrelativistic limit of the particle with torsion [35]. However , in this case we\nhave a pseudo-Euclidean metric and the RD constant ( γ) which act as a coupling constant\nof a Chern-Simons-like term. The RD constant γplay the same mathematical role of the\n“exotic” parameter κin (2) [21, 27]. However, there is an underlying physical difference\nbetween both γandκ.\nIt is important to reinforce that the difference between the result s that will be obtained\nhere and the ones in [28] is that, besides the metric, the physical sy stems are different, where\nthe RD constant γis not a simple coupling constant. It depends on the physical proper ties\nof the charged particle, like the charge eand massmwhich are related to the objects in its\nequations of motion which depicts an interaction between the charg e and its own radiation\nfield. We will see that the RD constant introduced the NCY into the sy stem,.\nIII. DUALITY THROUGH DUALIZATION\nThe dualization technique [36, 37] is based on the traditional idea of a local lifting of a\nglobal symmetry and may be realized by an iterative embedding of Noe ther counter terms.\nThis technique was originally explored in the soldering formalism contex t [38–40] and was\nexplored in [41–43] since it seems to be the most appropriate techniq ue for non-Abelian\ngeneralization of the dual mapping concept.\nHidden symmetries may be revealed by a direct construction of a gau ge invariant theory\nout of a non-invariant one [44–47]. The advantage in having a gauge t heory lies in the\nfact that the underlying gauge invariant theory allows us to establis h a chain of equivalence\namong different models by choosing different gauge fixing conditions. Clearly, the resulting\nembedded theory is dynamically equivalent to the original one [42]. This is the meaning of\nthe “duality” expression, namely dual equivalence.\nAs the first step, let us rewrite our RD model equation (3),\nLRD\n0=1\n2mgij˙xi˙xj−γ\n2εij˙xi¨xj, i,j= 1,2, (5)\nhence the variation of this action is\nδLRD\n0=Ji\n1˙ηi+Ji\n2¨ηi (6)Symmetry considerations on radiation damping 5\nwhereδxi=ηiand the Noether currents are\nJ1i=mgij˙xj−1\n2γεij¨xj (7)\nJ2i=1\n2γεij˙xj. (8)\nThe second step in the iterative method [36, 37] is to construct the action with two new\nfields, i.e., the so-called auxiliary fields which will be eliminated through th e equations of\nmotion.\nHence the new action is\nLRD\n1=LRD\n0−Di\n1J1i−Di\n2J2i (9)\nwhereDi\n1andDi\n2are auxiliary fields.\nLet us establish another symmetry, namely,\nδDi\n1= ˙ηi\nδDi\n2= ¨ηi. (10)\nWith equations (6) and (10) we can carry out the variation of equat ion (9), which results in,\nδLRD\n1=−Di\n1(δJ1i)−Di\n2(δJ2i). (11)\nUsing the variations of the Noether currents, equations (7) and ( 8), and that δxi=ηiwe\nsubstitute these results in (11) and we have that,\nδLRD\n1=−mgijDi\n1δDj\n1+1\n2γεijδ(Di\n1Dj\n2). (12)\nTo construct a gauge invariant action we have to use (12) convenie ntly. We can see directly\nthat,\nLRD\n2=LRD\n1+1\n2mgijDi\n1Dj\n1−1\n2γεijDi\n1Dj\n2 (13)\nand accomplishing the variation of (13), using (6), (9) and (12) we c an see clearly that it is\ngauge invariance so that δL2= 0 finishing the iterative chain.\nFrom Eq. (13) we can calculate the equations of motion for the auxilia ry fieldsD1\n1,D2\n1,\nD1\n2andD2\n2and the results are\nD1\n1=−1\nγJ22\nD2\n1=1\nγJ21\nD1\n2=2\nγ/bracketleftBig\nJ12+m\nγJ21/bracketrightBig\n(14)\nD2\n2=−2\nγ/bracketleftBig\nJ11+m\nγJ22/bracketrightBig\n,Symmetry considerations on radiation damping 6\nwhere the Noether currents J11, J12, J21andJ22are given by Eqs. (7) and (8). The next\nstep is to substitute Eqs. (14) into the Lagrangian in (13). The res ult is\nLRD\n2=5\n8m(˙x2\n1−1\n5˙x2\n2) +1\n4γ(˙x1¨x2−˙x2¨x1) (15)\nwhich can be rewritten as\nLRD\n2=1\n2m(˙X2\n1−˙X2\n2)−1\n2Γ(˙X1¨X2−˙X2¨X1) (16)\nwhere\nX1=√\n5\n2x1\nX2=1\n2x2 (17)\nΓ =−2√\n5γ\n(18)\nandLRD\n2in Eq. (17) is invariant under δXi=ηiso we can say that LRDin Eq. (3) is\nself-dual.. Of course, LRD\n2is the same as our original Lagrangian in Eq. (3). We will talk\nabout what this result means in a few moments. Before that we would like to explore an\nconnection involving LLSZin Eq. (2), which can be explicitly written as\nLLSZ=1\n2m(˙x2\n1+ ˙x2\n2)−κ(˙x1¨x2−˙x2¨x1) (19)\nwhich is analogous to LRD. If we substitute\nx2→ix2 (20)\nwe have that\nLLSZ=1\n2m(˙x2\n1−˙x2\n2)−1\n2κ′(˙x1¨x2−˙x2¨x1) (21)\nwhereκ′=−2iκinLLSZabove, which is equal to LRD. This result is a surprise because\nlead us to conclude directly that the Lagrangian LLSZis dual toLRD. However, LRDis\ninvariant under δxi=ηi, butLLSZis not. Both Lagrangians are invariant under different\ngauge transformations. This fact happens because, although in LLSZ,λis simply a constant,\ninLRDγcannot be rewritten as in (19) and (21). Since γandλare not connected, to find\na Lagrangian dual to LLSZwe have to apply the Noether procedure from the beginning in\nEq. (1). This calculation is beyond the RD’s scope of this work. These results show us that\nalthough, naively, we can think that both LRDandLLSZare connected by the metric, i.e.,\nLRDuses a pseudo-Euclidean metric, the Noether procedure confirms , once more, that they\nrepresent two completely different physical systems. This fact is w ell known together with\nthe fact that γin Eq. (3) is not a simple coupling constant. It depends on the physica l\nproperties of the charged particle that is being analyzed. To clarify , we can cite its charge\n(e) and mass (m). Their relation with the terms is through the equat ion of motion, which\ndepicts an interaction between the charge and its own radiation field .Symmetry considerations on radiation damping 7\nHaving said that, we can claim that what is new here is that γspoiled the connection\nbetweenLRDandLLSZthrough dualization, since one can think that this task (to obtain a\ndual actionofonedirectly fromtheother’sdual) would beaneasy on e. Another consequence\nofNoether’sprocedureisthatitalsocannotrelate x2RDandx2LSZviaEq. (20)aswethought\nnaively.\nTo sum up, we can say that although both LRDandLLSZare mathematically analogous\nbut the Noether approach showed precisely that they are physica lly different.\nIV. NONCOMMUTATIVITY\nThe study of NC theories has received a special attention through the last years thanks\nto the possibility that noncommutativity can explain the physics of th e Early Universe. In\nother words the spacetime of the Early Universe can be a NC one. It has been used in many\nareas of theoretical physics [48], cosmology [49] and with Lorentz in variance [50].\nIntroducing a Lagrangian multiplier which connects ˙ xtoz, and substituting all differen-\ntiatedx-variables in the Lagrangian (3) by z-variables, one has a first-order Lagrangian\nL(0)=gijpi(˙xj−zj)+m\n2gijzizj−γ\n2εijzi˙zj. (22)\nThe equations of motion can be written, using the symplectic struct ure [51], as\nωij˙ξj=∂H(ξ)\n∂ξi(23)\nwhere the symplectic two form is\n(ω) =\n0 g 0\n−g 0 0\n0 0−γε\n (24)\nwith\nε=/parenleftbigg\n0 1\n−1 0/parenrightbigg\n, (25)\nwheregwas given in Eq. (4) and 0denotes the 2 ×2 null matrix. H(ξl) is the Hamiltonian\nandξiare the symplectic variables.\nUsing the variables introduced in [22, 23] modified as\nQi=γgij(mzj−pj),\nXi=xi+εijQj,\nPi=pi, (26)\nwe can write that our Lagrangian can be separated into two disconn ected terms in order to\ndescribe the “external” and “internal” degrees of freedom. So,\nL(0)=L(0)\next+L(0)\nint (27)Symmetry considerations on radiation damping 8\nwhere\nL(0)\next=gijPi˙Xj+γ\n2εijPi˙Pj−1\n2mgijPiPj, (28)\nand\nL(0)\nint=1\n2γεijQi˙Qj+1\n2mγ2gijQiQj. (29)\nThe internal coordinates, /vectorQ, and the external ones, /vectorX, are decoupled [22] and we can see\nthat they do not commute, with the following nonvanishing Poisson br ackets,\n{Xi,Xj}=γεij,{Xi,Pj}=gij,\n{Qi,Qj}=γεij. (30)\nWe can see in (27) that our Lagrangian can be written as two separa ted and disconnected\nparts describing the “external” and “internal” degrees of freedo m in a NC phase space,\nparameterized by the variables ( Xi,Pi) (external structure) and Qi(internal structure) [22,\n23].\nNow we introduce an interaction term to the “external” sector, eq uation (28), which do\nnot modify the internal sector, represented by a potential ener gy termU(X) involving NC\nvariables, as follows\nLext=gijPi˙Xj+γ\n2εijPi˙Pj−1\n2mgijPiPj−U(X). (31)\nThis leads to a deformation of the constraint algebra, since the con straint now involves a\nderivative of the potential [30].\nTo end this section, notice the L(0)has the double of the phase-space dimension. The “ex-\nternal”and“internal”dynamicsshouldbeinterpretedintermsofun derlyingone-dimensional\ndissipative dynamics. The same observation can be made for the res ults below.\nNotice that the Lagrangian in Eqs. (27), (28) and (30) are formed by the objects that\nshows a NC algebra described in Eq. (29). The standard NC procedu re is to recover the\ncommutative algebra in (29) using the so-called Bopp shift.\nxi= ˆxi+1\n2γǫijpj (32)\nwhere the hat defines a NC variable. Using (32) in (29) we will have tha t{xi,xj}= 0. The\nsame can be made with Qiso that\nQi=ˆQi+1\n2γǫijPj (33)\nwhere ˆpi=piandPi=ˆPiand consequently {pi,pj}={Pi,Pj}= 0. Substituting (32) and\n(33)intheLagrangians(27), (28)and(30)resultsinaLagrangian definedinNCphase-space.\nBut, to go further in this analysis is an ongoing research.Symmetry considerations on radiation damping 9\nV. THE SUPERSYMMETRIC MODEL IN N= 2\nTo obtain the supersymmetric extension of the model described by the Lagrangian (31),\nfor each commuting space coordinate, representing the system d egrees of freedom, we will\nassociate one anti-commuting variable, which are the well known Gra ssmannian variables.\nWe are considering only the N= 2 BUSY for a non-relativistic particle, which is described\nby the introduction of two real Grassmannian variables Θ and ¯Θ (the Hermitian conjugate\nof Θ) in the configuration space, but all the dynamics are represen ted by the time t[52, 53].\nLet us carry out the Taylor expansion for the real scalar superco ordinate as\nXi→ Xi(t,Θ,¯Θ) =\n=Xi(t)+iψi(t)Θ+i¯Θ¯ψi(t)+¯ΘΘFi(t) (34)\nand their canonical supermomenta\nPi(t)→ Pi(t,Θ,¯Θ) =\n=iηi(t)−iΘ(Pi(t)+ifi(t))−¯ΘΘ˙ηi(t), (35)\nwhich under the infinitesimal supersymmetry transformation law\nδt=i¯ǫθ+i¯ǫΘ, δΘ =ǫandδ¯Θ = ¯ǫ, (36)\nwhereǫis a complex Grassmannian parameter, we can write that\nδXi= (ǫ¯Q+¯ǫQ)Xi (37)\nandδPi= (ǫ¯Q+¯ǫQ)Pi, (38)\nwhere both Qand¯Qare the two SUSY generators\nQ=∂\n∂¯Θ+iΘ∂\n∂t,¯Q=−∂\n∂Θ−i¯Θ∂\n∂t. (39)\nIn terms of ( Xi(t),Pi(t),Fi,fi), the bosonic (even) components and ( ψi(t),¯ψi(t),ηi(t)),\nthe fermionic (odd) components, we obtain the following supersymm etric transformations,\nδXi=i(¯ǫ¯ψi+ǫψi) ;δψi=−¯ǫ(˙Xi−iFi)\nδ¯ψi=−ǫ(˙Xi+iFi) ;δFi=ǫ˙ψi−¯ǫ˙¯ψi, (40)\nand\nδηi=ǫ(Pi+ifi);δPi= 0;δfi= 2¯ǫ˙ηi. (41)\nThe super-Lagrangian for the super point particle with N= 2, invariant under the trans-\nformations (40) and (41), can be written as the following integral ( we use for simplicity that\nm= 1)\n¯Lext=1\n2/integraldisplay\ndΘd¯Θ/bracketleftBig\ngij/parenleftbig¯DXi¯Pj+PjDXi/parenrightbig\n+γ\n2εij/parenleftBig\nPi˙¯Pj+˙Pj¯Pi/parenrightBig\n−1\n2gij/parenleftbig\nPi¯Pj+Pj¯Pi/parenrightbig/bracketrightbigg\n−/integraldisplay\ndΘd¯ΘU[X(t,Θ,¯Θ)] (42)Symmetry considerations on radiation damping 10\nwhereDis the covariant derivative ( D=∂Θ−i¯Θ∂t) and¯Dis its Hermitian conjugate. The\nU[X] is a polynomial function of the supercoordinate\nExpanding the superpotential U[X] in Taylor series and maintaining Θ ¯Θ (because only\nthese terms survive after integrations on Grassmannian variables Θ and¯Θ), we have that\nU[X] =Xi∂U[X(t)]\n∂Xi+XiX∗\nj\n2∂2U[X(t)]\n∂Xi∂Xj+... (43)\n=Fi¯ΘΘ∂iU[X(t)]+¯ΘΘψi¯ψj∂i∂jU[X(t)]+...\nwhere the derivatives ∂i=∂\n∂Xiare such that Θ = 0 = ¯Θ, which arefunctions only of the X(t)\neven coordinate. Substituting equation (43) in equation (42), we o btain after integrations\n¯Lext=L(0)\next−1\n2gijfifj−gijFifj+γ\n2εijfi˙fj\n−bigij/parenleftbig¯ψi˙¯ηj−˙ηjψi/parenrightbig\n−bigij˙ηi¯ηj+iγεij˙ηi˙¯ηj\n−Fi∂iU[X(t)]−ψi¯ψj∂i∂jU[X(t)], (44)\nwhich is the complete Lagrangian for N= 2.\nThe bosonic component Fiis not a dynamic variable. In this case, using the Euler-\nLagrange equations for the auxiliary variables fiandFi, we obtain\nfi(t) =gij∂jU[X(t)], (45)\nFi(t) =fi+γgailεj˙fj\n=gij∂jU[X(t)]−γεij∂j∂kU[X]˙Xk(t), (46)\nwhere we have to eliminate the variable fias well as its derivative in Fi. Now, substituting\n(45) and (46) in (44) the auxiliary variables can be completely eliminate d, hence\n¯L(N=2)ext=L(0)\next−1\n2gij∂iU∂jU+γ\n2εij∂iU∂j∂kU˙Xk\n−bigij/parenleftbig¯ψi˙¯ηj−˙ηjψi/parenrightbig\n−bigij˙ηi¯ηj+iγεij˙ηi˙¯ηj\n−ψi¯ψj∂i∂jU . (47)\nNote that, as in [27], we can rewrite equation (47) as\n¯L(N=2)ext=L(0)\next+Ak(X,t)˙Xk+A0(X,t)+\n−bigij/parenleftbig¯ψi˙¯ηj−˙ηjψi/parenrightbig\n−bigij˙ηi¯ηj+iγεij˙ηi˙¯ηj\n−ψi¯ψj∂i∂jU, (48)\nwhich is invariant under standard gauge transformations Aµ→A′\nµ=Aµ+∂µΛ, where\nA0(X,t) =−1\n2gij∂iU∂jU (49)\nand\nAk(X,t) =γ\n2εij∂iU∂j∂kU (50)Symmetry considerations on radiation damping 11\nwere both identified in [27] with the scalar potential A0(that in this case have a pseudo-\nEuclidean metric) and the vector potential Ak. Notice that both potentials above are not\nindependent. The vector potential introduce a magnetic field B=εij∂iAjgiven by\nB(X) =γ\n2εAKεj(∂i∂lU)(∂j∂kU) (51)\nwhere we can see that the noncommutativity introduced by the par ameterγgenerates a\nconstant magnetic field [27] and an electric field given by Ei=∂iA0which can be written\nas\nEi(X) =−gj∂i∂lU∂jU . (52)\nThe Euler-Lagrange equations, in this case, are\nm∗˙Xi=Pi−meγεijEj\n+mγεijψl¯ψk∂l∂k∂jU, (53a)\n˙Pi=egijεAl˙XlB+begijEj\n−gijψl¯ψk∂l∂k∂jU, (53b)\nwhereEiandBare the electric and magnetic field, respectively, and m∗=m(1−eγB) is\nan effective mass. However, this way of introducing electromagnet ic interaction modifies the\nsymplectic structure of the system which determines the NC phase -space geometry, for the\nbosonic sector, equation (30), we have\n{Xi,Xj}=m\nm∗γεij,{Xi,Pj}=m\nm∗gij,\n{Pi,Pj}=m\nm∗bεij, (54)\nwhich imply an analysis of the value eγB/ne}ationslash= 1 in order to avoid a singularity [24, 54]. Notice\nthat the algebra in (54) is different from the one in (29) where the mo menta commute.\nConcerning the fermionic sector, the Euler-Lagrange equations a re\niγεij¨¯ηj+bigij˙¯ηj−i˙ψi= 0,\niγεij¨ηj+bigij˙ηj−i˙¯iψ= 0, (55)\nfor the fermionic variables ( η,¯η). For the fermionic variables ( ψi,¯ψi) the Euler-Lagrange\nequations are\ni˙ηi+gAK¯ψj∂k∂jU= 0\ni˙¯ηi−gAKψj∂j∂kU= 0. (56)\nwhere the fermionic variables ( ψi,¯ψi) do not have dynamics.\nSo, analogously to [27] we have here that the noncommutativity orig inates electric and\nmagnetic fields. In the case of RD, studied here, in the NC hyperbolic phase-space, the\nmovement of the charged particle has an extra electromagnetic en ergy that did not appear\nin anN= 1 SUSY analysis [34]. This result agree with the fact that noncommut ativity does\nnot change the physics of the system. However, we understand t hat this electromagnetic\nenergy is an extra one due to the NC feature of the phase-space. This result is also different,\nas it should be expected, from the one obtained in [27] where only a ma gnetic interaction\nappear.Symmetry considerations on radiation damping 12\nA. The harmonic oscillator solutions\nIn order to obtain an interesting solution of equations (53) let us co nsider a particular\nform for the superpotential like,\nU(X) =ω\n2gijXiXj, (57)\nwhich has clearly an harmonic-like form.\nIt iseasy to see that in bothequations (53a) and(53b) thelast ter mwith three derivatives\ndisappear and so we have two new equations with the fermionic and bo sonic sector separated\nso that,\nm∗˙Xi=Pi−eγǫijEj (58a)\nand\n˙Pi=egAlǫij˙XlB+egijEj. (58b)\nComputing a second time derivative of equation (58a) we have\nm∗¨Xi=˙Pi−eγǫij˙Ej. (59)\nSubstituting (58b) into (59) we have that\nm∗¨Xi=egijǫAl˙XlB+egijEj−eγǫij˙Ej, (60)\nwhich will disclose a very well known result in a moment.\nBack in (49) and (50) but now using (57) we can write that,\nA0(X,t) =−ω2\n2gijXiXj, (61a)\nAk(X,t) =γ\n2ω2ǫjXj (61b)\nrespectively. Substituting these equations in (51) and (52) we hav e that\nB=γω2(62)\nand\nEi=−ω2gijXj, (63)\nand finally, substituting these both equations in (60) it is easy to sho w that\n¨Xi−γeω2\n1−γ2ω2e(gijǫAK+gAKǫij)˙Xk+eω2\n1−γ2ω2eXi= 0, (64)\nwhich is the equation of a damped harmonic oscillator and we see clearly that the second-\nterm of (64) represents a dissipative force proportional to the v elocity and in the last term\nof (64), we have that\nω2\n0=eω2\n1−γ2ω2eSymmetry considerations on radiation damping 13\nrepresents the natural frequency of this oscillator ω0. Notice that the RD constant is re-\nsponsible for the dissipative force and affects the frequency also. The instantaneous rate of\nenergy of the oscillator in equation (64) can be written as\ndE\nat=m∗γeω2\n1−γ2ω2e˙Xi, (65)\nso that the RD constant also affects the energy rate. However, w e note that equation (64)\nhas a general metric so that this equation is a general case. Using t he metric for the pseudo-\nEuclidean plane given in (4) we see that the second term in (64) disapp ear, and we have\nthat\n¨Xi+eω2\n1−γ2ω2eXi= 0\n=⇒¨Xi+ω2\n0Xi= 0,\nwhich is the equation for the standard harmonic oscillator which has t he standard solutions.\nFrom (64) we can see that, since there is not a term which has three derivatives of X, one\ncan conclude that in the NC space, the non-physical solutions, nam ely the pre-acceleration\nsolutions (for...\nX), do not exist.\nB. The supercharge algebra\nNow, from the supersymmetric transformations, equations (40) and (41), and the La-\ngrangian (48), we can compute the supercharge, through the No ether’s theorem. The results\nfor the charge operator are given by\nQ=bigij(Pi−ii)ψjand¯Q=bigij(Pi+ii)¯ψj, (66)\nwhereWi(X) =∂iU(X).\nThe supercharge algebra is\n{Q,¯Q}={¯Q,Q}= 0, (67)\nand\n{Q,¯Q}=−2iH. (68)\nFurther, we easily carried out a canonical calculation of the Hamilton ian and we find that\nH=Hb+Hf, (69)\nwhere the bosonic Hamiltonian Hbis\nHb=1\n2gij(PiPj+WiWj), (70)\nand the fermionic part Hfis\nHf=m\nm∗/parenleftbig\nieB(X)εij¯ψiψj+gik∂jWk(X)¯ψiψj/parenrightbig\n. (71)Symmetry considerations on radiation damping 14\nNote that the second term in Hbis proportional to the scalar potential, equation (49), i. e.,\nthere is a potential energy term in Hb. We can say that the origin of this term is related to\nthe electric field.\nThere is an alternative way to introduce the minimal electromagnetic interaction. It can\nbe accomplished through the transformation Pi→ Pi=Pi+eAi(Xi,t) in the Hamiltonian,\nthat preserve the symplectic structure of equation (30). In [27] this transformation has been\nconsidered and it leads to the same expression for the magnetic field equation (51).\nVI. REMARKS AND CONCLUSIONS\nA fundamental property of all charged particles is that the electr omagnetic energy is\nradiatedwhenever theyareaccelerated. Therecoil momentum of thephotonsemitted during\nthis process is equivalent to a reaction force corresponding to self -interaction of the particle\nwith its own electromagnetic field, which originates the RD.\nHere the supersymmetric model was split into “external” and “inter nal” degrees of free-\ndom of the supersymmetric model in terms of new variables, where t he RD constant in-\ntroduced noncommutativity in the coordinate sector. We present ed a way to introduce an\nelectromagnetic coupling.\nWe performed the supersymmetric N= 2 extension of the RD model and realized that\nthe noncommutativity introduced by the parameter generates a c onstant magnetic field.\nWith this result, together with the electric field we obtained a genera l expression for the\ndamped harmonic oscillator which results in the standard harmonic os cillator in our pseudo-\nEuclidean space. We saw that in the NC space, the non-physical solu tions, namely the\npre-acceleration solutions disappear. After that we compute the supercharges algebra and\nthe total Hamiltonian of the system, separated in bosonic and ferm ionic parts.\nAlso in this work, we used an alternative way to construct a dual equ ivalent action to\nthe RD one, a dualization procedure. We showed that the RD action is self-dual and also\nthat, despite LSZ can be transformed in the RD action, both have d ifferent symmetries.\nThe dualization procedure showed precisely that although both act ions are mathematically\nequivalent, theyarephysically different thankstoitscouplingconst ant. TheRDonedepends\non each problem while for the LSZ action, it is simply a constant parame ter. Although it\ncan sound like an obvious thing, but it is not.\nWith this new features revealed here we hope that this work has impr oved the fathoming\nof this extended space, which we believe it is not closed in the current literature.\nA perspective for future analysis is to study some typical problems of dissipative systems,\nlike self-acceleration and pre-acceleration, for example. To accom plish this, in our N= 2\nsupersymmetric case, we have to begin analyzing the Euler-Lagran ge equations (53), (55)\nand (56).\nVII. 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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": "1907.07051v1.Damping_of_slow_magnetoacoustic_oscillations_by_the_misbalance_between_heating_and_cooling_processes_in_the_solar_corona.pdf",        "content": "arXiv:1907.07051v1  [astro-ph.SR]  16 Jul 2019Astronomy&Astrophysics manuscript no. Kolotkov_aanda_slow_R2 c∝circleco√yrtESO 2019\nJuly 17, 2019\nDamping of slow magnetoacoustic oscillations by the misbal ance\nbetween heating and cooling processes in the solar corona\nD. Y . Kolotkov1⋆, V . M. Nakariakov1,2, and D. I. Zavershinskii3,4\n1Centre for Fusion, Space and Astrophysics, Department of Ph ysics, University of Warwick, CV4 7AL, UK\n2St. Petersburg Branch, Special Astrophysical Observatory , Russian Academy of Sciences, 196140, St. Petersburg, Russ ia\n3Samara National Research University, Department of Physic s, Samara 443086, Russia\n4Lebedev Physical Institute of Russian Academy of Sciences, Samara Branch, Department of Theoretical physics\nReceived July 17, 2019 /Accepted dd mm yyyy\nABSTRACT\nContext. Rapidly decaying slow magnetoacoustic waves are regularly observed in the solar coronal structures, o ffering a promising\ntool for a seismological diagnostics of the coronal plasma, including its thermodynamical properties.\nAims. The effect of damping of standing slow magnetoacoustic oscillatio ns in the solar coronal loops is investigated accounting for\nthe field-aligned thermal conductivity and a wave-induced m isbalance between radiative cooling and some unspecified he ating rates.\nMethods. The non-adiabatic terms were allowed to be arbitrarily larg e, corresponding to the observed values. The thermal conduc tiv-\nity was taken in its classical form, and a power-law dependen ce of the heating function on the density and temperature was assumed.\nThe analysis was conducted in the linear regime and in the infi nite magnetic field approximation.\nResults. The wave dynamics is found to be highly sensitive to the chara cteristic time scales of the thermal misbalance. Depending on\ncertain values of the misbalance time scales three regimes o f the wave evolution were identified, namely the regime of a su ppressed\ndamping, enhanced damping where the damping rate drops down to the observational values, and acoustic over-stability. The specific\nregime is determined by the dependences of the radiative coo ling and heating functions on thermodynamical parameters o f the plasma\nin the vicinity of the perturbed thermal equilibrium.\nConclusions. The comparison of the observed and theoretically derived de cay times and oscillation periods allows us to constrain\nthe coronal heating function. For typical coronal paramete rs, the observed properties of standing slow magnetoacoust ic oscillations\ncould be readily reproduced with a reasonable choice of the h eating function.\nKey words. Sun: oscillations - Waves - Radiation mechanisms: thermal\n1. Introduction\nThe study of wave and oscillatory processes in the plasma of\nthe solar corona is one of the most rapidly developing resear ch\ntopics of modern solar physics (e.g. De Moortel & Nakariakov\n2012; Wang 2016). The interest in coronal oscillations is co n-\nnected, in particular, with their seismological potential , i.e. with\nthe use of the oscillations as natural probes of the plasma an d\nphysical processes operating there (e.g. Liu & Ofman 2014).\nMoreover, the striking similarity between the properties o f os-\ncillations detected in solar and stellar flares (see, e.g. Ch o et al.\n2016), suggests interesting perspectives for the exploita tion of\nthe solar-stellar analogy.\nSlow magnetoacoustic waves are often detected in coronal\nplasma non-uniformities, such as coronal loops, and plumes\nand the interplume regions, as propagating periodic distur -\nbances of the EUV emission, (see, e.g. De Moortel 2009;\nBanerjee & Krishna Prasad 2016, respectively). Another com -\nmon manifestation of slow waves in the corona are standing\nwaves in loops, detected as rapidly decaying periodic Doppl er\nshifts of coronal emission lines (see, e.g. Wang 2011). Stan ding\nslow waves are usually refereed to as SUMER oscillations, af -\nter the instrument used in their first detection (SoHO /SUMER,\nsee Wang et al. 2002) and interpretation (Ofman & Wang 2002).\nSUMER oscillations still remain a subject of intensive stud ies.\n⋆Corresponding author: D. Y . Kolotkov, D.Kolotkov.1@warwi ck.ac.ukFor example, standing slow waves in non-flaring fan loops, wi th\nthe periods of 27 min, damping time about 45 min, and the\nphase speed corresponding to the plasma temperature of abou t\n0.6 MK, have been studied by Pant et al. (2017). A 10-min pe-\nriodicity has been identified in the time series of Doppler sh ift\nand line-integrated intensity of the Fe xxiemission line, soft X-\nray flux, and EUV light curves (Li et al. 2017). A 2-min os-\ncillation of the thermal component of the microwave emissio n\nof a solar flare has been interpreted in terms of the emission\nmodulation by a standing slow wave. An 80 s oscillation of\nthe X-ray and microwave emissions in a solar flare has been\nassociated with second harmonic of standing slow wave in a\nflaring arcade (Kupriyanova et al. 2019). The 8–30 min peri-\nodic pulsations of the soft X-ray emission generated in an ac -\ntive region before a flare could also be associated with stand -\ning slow waves (Tan et al. 2016). Seismological application s\nof slow waves include the estimation of the polytropic index\n(Van Doorsselaere et al. 2011; Krishna Prasad et al. 2018), a v-\nerage magnetic field (Wang et al. 2007) in the oscillating loo p,\nand transport coefficients (Wang et al. 2015, 2018). An impor-\ntant foundation of the interpretations and seismology is pr ovided\nby the forward modelling of imaging and spectroscopic obser v-\nables (Yuan et al. 2015).\nRecent theoretical studies of standing slow waves in corona l\nloops include accounting for weakly-nonlinear e ffects that are\nfound to manifest as an appearance of higher parallel harmon ics\nArticle number, page 1 of 6(e.g. Kumar et al. 2016); full-MHD numerical simulations wi th\nvarious scenario of transport processes, which aim at revea l-\ning the reason for the unexpected linear scaling of the obser ved\ndamping time with the oscillation period (e.g. Wang et al. 20 18),\nand the excitation mechanism (e.g. Provornikova et al. 2018 ).\nAn important physical process that should be taken into ac-\ncount in the modelling of compressive oscillations is the pe r-\nturbation of the thermal equilibrium by the oscillation, i. e.\nthe effect of the misbalance between radiative and, possibly,\nthermal conductive cooling, and an unspecified but definitel y\npresent heating. Similar e ffects are considered in the interstel-\nlar medium and molecular clouds, while mainly in the con-\ntexts of the plasma condensation caused by thermal instabil ity\n(e.g. Krasnobaev & Tagirova 2017), and basic theoretical st udies\nof the autowave regimes (e.g. Zavershinsky & Molevich 2013)\nand Alfvén wave amplification (e.g. Zavershinsky & Molevich\n2014). In the coronal context, it has been shown that the e ffect\nof thermal misbalance can either strengthen the damping or s up-\npress it (e.g. Nakariakov et al. 2017). However, this conclu sion\nwas reached in the limit of weak non-adiabaticity, using the as-\nsumption that the imaginary part of the oscillation frequen cy is\nmuch smaller than the real part. On the other hand, for exam-\nple, the damping time of SUMER oscillations is known to be\ncomparable with the oscillation period. It justifies the nee d for\nsoftening this assumption.\nThe aim of this paper is to develop a theory of linear standing\nslow magnetoacoustic oscillations in coronal loops with th ermal\nmisbalance. In Section 2 we describe the model, and derive di s-\npersion relations that are analysed in Section 3. The finding s are\nsummarised and discussed in Section 4.\n2. Governing equations, time scales, and\ndispersion relation\nWe consider evolution of slow magnetoacoustic waves in the\ninfinite magnetic field approximation, upon which the set of\ngoverning equations reduces to the usual hydrodynamic Eu-\nler equation, continuity equation, ideal gas state equatio n, and\nthe energy equation (see Eqs. (1)–(4), respectively). This ap-\nproximation is extensively used for modelling slow waves in\nthe corona, see e.g. Nakariakov et al. (2000); Ofman & Wang\n(2002); De Moortel & Hood (2004); Verwichte et al. (2008);\nRuderman (2013); Kumar et al. (2016). Under this approxima-\ntion, the waves are assumed to propagate strictly along the a mbi-\nent infinitely stiffmagnetic field lines, hence do not perturb the\nfield and their speed is independent of it.\nAccounting for the e ffects of the optically thin radiation, un-\nspecified heating, and thermal conductivity, the governing equa-\ntions are\nρdVz\ndt=−∂P\n∂z, (1)\n∂ρ\n∂t+∂\n∂z(ρVz)=0, (2)\nP=kBTρ\nm, (3)\nCVdT\ndt−kBT\nmρdρ\ndt=−Q(ρ,T)+κ\nρ∂2T\n∂z2, (4)\nwhereρ,T, and Pare the density, temperature, and pressure, re-\nspectively; Vzis the velocity component along the z-axis which\ncoincides with the magnetic field direction, kBis Boltzmann con-\nstant, mis the mean particle mass, CV=(γ−1)−1kB/mis thespecific heat capacity at constant volume with γ=5/3 being\nthe standard adiabatic index, κis the field-aligned thermal con-\nductivity, and the function Q(ρ,T)=L(ρ,T)−H(ρ,T) com-\nbines the effects of radiative losses L(ρ,T) and some unspeci-\nfied heating H(ρ,T). For the energy equation in form (4), the\nheating/cooling function Q(ρ,T) is measured in W kg−1. For\nexample, numerous observational studies demonstrated tha t the\ntemperature across and along the loop remains almost consta nt\n(see e.g. Reale (2014) for the detailed review of the coronal\nloop properties, and Gupta et al. (2019) and references ther ein\nfor the most recent results). Hence, we consider the plasma t o\nbe in a uniform isothermal equilibrium. Thus, in the equilib rium\nQ(ρ0,T0)=0, where the index 0 indicates equilibrium quanti-\nties. In general, the equilibrium thermal structure of the l oop is\nalso determined by thermal conduction at the footpoints. Bu t, as\nwe consider waves in the coronal, almost isothermal part of a n\nactive region, this e ffect is omitted. For the slow waves prop-\nagating upwards along loops and plumes this omission is natu -\nrally justified. For standing slow waves this omission could be\njustified by the structure of the pressure, density and tempe rature\nperturbations along the loop. In contrast with the perturba tions\nof the parallel velocity that have nodes at the footpoints, p ertur-\nbations of thermodynamical parameters in standing slow wav es\nhave anti-nodes at the footpoints (e.g. Reale 2016; Wang et a l.\n2018). Hence, near the footpoints the derivative of the temp er-\nature perturbation in the wave with respect to the field-alig ned\ncoordinate could be taken as zero, thus suppressing the wave\ndamping by the thermal conduction in these regions. Thus, in\nour analysis the chromosphere and transition region act onl y\nas the solid-wall perfectly reflecting boundaries for slow w aves\nand are not involved in the wave evolution by any other mean\n(see e.g. Ofman & Wang 2002; Selwa et al. 2005; Taroyan et al.\n2007, where a similar approach was employed for the coronal\nslow wave modelling). In other words, our simple reflecting\nboundary conditions mimic a more realistic model of the tran si-\ntion region and the chromosphere used by e.g. Nakariakov et a l.\n(2004) or Reale (2016), in which slow waves are found to natu-\nrally reflect at the lower boundary because they hit the trans ition\nregion. We need to stress that in the considered scenario the\nwaves do not contribute to the heating themselves, but pertu rb\nthe physical parameters of the plasma that may a ffect the effi-\nciency of the heating.\nFor the solar corona, the optically thin radiation loss func -\ntion can be modelled as L(ρ,T)=χρTβ, whose tempera-\nture dependence is illustrated in Fig. 1, determined from th e\nCHIANTI atomic database (Dere et al. 1997; Del Zanna et al.\n2015). Function L(ρ,T) represents the radiative losses per\nunit mass (W kg−1), which is obtained from the radia-\ntive losses per unit volume (W m−3) divided by the plasma\ndensityρ. Likewise, the unknown coronal heating func-\ntion can be locally parametrised as H(ρ,T)=hρaTb\n(see e.g. Rosner et al. 1978; Ibanez S. & Escalona T. 1993;\nDahlburg & Mariska 1988), where a certain combination of the\npower law indices aandbcould be associated with a specific\nheating mechanism. The proportionality coe fficient hcan in\nturn be determined applying the thermal equilibrium condit ion\nQ(ρ0,T0)=0. More recent observational and theoretical works\nsuggested that the coronal heating function may also have an in-\ntermittent time-dependent component (see e.g. Klimchuk 20 06;\nReale 2016). Characteristic times of such a time-varying he at-\ning are shown to be predominantly short, shorter than a minut e\n(e.g. Testa et al. 2014; Tajfirouze et al. 2016). On the time sc ales\nof the considered slow coronal waves (with periods from sev-\neral minutes to several tens of minutes), the chosen form of t he\nArticle number, page 2 of 6Kolotkov et al.: Damping of slow waves by the thermal misbala nce\nFig. 1. Left: A piecewise dependence of the optically thin radiatio n\nlosses per unit mass L(ρ,T)=χρTβon temperature, where the specific\nvalues of the parameters χandβare determined from the CHIANTI\natomic database v. 8.0.7 for the plasma concentration 1016m−3, and\nvary with the temperature interval considered. Right: Vari ation ofτ1\n(red) andτ2(blue) determined by Eq. (6) with temperature, for the ra-\ndiative cooling shown in the left-hand panel and some heatin g model\nwith the density and temperature power indices a=−0.5 and b=−3,\nrespectively. The green dashed lines indicate the SUMER obs ervational\nchannels 6.3 MK and 8.9 MK.\nfunction H(ρ,T) thus represents a time-averaged steady heating,\nsustaining the oscillating loop at approximately the same m ean\ntemperature. Thus, we determine a misbalance between the he at-\ning and cooling processes in the solar corona, caused by slow\nwaves, through different dependences of the functions L(ρ,T)\nandH(ρ,T) on the plasma density and temperature perturbed by\nthe wave. As a specific heating scenario has not been revealed\nyet, the power law indices aandbin the parametric dependence\nof the heating function are treated as free parameters.\nWe linearise the governing equations around the initial equ i-\nlibrium, obtaining energy equation (4) in the form\n∂˜T\n∂t−(γ−1)T0\nρ0∂˜ρ\n∂t=κ\nρ0CV∂2˜T\n∂z2−˜T\nτ2−/parenleftBigg1\nτ2−γ\nτ1/parenrightBiggT0\nρ0˜ρ, (5)\nwhere the symbol “ ∼” indicates the linear perturbations, and\nτ1=γCV//bracketleftBig\nQT−(ρ0/T0)Qρ/bracketrightBig\n, τ 2=CV/QT (6)\nare characteristic time scales of the thermal misbalance, f ully de-\ntermined by the parameters of the equilibrium and by the rate s\nof change of the heating /cooling function Q(ρ,T) with density,\nQρ≡(∂Q/∂ρ)T, and temperature QT≡(∂Q/∂T)ρ. In the fol-\nlowing analysis, we consider only positive values of both τ1and\nτ2, thus focusing on the e ffect of the slow wave (isentropic)\ndamping or over-stability (see Field 1965, for details). Ty pi-\ncal values of the misbalance time scales τ1andτ2for the ra-\ndiative cooling determined by CHIANTI and a guessed heat-\ning function (determined by the specific values of the densit y\nand temperature power indices aandb) in a dense loop (Reale\n2014) are illustrated in the right-hand panel of Fig. 1. For e xam-\nple, for the temperatures associated with SUMER oscillatio ns,\n6.3 MK and 8.9 MK, we obtain τ1≈37 min andτ2≈12 min,\nandτ1≈65 min andτ2≈19 min, respectively, for a=−0.5 and\nb=−3. This example is provided for the illustrative purposes\nonly, while a more comprehensive analysis of the behaviour o f\nτ1andτ2with aandband their effect on the slow wave dy-\nnamics are given in Sec. 3. No further assumptions on the val-\nues of the characteristic times τ1andτ2are made in the follow-\ning analysis, implying the non-adiabatic terms on the right -hand\nside of energy equation (5) are allowed to be arbitrarily lar ge (in\ncontrast with Kumar et al. 2016; Nakariakov et al. 2017, wher ethe effect of the thermal misbalance on slow waves is investi-\ngated under the assumption of a weak non-adiabaticity). We\nwould also like to stress that in contrast to previous works ( e.g.\nDe Moortel & Hood 2004), investigating e ffects of the radiative\ncooling on the damping of slow waves keeping the heating term\nconstant, i.e. not affected by the perturbations of the plasma pa-\nrameters by a wave and hence not contributing into the wave\ndynamics, we account for the variation of both heating and ra -\ndiative cooling by the wave. Therefore, the heating /cooling mis-\nbalance timesτ1,2(6) are not associated with the corresponding\ntime scales of the cooling or heating processes considered s epa-\nrately of each other.\nWe seek a solution of the linearised set of governing equa-\ntions in the form ei(kz−ωt), which yields the following dispersion\nrelation between the cyclic frequency ωand the wavenumber k,\nω3+A(k)ω2+B(k)ω+C(k)=0, (7)\nwhere the coefficients are\nA=i/bracketleftBiggk2κ\nρ0CV+1\nτ2/bracketrightBigg\n,B=−C2\nsk2,C=−ikBT0\nmk2/bracketleftBiggk2κ\nρ0CV+γ\nτ1/bracketrightBigg\n,\nwhere Cs=/radicalbig\nγkBT0/mis a standard definition of the sound\nspeed. We need to mention here that as the plasma gets per-\nturbed by the wave, the condition of the initial isothermali ty\ndiscussed above is violated, allowing the plasma temperatu re\nto vary with both space and time. Thus, Csis the sound speed\nin a non-isothermal medium with the adiabatic index γ=5/3.\nEquation (7) is found to be asymmetric with respect to space a nd\ntime, being a fourth- and third-order equation with respect tok\nandω, respectively. Similarly to De Moortel & Hood (2003),\na wavelength-dependent term in the coe fficients A(k) and C(k)\ncould be associated with the characteristic time scale of th e field-\naligned thermal conductivity, so that\nτcond=ρ0CVλ2/κ, (8)\nwithλ=2π/kbeing the wavelength. In the regime of a weak\nnon-adiabaticity, i.e. assuming the parameters 1 /ωτ cond and\n1/ωτ 1,2are small, dispersion relation (7) reduces to\nω2=C2\nsk2/braceleftBigg\n1−iω−1/bracketleftBiggγ−1\nγ4π2\nτcond+τ1−τ2\nτ1τ2/bracketrightBigg/bracerightBigg\n, (9)\nWeakly non-adiabatic dispersion relation (9) is a limiting case of\nEq. (21) in Nakariakov et al. (2017) in neglecting the e ffects of\nthe viscosity and oblique propagation. In the following ana lysis,\nwe study full dispersion relation (7). Thus, we allow the ima gi-\nnary part of the frequency to be of the same order of magnitude\nas the real part. This regime is motivated by the apparently h igh\ndamping rates of coronal slow oscillations usually observe d (see\nSec. 3 for references).\n3. Stability analysis\nProcesses described by dispersion relation similar to (7) h ave\nbeen previously shown to a ffect both the phase speed and the\ndamping/amplification length of propagating magnetoacoustic\nwaves (Ibanez S. & Sanchez D. 1992; Ibanez S. & Escalona T.\n1993). In this section we analyse these e ffects on standing slow\nmagnetoacoustic waves in hot coronal loops (SUMER oscilla-\ntions), addressing recent advances in observational detec tions of\nthese waves (Wang 2011). In particular, SUMER oscillations are\nusually seen to rapidly damp, with the quality factor ( q-factor )\nArticle number, page 3 of 6Fig. 2. Variation ofωIobtained forτ1=15 min, andτ2=8.2 min\n(green) andτ2=6 min (red), left; and τ1=10 min, andτ2=13 min\n(green) andτ2=22 min (red), right. The grey lines in both panels\nindicateωcond\nIobtained withτ1,2→∞ . The dashed lines in both panels\nindicateωM\nIobtained withτcond→∞ .\nthat is the ratio of the damping time to the oscillation perio d,\nbeing less than 2–3 (Wang et al. 2003; Mariska 2006; Cho et al.\n2016; Nakariakov et al. 2019).\nDictated by the observational properties of standing slow o s-\ncillations in the corona, we choose the following set of phys ical\nparameters\nT0=6.3×106K,\nρ0=10−11kg m−3,\nL=180×106m,\nκ=10−11T5/2\n0W m−1K−1,\nm=0.6×1.67×10−27kg,\nkB=1.38×10−23m2kg s−1K−1,\nγ=5/3,(10)\nwhere Lis the loop length, and the chosen value of the temper-\nature T0corresponds to a typical detection of a SUMER oscil-\nlation (see Nakariakov et al. 2019, for the most recent revie w).\nThe set of parameters (10) corresponds to the observations o f\ndense loops (e.g. Nisticò et al. 2017), providing the sound s peed\nCs≈152√T0[MK]≈382 km s−1, acoustic oscillation period\nP=2L/Cs≈15.7 min, and the characteristic time scale of the\nthermal conduction τcond≈448 min (obtained by substitution\nof the set of parameters (10) into Eq. (8) and taking λ=2L).\nThe ratio of the oscillation period to thermal conduction ti me,\nP/τcond≈0.035, coincides by an order of magnitude with the es-\ntimation in e.g. De Moortel & Hood (2003) for the chosen value\nofρ0. Such a ratio of the oscillation period to the thermal con-\nduction time justifies a non-isothermal nature of the discus sed\nwaves, implying that in the considered physical conditions (10)\nthe thermal conduction mechanism is insu fficient to smooth out\nthe temperature perturbation on the wave period. However, i n\nshorter and hotter loops the thermal conduction time could b e\nsignificantly shorter, making the waves almost isothermal. In\nturn, the heating/cooling timesτ1,2(6) are treated as free param-\neters in this analysis, being mainly determined by the prope rties\nof an unknown heating function.\nWe seek a solution to dispersion relation (7) in a standing\nwave form, i.e. assuming the cyclic frequency ωto be complex,\nω=ωR+iωI, while the wavenumber kis real. Substituting this\ninto Eq. (7), we solve the polynomial equation for ωInumeri-\ncally using Maple 20161environment. Variation of ωIwith kis\nshown in Fig. 2 for di fferent values of the heating /cooling times\nτ1,2, including the case with τ1,2→∞ which corresponds to the\n1https://www.maplesoft.com/support/help/Fig. 3. Left: Parametric regions of the wave damping enhancement\n(I), suppression (II), and thermal over-stability (III). G rey-shaded re-\ngions indicate the values of τ1,2where the q-factor is in between 1 (the\ngreen line) and 2 (the blue line, Ia), and in between 2 and 3 (th e purple\nline, Ib). The red, green, and blue symbols indicate some arb itrary val-\nues ofτ1,2chosen for the numerical solutions shown in Fig. 4. Right:\nHeating/cooling timesτ1,2(see Eq. (6), black and red contours, respec-\ntively) determined for the CHIANTI radiative cooling, and t he heating\nfunction in the form H(ρ,T)∝ρaTbfor the varying temperature and\ndensity power indices aandb. The grey-shaded areas indicate the val-\nues of aandbwhere 1<q-factor<2 (light grey) and 2 <q-factor<3\n(dark grey). The green, blue, and purple lines show q-factor equals 1, 2,\nand 3, respectively.\nTable 1. Coronal heating functions modelled as H(ρ,T)∝ρaTb: Ohmic\nheating (1), constant heating per unit volume (2) and mass (3 ), and by\nAlfvén waves/mode conversion (4) (see Ibanez S. & Escalona T. 1993);\nand the corresponding τ1,2(6) in minutes with the radiative cooling de-\ntermined by CHIANTI (see Fig. 1).\nModel a bτ1τ2 Model a bτ1τ2\n1 0 1−42.3−64.6 3 0 0−107.6 118.9\n2 -1 0−42.3 118.9 4 1/6 7/6−42.3−51.4\ndamping by thermal conduction only, ωcond\nI, and withτcond→∞\nindicating a pure thermal misbalance case, ωM\nI. Depending on\nthe values ofτ1,2, the imaginary value ωM\nIcan contribute ei-\nther positively or negatively into ωcond\nI, revealing regimes of\nthe enhanced damping ( ωI< ωcond\nI) or suppressed damping\n(ωcond\nI<ω I<0) and over-stability ( ωI>0). These regimes have\nbeen discussed in, e.g. Kumar et al. (2016) and Nakariakov et al.\n(2017). However, in those works the non-adiabatic e ffects were\nweak, thus not describing the strong damping detected in obs er-\nvations (e.g. Wang et al. 2003; Mariska 2006; Cho et al. 2016;\nNakariakov et al. 2019).\nThe left-hand panel of Fig. 3 illustrates regions of the damp -\ning enhancement, suppression, and over-stability in the tw o-\ndimensional parametric space ( τ1,τ2), for the fundamental mode\nof the oscillation, i.e. with k=π/L. Here, we treat the character-\nistic timesτ1andτ2as free parameters. The damping enhance-\nment occurs when τ1>τ 2(see e.g. the last term on the right-\nhand side of Eq. (9)), where the q-factor drops down to the ob-\nservational values of about 1–3 (e.g. Wang et al. 2003; Maris ka\n2006; Cho et al. 2016; Nakariakov et al. 2019). We calculated\nthe values of the heating /cooling timesτ1,2adapting four heat-\ning models from Ibanez S. & Escalona T. (1993) (see Table 1).\nFor the chosen set of parameters (10), the obtained values of τ1,2\nfor those heating models are found to be either of di fferent signs\nor both negative, which would result into the development of\nthermal instabilities of a non-acoustic nature (see Field 1 965).\nTherefore, neither of them is found to be suitable for the obs er-\nvational damping of SUMER oscillations.\nArticle number, page 4 of 6Kolotkov et al.: Damping of slow waves by the thermal misbala nce\nAsτ1,2depend on the parameters aandbof the heating func-\ntion (6), we calculate τ1,2foraandbboth ranging from e.g. −5\nto 5 (see the right-hand panel of Fig. 3). The obtained values of\nτ1,2are seen to depend strongly on aandb, varying from sev-\neral to a hundred of minutes and longer for the chosen values o f\nthe plasma density and temperature. They both have the verti cal\nasymptote at a≈1 and b≈0.4, above which they both become\nnegative. The blank regions in the right-hand panel of Fig. 3 and\nwhere the contour lines do not intersect correspond to the ne g-\native or different signs ofτ1,2, respectively, which give raise to\nother thermal instabilities (see Field 1965) which are out o f the\nscope of this study. We now compare this diagram to the values\nofτ1,2, for which the oscillation q-factor was found to vary from\n1 to 2 (see the left-hand panel of Fig. 3), constraining the he ating\nfunctions which are able to reproduce the observational dam ping\n(see the grey-shaded area in the right-hand panel of Fig. 3). For\nlower plasma densities, the values of indices aandb, which give\nthe misbalance times τ1,2about the observed periods, would be\neven lower.\nChoosing three different pairs of the heating /cooling times\nτ1andτ2, which provide the q-factor to be lower than 1, from\n1 to 2, and from 2 to 3, and using parameters (10), we solve\nthe linearised set of governing equations numerically in Maple\n2016 , in a closed resonator located between z=0 and z=L\nand with the initial broadband Gaussian-shaped acoustic pe r-\nturbation of the width w=0.12L, shifted towards one of the\nboundaries. The cross-sections of the obtained standing so lu-\ntions at z=L/2 are shown in the left-hand panel of Fig. 4. As\nexpected from the dispersion relation (see Eq. (7) and Fig. 2 ),\nthe higher harmonics decay faster, so that after about one cy -\ncle of the oscillation the initial broadband pulse develops into\na pure fundamental mode which then also decays. This exam-\nple illustrates how sensitive the damping of standing slow w aves\nis to the parameters of the heating /cooling function, and it repre-\nsents the rapidly decaying oscillations of the SUMER-oscil lation\ntype. In a more exotic case, when the values of τ1andτ2ap-\npear to be just near the boundary ωI=0 (see e.g. the red line\nin the left-hand panel of Fig. 3), the damping could be highly\nsuppressed by the thermal misbalance (see e.g. the apparent ly\nnon-decaying oscillation observed in the Fe xvemission line in\nFig. 3 of Mariska et al. 2008). Adapting the physical parame-\nters corresponding to this observation, namely T0=106.32K and\nL=342 Mm, and choosing ρ0=10−12kg m−3, we can repro-\nduce the observed non-decaying oscillation within the deve loped\nmodel for, e.g.τ1=19.5 min andτ2=22.3 min.\n4. Summary and conclusions\nWe investigated the mechanism for damping of linear standin g\nslow magnetoacoustic waves in the solar corona through the m is-\nbalance of some heating and radiative cooling processes. We\naddressed the coronal part of a loop with an isothermal equi-\nlibrium. This is a standard approach for modelling slow wave s\nin the corona. However, we consider the wave dynamics in the\npresence of a temperature- and pressure-depend heating and ra-\ndiative cooling and thermal conduction, addressing a misba lance\nof those processes caused by the waves. The wave dynamics was\nfound to be highly sensitive to the parameters of the misbala nce,\nexpressed in terms of the characteristic times τ1,2of the heat-\ning/cooling function change with the plasma density and tem-\nperature perturbed by the wave (see Sec. 2). Depending upon t he\nvalues ofτ1,2, we found three di fferent regimes of the wave evo-\nlution, which are the enhanced and suppressed damping (with\nrespect to the one caused by the field-aligned thermal conduc tiv-Fig. 4. Left: Cross-sections of the perturbed velocity as a functio n\nof time, obtained for τ1=40 min andτ2=23 min (blue),τ1=61 min\nandτ2=11.5 min (red), andτ1=30 min andτ2=5 min (green).\nRight: Similar to the left-hand panel, but for the set of phys ical pa-\nrameters from Mariska et al. (2008), see Sec. 3 for details, a nd with\nτ1=19.5 min andτ2=22.3 min.\nity), and the thermal over-stability. Unlike the previous a nalyt-\nical works, we did not treat the non-adiabatic terms small, t hat\nallowed us to obtain the enhanced damping rates matching tho se\ndetected in observations.\nOur findings allow one to reproduce the observed behaviour\nof SUMER oscillations, keeping the thermal conduction coef -\nficient in its standard estimation, but accounting for the he at-\ning/cooling misbalance. For the set of physical parameters\ncorresponding to the observations of SUMER oscillations (s ee\nSec. 3), the characteristic time scale of the thermal conduc tion\nwas found to be at least an order of magnitude longer than the\noscillation period. This indicates a low e fficiency of the field-\naligned thermal conductivity in damping these oscillation s. In\nturn, typical heating /cooling timesτ1,2were found to be compa-\nrable to the observed periods of SUMER oscillations (from a f ew\nminutes to a few tens of minutes, see Fig. 3), for a su fficiently\nbroad range of the heating function parameters and for the CH I-\nANTI radiative cooling. For τ1>τ 2, this results into a domina-\ntion of the damping by the heating /cooling misbalance over con-\nductive damping. Moreover, the discussed e ffect persists even\nin the limiting case of isothermal waves, which are not subje ct\nto the damping by thermal conduction at all, occurring in the\ncase of the dominant thermal conduction (De Moortel & Hood\n2003). In this regime, the cooling and heating functions, an d\nhence their misbalance, are still a ffected by the perturbations of\ndensity in the wave and hence contribute to its damping.\nUsing the CHIANTI model for the radiative cooling and fix-\ning other parameters of the equilibrium, the values of τ1,2be-\ncome fully determined by the heating function. This suggest s\na new way for the diagnostics of the coronal heating mecha-\nnism via damping of SUMER oscillations. For example, neithe r\nof four heating models considered by Ibanez S. & Escalona T.\n(1993) (see Table 1) was found to reproduce the observed damp -\ning of SUMER oscillations. On the other hand, we determined\nthe range of the power-law indices aandb, which give the ob-\nserved damping times. Moreover, the developed theory could\nalso address a more exotic case of an apparently non-decayin g\nSUMER type oscillation detected by Mariska et al. (2008), by\nchoosing the values of τ1andτ2which giveωI≈0. In addition,\nacoustic over-stability could be considered as a mechanism for\nthe excitation of 8–30 min oscillations of the soft X-ray emi ssion\ngenerated in pre-flaring active region (Tan et al. 2016).\nThe need to comply with observational properties of coro-\nnal slow waves may put additional constraints on the empiric al\nArticle number, page 5 of 6determination of the dependence of the heating function on t he\nplasma parameters. This seismological information about t he ac-\nceptable ranges of the parameters aandb, together with the in-\nformation obtained by other methods, could be used for revea l-\ning the heating function. In particular, our study suggests that\n−2/lessorsimilara/lessorsimilar2 and b/lessorsimilar0 for the chosen values of the equilibrium\ndensity and temperature. Those intervals should be subject to a\ndedicated follow-up analysis. In particular, the e ffect of differ-\nent parametric forms of the heating function dependence on t he\ndensity and temperature, e.g. polynomial, should be consid ered.\nLikewise, the time-dependence of the coronal heating funct ion,\nneglected in this study on the time scale of a slow wave, could\nbe more important for shorter-period coronal MHD waves, e.g .\nthe fast waves with about 1-min periodicity. Also, this negl ec-\ntion does not allow us to address the transient events in whic h\nthe loop is impulsively heated and rapidly cools down at the t ime\nscale comparable to the wave period (e.g. Reale et al. 2019), thus\nmaking the developed theory restricted to the loops sustain ed\nat approximately the same mean temperature during the whole\nwave evolution. In addition, the future development of the t he-\nory needs to soften certain assumptions made in this paper. I n\nparticular, we neglect the e ffects of the oblique wave propaga-\ntion, i.e. the departure of the slow wave speed from the sound\nspeed in the case of finite β, and viscosity, which could bring ad-\nditional time scales into the problem. This could be importa nt if\nthe coronal heating depends on the magnetic field (Hood 1992;\nNakariakov et al. 2017). We also do not consider the e ffect of ge-\nometrical dispersion (Edwin & Roberts 1983; Yuan et al. 2015 )\nthat is usually weak for slow waves in coronal loops. Likewis e,\nwe do not account for nonlinear e ffects. Another interesting\ndevelopment of this study could be the inclusion of a chromo-\nsphere. Accounting for these e ffects should be addressed in a\nfollow up study.\nAcknowledgements. This work was supported by STFC consolidated grant\nST/P000320/1 (V .M.N., D.Y .K.), and the Russian Foundation for Basic Res earch\ngrant No. 18-29-21016 (V .M.N.). Calculations presented in the reported study\nwere also funded by RFBR according to the research project No . 18-32-00344\n(D.I.Z.). The study was supported in part by the Ministry of E ducation and\nScience of Russia under the public contract with educationa l and research insti-\ntutions within the project 3.1158.2017 /4.6. CHIANTI is a collaborative project\ninvolving George Mason University, the University of Michi gan (USA) and the\nUniversity of Cambridge (UK). Maple is a trademark of Waterl oo Maple Inc.\nReferences\nBanerjee, D. & Krishna Prasad, S. 2016, Washington DC Americ an Geophysical\nUnion Geophysical Monograph Series, 216, 419\nCho, I.-H., Cho, K.-S., Nakariakov, V . M., Kim, S., & Kumar, P . 2016, ApJ, 830,\n110\nDahlburg, R. B. & Mariska, J. T. 1988, Sol. Phys., 117, 51\nDe Moortel, I. 2009, Space Sci. Rev., 149, 65\nDe Moortel, I. & Hood, A. W. 2003, A&A, 408, 755\nDe Moortel, I. & Hood, A. W. 2004, A&A, 415, 705\nDe Moortel, I. & Nakariakov, V . 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E. 2014, Technical Physic s Letters, 40, 701\nArticle number, page 6 of 6"    },    {        "title": "1602.06673v3.Effects_of_Landau_Lifshitz_Gilbert_damping_on_domain_growth.pdf",        "content": "arXiv:1602.06673v3  [cond-mat.stat-mech]  1 Dec 2016Effects of Landau-Lifshitz-Gilbert damping on domain growt h\nKazue Kudo\nDepartment of Computer Science, Ochanomizu University,\n2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan\n(Dated: May 25, 2021)\nDomain patterns are simulated by the Landau-Lifshitz-Gilb ert (LLG) equation with an easy-axis\nanisotropy. If the Gilbert damping is removed from the LLG eq uation, it merely describes the\nprecession of magnetization with a ferromagnetic interact ion. However, even without the damping,\ndomains that look similar to those of scalar fields are formed , and they grow with time. It is demon-\nstrated that the damping has no significant effects on domain g rowth laws and large-scale domain\nstructure. In contrast, small-scale domain structure is aff ected by the damping. The difference in\nsmall-scale structure arises from energy dissipation due t o the damping.\nPACS numbers: 89.75.Kd,89.75.Da,75.10.Hk\nI. INTRODUCTION\nCoarseningorphase-orderingdynamicsisobservedina\nwidevarietyofsystems. Whenasystemisquenchedfrom\na disordered phase to an ordered phase, many small do-\nmainsareformed, andtheygrowwithtime. Forexample,\nin the case of an Ising ferromagnet, up-spin and down-\nspin domains are formed, and the characteristic length\nscale increases with time. The Ising spins can be inter-\npreted as two different kinds of atoms in the case of a\nbinary alloy. At the late stage of domain growth in these\nsystems, characteristic length L(t) follows a power-law\ngrowth law,\nL(t)∼tn, (1)\nwherenis the growth exponent. The growth laws in\nscalarfieldshavebeenderivedbyseveralgroups: n= 1/2\nfornon-conservedscalarfields, and n= 1/3forconserved\nscalar fields [1–8].\nSimilar coarsening dynamics and domain growth have\nbeen observed alsoin Bose-Einstein condensates (BECs).\nThe characteristic length grows as L(t)∼t2/3in two-\ndimensional (2D) binary BECs and ferromagnetic BECs\nwith an easy-axis anisotropy [9–11]. The same growth\nexponent n= 2/3 is found in classical binary fluids in\nthe inertial hydrodynamic regime [1, 12]. It is remark-\nable that the same growth law is found in both quan-\ntum and classical systems. It should be also noted that\ndomain formation and coarsening in BECs occur even\nwithout energy dissipation. The dynamics in a ferro-\nmagnetic BEC can be described not only by the so-\ncalled Gross-Pitaevskii equation, which is a nonlinear\nSchr¨ odinger equation, but also approximately by a mod-\nified Landau-Lifshitz equation in which the interaction\nbetween superfluid flow and local magnetization is incor-\nporated[13–15]. Ifenergydissipationexists, theequation\nchanges to an extended Landau-Lifshitz-Gilbert (LLG)\nequation [9, 15, 16]. The normal LLG equation is usu-\nally used to describe spin dynamics in a ferromagnet.\nThe LLG equation includes a damping term which is\ncalled the Gilbert damping. When the system has an\neasy-axis anisotropy, the damping has the effect to directa spin to the easy-axis direction. The Gilbert damping\nin the LLG equation corresponds to energy dissipation\nin a BEC. In other words, domain formation without en-\nergy dissipation in a BEC implies that domains can be\nformed without the damping in a ferromagnet. However,\nthe LLG equation without the damping describes merely\nthe precession of magnetization with a ferromagnetic in-\nteraction.\nInthispaper, wefocusonwhateffectsthedampinghas\nondomainformationanddomaingrowth. UsingtheLLG\nequation (without flow terms), we investigate the mag-\nnetic domain growth in a 2D system with an easy-axis\nanisotropy. Since our system is simpler than a BEC, we\ncan also give simpler discussions on what causes domain\nformation. When the easy axis is perpendicular to the\nx-yplane, the system is an Ising-like ferromagnetic film,\nand domains in which the zcomponent of each spin has\nalmostthesamevalueareformed. Inordertoobservedo-\nmain formation both in damping and no-damping cases,\nwe limit the initial condition to almost uniform in-plane\nspins. Actually, without the damping, domain formation\ndoes not occur from an initial configuration of spins with\ntotally random directions. Without the damping, the z\ncomponent is conserved. The damping breaks the con-\nservation of the zcomponent as well as energy. Here,\nwe should note that the growth laws for conserved and\nnonconserved scalar fields cannot simply be applied to\nthe no-damping and damping cases, respectively, in our\nsystem. Although the zcomponent corresponds to the\norderparameterofascalarfield, oursystemhastheother\ntwo components. It is uncertain whether the difference\nin the number of degrees of freedom can be neglected in\ndomain formation.\nThe restofthe paperis organizedas follows. In Sec. II,\nwe describe the model and numerical procedures. Ener-\ngies and the characteristic length scale are also intro-\nduced in this section. Results of numerical simulations\nare shown in Sec. III. Domain patterns at different times\nand the time evolution of energies and the average do-\nmain size are demonstrated. Scaling behavior is con-\nfirmed in correlation functions and structure factors at\nlate times. In Sec. IV, we discuss why domain formation2\ncan occur even in the no-damping case, focusing on an\nalmost uniform initial condition. Finally, conclusions are\ngiven in Sec. V.\nII. MODEL AND METHOD\nThe model we use in numerical simulations is the LLG\nequation, which is widely used to describe the spin dy-\nnamics in ferromagnets. The dimensionless normalized\nform of the LLG equation is written as\n∂m\n∂t=−m×heff+αm×∂m\n∂t, (2)\nwheremis the unit vector of spin, αis the dimensionless\nGilbert damping parameter. We here consider the 2D\nsystemlyinginthe x-yplane,andassumethatthesystem\nhas a uniaxial anisotropy in the zdirection and that no\nlong-range interaction exists. Then, the dimensionless\neffective field is given by\nheff=∇2m+Canimzˆz, (3)\nwhereCaniis the anisotropy parameter, and ˆzis the unit\nvector in the zdirection.\nEquation (2) is mathematically equivalent to\n∂m\n∂t=−1\n1+α2m×heff+α\n1+α2m×(m×heff).\n(4)\nIn numerical simulations, we use a Crank-Nicolson\nmethod to solve Eq. (4). The initial condition is given as\nspins that are aligned in the xdirection with a little ran-\ndom noises: mx≃1 andmy≃mz≃0. Simulations are\nperformed in the 512 ×512 lattice with periodic bound-\nary conditions. Averages are taken over 20 independent\nruns.\nThe energy in this system is written as\nE=Eint+Eani\n=1\n2/integraldisplay\ndr(∇m(r))2−1\n2Cani/integraldisplay\ndrmz(r)2,(5)\nwhich gives the effective field as heff=−δE/δm. The\nfirst and second terms are the interfacial and anisotropy\nenergies, respectively. When Cani>0, thezcomponent\nbecomes dominant since a large m2\nzlowers the energy.\nWe take Cani= 0.2 in the simulations. The damping\nparameter αexpresses the rate of energy dissipation. If\nα= 0, the spatial average of mzas well as the energy E\nis conserved.\nConsidering mzas the order parameter of this system,\nwe here define the characteristic length scale Lof a do-\nmain pattern from the correlation function\nG(r) =1\nA/integraldisplay\nd2x/angb∇acketleftmz(x+r)mz(x)/angb∇acket∇ight,(6)\nwhereAis the area of the system and /angb∇acketleft···/angb∇acket∇ightdenotes an\nensemble average. The average domain size Lis defined\nby the distance where G(r), i.e., the azimuth average of\nG(r), first drops to zero, and thus, G(L) = 0.\nFIG. 1. (Color online) Snapshots of z-component mzat time\nt= 102((a) and (b)), 103((c) and (d)), and 104((e) and\n(f)). Snapshots (g) and (h) are enlarged parts of (e) and (f),\nrespectively. Profiles (i) and (j) of mzare taken along the\nbottom lines of snapshots (g) and (h), respectively. Left an d\nright columns are for the no-damping ( α= 0) and damping\n(α= 0.03) cases, respectively.\nIII. SIMULATIONS\nDomain patterns appear, regardless of the damping\nparameter α. The snapshots of the no-damping ( α= 0)\nand damping ( α= 0.03) cases are demonstrated in the\nleft and right columns of Fig. 1, respectively. Domain\npatterns at early times have no remarkable difference be-\ntweenthe twocases. Thecharacteristiclengthscalelooks\nalmostthesamealsoatlatertimes. However,asshownin\nthe enlarged snapshots at late times, difference appears\nespecially around domain walls. Domain walls, where\nmz≃0, are smooth in the damping case. However, in\nthe no-damping case, they look fuzzy. The difference ap-\npears more clearly in profiles of mz(Figs. 1(i) and 1(j)).\nWhile the profile in the damping case is smooth, that\nin the no-damping case is not smooth. Such an uneven\nprofile makes domain walls look fuzzy.\nThe difference in domain structure is closely connected\nwith energydissipation, which is shownin Fig. 2. The in-\nterfacial energy, which is the first term of Eq. (5), decays\nforα= 0.03 but increases for α= 0 in Fig. 2 (a). In con-3\n0 2000 4000 6000 8000 10000t00.020.040.060.080.1Eint α = 0\nα = 0.03(a)\n0 2000 4000 6000 8000 10000t-0.1-0.08-0.06-0.04-0.020Eaniα = 0\nα = 0.03(b)\nFIG. 2. (Color online) Time dependence of (a) the inter-\nfacial energy Eintand (b) the anisotropy energy Eani. The\ninterfacial energy increases with time in the no-damping ca se\n(α= 0) and decreases in the damping case ( α= 0.03). The\nanisotropy energy decreases with time in both cases.\ntrast, the anisotropy energy, which comes from the total\nofm2\nz, decreases with time for both α= 0 and α= 0.03.\nIn other words, the energy dissipation relating to the in-\nterfacial energy mainly causes the difference between the\ndamping and no-damping cases. In the damping case,\nthe interfacial energy decreases with time after a shot-\ntime increase as domain-wall structure becomes smooth.\nHowever, in the no-damping case, the interfacial energy\nincreases with time to conserve the total energy that is\ngiven by Eq. (5). This corresponds to the result that\nthe domain structure does not become smooth in the no-\ndamping case.\nBeforediscussinggrowthlaws, we shouldexaminescal-\ning laws. Scaled correlation functions of mzat different\ntimes are shown in Fig. 3. The functions look pretty\nsimilar in both damping and no-damping cases, which\nreflects the fact that the characteristic length scales in\nboth cases looks almost the same in snapshots. At late\ntimes, the correlation functions that are rescaled by the\naverage domain size L(t) collapse to a single function.\nHowever, the scaled correlation functions at early times\n(t= 100 and 1000) do not agree with the scaling func-\ntion especially in the short range. The disagreement at\nearly times is related with the unsaturation of mz. How\nmzsaturates is reflected in the time dependence of the0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(a)\n0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(b)\nFIG.3. (Color online) Scaledcorrelation functions atdiffe rent\ntimes in (a) no-damping ( α= 0) and (b) damping ( α= 0.03)\ncases. The correlation functions at late times collapse to a\nsingle function, however, the ones at early times do not.\nanisotropy energy which is shown in Fig. 2(b). At early\ntimes (t/lessorsimilar1000),Eanidecays rapidly. This implies that\nmzis not saturated enough in this time regime. The de-\ncreasein theanisotropyenergyslowsatlatetimes. In the\nlate-time regime, mzis sufficiently saturated except for\ndomain walls, and the decrease in the anisotropy energy\nis purely caused by domain growth. This corresponds to\nthe scaling behavior at late times.\nIn Fig. 4, the average domain size Lis plotted for\nthe damping and no-damping cases. In both cases,\nthe average domain size grows as L(t)∼t1/2at late\ntimes, although growth exponents at early times look\nliken= 1/3. Since scaling behavior is confirmed only\nat late times, the domain growth law is considered to be\nL(t)∼t1/2rather than t1/3in this system. In our pre-\nvious work, we saw domain growth as L(t)∼t1/3in a\nBEC without superfluid flow [9], which was essentially\nthe same system as the present one. However, the time\nregion shown in Ref. [9] corresponds to the early stage\n(t/lessorsimilar1830) in the present system.\nAlthough the growth exponent is supposed to be n=\n1/3 for conserved scalar fields, the average domain size\ngrows as L(t)∼t1/2, in our system, at late times even\nin the no-damping case. This implies that our system\nwithout damping cannot be categorized as a model of a4\n100 1000 10000t10100 Lα = 0\nα = 0.03\nt1/2\nt1/3\nFIG. 4. (Color online) Time dependence of the average do-\nmain size Lforα= 0 and 0 .03. In both damping and no-\ndamping cases, domain size grows as L(t)∼t1/2at late\ntimes. Before the scaling regime, early-time behavior look s\nas ifL(t)∼t1/3.\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3S(k)/L(t)2\nt =  6000\nt =  8000\nt = 10000\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3\nt =  6000\nt =  8000\nt = 10000(a) (b)\nk-3k-3\nFIG. 5. (Color online) Scaling plots of the structure factor\nscaled with L(t) at different times in (a) no-damping ( α= 0)\nand (b) damping ( α= 0.03) cases. In both cases, S(k)∼k−3\nin the high- kregime. However, they gave different tails in the\nultrahigh- kregime.\nconserved scalar field. Although we consider mzas the\norder parameter to define the characteristic length scale,\nthe LLG equation is described in terms of a vector field\nm.\nScaling behavior also appears in the structure factor\nS(k,t), which is given by the Fourier transformation of\nthe correlation function G(r). According to the Porod\nlaw, the structure factor has a power-law tail,\nS(k,t)∼1\nL(t)kd+1, (7)\nin the high- kregime [1]. Here, dis the dimension of\nthe system. Since d= 2 in our system, Eq. (7) leads\ntoS(k,t)/L(t)2∼[kL(t)]−3. In Fig. 5, S(k,t)/L(t)2is\nplotted as a function of kL(t). The data at different late\ntimes collapse to one curve, and they show S(k)∼k−3in the high- kregime (kL∼10) in both the damping and\nno-damping cases. In the ultrahigh- kregime (kL∼100),\ntails are different between the two cases, which reflects\nthe difference in domain structure. Since domain walls\nare fuzzy in the no-damping case, S(k) remains finite.\nHowever, in the damping case, S(k) decays faster in the\nultrahigh- kregime, which is related with smooth domain\nwalls.\nIV. DISCUSSION\nWe here have a naive question: Why does domain\npattern formation occur even in the no-damping case?\nWhenα= 0, Eq. (2) is just the equation of the pre-\ncession of spin, and the energy Eas well as mzis con-\nserved. We here discuss why similar domain patterns are\nformed from our initial condition in both damping and\nno-damping cases.\nUsing the stereographic projection of the unit sphere\nof spin onto a complex plane [17], we rewrite Eq. (4) as\n∂ω\n∂t=−i+α\n1+α2/bracketleftbigg\n∇2ω−2ω∗(∇ω)2\n1+ωω∗−Caniω(1−ωω∗)\n1+ωω∗/bracketrightbigg\n,\n(8)\nwhereωis a complex variable defined by\nω=mx+imy\n1+mz. (9)\nEquation (8) implies that the effect of the Gilbert damp-\ning is just a rescaling of time by a complex constant [17].\nThe fixed points of Eq. (8) are |ω|2= 1 and ω= 0.\nThelinearstabilityanalysisaboutthesefixedpointsgives\nsome clues about domain formation.\nAt the fixed point ω= 1,mx= 1 and my=mz= 0,\nwhich corresponds to the initial condition of the numer-\nical simulation. Substituting ω= 1 +δωinto Eq. (8),\nwe obtain linearized equations of δωandδω∗. Perform-\ning Fourier expansions δω=/summationtext\nkδ˜ωkeik·randδω∗=/summationtext\nkδ˜ω∗\n−keik·r, we have\nd\ndt/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n=/parenleftbigg\n˜α1(Cani−k2) ˜α1Cani\n˜α2Cani˜α2(Cani−k2)/parenrightbigg/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n,\n(10)\nwhere ˜α1=1\n2(−i+α)/(1+α2), ˜α2=1\n2(i+α)/(1+α2),\nk= (kx,ky), andk=|k|. The eigenvalues of the 2 ×2\nmatrix of Eq. (10) are\nλ(k) =α\n2(1+α2)(Cani−2k2)±/radicalbig\n4k2(Cani−k2)+α2C2\nani\n2(1+α2).\n(11)\nEven when α= 0,λ(k) has a positive real part for\nk <√Cani. Thus, the uniform pattern with mx= 1\nis unstable, and inhomogeneous patterns can appear.\nThe positive real parts of Eq. (11) for α= 0 and\nα= 0.03 have close values, as shown in Fig. 6. This cor-\nresponds to the result that domain formation in the early5\n0 0.1 0.2 0.3 0.4 0.5\nk00.020.040.060.080.1λ(k)α = 0\nα = 0.03\nFIG. 6. (Color online) Positive real parts of λ(k) that is given\nby Eq. (11), which has a positive real value for k <√Cani.\nThe difference between α= 0 and α= 0.03 is small.\nstage has no remarkable difference between the damping\n(α= 0.03) and no-damping ( α= 0) cases (See Fig. 1).\nFrom the view point of energy, the anisotropy energy\ndoes not necessarily keep decaying when α= 0. For con-\nservation of energy, it should be also possible that both\nanisotropy and interfacial energies change only a little.\nBecause of the instability of the initial state, mzgrows,\nand thus, the anisotropy energy decreases.\nThe initial condition, which is given as spins aligned in\nonedirection with somenoisesin the x-yplane, is the key\nto observe domain pattern formation in the no-damping\ncase. Actually, if spins have totally random directions,\nno large domains are formed in the no-damping case,\nalthough domains are formed in damping cases ( α >0)\nfrom such an initial state.\nWhenω= 0,mx=my= 0 and mz= 1, which is also\none of the fixed points. Substituting ω= 0 +δωinto\nEq. (8) and performing Fourier expansions, we have the\nlinearized equation of δ˜ωk,\nd\ndtδ˜ωk=i−α\n1+α2(k2+Cani)δ˜ωk. (12)\nThis implies that the fixed point is stable for α >0 andneutrally stable for α= 0. Although mz=−1 corre-\nsponds to ω→ ∞, the same stability is expected for\nmz=−1 by symmetry.\nSincetheinitialconditionisunstable, the z-component\nof spin grows. Moreover, linear instability is similar for\nα= 0 and α= 0.03. Since mz=±1 are not unstable,\nmzcan keep its value at around mz=±1. This is why\nsimilar domain patters are formed in both damping and\nno-damping cases. The main difference between the two\ncases is that mz=±1 are attracting for α >0 and neu-\ntrally stable for α= 0. Since mz=±1 are stable and at-\ntractingin the dampingcase, homogeneousdomainswith\nmz=±1 are preferable, which leads to a smooth profile\nofmzsuch as Fig. 1(j). In the damping case, mz=±1\nare neutrally stable (not attracting) fixed points, which\ndoes not necessarily make domains smooth.\nV. CONCLUSIONS\nWe have investigated the domain formation in 2D vec-\ntor fields with an easy-axis anisotropy, using the LLG\nequation. When the initial configuration is given as al-\nmost uniform spins aligned in an in-plane direction, sim-\nilar domain patterns appear in the damping ( α/negationslash= 0) and\nno-damping ( α= 0) cases. The average domain size\ngrows as L(t)∼t1/2in late times which are in a scal-\ning regime. The damping gives no remarkable effects\non domain growth and large-scale properties of domain\npattern. In contrast, small-scale structures are different\nbetween the two cases, which is shown quantitatively in\nthe structure factor. This difference is induced by the re-\nduction of the interfacial energy due to the damping. It\nshould be noted that the result and analysis especially\nin the no-damping case are valid for a limited initial\ncondition. Although domains grow in a damping case\neven from spins with totally random directions, domain\ngrowth cannot occur from such a random configuration\nin the no-damping case.\nACKNOWLEDGMENTS\nThis work was supported by MEXT KAKENHI\n(No. 26103514, “Fluctuation & Structure”).\n[1] A. Bray, Adv. Phys. 43, 357 (1994)\n[2] I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids\n19, 35 (1961)\n[3] C. Wagner, Z. Elektrochem 65, 581 (1961)\n[4] T. Ohta, D. Jasnow, and K. Kawasaki, Phys. Rev. Lett.\n49, 1223 (1982)\n[5] D. A. Huse, Phys. Rev. B 34, 7845 (1986)\n[6] A. J. Bray, Phys. Rev. Lett. 62, 2841 (1989)\n[7] A. J. Bray, Phys. Rev. B 41, 6724 (1990)\n[8] A. J. Bray and A. D. Rutenberg, Phys. Rev. E 49, R27\n(1994)[9] K. Kudo and Y. Kawaguchi, Phys. Rev. A 88, 013630\n(2013)\n[10] J. Hofmann, S. S. Natu, and S. Das Sarma, Phys. Rev.\nLett.113, 095702 (2014)\n[11] L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 116,\n025301 (2016)\n[12] H. Furukawa, Phys. Rev. A 31, 1103 (1985)\n[13] A. Lamacraft, Phys. Rev. A 77, 063622 (2008)\n[14] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85,\n1191 (2013)\n[15] Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012)6\n[16] K. Kudo and Y. Kawaguchi, Phys. Rev. A 84, 043607\n(2011)\n[17] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53,\n2497 (1984)"    },    {        "title": "0706.1736v1.Gilbert_and_Landau_Lifshitz_damping_in_the_presense_of_spin_torque.pdf",        "content": "Gilbert and Landau-Lifshitz  damping in the presense of spin-torque \n \nNeil Sm ith \nSan J ose Reserch Center, Hitachi Global Strage Tec hnologies, Sa n Jose, CA 95135 \n(Dated 6/12/07) \n \nA recent arti cle by Stiles  et al.  (cond-mat/0702020) argued  in favor of th e Land au-Lifshitz dampin g term in  the \nmicromagnetic equations of  motion over that of  the more commo nly accepted  Gilbert d amping for m. Much of \ntheir argument r evolved  around  spin-torque dr iven domai n wall motion in n arrow magnetic wires, since th e \npresence of spin -torques can mor e acut ely draw a distinct ion b etween the two form s of dam ping. In this ar ticle, \nthe author  uses simple argum ents and exam ples to offer  an alterna tive po int of  view favoring  Gilb ert.  \n \nI. PREL IMINARIES \n \n       The Gilbert1 (G) or La ndau-Li fshitz2 (LL) equations of  \nmotion for unit magnetization vect or  \nare formally descri bed by the gene ric form sMt t /),( ),(ˆ rM rm≡\n \n) derivativeal (variationˆ/ /1)] (ˆ[ /ˆ\neffeff totdamp tot\nNC\nm HH H HH Hm m\n∂∂−≡+≡+×γ−=\nE Mdt d\ns      (1) \n \nwhere  the satu ration  magnetizatio n, and  sM γ is th e \ngyromagnetic ratio (tak en here to  be a po sitive constant). \nThe total (p hysical) field  by h as con tributions fro m the \nusual \"effecti ve field\" term , pl us t hat of a   \n\"nonconservative-field\"  that is supp osed not to be \nderivable from the -gradient of the (internal) free-energy \ndensity functional . Although also non conservative \nby defi nition, the \"dam ping-field\"  is p rimarily a \nmath ematica l vehicle for  describing a physical d amping \ntorque , and i s properly treated separat ely. \nFor most of the rem ainder of th is article, an y sp atial \ndepe ndence of  will be im plicitly unde rstood.  effH\nNCH\nmˆ\n)ˆ(mE\ndampH\ndampˆHm×sM\n),(ˆtrm\n       As was described by Brown,3 the Gilb ert eq uations of \nmotion m ay be de rived using st andard techni ques of \nLagra ngian mechanics .4 In particular, a phenomenological \ndamping of the motion in included via the use of a R ayleigh \ndissipation function : )/ˆ( dtdmℜ\ndt d dt d M/dtdM\nGG\nss\n/ˆ)/ ( )/ˆ(/ /1ˆ )2/ (\nG\ndamp2\nm m Hm\nγα−= ∂∂ℜ−≡γα=ℜ\n     (2) \n \nwhere dimensionless  is th e Gilb ert damping parameter. \nBy d efinition,Gα\n4 2\ndampˆ / /ˆ 2G/dtd M dtds G m m H γα=⋅−=ℜ  \nis th e in stantaneous rate o f energy lost fro m the \nmagnetizatio n syste m to its thermal environment (e.g ., to \nthe lattice) d ue to the viscous  \"friction\" re present ed by the \ndamping field .   dt d/ˆG\ndampm H−∝\n        The Lag rangian method is well su ited to include \nnonconservative fields , which can be  generally \ndefined using the principles of virtual work:0NC≠H\n3,4 \n \n)ˆ ( /1 ) ˆ() ˆ( ˆ\nNC NCNC NC NC\nmN H HmHm m H\n× =⇔×=⇒δ⋅×=δ⋅ =δ\ns ss s\nM MNM M W θ       (3) \n \nThe latter exp ressio n is useful in cases (e.g.,  spin-torques) \nwhere the torque density funct ional  is specified. \nTreatin g  as fi xed, the  (virtual) dis placem ent )ˆ(mN\nsM mˆδ is of \nthe fo rm m m ˆ ˆ×δ=δθ , and only the orthogonal compone nts \nof the torque mNmN ˆ ˆ××↔  are physically signi ficant.  \n       Combining (1) an d (2) gives the Gilbert equations: \n \n)/ˆ ˆ( ) ˆ( /ˆtot dtd dt dG mm Hm m ×α+×γ−=           (4) \n \nAs is well known, the G eq uations o f (4) may be rearra nged \ninto their equivalent (and perhaps m ore common) f orm:  \n \n)] ˆ(ˆ ˆ[\n1/ˆtot tot2\nGHmm Hm m ××α+×\nα+γ−=G dt d      (5) \n \n       With re gard to the LL e quations, the form  of  is \nnot uniquely  defined in problems where LL\ndampH\n0NC≠H , whic h \nhave only c ome to the forefront with the  recent interest in \nspin-torque phen omena. Two d efinitions  conside red are \n \n) ˆ(eff damp LLLLHm H ×α≡ ,                       (6a) \n                        (6b) ) ˆ(tot damp LLLLHm H ×α≡\n \nThe fi rst de finition of (6a) is the historical/conventional \nform  of LL, and is that em ployed by Stiles et al.5 Howe ver, \nin this a uthor's view, the re is no a-priori reason, other than \nhistorical,  to not replace  as in (6b). Doing so \nyields a form  of LL that reta ins it \"usual\" e quivalence (i.e., \nto first o rder in tot eff H H→\nα) to G w hether or not 0NC≠H , as is \nseen by com paring (5) and (6b). The form o f (6b) treats \nboth  and  on an equal footin g. effHNCH       Nonet heless, to facilita te a com parative discussi on with \nthe analysis of Stiles et al. ,5  (6a) will he ncefort h be used to \ndefine what will be re ferred to below as the LL eq uatio ns \nof motion: \n \n)] ˆ(ˆ ˆ[ /ˆeff tot LL Hmm Hm m ××α+×γ−=dt d        (7) \n \nIn cases of pre sent interest where  , the difference \nbetwee n G in (4) (or (5)) and the f orm of LL give n in ( 7) \nare first orde r in the dam ping param eter, and  thus o f a more \nfundam ental nature. T hese differences a re the subj ect of the  \nremainder of t his article. 0NC≠H\n \nII. SPIN-TOR QUE  EXAM PLES \n \n       Two distinct situations where spin-torque effects have \ngarnere d substantial intere st are those  of CPP-GMR \nnanopillars, and spin-torque driven dom ain wall motion i n \nnanowires as was conside red in R ef. 5. The spi n-torque \nfunctio n  is taken t o have a \npredominant \"adiabatic\" c omponent , alon g with a \nsmall \"nona diabatic\" com pone nt   described \nphenomenolog ically  by the relation )ˆ( )ˆ( )ˆ(nad ad ST m NmNmN + =\n)ˆ(admN\n)ˆ(nadm N\nad nadˆNm N ×β−≡ , \nwith . In t he case of a narrow nanowire along the -\naxis, with m agnetization  and electron curre nt density  \n, the torque function  and associate d \nfield (see ( 2)) are descri bed by1<<β xˆ\n)(ˆxm\nx J ˆe eJ= )ˆ(STmN\n)ˆ(STmH5 \n \n)/ˆ /ˆ ˆ() 2/ ()/ˆ()2/ ()ˆ(\nSTad\ndxd dxd eM PJdxde PJ\ns ee\nm mm Hm mN\nβ+× −==\nhh   (8) \n \nwhere P is the  spin-polarization of t he electron curre nt.  \n       To check if  is conse rvative, one ca n \"discretize\"  \nthe spatial derivatives app earing in (8) in the form  STH\nx dxdi iixx∆ −→−+=2/)ˆ ˆ( /ˆ1 1m m m , whe re )(ˆ ˆi i xmm≡  \nand , not unlik e the com mon m icrom agnetics  \napproxim ation. For a c onservative H-field where \n, the set of  Cartesian tensorsi i x xx−≡∆+1\ni i Em H ∂∂∝ / 33×6 \nj i j iuv\nji E H mm mH ∂∂∂∝∂∂≡ /2/t\n will be sym metric, i.e., \nvu\nijuv\nji H Htt\n= , under sim ultaneo us reversal of s patial indices \n and vect or in dices ji, z yxvu or,, ,= . For the adiabatic \nterm in (6), it can be readily  shown that the  uv\njiHt\n are in \ngene ral asymmetric , i.e., always antisym metric in vect or \nindices (du e to cross pr oduct) , but asy mmetr ic in  spatial \nindices , being antisym metric he re only for  \nlocally uniform  magnetization . The  \nnonadiabatic term  yields an -inde pendent 1 ,±=iji\ni im m ˆ ˆ1=±\nmˆuv\njiHt\n that is  \nalways antisym metric, i.e., symmetric in ve ctor i ndices, but antisym metric in spatial indic es . The concl usion \nhere t hat  is in ge neral nonconservative a grees with \nthat f ound in Ref. 5, by way of a rathe r diffe rent argument.  1 ,±=iji\nSTH\n       Anot her well known example is a nanopillar stack wit h \nonly two fe rrom agnetic (FM ) layers, the \" refere nce\" layer \nhaving a m agnetization  rigidly fixe d in time, and  a \ndynamically  varia ble \"free \" lay er refˆm\n)(ˆ)( ˆfree t tm m= . As  \ndescri bed by  Sloncze wski,7 the (adiaba tic) spin-t orque \ndensity function a nd field is given by: )ˆ(STmH\n \n]ˆ )ˆ ˆ[() 4/ ()ˆ ˆ(]ˆ ˆˆ[) 4/ ()ˆ ˆ(\nref reffree refref free ref ad\nST\nm m mm m Hm mm m m N\nβ+××⋅−=×× ⋅−=\ntMe PJ gte PJ g\ns ee\nhh\n   (9) \n \nwhere  is the free layer thickness, and freet )ˆ ˆ(refm m⋅ g  is a \nfunctio n of order unity, the de tails of which are not relevant  \nto the present discu ssion. From  the the -tenso r, or by  \nsimple inspection, t he adiabatic  term  in (7) is \nmanifestly  nonc onservative . However, app roxim ating uvHt\nm m ˆ ˆref×\n)ˆ ˆ(refm m⋅ g ~ consta nt, the conse rvative nonadiabatic ter m \nresem bles a magnetic field d escribe d by the -gradient  of \nan Zeem an-like ene rgy function mˆ\nm m ˆ ˆref nad ⋅∝ E . The  \nremaining discussion will restrict attention to \nnonconservative contrib utions. \n \nIII. STATIO NARY SOLU TIONS OF G AND LL \n \n       With , stationary  (i.e, ST NC H H→ 0 /ˆ=dt dm ) \nsolutions  of G-equatio ns (4) satisfy  the c onditions  that 0ˆm\n \nST STST\n0 eff 0 0G\ndamp eff 0\nˆ ˆ 0 ˆ0 /ˆ ;0) ( ˆ\nHm Hm Hmm H H H m\n×−=×⇒≠×=∝ =+× dt d      (10) \n \nThe clear and physically intuitiv e interpreta tion of (10) is \nthat stationary  state  satisfies a condition of zero \nphysical tor que, 0ˆm\n0 ˆtot 0=×Hm , includin g bot h \nconservative ( ) an d nonconservative s pin-torque \n( ) fields.  Being visco us in nature, the G dam ping \ntorque  inde pendently vanishes..  effH\nSTH\n0 /ˆ ˆG\ndamp 0 ≡∝× dt dm Hm\n       Previous measurem ents6 of the angular depe ndence of \nspin-torque critical curre nts  in CPP-GMR \nnanopillar syste ms by this  author and colleagues  \ndemonstrated t he existence of such stationary states with \nnon-collinear )ˆ ˆ(refcritm m⋅eJ\n0 ˆ ˆ0 ref≠×m m  and crit0e eJ J<< . In t his \nsituation, it follows from (9) an d (10) that the stationa ry \nstate  satisfies 0ˆm 0 ˆ ˆeff 0 0 ST ≠×=×− Hm Hm . It is  \nnoted that the last result i mplies that  is not a (therm al) 0ˆmequilibrium  state which m inimizes the free energy , \ni.e., )ˆ(mE\n0 ) ˆ()ˆ ()ˆ/(ˆ/eff 0 0 ≠δ⋅×∝×δ⋅∂∂=δδ θ θ Hm m m m E E  \nfor arbitra ry .  θδ\n       In the present described circum stance of  stationary   \nwith , the LL equatio ns of (7) differ from G \nin a fundam ental respect. Setting  in (7) yield s  0ˆm\n0 ˆST 0≠×Hm\n0 /ˆ=dt dm\n \n) ˆ( ˆ ) ( ˆeff 0 0 eff 0 LL ST Hm m H H m ××α−=+×         (11) \n \nLike (10), (11) im plies  that 0 ˆeff 0≠×Hm  whe n \n. However, (11)  also  imply a static , nonzero \nphysical tor que , alon g with a static,  \nnonzero damping tor que  (see (6a) ) to \ncancel it out . In sim ple mechanical term s, the latte r \namounts to non-visco us \"static-frictio n\". It has n o anal ogue \nwith G in a ny circum stance, or with LL in conventional  \nsituations with  and  equilibrium \nfor which LL  dam ping wa s origi nally  develo ped as a  \nphenomenolog ical dam ping f orm. It furth er contra dicts th e \nviscous (o r -depe ndent) nature o f the damping \nmechanism s desc ribed by  physical (rathe r than  \nphenomenolog ical) base d theoretical m odels0 ˆST 0≠×Hm\n0 ˆtot 0≠×Hm\n0 ˆLL\ndamp 0 ≠×Hm\n0ST NC =↔H H ↔0ˆm\ndtd/ˆm\n,8,9. \n       The above arguments ignored therm al fluctuatio ns of \n. However, thermal fluctuations mˆ10 scale approxim ately a s \n, while ( 10) or (11) are scale-inva riant \nwith 2\neff 0 ) /( Hm⋅ kT\nH.  In t he simple CPP nanopillar exam ple of (9), one \ncan (conce ptually at least)  continually increase both eJ \nand a n applied  field  contri bution to  to scale up appHeffH\nST 0ˆHm×  and eff 0ˆHm×  while approxim ately keeping \na fixe d statio nary state  (satis fying 0ˆm 0 ˆ ˆref 0≠×mm  with \nfixed ).  However, unique to LL eq uatio ns (11) based \non (6a) is t he additional requi rement that the static dam ping \nmechanism  be able to produce an refˆm\neff dampˆLLHm H ×∝  \nwhich sim ilarly scales (without li mit). This author finds thi s \na physically unreasonable proposition.  \n \nIV. ENERGY A CCOUN TING  \n \nIf one ignores/forgets t he Lagrangian formulation3 of the \nGilbert e quatio ns (4), one may derive t he followin g energy \nrelationships, substitutin g the right side of (4) for evaluating \nvecto r products of form : dtd/ˆmH⋅\n \n)/ˆ ˆ( ) ˆ()/ˆ ˆ( ) ˆ(/ˆ )/ˆˆ/ /( /1\neff effeff effeff\nNCNC\ndtddtddtd dtd E dtdE Ms\nmm H Hm Hmm H Hm Hm H mm\n×⋅α−×⋅γ−=×⋅α−×⋅γ=⋅−≡⋅∂∂=\n   (12a) \n )/ˆ ˆ( ) ˆ(/ˆ / /1\nNC NCNC NC\neff dtddtd dt dWMs\nmm H Hm Hm H\n×⋅α+×⋅γ−=⋅≡   (12b) \n \n)/ˆ ˆ() () ˆ(/ˆ /ˆ\nNC efftot2\ndt ddt d dt d\nmm H HHm m m\n×⋅+γ=×γ−⋅=          (12c) \n \nSubtractin g (12b) from (12a), and usin g (12c) one  finds \n \ndt d M dt dWdt d M dt dW dtdE\nss\n/ˆ //ˆ / / / :G\nG\nNCG NC\ndamp2\nm Hm\n⋅ + =γα− =\n         (13) \n \nThe re sult o f (13) is essentially a state ment of energy \nconservation. Nam ely, that the rate of change of the internal \nfree e nergy (density) of the magnetic sy stem  is give by  the \nwork done on the system  by the (exte rnal) no nconservative \nforces/fields , minus the loss of energy (t o the lattice) \ndue t o dam ping. The G damping term  in ( 13) is ( not \nsurprisi ngly) t he sam e as expected from  (2). It is a strictly  \nlossy, negative-definite  contributio n to .  NCH\ndtdE/\n       Over a finite interval of motion from  time  to , the \nchange 1t2t\n)ˆ( )ˆ(1 2 m m E EE −=∆  is, from (12b ) and (13):  \n \n∫⋅γα− =∆2\n1G NCˆ)/ˆ / )ˆ( (t\ntsdtddt d dt MEmm m H       (14) \n \nSince  is nonc onservative, t he work NCHNCW∆  is pat h-\ndepe ndent, and so use of (14) requires indepe ndent \nknowledg e of the solution  of (4). Sin ce \n itself depe nds on ) (ˆ2 1 ttt≤≤m\n)(ˆtmGα, the  term's contribution  \nto (14) also can vary with . Regardless, NCH\nGα 0>∆E  can \nonly result in the case of a positive  amount of work \n done by .  ∫⋅ =∆2\n1NC NC )/ˆ (t\nts dtdt d M W m HNCH\n       Working out the results  analogous to (12 a,b) for the LL \nequatio ns of (6a) and (7), one finds \n \n) ˆ() ˆ( /ˆ/ˆ / / :LL\ntot eff dampdamp\nLLLL\nNC\nHm Hm m Hm H\n×⋅×αγ−=⋅⋅ + =\ndtddtd M dt dW dtdEs\n      (15) \n \nThe form of (15) is the sam e as the latter result in (13). \nHowever, unlike G, the LL damping term in (15) is not \nmanifestly negative-definite, except when 0NC=H .  \n       The results of (13)-(15) apply equally to situations \nwhere one inte grates over the spatial distribution of  \nto evaluate the total syste m free energy, rat her tha n (local ) \nfree e nergy density . Total time derivatives  may be \nreplace d by partial deri vative s  whe re appropriate. ),(ˆtrm\ndtd/\nt∂∂/      Dropping terms of order  (and sim plifying notation \n), (7) is easily transfor med to a Gilbert-like form : 2\nLLα\nα→αLL\n \n)ˆˆ( )] ˆ( [ˆˆ:LLNC totdtd\ndtd mm Hm Hmm×α+×α−×γ−=  (16) \n \nwhic h differs from G in  (4) by the term  ) ˆ(NCHm×α  \nwhich is first order in both  and . For the \"wire \nproblem\" described by  (8), the equation s of m otion bec ome αNCH\n \n)ˆ ˆ(ˆ ˆˆ ˆ:LL)ˆ ˆ(ˆ ˆˆ ˆ\neffeff :G\ndxdv\ndtd\ndxdv\ndtddxdv\ndtd\ndxdv\ndtd\nm mm Hmm mm mm Hmm m\nαβ+α+×α+×γ−=+αβ+×α+×γ−=+\n (17) \n \nwhere , and terms of or der eM PJ vs e2/γ=h βα are \ndropped for LL. A s noted previously,5,9,11 (17) permits \n\"translational\" solutions ) (ˆ)(ˆeq vtx x,t −=m m  whe n α=β  \n(G) or  (LL), with  the static , equilibrium \n(minimum E) solution of . Evaluatin g \n by takin g  from (8), and \n with , one finds \nthat 0=β )(ˆeqxm\n0) ˆ(eff eq =×H m\ndtd M dt dWs /ˆ /ST ST m H⋅ =STH\n) (ˆ /ˆeq vtx v dtd −′−→m m dqd q /ˆ )(ˆ m m≡′\n2\neq2) (ˆ)/ ( /ST vtx Mv dt dWs −′γβ= m . In transl ational \ncases where  is exactly  collinear  to , only \nthe nonadiabatic term  does work on t he -system.  dtd/ˆm dx d/ˆm\nmˆ\n       Interestingly, the energy interpretation of these \ntranslational solutions is very diffe rent fo r G or LL. For G, \nthe positive rate of work  when dt dW /ST α=β  exactly \nbalances t he negative damping l oss as  given in (13), the \nlatter alway s nonzero and scaling as . For LL by  \ncontrast, the wo rk done by  vanishes when 2v\nSTH 0=β , \nmatching t he damping l oss whic h, from (6a)  or (15), is \nalways zero since  regardle ss of 0) ˆ(eff eq =×H m v. If \n is a sharp domain wall, )(ˆeqxm ) (ˆ /ˆeq vtx v dtd −′−=m m  \nrepresents, from a spatially local  perspective at a fixed  point \nx, an abrupt, irreversi ble, non-equilibrium  reor ientation of \n at/near tim e  whe n the wall core passes by. The  \nprediction of LL/(6a) that t his magnetization re versal c ould \ntake place locally (at arbitrarily large v), with the com plete \nabsence of the spin-orbit couple d, ele ctron scatteri ng \nprocessesmˆ vxt /≈\n8 that lead to spin-latti ce dam ping/r elaxation in all \nother known circum stances (e. g., external field-driven \ndomain wall motion) is, in the vi ew of this author, a rather \ndubious, nonphysical aspect of (6a) when 0ST≠H . \n       Stiles et al.5 repo rt that micromagnetic com putation s \nusing G in the case  sho w (non-translational)  \ntime/distance l imited dom ain wall displacement, resulting  in a net positive  increase \n0=βE∆. They claim  that 1) \"spi n \ntrans fer to rques do not cha nge the ene rgy of the sy stem\", \nand that 2) \"Gilbert dam ping to rque is the only  torque th at \nchanges the e nergy\". Acce pting  as accurate, it is \nthis aut hor's view that t he elementary physics/ mathematics \nleading t o (13) and (14) demonstrably  prove t hat bot h of \nthese claim s must be incorrect  (err or in the first per haps \nleading t o the misinterpretation of the second). On a related \npoint, the res ults of (1 3) and (1 5) shows that excludi ng \nwork  or 0>∆E\ndt dW /ST STW∆ , only LL- damping may possibly  \nlead to a positive contri bution to  or dtdE/ E∆ when \n0ST≠H , in a pparent contradictio n to the claim  in R ef. 5 \nthat LL damping \"uniquely and  irre versibly  reduces \nmagnetic free energy whe n spin-transfer torque is prese nt\". \n \nACK NOWLEDGM ENTS \n \nThe aut hor would like to ackno wledge em ail discussions on \nthese or related topics with W. Sa slow and R. Duine, as \nwell as an  extende d series of friendly discussio ns with  \nMark Stiles. Obviously, the latter have not (as of yet) \nachieve d a mutually agree d viewpoint on thi s subj ect.  \n \nREFERENCES \n \n1 T. L. Gilber t, Armour Research Report, M ay 1956; IEEE Tran s. \nMagn., 40, 3343  (2004).  \n2 L. Landau  and E. Lifshit z, Phys. Z. Sow jet 8, 153 (1935). \n3 W. F. Brown, Micromagnetics  (Krieger , New Y ork 1978). \n4 H. Gol dstein, Classical Me chanics , (Addison  Wesley, Reading \nMassachusetts, 1 950). \n5M. D. Stiles, W. M. Saslow, M. J. Donahue,  and A . Zangwill, \narXiv:cond-m at/0702020. \n6 N. Smith, J. A. Katine, J. R. Childress, and M. J. Carey, IEEE \nTrans. M agn. 41, 2935 (2005) ; N. Sm ith, J Appl. Ph ys. 99, \n08Q703 (2006). \n7 J. C. Slonczewski, J.  Magn. Magn. Mater. 159, L1 (1996); J. \nMagn. Magn . Mater. 247, 324 (2 002) \n8 V. Kambersky, Can. J. Phys. 48, 2906 (1970) ; V. Kam bersky and \nC. E. Patton , Phys. Rev . B 11, 2668 (1975). \n9 R. Duine, A. S. Nunez, J.  Sinova, and A. H. MacDonald, \narXiv:cond-m at/0703414. \n10 N. Sm ith, J. A ppl. Ph ys. 90, 57 68 (2001). \n11 S. E Barnes and S . Maekaw a, Phys. R ev. Lett. 95, 10720 4 \n(2005). \n "    },    {        "title": "1507.06733v1.Effect_of_Landau_damping_on_alternative_ion_acoustic_solitary_waves_in_a_magnetized_plasma_consisting_of_warm_adiabatic_ions_and_non_thermal_electrons.pdf",        "content": "arXiv:1507.06733v1  [physics.plasm-ph]  24 Jul 2015Effect of Landau damping on alternative ion-acoustic solita ry\nwaves in a magnetized plasma consisting of warm adiabatic io ns\nand non-thermal electrons\nJayasree Das\nDepartment of Mathematics, Chittaranjan College,\n8A Beniatola Lane, Kolkata - 700 009, India.\nAnup Bandyopadhyay∗\nDepartment of Mathematics, Jadavpur University, Kolkata - 700 032, India.\nK. P. Das\nDepartment of Applied Mathematics, University of Calcutta ,\n92-Acharya Profulla Chandra Road, Kolkata - 700 009,India.\nAbstract\nBandyopadhyay and Das [Phys. Plasmas, 9, 465-473, 2002] have derived anonlinear macroscopic\nevolution equation for ion acoustic wave in a magnetized pla sma consisting of warm adiabatic ions\nand non-thermal electrons including the effect of Landau damp ing. In that paper they have also\nderived the corresponding nonlinear evolution equation wh en coefficient of the nonlinear term of\nthe above mentioned macroscopic evolution equation vanish es, the nonlinear behaviour of the ion\nacoustic wave is described by a modified macroscopic evoluti on equation. But they have not\nconsidered the case when the coefficient is very near to zero. T his is the case we consider in this\npaper and we derive the corresponding evolution equation in cluding the effect of Landau damping.\nFinally, a solitary wave solution of this macroscopic evolu tion is obtained, whoseamplitudeis found\nto decay slowly with time.\n∗abandyopadhyay1965@gmail.com\n1I. INTRODUCTION\nBandyopadhyay and Das [1] derived a macroscopic evolution equatio n to investigate the\nnonlinearbehaviour oftheionacousticwaves inamagnetizedplasma c onsisting ofwarmadi-\nabaticionsandnon-thermal electronsincluding theeffect ofLanda udamping. Thisequation\nis a Korteweg-de Vries-Zakharov-Kuznetsov (KdV-ZK) equation except for an extra term\nthat accounts for the effect of Landau damping. Bandyopadhyay and Das [1] reported that\nthis macroscopic evolution equation admits solitary wave solution pro pagating obliquely to\nthe external uniform static magnetic field and having a sech2-profile. But the amplitude of\nsolitary wave does not remain constant; it varies slowly with time τas (1+τ/τ′)−2, whereτ′\nis a constant depending on the initial amplitude of the solitary wave, t he angle between the\ndirection of propagation of the solitary wave and the external unif orm static magnetic field\nand the parameters involved in the system. This evolution equation, which we are discussing\nabout, losses its validity when the coefficient of the nonlinear term of the macroscopic evolu-\ntion equation vanishes and this vanishes along a particular curve in th eβσ-parametric plane\nas shown in Fig.1, where βis the nonthermal parameter associated with the nonthermal dis-\ntributionofelectronsand σistheratiooftheaveragetemperatureofionstothatofelectron s.\nIn this situation, in the same paper, they have derived a modified mac roscopic evolution\nequation when the coefficient of the nonlinear term of the macrosco pic evolution equation\nvanishes. This equation is a modified Korteweg-de Vries-Zakharov- Kuznetsov (MKdV-ZK)\nequation except for an extra term that accounts for the effect o f Landau damping. Bandy-\nopadhyay and Das [1] reported that this modified macroscopic evolu tion equation admits\nsolitary wave solution propagating obliquely to the external uniform static magnetic field\nand having a sech-profile. But the amplitude of solitary wave does not remain constan t; it\nvaries slowly with time τas (1 +τ/τ′)−1, whereτ′is a constant depending on the initial\namplitude of the solitary wave, the angle between the direction of pr opagation of the soli-\ntary wave and the external uniform static magnetic field and the pa rameters involved in the\nsystem. But again this modified macroscopic evolution equation is una ble to describe the\nnonlinear behaviour of the ion acoustic waves including the effect of L andau damping if the\ncoefficient of the nonlinear term of the macroscopic evolution equat ion approaches to zero,\nbut not exactly equal to zero. In such situation, i.e., when the coeffi cient of the nonlinear\nterm of the macroscopic evolution equation approaches to zero, b ut not exactly equal to\n2zero, Das et al.[2] derived a combined MKdV-KdV-ZK equation to describe the nonline ar\nbehaviour of the ion acoustic wave when the Landau damping effect h as not been taken\ninto account. Following Ott and Sudan [3], in the present paper, we d erive a macroscopic\nevolution equation to study the nonlinear behaviour of the ion acous tic waves including the\neffect of Landau damping. This equation is a combined MKdV-KdV-ZK e quation except for\nan extra term that accounts for the effect of Landau damping. Th e solitary wave solution\nof this further modified macroscopic evolution has been obtained. I t is found that due to\ninclusion of the effect of Landau damping the amplitude of the alterna tive solitary wave\nsolution of this equation is a slowly varying function of time.\nIn the investigations made by Das et al.[2], the Landau damping effect has not been\ntaken into account. In the present paper, we include this effect on the problem considered\nin Daset al.[2]. Starting from the same governing equations but replacing the ex pression\nfor the number density of non-thermal electrons by the Vlasov-B oltzmann equation for\nelectrons, an appropriate macroscopic evolution equation corres ponding to the combined\nmodified Korteweg-de Vries-Zakharov-Kuznetsov (combined MKd V-KdV-ZK) equation of\nDaset al.[2] is derived, which describes the long-time evolution of weakly nonline ar long\nwave-length ion acoustic waves in a magnetized plasma consisting of w arm adiabatic ions\nand non-thermal electrons including the effect of Landau damping.\nThe physics of nonlinear Landau damping is of interest for two major reasons. First, it is\na fundamental and distinctive plasma phenomenon that links collectiv e and single-particle\nbehaviour. Second, the derivation of reduced fluid models that inco rporate accurately such\nkinetic effects, is of great importance for plasma transport studie s. For instance, some\nauthors have proposed a k-dependent dissipation term, which correctly reproduces linear\nLandaudamping withintheframework offluidmodels [4]. However, th elongtimebehaviour\nof Landau damping is intrinsically nonlinear, and, in order to assess th e validity of the above\nmodels, it is important to understand whether the damping will contin ue indefinitely, or will\neventually be stopped by the nonlinearity.\nThe research works on solitary waves in plasmas have been done und er various physical\nconditions such as plasmas including multi-species ions [5], negative ions [6], and dust par-\nticles [7]. In many cases, the Korteweg-de Vries (KdV) equation is us ed to describe basic\ncharacters of the wave. Detailed properties of the solitary waves observed in experiments\nin plasmas are, however, slightly different from those predicted by t he equation. Using\n3Q-machine plasmas, Karpman et al.[8] have observed oscillations in the tail of solitary\nwaves, which are caused by resonant particles and have shown tha t the tail changes its\nshape depending on the strength of Landau damping.\nThis paper is an extension of the work of Bandyopadhyay and Das [1], where we derive a\nfurther modified macroscopic evolution equation which describe the non-linear behaviour of\nion-acoustic waves in fully ionized collisionless plasma consisting of warm adiabatic ions and\nnon-thermal electrons having vortex-like velocity distribution, imm ersed in a uniform static\nmagnetic field directed along z-axisincluding the effect of Landaudam ping. This equation is\ntrue only for the case when the coefficient of the nonlinear term of t he macroscopic evolution\nequation derived by Bandyopadhyay and Das [1] approaches to zer o and not exactly equal\nto zero. With the help of multiple time scale analysis of Ott and Sudan [3], we find a solitary\nwave solution of this equation. From the solution, we can conclude th at the amplitude of\nthe solitary wave slowly decreases with time.\nThis paper is organized as follows. The basic equation have been given in Section II.\nThe macroscopic evolution equations are given in Section III, in which the derivation of\nMKdV-KdV-ZK like macroscopic evolution equation is given in subsectio n IIIC. Solitary\nwave solutions of the combined MKdV-KdV-ZK like macroscopic evolut ion equation are\ninvestigated in Section IV. Finally, we have concluded our findings in Se ction V.\nII. BASIC EQUATION\nThe following are the governing equations describing the non-linear b ehaviour of ion-\nacoustic waves in fully ionized collisionless plasma consisting of warm adia batic ions and\nnon-thermal electrons having vortex-like velocity distribution, imm ersed in a uniform static\nmagnetic field directed along z-axis. Here it is assumed that the plasm a beta i.e., the ratio\nof particle pressure to the magnetic pressure is very small and the characteristic frequency\nis much smaller than ion cyclotron frequency (Cairns et al.[9], Mamun [10]).\n∂n\n∂t+∇.(nu) = 0, (1)\n∂u\n∂t+(u.∇)u=−∇ϕ+ωc(u׈z)−σ\nn∇p, (2)\n4∇2ϕ=ne−n, (3)\np=nγ. (4)\nwhere\nne=/integraldisplay∞\n−∞fdv/parallelshort, (5)\nand the velocity distribution function of electrons fmust satisfy the Vlasov- Boltzmann\nequation\n/radicalbiggme\nmi∂f\n∂t+v/parallelshort∂f\n∂z+∂ϕ\n∂z∂f\n∂v/parallelshort= 0. (6)\nHeren,ne,u,p,ϕ,(x,y,z) andtare respectively the ion number density, electron number\ndensity, ion fluid velocity, ion pressure, electrostatic potential, sp atial variables and time,\nand they have been normalized respectively by n0(unperturbed ion number density), n0,\ncs/parenleftBig\n=/radicalBig\nKBTe\nm/parenrightBig\n(ion-acoustic speed), n0KBTi,KBTe\ne,λD/parenleftBig\n=/radicalBig\nKBTe\n4πn0e2/parenrightBig\n(Debye length) and\nω−1\np(ion plasma period), where σ=Ti\nTe,ωcis the ion cyclotron frequency normalized by\nωp/parenleftBig\n=/radicalBig\n4πn0e2\nm/parenrightBig\nandγ/parenleftbig\n=5\n3/parenrightbig\nis the ratio of two specific heats. Here KBis the Boltzmann\nconstant; Te,Tiare respectively the electron and ion temperatures; mis the mass of an ion\nandeis the electronic charge v/parallelshortis the velocity of electrons in phase space normalized to\nve=/radicalBig\nKBTe\nme. In (4), the adiabatic law has been taken on the basis of the assump tion that\nthe effect of viscosity, thermal conductivity and the energy tran sfer due to collision can be\nneglected.\nSince the electrons are assumed to be nonthermally distributed, th e electron velocity\ndistribution function can be taken as (Cairns et al.[9])\nf0(v/parallelshort) =1√\n2π(1+3α1)(1+α1v4\n/parallelshort)e(−1\n2v2\n/parallelshort)(7)\nTo discuss the effect of Landau damping on ion-acoustic solitary wav es, we follow the\nmethod of Ott and Sudan [3] and following them, we replace/radicalBig\nme\nmiby/radicalBig\nme\nmiεwhereεis a\nsmall parameter. The equation (6) then assumes the following form\nα2ε∂f\n∂t+v/parallelshort∂f\n∂Z+∂ϕ\n∂Z∂f\n∂v/parallelshort= 0, (8)\nwhereα2=/radicalBig\nme\nmi.\n5Again using Eq. (4), the equation (2) becomes\n∂u\n∂t+(u.∇)u=−∇ϕ+ωc(u׈z)−σn1\n3∇n, (9)\nand the equation (3), which is the Poisson equation becomes\nε∇2\nξϕ+n−/integraldisplay∞\n−∞fdv/parallelshort= 0, (10)\nTherefore Eqs. (1), (9), (10) and (8) are our governing equatio ns.\nIII. EVOLUTION EQUATIONS\nA. Macroscopic evolution equation\nBefore deriving the nonlinear evolution equation for ion-acoustic wa ve in a magnetized\ncollisionless plasma consisting of warm adiabatic ions and non-thermal electrons including\nthe effect of Landau damping for a particular case not considered in the paper of Bandy-\nopadhyay and Das [1], we give below in short a summary of the results o btained in that\npaper. The macroscopic evolution equation obtained is the following:\n∂ϕ(1)\n∂τ+AB′ϕ(1)∂ϕ(1)\n∂ζ+1\n2A∂3ϕ(1)\n∂ζ3+1\n2AD∂\n∂ζ/parenleftBigg\n∂2ϕ(1)\n∂ξ2+∂2ϕ(1)\n∂η2/parenrightBigg\n+1\n2AEα2P/integraldisplay∞\n−∞∂ϕ(1)\n∂ζ′dζ′\nζ−ζ′= 0, (11)\nwhere\nA=1\nV/parenleftBigg\nV2−5σ\n3/parenrightBigg2\n, (12)\nB′=1\n2/bracketleftBigg/parenleftBigg\n3V2−5\n9σ/parenrightBigg\n/parenleftBigg\nV2−5\n3σ/parenrightBigg3−1/bracketrightBigg\n, (13)\nD= 1+V4\nω2\nc/parenleftBigg\nV2−5\n3σ/parenrightBigg−2\n, (14)\n6E=V\n4√\n2π(4−3β), (15)\nand the constant Vis given by\n(1−β)/parenleftBigg\nV2−5\n3σ/parenrightBigg\n= 1. (16)\nThe equation (11) is a KdV-ZK equation except for an extra term/parenleftbig\nlast term of the left hand\nside of (11)/parenrightbig\nthat accounts for the effect of Landau damping. The solitary wave solution of\nthe equation (11) has been obtained in that paper of Bandyopadhy ay and Das [1]. They\nhave found that the solitary wave solution of the equation (11) has the same sech2- profile\nas in the case of KdV-ZK equation. But, here the amplitude as well as the width of the\nsolitary wave varies slowly with time. In particular, the amplitude ( a) of the solitary wave\nsolution of the equation (11) is given by the following equation.\na=a0/parenleftbigg\n1+τ\nτ′/parenrightbigg−2\n, (17)\nwherea0is the value of aatτ= 0 and τ′is given by the following equation\nτ′=/bracketleftBigg\n1\n4cosδ/radicalBigg\nA2E2α2\n2B′a0\n6(cos2δ+Dsin2δ)P/integraldisplay∞\n−∞/integraldisplay∞\n−∞sech2Z∂(sech2Z′)\n∂Z′dZ′dZ\nZ−Z′/bracketrightBigg−1\n.(18)\nUsing (16), the expression for B′can be simplified as\nB′=20(1−β)3\n9(σ−σβ), (19)\nwhere\nσβ=9{1−3(1−β)2}\n40(1−β)3. (20)\nFrom this expression of B′, it is easy to see that the coefficient of the non-linear term\nin (11) vanishes along a particular curve Fig.1 in the βσplane, and consequently, it is not\npossibletodiscussthenonlinearbehaviourofionacousticwaveinclud ingtheeffectofLandau\ndamping with the help of Eq. (11). In this situation, i.e., when B′= 0, Bandyopadhyay\nand Das [1] have also derived a modified macroscopic evolution equatio n.\n7B. Modified macroscopic evolution equation\nFor this case, i.e., when B′= 0, giving appropriate stretching of space coordinates and\ntime, and appropriate perturbation expansions of the dependent variables Bandyopadhyay\nand Das [1] in the same paper have derived the following modified macro scopic evolution\nequation for ion acoustic waves in a fully ionized collisionless plasma cons isting of warm ions\nand non-thermal electrons immersed in a uniform static magnetic fie ld directed along the\nz-axis:\n∂ϕ(1)\n∂τ+AB′′[ϕ(1)]2∂ϕ(1)\n∂ζ+1\n2A∂3ϕ(1)\n∂ζ3+1\n2AD∂\n∂ζ/parenleftBigg\n∂2ϕ(1)\n∂ξ2+∂2ϕ(1)\n∂η2/parenrightBigg\n+1\n2AEα2P/integraldisplay∞\n−∞∂ϕ(1)\n∂ζ′dζ′\nζ−ζ′= 0. (21)\nHereA,DandEare same as given by the equations (12), (14) and (15) respective ly and\nB′′is given by the following equation:\nB′′=1\n2/bracketleftbigg/parenleftbigg\nV2−5\n3σ/parenrightbigg−4/braceleftbigg10\n27σ−6V2+3\n2/parenleftbigg\nV2−5\n3σ/parenrightbigg−1/parenleftbigg\n3V2−5\n9σ/parenrightbigg/bracerightbigg\n−1\n2(1+3β)/bracketrightbigg\n, (22)\nand the constant Vis determined from equation (16).\nThe equation (21) is a MKdV-ZK equation except for an extra term/parenleftbig\nlast term of the\nleft hand side of (21)/parenrightbig\nthat accounts for the effect of Landau damping. The solitary wave\nsolution of the equation (21) has been investigated by Bandyopadh yay and Das [1] in the\nsame paper. They have found that the solitary wave solution of the equation (21) has the\nsamesech-profile as in the case of MKdV-ZK equation. But, here the amplitude as well as\nthe width of the solitary wave varies slowly with time. In particular, th e amplitude ( a) of\nthe solitary wave solution of the equation (21) is given by the following equation.\na=a0/parenleftbigg\n1+τ\nτ′/parenrightbigg−1\n, (23)\nwherea0is the value of aatτ= 0 and τ′is given by the following equation\nτ′=/bracketleftBigg\n1\n2cosδ/radicalBigg\nA2E2α2\n2a2\n0B′′\n3(cos2δ+Dsin2δ)P/integraldisplay∞\n−∞/integraldisplay∞\n−∞sechZ∂(sechZ′)\n∂Z′dZ′dZ\nZ−Z′/bracketrightBigg−1\n.(24)\nBut both the evolution equations (11) and (21) are unable to descr ibe the nonlinear\nbehaviour of the ion acoustic wave along with the effect of Landau da mping in the neigh-\nbourhood of the curve σ=σβin theβσ- parametric plane along which B′= 0 (Fig.1).\n8This is the situation we are considering here. We have derive in this cas e, a further modified\nmacroscopic evolution equation which describes the nonlinear behav iour of the ion acoustic\nwave in the neighbourhood of the curve σ=σβin theβσ- parametric plane along which\nB′= 0.\nC. Further Modified Macroscopic evolution equation\nTo discuss the nonlinear behaviour of the ion acoustic wave in the neig hbourhood of the\ncurve in the βσparametric plane along which B′= 0, we assume B′≈ /ci∇cleco√y∇t(ǫ1\n2) (Nejoh [11]),\nand we take the following stretching of space coordinates and time.\nξ=ε1\n2x,η=ε1\n2y,ζ=ε1\n2(z−Vt),τ=ε3\n2t, (25)\nwhereεis a small parameter measuring the weakness of the dispersion and Vis a constant.\nWiththe stretching given by (25), theequations (1), (9), (10) an d (8) respectively assume\nthe following form:\n−ε1\n2V∂n\n∂ζ+ε3\n2∂n\n∂τ+ε1\n2∇ξ.(nu) = 0, (26)\n−ε1\n2V∂u\n∂ζ+ε3\n2∂u\n∂τ+ε1\n2(u.∇ξ)u=−ε1\n2∇ξϕ+ωc(u×/hatwidez)−5\n3σε1\n2n−1\n3∇ξn,(27)\nε∇2\nξϕ+n−/integraldisplay∞\n−∞fdv/parallelshort= 0, (28)\n−Vα2ε3\n2∂f\n∂ζ+α1ε5\n2∂f\n∂τ+ε1\n2v/parallelshort∂f\n∂ζ+ε1\n2∂ϕ\n∂ζ∂f\n∂v/parallelshort= 0. (29)\nHere\n∇ξ=/hatwideex∂\n∂ξ+/hatwideey∂\n∂η+/hatwideez∂\n∂ζ,u= (u,v,w). (30)\nNext we use the following perturbation expansions of the dependen t variables to make a\nbalance between nonlinear and dispersive terms.\n(n,ϕ,w,f ) = (1,1,0,0,f0)+/summationtext∞\ni=1εi\n2(n(i),ϕ(i),w(i),f(i)),\n(u,v) =/summationtext∞\ni=1εi+1\n2(u(i),v(i)).\n\n(31)\n9Substituting (31) into the equations (26)-(29) and then equating coefficient of different\npowers of εon both sides, we get a sequence of equations. From the lowest ord er equations\nobtained from (26)-(28), which are at the order ǫ, we get the following equations.\nn(1)=/parenleftbigg\nV2−5\n3σ/parenrightbigg−1\nϕ(1),\nw(1)=V/parenleftbigg\nV2−5\n3σ/parenrightbigg−1\nϕ(1),\nu(1)=−V2\nωc/parenleftbigg\nV2−5\n3σ/parenrightbigg−1\n∂ϕ(1)\n∂η,\nv(1)=V2\nωc/parenleftbigg\nV2−5\n3σ/parenrightbigg−1\n∂ϕ(1)\n∂ξ,\nn(1)=/integraltext∞\n−∞f(1)dv/parallelshort.\n\n(32)\nFromtheVlasovequation(29)atthelowest order, i.e., attheorder ǫ1/2, wegetthefollowing\nequation\nv/parallelshort∂f(1)\n∂ζ+∂f0\n∂v/parallelshort∂ϕ(1)\n∂ζ= 0. (33)\nAs this equation does not have a unique solutions, we include an extra higher order term\nε3α2∂f(1)\n∂τ=εε2α2∂f(1)\n∂τoriginated from the Vlasov equation at the order ε2. Let us write the\nequation (33) as follows.\nε2α2∂f(1)\nε\n∂τ+v/parallelshort∂f(1)\nε\n∂ζ+∂f0\n∂v/parallelshort∂ϕ(1)\n∂ζ= 0. (34)\nThenf(1)can be obtained as unique solution of this equation by imposing the nat ural\nrelation of the form\nf(1)= lim\nε→0f(1)\nε. (35)\nAssuming τdependence of f(1)\nεandϕ(1)to be of the form exp(iωτ), the equation (35) can\nbe written as\niωε2α2f(1)\nε+v/parallelshort∂f(1)\nε\n∂ζ+∂f0\n∂v/parallelshort∂ϕ(1)\n∂ζ= 0. (36)\nNow taking Fourier transform of this equation with respect to the v ariableζaccording to\nthe definition,\nˆg= ˆg(k) =1√\n2π/integraldisplay∞\n−∞g(ζ)exp(−ikζ)dζ, (37)\n10we get\niωε2α2ˆf(1)\nε+ikv/parallelshortˆf(1)\nε+ik∂f0\n∂v/parallelshortˆϕ(1)= 0. (38)\nThis equation gives the following expression for ˆf(1)\nε:\nˆf(1)\nε=−k∂f0\n∂v/parallelshort\nkv/parallelshort+α2ωε2ˆϕ(1). (39)\nNow whenever the factor 1 /(kv/parallelshort+α2ωε2) comes under an integration over v/parallelshortalong the real\naxis, the general prescription is to replace this integration accord ing to Landau, along a\ncontour in the complex v/parallelshort-plane known as Landau contour. This is equivalent to replacing\nthe factor 1 /(kv/parallelshort+α2ωε2) by the following\n1\nkv/parallelshort+α2ωε2=P1\nkv/parallelshort+α2ωε2+iπδ(kv/parallelshort+α2ωε2). (40)\nSubstituting this relation into the equation (39) and then proceedin g to the limit ε−→0+,\nwe get according to (35) the following expression for ˆf(1):\nˆf(1)=−2∂f0\n∂v2\n/parallelshortˆϕ(1)−2πi∂f0\n∂v2\n/parallelshortkv/parallelshortδ(kv/parallelshort). (41)\nDue to the relation xδ(x) = 0, this equation assumes the following form:\nˆf(1)=−2∂f0\n∂v2\n/parallelshortˆϕ(1). (42)\nTaking Fourier inverse transform of (42), we get\nf(1)=−2∂f0\n∂v2\n/parallelshortϕ(1). (43)\nSubstituting (43) in the last equation of (32) and then performing t he integration we get\nn(1)= (1−β)ϕ(1)withβ=4α1\n1+3α1. (44)\nThis equation along with the first equation of (32) gives the following d ispersion relation to\ndetermine the constant V\n(1−β)/parenleftbigg\nV2−5\n3σ/parenrightbigg\n= 1. (45)\nThis equation is same as the equation (16) as well as the equation (20 ) of Das et. al. [2].\nIn the next order, i.e., at the order ε3/2, solving the ion continuity equation and the parallel\ncomponent (i.e., the component parallel to the ambient magnetic field , i.e., z-component or\n11ζ-component) of ion fluid equation of motion for n(2)andw(2)to express them in terms of\nϕ(1)andϕ(2), we get the following equations:\nn(2)=1/parenleftbigg\nV2−5\n3σ/parenrightbiggϕ(2)+1\n2/parenleftbigg\n3V2−5\n9σ/parenrightbigg\n/parenleftbigg\nV2−5\n3σ/parenrightbigg3[ϕ(1)]2,\nw(2)=V/parenleftbigg\nV2−5\n3σ/parenrightbiggϕ(2)+1\n2V/parenleftbigg\nV2+25\n9σ/parenrightbigg\n/parenleftbigg\nV2−5\n3σ/parenrightbigg3[ϕ(1)]2.\n\n(46)\nFrom the perpendicular component (i.e., the component perpendicu lar to the ambient mag-\nnetic field, i.e., the components along x-axis and y-axis) of the ion fluid equation of motion\nat the order ε3/2, we get the following equation:\n∂u(2)\n∂ξ+∂v(2)\n∂η=V3\nω2\nc/parenleftbigg\nV2−5\n3σ/parenrightbigg−1∂\n∂ζ/parenleftbigg∂2ϕ(1)\n∂ξ2+∂2ϕ(1)\n∂η2/parenrightbigg\n. (47)\nFrom the Poisson equation at the order ε, we get\nn(2)−/integraldisplay∞\n−∞f(2)dv/parallelshort= 0. (48)\nTo findf(2), we again consider the Vlasov equation at the order ε3/2. The Vlasov equation\nat the order ε3/2is the following, in which as mentioned in the lowest order Vlasov equatio n,\nan extra time derivative term ε7/2α2∂f(2)\n∂τ=ε3/2ε2α2∂f(2)\n∂τhas been included and f(2)has been\nreplaced by f(2)\nε.\nε2α2∂f(2)\nε\n∂τ+v/parallelshort∂f(2)\nε\n∂ζ+∂f0\n∂v/parallelshort∂ϕ(2)\n∂ζ= 2v/parallelshort∂2f0\n∂(v2\n/parallelshort)2∂\n∂ζ[ϕ(1)]2. (49)\nThenf(2)can be obtained as unique solution of this equation by imposing the nat ural\nrelation of the form\nf(2)= lim\nε→0+f(2)\nε. (50)\nAssuming τdependence of f(2)\nεandϕ(2)to be of the form exp(iωτ), taking Fourier transform\nof this equation with respect to the variable ζ, using the causality condition (40) and finally\nproceeding to the limit ε→0+, we get according to (50) the following expression for ˆf(2):\nˆf(2)=−2∂f0\n∂v2\n/parallelshortˆϕ(2)+2∂2f0\n∂(v2\n/parallelshort)2ˆd3, (51)\n12where\nˆd3= [ˆϕ(1)]2, (52)\nand we have used the relations xδ(x) = 0 and xP(1\nx) = 1 to simplify the equation (51).\nTaking Fourier inverse transform of (51), we get\nf(2)=−2∂f0\n∂v2\n/parallelshortϕ(2)+2∂2f0\n∂(v2\n/parallelshort)2[ϕ(1)]2. (53)\nSubstituting for f(2)andn(2)given respectively by (53) and the first equation of (46) into\nthe equation (48), we get the following equation after simplification\n−/parenleftbigg\nV2−5\n3σ/parenrightbigg−1/bracketleftbigg\n(1−β)/parenleftbigg\nV2−5\n3σ/parenrightbigg\n−1/bracketrightbigg\nϕ(2)+B′[ϕ(1)]2= 0. (54)\nNow the first term of the left hand side of the equation (54) is identic ally equal to zero due\nto the dispersion relation as given by the equation (45) and as B′≈O(ε1/2), the second\nterm of the left hand side of the equation (54) along with its sign has t o be included in the\nnext higher order Poisson equation, i.e., this term along with its sign mu st be included in\nthe left hand side Poisson equation of order ε3/2and consequently the equation (54), i.e., the\nPoisson equation of order ε2/2(=ε) is identically satisfied. So, including the term + B′[ϕ(1)]2\nin the Poisson equation of order ε3/2, we can write the Poisson equation at the order ε3/2as\nfollows\n∂2ϕ(1)\n∂ξ2+∂2ϕ(1)\n∂η2+∂2ϕ(1)\n∂ζ2+n(3)−/integraldisplay∞\n−∞f(3)dv/parallelshort+B′[ϕ(1)]2= 0. (55)\nNow, in the next order, i.e., at the order ε2, solving the ion continuity equation and the\nparallel component of ion fluid equation of motion for the variable∂n(3)\n∂ζto express it in terms\nofϕ(3),ϕ(2),ϕ(1), we get the following equation.\n∂n(3)\n∂ζ=/parenleftbigg\nV2−5\n3σ/parenrightbigg−1∂ϕ(3)\n∂ζ+2V/parenleftbigg\nV2−5\n3σ/parenrightbigg−2∂ϕ(1)\n∂τ\n+V4\nω2c/parenleftbigg\nV2−5\n3σ/parenrightbigg2∂\n∂ζ/bracketleftbigg∂2ϕ(1)\n∂ξ2+∂2ϕ(1)\n∂η2/bracketrightbigg\n+/parenleftbigg\nV2−5\n3σ/parenrightbigg−4/bracketleftBigg\n10\n27σ−6V2+3\n2/parenleftbigg\n3V2−5\n9σ/parenrightbigg\n/parenleftbigg\nV2−5\n3σ/parenrightbigg/bracketrightBigg\n[ϕ(1)]2∂ϕ(1)\n∂ζ\n+/parenleftbigg\n3V2−5\n9σ/parenrightbigg\n/parenleftbigg\nV2−5\n3σ/parenrightbigg3∂\n∂ζ[ϕ(1)ϕ(2)], (56)\n13where we have used equations (32), (46) and (47) to get this equa tion in this present form.\nNow our task is to find f(3)that determines n(3)from the Poisson equation (55) at the\norderε3/2. To find f(3)we consider theVlasov equationat the order ε2. The Vlasov equation\nat the order ε2is the following, in which as in the lowest order case an extra higher ord er\ntermε4α2∂f(3)\n∂τhas been included and f(3)has been replaced by f(3)\nεand where we have\nsubstituted the expressions for f(1)andf(2)given by equations (43) and (53) respectively.\nε2α2∂f(3)\nε\n∂τ+v/parallelshort∂f(3)\nε\n∂ζ+∂f0\n∂v/parallelshort∂ϕ(3)\n∂ζ=−2Vα2∂f0\n∂v2\n/parallelshortx2+4v/parallelshort∂2f0\n∂(v2\n/parallelshort)2y2−4v/parallelshort∂3f0\n∂(v2\n/parallelshort)3z2,(57)\nwhere\nx2=∂ϕ(1)\n∂ζ,\ny2=∂\n∂ζ[ϕ(1)ϕ(2)],\nz2= [ϕ(1)]2∂ϕ(1)\n∂ζ.\n\n(58)\nTherefore f(3)is obtained from the unique solution of the equation (57) by the relat ion\nf(3)= lim\nε→0+f(3)\nε. (59)\nAs in the earlier cases, assuming τdependence of f(3)\nεandϕ(3)to be of the form exp(iωτ),\ntaking Fourier transform of this equation with respect to the varia bleζ, using the causality\ncondition (40) and finally proceeding to the limit ε→0+, we get according to (50) the\nfollowing equation determining ˆf(3):\nik/bracketleftbigg\nˆf(3)+2∂f0\n∂v2\n/parallelshortˆϕ(3)/bracketrightbigg\n=−2Vα2∂f0\n∂v2\n/parallelshort/bracketleftbigg\nkP/parenleftbigg1\nkv/parallelshort/parenrightbigg\n+iπsgn(k)δ(v/parallelshort)/bracketrightbigg\nˆx2\n+4∂2f0\n∂(v2\n/parallelshort)2ˆy2−4∂3f0\n∂(v2\n/parallelshort)3ˆz2. (60)\nIntegrating (60) over the entire range of v/parallelshort, we get the following equation.\nik[ˆn(3)\ne−(1−β)ˆϕ(3)] =−1\n4iVα2(4−3β)/radicalbiggπ\n2sgn(k)ˆx2+ ˆy2+1\n2(1+3β)ˆz2,(61)\nwhere we set\nn(3)\ne=/integraldisplay∞\n−∞f(3)dv/parallelshort. (62)\n14Taking inverse Fourier transform of the above equation, we get\n∂n(3)\ne\n∂ζ= (1−β)∂ϕ(3)\n∂ζ+∂\n∂ζ(ϕ(1)ϕ(2))+1\n2(1+3β)[ϕ(1)]2∂ϕ(1)\n∂ζ\n−V\n4√\n2πα2(4−3β)P/integraldisplay∞\n−∞∂ϕ(1)\n∂ζ′dζ′\nζ−ζ′, (63)\nin which the convolution theorem has been used to find the inverse Fo urier transform of\nsgn(k)ˆx2. Now using the equations (62) and (63), we get the following equatio n.\n∂\n∂ζ/bracketleftbigg/integraldisplay∞\n−∞f(3)dv/parallelshort/bracketrightbigg\n= (1−β)∂ϕ(3)\n∂ζ+∂\n∂ζ(ϕ(1)ϕ(2))+1\n2(1+3β)[ϕ(1)]2∂ϕ(1)\n∂ζ\n−1\n4√\n2πVα2(4−3β)P/integraldisplay∞\n−∞∂ϕ(1)\n∂ζ′dζ′\nζ−ζ′, (64)\nSubstituting (64) into the equation obtained by differentiating the P oisson equation (55) at\nthe order ε3/2with respect to ζ, we get the following equation\n∂n(3)\n∂ζ−(1−β)∂ϕ(3)\n∂ζ+2B′ϕ(1)∂ϕ(1)\n∂ζ−1\n2(1+3β)[ϕ(1)]2∂ϕ(1)\n∂ζ\n+∂\n∂ζ/bracketleftbigg∂2ϕ(1)\n∂ξ2+∂2ϕ(1)\n∂η2+∂2ϕ(1)\n∂ζ2/bracketrightbigg\n+1\n4√\n2πVα2(4−3β)P/integraldisplay∞\n−∞∂ϕ(1)\n∂ζ′dζ′\nζ−ζ′\n−∂\n∂ζ(ϕ(1)ϕ(2)) = 0. (65)\nNow substituting for∂n(3)\n∂ζgiven by (56) into the equation (65), we get the following\nfurther modified macroscopic evolution equation, where the term + 2B′∂\n∂ζ(ϕ(1)ϕ(2)) being of\nhigher order since B′=/ci∇cleco√y∇t(ǫ1/2) has been omitted.\n∂ϕ(1)\n∂τ+AB′ϕ(1)∂ϕ(1)\n∂ζ+AB′′[ϕ(1)]2∂ϕ(1)\n∂ζ+1\n2A∂3ϕ(1)\n∂ζ3\n+1\n2AD∂\n∂ζ/parenleftBigg\n∂2ϕ(1)\n∂ξ2+∂2ϕ(1)\n∂η2/parenrightBigg\n+1\n2AEα2P/integraldisplay∞\n−∞∂ϕ(1)\n∂ζ′dζ′\nζ−ζ′= 0. (66)\nHereA,B′,DandB′′are respectively given by the equations (12)-(15) and (22) and th e\nconstant Vis determined by the equation (16). The equation (66) is a combined M KdV-\nKdV-ZK equation except for an extra term/parenleftbig\nlast term of the left hand side of (66)/parenrightbig\nthat\naccounts for the effect of Landau damping. In the next section, w e find the solitary wave\nsolution of this further modified macroscopic evolution equation.\n15IV. SOLITARY WAVE SOLUTION OF THE FURTHER MODIFIED MACRO-\nSCOPIC EQUATION\nIf we neglect the electron to ion mass ratio, i.e., if we set α2= 0, the equation (66)\nreduce to acombined MKdV-KdV-ZKequation. The solitarywave solu tionofthis combined\nMKdV-KdV-ZK equation has been studied in Das et al.[2]. In this paper, our aim is to\nfind the solitary wave solution of the equation (66).\nThe solitary wave solution of the equation (66) with α2= 0 propagating at an angle\nδwith the external uniform static magnetic field is the following, which h as already been\nobtained in section IV of Das et al. [2] ,\nϕ(1)=ϕ0(Z) =aS\nΨ. (67)\nwhere\nS=sech[2pZ], (68)\nΨ =S+λ√\nM, (69)\nλ=±1, (70)\na=12p2(cos2δ+Dsin2δ)\nB′, (71)\nM= 1+12p2B′′(cos2δ+Dsin2δ)\nB′2, (72)\nZ=ξsinδ+ζcosδ−Uτ. (73)\nFortheexistenceofthesolitarywavesolution(67), itisnecessary thatthefollowingcondition\nis satisfied.\nL=MB′2=B′2+12B′′p2(cos2δ+Dsin2δ)>0. (74)\nIf the condition (74) holds good, Uis given by the equation\nU= 4p2a3, (75)\n16where\na3=1\n2Acosδ(cos2δ+Dsin2δ). (76)\nWith the help of the equations (71), (72), (75) and (76), we get th e following expressions of\np,UandMto express them in terms of a.\np=1\n2/radicalbiggaa1\n6a3, (77)\nU=1\n6aa1, (78)\nM= 1+aa2\na1, (79)\nwhere\na1=AB′cosδ, (80)\na2=AB′′cosδ. (81)\nUsing (77), we can write the Eq.(67) as\nϕ(1)=ϕ0(Z)\n=asech/bracketleftbigg/radicalBig\naa1\n6a3(ξsinδ+ζcosδ−1\n6aa1τ)/bracketrightbigg\nsech/bracketleftbigg/radicalBig\naa1\n6a3(ξsinδ+ζcosδ−1\n6aa1τ)/bracketrightbigg\n+λ/radicalBig\n1+aa2\na1. (82)\nAssuming that ato be a slowly varying function of time, following Ott and Sudan [3], we\nintroduced the following space coordinate in a frame moving with the s olitary wave.\nZ=/radicalbiggaa1\n6a3/parenleftBigg\nξsinδ+ζcosδ−1\n6a1/integraldisplayτ\n0adτ/parenrightBigg\n. (83)\nIt is important to note that if ais a constant, then Z= 2pZand consequently,\nϕ(1)=ϕ0(Z)\n=asechZ\nsechZ+λ√\nM\n=asech/bracketleftbigg/radicalBig\naa1\n6a3(ξsinδ+ζcosδ−1\n6a1/integraltextτ\n0adτ)/bracketrightbigg\nsech/bracketleftbigg/radicalBig\naa1\n6a3(ξsinδ+ζcosδ−1\n6a1/integraltextτ\n0adτ)/bracketrightbigg\n+λ/radicalBig\n1+aa2\na1. (84)\n17is the solitary wave solution of the combined MKdV-KdV-ZK equation p ropagating at an\nangleδto the external uniform static magnetic field. Now dropping “overlin e” onZ, we can\nwrite the equation (84) as\nϕ(1)=ϕ0(Z)\n=asechZ\nsechZ+λ√\nM\n=asech/bracketleftbigg/radicalBig\naa1\n6a3(ξsinδ+ζcosδ−1\n6a1/integraltextτ\n0adτ)/bracketrightbigg\nsech/bracketleftbigg/radicalBig\naa1\n6a3(ξsinδ+ζcosδ−1\n6a1/integraltextτ\n0adτ)/bracketrightbigg\n+λ/radicalBig\n1+aa2\na1, (85)\nwhereZis given by the following equation:\nZ=/radicalbiggaa1\n6a3/parenleftBigg\nξsinδ+ζcosδ−1\n6a1/integraldisplayτ\n0adτ/parenrightBigg\n. (86)\nNow our aim is to find the condition for which ϕ(1)given by the equation (85) is a solitary\nwave solution of the further modified macroscopic equation (66).\nWith the change of variable defined by the equation (86) and assumin g thatϕ(1)is a\nfunction of Z,τonly, Eq.(66) can be written as\n∂ϕ(1)\n∂τ+/parenleftbigg\n−1\n3a1pa+Z\n2a∂a\n∂τ/parenrightbigg∂ϕ(1)\n∂Z+2pa1ϕ(1)∂ϕ(1)\n∂Z+2pa2(ϕ(1))2∂ϕ(1)\n∂Z\n+8p3a3∂3ϕ(1)\n∂Z3+AEα2pcosδP/integraldisplay∞\n−∞∂ϕ(1)\n∂Z∂Z′\nZ−Z′= 0. (87)\nTo investigate the solution of Eq. (87), we follow Ott and Sudan [3] an d generalizing the\nmultiple-time scale analysis with respect to α2, by setting\nϕ(1)(Z,τ) =q(0)+α2q(1)+α2\n2q(2)+α3\n2q(3)+......... (88)\nwhere each q(j)(j= 0,1,2,3,....) are the function of τ=τ0,τ1,τ2...... Hereτjis given by\nτj=αj\n2τ,j= 0,1,2,3,........ (89)\nSubstituting (88) into (87) and then equating the coefficient of diffe rent power of α2on each\nside of Eq. (87), we get a sequence of equations. The zeroth and t he first order equation of\nthis sequence are respectively, given by the following equations.\nρ/bracketleftbigg∂\n∂τ+Z\n2a∂a\n∂τ∂\n∂Z/bracketrightbigg\nq(0)+L∂\n∂Zq(0)= 0, (90)\n18ρ/bracketleftbigg∂\n∂τ+Z\n2a∂a\n∂τ∂\n∂Z/bracketrightbigg\nq(1)+∂\n∂ZLq(1)=ρMq(0), (91)\nwhere\nL=∂2\n∂Z2+6\naq(0)+6(M−1)\na2[q(0)]2−1, (92)\nρ= 6/radicalBigg\n6a3\na3\n1a−3\n2, (93)\nMq(0)=−/bracketleftbigg∂q(0)\n∂τ1+Z\n2a∂a\n∂τ1∂q(0)\n∂Z+AEpcosδP/integraldisplay∞\n−∞∂q(0)\n∂Z′∂Z′\nZ−Z′/bracketrightbigg\n. (94)\nNow it can be easily verified that q(0)=asechZ\nsechZ+λ√\nMis the soliton solution of the zeroth\norder equation if\n∂a\n∂τ= 0, (95)\nwhich implies that ais independent of time, i.e., at the lowest order, the solitary wave solu -\ntion of the further modified macroscopic evolution equation is same a s that of the combined\nMKdV-KdV-ZK equation.\nUsing (95), Eq.(91) can be written as\nρ∂q(1)\n∂τ+∂\n∂ZLq(1)=ρMq(0). (96)\nNow forthe existence of a solutionof the equation (96), itsright ha ndmust beperpendicular\nto the kernel of the operator adjoint to the operator∂\n∂ZL; this kernel, which must tend to\nzero as|Z| → ∞issechZ\nsechZ+λ√\nM. Thus we get the following consistency condition for the\nexistence of a solution of the equation (96).\n/integraldisplay∞\n−∞sechZ\nsechZ+λ√\nMMq(0)dZ= 0. (97)\nFromequation(97), wegetthefollowingdifferential equationforth esolitarywave amplitude\na.\n∂a\n∂τ1+AEa3/2(B′+aB′′)cosδ/radicalbig\n3B′(cos2δ+Dsin2δ)P/integraldisplay∞\n−∞/integraldisplay∞\n−∞Ψ(Z)∂\n∂Z′[Ψ(Z′)]dZ′dZ\nZ−Z′= 0,(98)\nwhere\nΨ(Z) =sechZ\nsechZ+λ√\nM. (99)\n19Using the relation τ1=α2τ, the equation (98) can be written in the following simplified\nform:\n∂a\n∂τ+AEα2a3/2(B′+aB′′)cosδ/radicalbig\n3B′(cos2δ+Dsin2δ)P/integraldisplay∞\n−∞/integraldisplay∞\n−∞Ψ(Z)∂[Ψ(Z′)]\n∂Z′dZ′dZ\nZ−Z′= 0.(100)\nHere it is important to note that M(= 1+aa2\na1) appearing in Ψ( Z) is a function of a. So,\nit is not possible to find the exact analytical dependence of aonτ. But we can solve the\nabove equation by using the Taylor series expansion for the terms o f the form1\nsechx+λ√\nMin\npowers of a. Keeping terms upto the order a5/2, we get the following differential equation\nforafrom equation (100).\n∂a\n∂τ+AEα2a3\n2/radicalBigg\nB′\n3(cos2δ+Dsin2δ)cosδ γ1\n−AEα2a5\n2λ\n2B′′\n/radicalbig\n3B′(cos2δ+Dsin2δ)cosδ(γ2+γ3)\n+AEα2a5\n2B′′\n/radicalbig\n3B′(cos2δ+Dsin2δ)cosδ γ1= 0, (101)\nwhereγ1,γ2,γ3are given by the following integrals.\nγ1=P/integraltext∞\n−∞/integraltext∞\n−∞Φ1(Z)∂\n∂Z′[Φ1(Z′)]dZ′dZ\nZ−Z′,\nγ2=P/integraltext∞\n−∞/integraltext∞\n−∞Φ1(Z)∂\n∂Z′[Φ2(Z′)]dZ′dZ\nZ−Z′,\nγ3=P/integraltext∞\n−∞/integraltext∞\n−∞Φ2(Z)∂\n∂Z′[Φ1(Z′)]dZ′dZ\nZ−Z′.\n\n(102)\nΦ1(Z) and Φ 2(Z) appearing in the above are given by\nΦ1(Z) =sechZ\nsechZ+λ, (103)\nΦ2(Z) =sechZ\n(sechZ+λ)2. (104)\nNow solving the above differential equation (101) for aby the use of the initial condition,\na=a0whenτ= 0, we get the following equation for a:\nµtan−1/bracketleftBigg\nµ(√a−√a0)\n1+µ2√aa0/bracketrightBigg\n−√a−√a0√aa0=τ\nτ′,\n(105)\nwhere\nτ′=/bracketleftbigg1\n2AEα2/radicalBigg\nB′\n3(cos2δ+Dsin2δ)cosδ γ1/bracketrightbigg−1\n, (106)\n20µ=/radicalBigg\nB′′\nB′/bracketleftbigg\n1−λ\n2(γ2+γ3)\nγ1/bracketrightbigg\n. (107)\nFrom equation (105), we see that ais implicitly depends on τand consequently, from this\nequation it is not possible to predict the nature (decreasing or incre asing) of dependence of\naonτ. But plotting aagainstτfor the appropriate set of values of the parameters involved\nin the system, we find that ais slowly varying function of time. By the phrase “ appropriate\nset of values of the parameters”, we mean that those values of th e parameters of the system\nfor which the condition for existence of alternative solitary wave so lution of the combined\nMKdV-KdV-ZK equation holds good, i.e., for those values of the para meters of the system\nfor which L >0. Taking a0= 0.5 (arbitrary) and the values of the parameters as mentioned\nin the figure 2, we plot aagainstτin Fig.2. This figure clearly shows that the amplitude\n(a) decays slowly with time ( τ) and consequently, the amplitude of the alternative solitary\nwave solution of the combined MKdV-KdV-ZK equation is a slowly varyin g function of time\nwhen the effect of Landau damping is considered.\nV. CONCLUSIONS\nA macroscopic evolution equation corresponding to the combined MK dV-KdV-ZK equa-\ntion has been derived to include the effect of Landau damping. This ma croscopic evolution\nequation admits the same alternative solitary wave solution of the co mbined MKdV-KdV-\nZK equation except the fact that the amplitude of the solitary wave solution of the combined\nMKdV-KdV-ZKlikemacroscopic equation isaslowly varying functionof time. The multiple\ntime scale method of Ott and Sudan [3] has been generalized here to s olve the said evolution\nequation. In small amplitude limit, we have observed the following resu lt.\nResult:: Due to inclusion of the effect of Landau damping, the amplitude of the alternative\nsolitary wave solution having profile different from sech2orsechof the macroscopic\nevolution equation decays slowly with time.\n[1] A. Bandyopadhyay and K. P. Das, Phys. Plasmas 9, 465 (2002).\n[2] J. Das, A. Bandyopadhyay, and K. P. Das, Phys. Plasmas 14, 092304 (2007).\n21[3] E. Ott and R. N. Sudan, Phys. Fluids 12, 2388 (1969).\n[4] G. W. Hammett and F. W. Perkins, Phys. Rev. Lett. 64, 3019 (1990).\n[5] J. F. McKenzie, F. Verheest, T. B. Doyle, andM. A. Hellber g, Phys. Plasmas 11, 1762 (2004).\n[6] T. S. Gill, H. Kaur, and N. S. Saini, Phys. Plasmas 10, 3927 (2003).\n[7] J.-K. Xue, Phys. Rev. E 69, 016403 (2004).\n[8] V. I. Karpman, J. P. Lynov, P. Michelsen, H. L. Pcseli, and J. J. Rasmussen, Phys. Fluids\n23, 1782 (1980).\n[9] R. A. Cairns, A. A. Mamun, R. Bingham, and P. K. Shukla, Phy sica Scripta T63, 80 (1995).\n[10] A. A. Mamun and R. A. Cairns, J. Plasma Phys. 56, 175 (2000a).\n[11] Y. Nejoh, Phys. Plasmas 5, 2830 (1992).\n220 0.2 0.4 0.6−0.4−0.200.20.40.60.81\n  \n0<σβ<1\nβ=0.552324\nβ=0.42265\nβσβ\nFIG. 1. Variation of σβagainstβ\n230 500 1000 1500 2000 2500 3000 3500 40000   0.1 0.2 0.3 0.4 0.5 \nτaωc=0.2,σ=0.0001, β=0.41,δ=0,λ=1\nFIG. 2. Variation of aagainstτ\n24"    },    {        "title": "1210.7796v1.Magnet_traveling_through_a_conducting_pipe__a_variation_on_the_analytical_approach.pdf",        "content": "arXiv:1210.7796v1  [physics.class-ph]  27 Oct 2012Magnet traveling through a conducting pipe:\na variation on the analytical approach\nBenjamin Irvine1, Matthew Kemnetz2Asim Gangopadhyaya3, and Thomas Ruubel4\nDepartment of Physics, Loyola University Chicago, Chicago , Illinois 60626\nAbstract\nWe present an analytical study of magnetic damping. In parti cular, we investigate the dynamics of a\ncylindrical neodymium magnet as it moves through a conducti ng tube. Owing to the very high degree of\nuniformity of the magnetization for neodymium magnets, we a re able to provide completely analytical\nresults for the EMF generated in the pipe, and the consequent retarding force. Our analytical expressions\nare shown to have excellent agreement with experimental obs ervations.\nPACS: 41.20.Gz; 75.50.Ww; 75.50.Dd\nKeywords: Faraday’s Law, Electromagnetic Damping, Regenerativ e Braking\nI Introduction\nMagnetic braking plays a significant role in industry. It is used to slow d own the moving parts of systems\nwithout losing energy to friction. In addition, the absence of frictio nal forces and direct physical contact\nbetween moving parts helps these parts last longer. Thus, an impro ved understanding of magnetic damping\nis important to the development of future technology in regenerat ive braking. In industry, complex compu-\ntational models are often used to simulate realistic scenarios of mag netic braking. We have developed a fully\ntheoretical model for a cylindrically symmetric system, which can be used to benchmark these computational\nmodels.\nWe present here an analysis of a common demonstration that compr ises a cylindrical magnet and a\nnon-ferromagnetic conducting tube in relative motion to each othe r. [1–18,20–22]. Owing to the interaction\nbetween the moving magnet and the induced current in the pipe, the magnet falls very slowly through the\ntube, always generating a sense of amazement in students and tea chers alike. This area has been explored\nby many researchers [1,2,4,5,8–18,20–22].\n1e-mail: birvine@luc.edu\n2e-mail: mkemnetz@luc.edu\n3e-mail: agangop@luc.edu\n4e-mail: truubel@luc.edu\n1In this paper we study the motion of a cylindrical neodymium magnet t hrough a copper pipe of circular\ncross-section. The azimuthal symmetry of the problem keeps the mathematics tractable and allows us to\ngenerate an analytical expression for the EMF generated in an arb itrary segment of the tube, and the\nresulting retarding force.\nOur paper is organizedas follows. In Sec. II, we will describe the exp erimental setup used for this demon-\nstration. In Sec. III, we develop our model assuming the near-un iformity of magnetization of neodymium\nmagnets, and then show that the resulting prediction of the magne tic field strength has excellent agreement\nwith the measured values of the field on the axis of the magnet. We als o compare the experimental results\nwith the often used point dipole approximation. In Sec. IV, from the model constructed in the previous\nsection, we compute the flux through circular loops of the conduct ing pipe and generate an expression for\nthe current in a section of pipe of arbitrary length. As a special cas e, in Sec. V, we also compute the current\ngeneratedin the forwardhalfofthe pipe (oralternativelyin the wak eofthe magnet). In Sec. VI, we compute\nthe force on the magnet due to the interaction between the magne t and the pipe. Our analytical results\nmatch extremely well with experimental observations. In the next section, we describe our experimental\nsetup.\nII Experimental Setup\nAs shown in Fig. (1a), we used hanging masses, mandM, to pull a cylindrical neodymium magnet through\na copper pipe with varying terminal velocities. We used smart pulleys f rom PASCO to record the position,\nvelocity, and acceleration of the magnet as it traveled into, throug h, and out of the pipe. Fig. (1b) shows\nthat for a significant segment of each individual trajectory of the magnet, the velocity remains constant.\nMm\n(a) Experimental setup of copper pipe\nand neodymium magnet attached to pul-\nley system.0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.8\nPosition/LParen1m/RParen1Velocity/LParen1m/Slash1s/RParen1\n(b) Velocity of magnet as it enters and\ntravels through the conducting pipe.\nFigure 1\n2We also find that the dependence of the resistive force on the term inal velocity can be accurately modeled\nby a linear relation. As we show in Sec. VI, this linear behavior is replicat ed by our theoretical analysis\nas well. Researchers have studied the damped oscillatory motion of a magnet in a conducting tube [22].\nHowever, in this work, we have limited ourselves to an analytical stud y of the emf and the retarding force\nfor a magnet moving with different terminal velocities.\nIII Magnetic Field due to a Neodymium Magnet\nIn orderto quantitativelyexpressthe magnetic field, we need to de velop an appropriatemodel ofourmagnet.\nSeveral authors have considered the magnet to be a pure dipole [5 ,15–17,22]. This model works well for\nsmall magnets moving through wide pipes. Some have also considered a physical dipole constructed of two\npoint monopoles separated by an appropriate distance [4]. This too w ould be a good approximation when\nthe radius of the magnet is much smaller than the diameter of the pipe , and the monopoles are well inside\nthe magnet; i.e., not too close to the surface. Our aim is to keep the a nalysis general and accessible to\nundergraduate students. In particular, we specifically include the case where the dimension of the magnet\nis comparable to the diameter of the pipe and generates strong bra king. For such cases, as we will show in\nFig. (3), the dipole model does not accurately fit the data.\nNeodymium magnets have a very uniform magnetization. This uniform ity allows us to simulate the\n/vectorB-field of the cylindrical magnet by two circular disks with uniform magn etic surface charge densities, σm\nand−σm, whereσmis proportional to the magnetization density Moof the magnet [19]. The method of\ndetermining the /vectorH-field is then identical to the case of finding the electric field due to tw o uniformly charged\nparallel disks of surface charge densities, σeand−σe. In [4], the authors had recognized that applicability of\nthe two-disk model for this case, however, they later chose to ap proximate it by a physical dipole consisting\nof two monopoles.\nA Magnetism in a polarizable medium\nThe magnetic field due to a current density /vectorJis given by\nAmpere′s Law :− →∇ ×/vectorB=µo/vectorJ . (1)\n/vectorJincludes the “free-currents” /vectorJfand the bound current density /vectorJb=− →∇×/vectorM, where/vectorMis the magnetization\ndensity (magnetic moment per unit volume). Thus, in the presence o f magnetization, we have\n− →∇ ×/vectorB=µo/parenleftBig\n/vectorJf+/vectorJb/parenrightBig\n=µo/parenleftBig\n/vectorJf+− →∇ ×/vectorM/parenrightBig\n. (2)\nFor a permanent magnet; i.e., /vectorJf= 0, eq. (2) yields:\n− →∇ ×/parenleftBig\n/vectorB−µo/vectorM/parenrightBig\n=− →∇ ×µo/vectorH= 0. (3)\n3Where we have defined the conservative field /vectorHsuch that /vectorB=µo/parenleftBig\n/vectorM+/vectorH/parenrightBig\n.Since− →∇ ·/vectorB= 0, we have\n− →∇ ·/vectorH=−− →∇ ·/vectorM . (4)\nComparing this equation with Gauss’ law− →∇ ·/vectorE=ρe\nǫo, we see that the /vectorH-field is generated by the source\nρm≡ −− →∇·/vectorMexactly in the same way as the electrostatic field /vectorEis found from the electrical charge density\nρe.\nB Magnetic Scalar Potential due to a magnet with uniform dens ityMoˆez\nSince/vectorHis a conservative field, we can write it as a gradient of a scalar field. I.e .,/vectorH=−− →∇Ψm.From Eq.\n(4), we have\n− →∇2Ψm=−ρm=− →∇ ·/vectorM . (5)\nθ\nRmr\nz−σ m σm\nFigure 2: Two disks of uniform magnetic charge density ±σmare atz=±L\n2respectively.\nFor a cylindrical magnet with uniform magnetization density Moˆez, the− →∇·/vectorMis zero at all points inside\nthe magnet, and receives non-zero contributions only at the two c ircular end surfaces. Hence, the /vectorH-field\ngenerated by the cylindrical magnet is the same as that of two disks of uniform magnetic surface charge\ndensities σmand−σmseparated by a distance L, whereσm=Mo. This expression for the /vectorH-field would be\nvalid both inside and outside the magnet. The /vectorB-field is then simply given by µo/vectorHoutside the magnet and\nµo/parenleftBig\n/vectorH+/vectorM/parenrightBig\ninside.\nThe/vectorH-field on the axis of the magnet can be readily derived by superimposit ion of scalar potentials due\nto a single disk of uniform magnetic surface charge density σm:\nΨm(z) =σm\n2/parenleftBig\nz−/radicalbig\nR2m+z2/parenrightBig\n=Mo\n2/parenleftBig\nz−/radicalbig\nR2m+z2/parenrightBig\n. (6)\n4The scalar potential due to the cylindrical magnet is then given by5\nΨ2−Disks\nm =Mo\n2\n\n/parenleftbigg\nz−L\n2/parenrightbigg\n−/radicalBigg\nR2m+/parenleftbigg\nz−L\n2/parenrightbigg2\n−\n/parenleftbigg\nz+L\n2/parenrightbigg\n−/radicalBigg\nR2m+/parenleftbigg\nz+L\n2/parenrightbigg2\n\n.(7)\n0.00 0.05 0.10 0.150.00.10.20.30.40.50.6\nPositionfromEdgeof Magnet /LParen1Meters/RParen1MagneticField/LParen1Tesla/RParen1\nExperimentalDataTwoDisk ModelDipoleModel\nFigure 3: Axial magnetic field for the Dipole model (dashed), Two-Dis k model (solid line) and the experi-\nmental data\nIn Fig. (3), we show a plot of the experimentally determined magnetic field against the values obtained\nfrom Eq. (7). For comparison, we also plot the field due to a pure dipo le with the net dipole moment\nequal to the dipole moment of the magnet/parenleftbig\nπR2\nmMo/parenrightbig\n. As is evident from Fig. (3), our experimental data\nis in excellent agreement with the predictions of the two-disk model, a nd hence verifies our assumption\nregarding the uniformity of the neodymium magnets. Henceforth, our theoretical analysis will assume that\nthe magnetization is uniform.\nC Computation of the Magnetic Field due to the Cylindrical\nNeodymium Magnet\nTo compute the off-axis /vectorB-field, we will start with the axial field given in Eq. (6). Except for poin ts on one\nof the circular end surfaces of the magnet, the magnetic scalar po tential Ψ msatisfies− →∇2Ψm=−ρm= 0.\n5The expression derived in Eq. (7) assumes that the origin is s et at the center of the magnet.\n5Hence, the general solution for Ψ mdue to one disk in spherical coordinates6is\nΨm(r,θ) =∞/summationdisplay\nℓ=0/parenleftbigg\naℓrℓ+bℓ\nrℓ+1/parenrightbigg\nPℓ(cosθ) (8)\nAs we will later see, for the calculation of flux, we will only need to work in the region r > Rm7, hence all\naℓ= 0, and the scalar potential is reduced to\nΨm(r,θ) =∞/summationdisplay\nℓ=0bℓ\nrℓ+1Pℓ(cosθ). (9)\nIn order to determine the values for constants bℓin Eq. (9), we note that the expression for Ψ m(r,0) must\nequal Ψ m(z) of Eq. (6) when zis replaced by rcos0o=r; i.e.,\n∞/summationdisplay\nℓ=0/parenleftbiggbℓ\nrℓ+1/parenrightbigg\n=σm\n2/bracketleftBig/parenleftbig\nR2\nm+r2/parenrightbig1\n2−r/bracketrightBig\n, (10)\nwhere we have used Pℓ(1) = 1 for all ℓ. By comparing the powers of ron both sides, we find that all b2ℓ+1\nare zero, and the even coefficients b2ℓare given by\nb2ℓ=/bracketleftbiggσmR2ℓ+2\nm\n2(ℓ+1)!/bracketrightbiggℓ/productdisplay\nk=0/parenleftbigg1\n2−k/parenrightbigg\n. (11)\nThus the magnetic scalar potential Ψ m(r,θ) is given by\nΨm(r,θ) =∞/summationdisplay\nℓ=0/bracketleftbiggσmR2ℓ+2\nm\n2(ℓ+1)!/bracketrightbigg/producttextℓ\nk=0/parenleftbig1\n2−k/parenrightbig\nr2ℓ+1P2ℓ(cosθ). (12)\nIn terms of Ψ m(r,θ), we can find the magnetic field, /vectorBoutside of the magnet by\n/vectorB=−µo− →∇Ψm, (13)\nand for inside the magnet, we will need to add an additional term:\n/vectorB=−µo/parenleftBig− →∇Ψm−/vectorM/parenrightBig\n. (14)\nThus, we have an exact expression for the magnetic field. The sum c an be computed to any desired level\nof accuracy by including a sufficiently large number of terms. In Ref. [14], Partovi et al. had carried out\na very comprehensive analysis for a uniformly magnetized cylinder as well. However, they considered the\nvector potential due to the moving magnet. Similarly, the authors o f [20] computed the magnetic field and\nthe flux due to a cylindrical magnet and reduced it to the computatio n of elliptical integrals that could be\ndone using Mathematica . We find that, due to the similarity with electrostatics, the scalar po tential method\nis much more accessible to undergraduate students. In addition, b y choosing to keep an appropriate number\nof terms in the expansion given in Eq. (12), students can compute t he scalar potential to any desired level\nof accuracy.\nIn the next section, we will use the expression of Eq. (12) to evalua te flux through a cross-section of the\npipe, a distance zfrom the face of the magnet.\n6For this azimuthally symmetric problem, we have set the orig in of the coordinates at the center of the disk, and z-axis\ncoincides with the axis of the magnet.\n7Rmis the radius of the magnetic disk; i.e., the same as the radiu s of the magnet.\n6IV Computation of Flux\nAs the magnet travels through the copper pipe, the changing magn etic flux causes eddy currents to form in\nthe pipe. We will assume that the pipe thickness is small compared to t he radius of the pipe. The authors\nof Refs. [14,15,17] have studied the effect of thickness more car efully. We also assume that the magnet\nfalls coaxially through the conducting pipe, and thus an azimuthal sy mmetry is maintained throughout the\nmotion. In this case, the eddy currents generated in the pipe would form perfect circles perpendicular to\nthe axis of symmetry. We will now carry out surface integrations of the magnetic field given by Eqs. (13)\nor (14) to determine the flux through a circular cross-section of t he pipe. However, instead of computing\nthe flux on a planar surface through the circle, we choose a spheric al surface that contains the circle, and is\ncentered at the center of the front-disk of the magnet. The flux Φm(z) through a circular loop at a distance\nFigure 4: Pipe diagram\nzfrom the front-disk is then given by\nΦm(z) =/integraldisplay\nS/vectorB·ˆrda=−µo/integraldisplay\nS∂Ψm(r,θ)\n∂rda\n=∞/summationdisplay\nℓ=0b2ℓ/parenleftBigg\n∂\n∂r1\nr2ℓ+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=√\nR2\np+z2/parenrightBigg/integraldisplayθp\n0P2ℓ(cosθ) sinθdθdφ\n= 2πµoN/summationdisplay\nℓ=0ℓ/productdisplay\nk=0/parenleftbigg1\n2−k/parenrightbigg/bracketleftbiggσmR2ℓ+2\n2(ℓ+1)!/bracketrightbigg(2ℓ+1)\n(R2p+z2)ℓ/integraldisplay1\nupP2ℓ(u)d(u), (15)\nwhere we havesubstituted u= cosθ,up=z√\nR2p+z2,b2ℓ=/bracketleftBig\nσmR2ℓ+2\nm\n2(ℓ+1)!/bracketrightBig/producttextℓ\nk=0/parenleftbig1\n2−k/parenrightbig\n, and haveused b2ℓ+1= 0\n7for allℓ. We can compute this integral using the identity P2ℓ(u) =1\n4ℓ+1/parenleftBig\ndP2ℓ+1\ndu−dP2ℓ−1\ndu/parenrightBig\nand get\nΦm(z) =2πµoσm\n2/radicalBig\nR2p+z2/bracketleftBig/radicalBig\nR2p+z2−z/bracketrightBig\n+2πµoN/summationdisplay\nℓ=1(2ℓ+1)b2ℓ\n(R2p+z2)ℓ/bracketleftbigg1\n4ℓ+1/bracketrightbigg\n×\nP2ℓ−1\nz/radicalBig\nR2p+z2\n−P2ℓ+1\nz/radicalBig\nR2p+z2\n\n. (16)\nPlease note that the above expression for Φ m(z) gives the flux due to one disk, measured from the center\nof that disk. To compute the flux due to the magnet, we need to con sider two disks with magnetic charge\ndensities σmand−σmseparated by a distance L. The net flux is then given by the summation of the\ncontributions from two disks situated at two planar faces of the ma gnet. In Figs. (5a,5b,5c), we have plotted\nthe contributions of the /vectorH-,/vectorM- and/vectorB-fields toward flux Φ m(z) through a circular cross-section of the pipe\nsituated at a distance zfrom the center of the pipe.. As expected, a superposition of Figs. (5a,5b) generates\nthe Fig. (5c).\n/Minus0.10 /Minus0.05 0.00 0.05 0.10/Minus0.0003/Minus0.0002/Minus0.00010.00000.00010.00020.0003\nPositionfromcenterof magnet /LParen1m/RParen1MagneticFlux/LParen1Wb/RParen1\n(a) Contribution of the /vectorH-field toward flux\nΦm(z). This part of the flux has a disconti-\nnuity across each face of the magnet./Minus0.10 /Minus0.05 0.00 0.05 0.100.00000.00010.00020.00030.00040.00050.0006\nPositionfromcenterof magnet /LParen1m/RParen1MagneticFlux/LParen1Wb/RParen1\n(b) Contribution of the magnetization /vectorM.\n/Minus0.10 /Minus0.05 0.00 0.05 0.100.00000.00010.00020.00030.00040.00050.0006\nPositionfromcenterof magnet /LParen1m/RParen1MagneticFlux/LParen1Wb/RParen1\n(c) Magnetic flux Φ m(z) given by/integraltext\nz/vectorB·d/vector a.\nFigure 5: Contributions of the various fields toward flux Φ m(z) through a cross-section of the pipe at a\ndistance zfrom the center of the magnet.\n8V Computation of EMF\nAssuming the magnet to be moving with a constant velocity voˆz, in this Sec., we will determine the time\nvariation of the flux Φ mthrough a loop as the magnet comes towards it, and then passes th rough it.\nIn order to compute the emf through a circular cross-section of t he conducting pipe at a distance zfrom\nthe center of the magnet, we need to determine the change in flux t hrough the loop during a time interval\n∆t. During this time interval the distance of the loop from the magnet c hanges from ( z−∆z) toz. Hence,\nthe change in flux ∆Φ mseen by a loop is: Φ m(z−∆z)−Φm(z) =−∆z/bracketleftBig\n∂Φm(z)\n∂z/bracketrightBig\n. Since, this change happens\nduring the time ∆ tin which the magnet moves a distance ∆ z=vo∆t, the emf is given by\nE=/contintegraldisplay\n/vectorE·d/vectorℓ=−∆Φm\n∆t=−−∆z∂Φm(z)\n∂z\n∆t=vo·∂Φm(z)\n∂z. (17)\nThe electric field in the wall of the pipe is then given by Eφ=vo\n2πRp·∂Φm(z)\n∂z, and hence the current density\nin the pipe will be given by Jφ=σcEφ=voσc\n2πRp·∂Φm(z)\n∂z. Here,Rpdenotes the average radius of the pipe.\nThe current dIthrough a section of the pipe of thickness δand length dzwill be given by\ndI=Jφδdz=voσcδ\n2πRp·∂Φm(z)\n∂zdz . (18)\nHence, the total current through a section of the pipe from z1toz2is then given by\nI=voσcδ\n2πRp(Φm(z2)−Φm(z1)). (19)\n(a) Setup for measuring current.\n (b) Plot of experimentally observed current and predicted\ncurrent (solid line).\nFigure 6: Experimental Setup\n9In order to verify the above expression for the current I, we took a small cylindrical slice from the middle\nof the pipe. We then cut a vertical slit down the spine of the above slic e and replaced it between two\nlonger segments of the pipe, as shown in Fig. (6a). We then wired the slice to an ammeter that recorded\nthe current generated as a function of time8. Fig. (6b) shows the current generated in a loop as the\nmagnet passes through it. The solid line, in the background of the ex perimental data points collected by the\nMyDAQ, represents the current predicted by our model. Please no te that while the general behavior of the\nsolid line is given by Eq. (19), the constants needed for the graph9were obtained by stipulating that two\npoints of the graph, namely the maximum and the minimum, matched wit h the corresponding points of the\nexperimentally obtained data set.\nIn a long pipe, the total current in the part of the pipe that the mag net is yet to travel through, is given\nby\nI=voσcδ\n2πRp[Φm(∞)−Φm(0)]. (20)\nIn the next section, we will use Eq. (18) to compute the energy loss through a circular section of pipe of\nthickness dz, and from it the energy lost through an arbitrary segment of the p ipe.\nVI Computation of Retarding Force\nSince the magnet travels with a constant velocity, conservation of energy stipulates that the thermal loss in\nthe conducting pipe per unit time will be equal to /vector v·/vectorF. Thus, if we know the power loss, we will be able\nto determine the force from the power loss. To compute the power loss in the pipe, we first determine the\ndifferential loss over an infinitesimal length ∆ zof the pipe. This loss will be given by\ndP= (dI)2(dR) = (Jφδ∆z)2×Resistance of length dz\n=/parenleftbiggvoσcδ\n2πRp·∂Φm(z)\n∂z∆z/parenrightbigg2\n·2πRp\nσcδ∆z,\n=v2\noσcδ\n2πRp·/parenleftbigg∂Φm(z)\n∂z/parenrightbigg2\n·∆z . (21)\nHence the total power loss is given by\nP=v2\noσcδ\n2πRp·/integraldisplay∞\n−∞/parenleftbiggdΦm(z)\ndz/parenrightbigg2\n·dz= 2v2\noσcδ\n2πRp·/integraldisplay∞\n0/parenleftbiggdΦm(z)\ndz/parenrightbigg2\n·dz . (22)\nThe retarding force Fcan then be derived using P=/vectorF·/vector v=voF. Thus, the force Fis given by\nF= 2voσcδ\n2πRp·/integraldisplay∞\n0/parenleftbiggdΦm(z)\ndz/parenrightbigg2\n·dz . (23)\n8We actually used the MyDAQ device made by National Instrumen ts to observe the generated current.\n9The horizontal and vertical ranges of the graph were determi ned by requiring that the crest and the trough of the theoreti cal\ngraph match with the experimental data.\n100.00 0.05 0.10 0.150.00.51.01.52.02.53.0\nVelocity/LParen1m/Slash1s/RParen1ResistiveForce/LParen1N/RParen1\nExperimentalDataTwo Disk Model\nFigure 7: Experimentally oberved and theoretically computed (solid lin e) values of the resistive force for\nvarious terminal velocities.\nThus, we find that the resistive force is proportional tovoσcδ\nRp. In particular, if all other parameters are\nkept constant, we find F∝vo. Fig. (7) clearly exhibits this behavior in both experimental data as w ell as\nthe theoretical model. It is important to point out that authors of Ref. [14] have shown that for speeds of\nless than 25 m/s, the linear-relation between the speed and the res istive force is an excellent model.\nForcomputation,wechosetousetheInternationalAnnealedCop perStandard(IACS)valueof5 .8X107S/m\nforσcin our model because we were not certain of the specific alloy our cop per pipe was made from. Rec-\nognizing that many commercially available copper pipes, like the one we u sed, have a conductivity closer to\n90% of the IACS, could explain why our predicted resistive force is slig htly higher than what we observed\nexperimentally.\nVII Conclusion\nWe studied the effect of a cylindrical neodymium magnet moving along t he axis of a cylindrical conducting\npipe. Using the symmetry of the setup and the excellent uniformity o f the magnetization density of a\nneodymium magnet, we were able to develop an analytical model for t he induced surface current density\nand resulting retarding force. The analytically predicted current d istribution and the retarding force show\nexcellent agreement with experimental observation. Since we used the scalar method that bears a close\nresemblance to electrostatics, our analysis is comparatively more a ccessible to undergraduates. In addition,\nstudents can compute the flux to a desired level of accuracy by ke eping a sufficiently large number of terms\nin the expansion of the scalar potential.\nForindustrial applications, sophisticatedcomputationalmodels ar eusedto understandthe eddycurrents,\nand the resulting magnetic braking. This analytical model could be us ed to verify the computational models.\n11VIII Acknowledgment\nTwo of the authors (BI and MK) would like to thank LoyolaUniversity C hicago for the Mulcahy scholarship,\nwhichhelped maketheirundergraduateresearchpossible. AG would liketothankthe CenterforExperiential\nLearning at Loyola University Chicago for an Engaged Learning Facu lty Fellowship that provided partial\nsupport for his research. We would also like to thank Mr. Christophe r Kabat for his help in designing the\nexperimental setups.\nReferences\n[1] H. D. Wiederick, N. Gauthier, D. A. Campbell, and P. Rochon, Magn etic braking: Simple theory and\nexperiment, Am. J. Phys. 55, 500-503, (1987).\n[2] M. A. Heald, Magnetic braking: Improved theory, Am. J. Phys. 56, 521-522, (1988).\n[3] Jhuules A. M. Clack and Terrence P. Toepker, Magnetic Inductio n Experiment, Phys. Teach. 28, 236\n(1990)\n[4] Y. Levin, F. L. da Silveira, and F.B. Rizzato, Electromagnetic Brak ing: a Simple Qualitatitive Model,\nAm. J. Phys. 74, 815 (2006).\n[5] J.´I˜ niguez and V. Raposo, Measurement of conductivity in metals: a s imple laboratory experiment on\ninduced currents, Eur. J. Phys. 28(2007).\n[6] J.´I˜ niguez and V. Raposo, Comment on ‘Magnetic Braking’: activities fo r Undergraduate Laboratory,\nEur. J. Phys. 30, L19-L21 (2007).\n[7] G. Ireson and J. Twidle, Magnetic braking revisited: activities for the undergraduate laboratory, Eur. J.\nPhys.29, (2008).\n[8] M. Marcuso, R. Gass, D. Jones, and C. Rowlett, Magnetic drag in the quasistatic limit: A computational\nmethod, Am. J. of Phys. 59, 1118-1123, (1991).\n[9] M. Marcuso, R. Gass, D. Jones, and C. Rowlett, Magnetic drag in the quasi-static limit: Experimental\ndata and analysis, Am. J. Phys. 59, 1123-1129, (1991).\n[10] C. MacLatchy, P. Backman, and L. Bogan, A quantitative magn etic braking experiment, Am. J. Phys.\n61, 12 (1993).\n[11] L. McCarthy, On the electromagnetically damped mechanical ha rmonic oscillator, Am. J. Phys. 64,\n885-891 (1996).\n[12] J. M. Aguirregabiria, A. Hernandez and M. Rivas, Magnetic Brak ing Revisited, Am. J. Phys. 65,\n851-856, (1997).\n12[13] J´I˜ niguez et al., Study of the conductivity of a metallic tube by analyzin g the damped fall of a magnet,\nEur. J. Phys. 25, 593 (2004).\n[14] M. Hossein Partovi and E. Morris, Electrodynamics of a magnet moving through a conducting pipe,\nCan. J. Phys. 84, 253-274, (2006)\n[15] B. Knyazev et al., Braking of a magnetic dipole moving with an arbitr ary velocity through a conducting\npipe, Physics-Uspekhi. 49, 8 (2006).\n[16] Jae-Sung Bae, Jai-Hyuk Hwang, Jung-Sam Park and Dong-Gi K wag, Modeling and experiments on\neddy current damping caused by a permanent magnet in a conductiv e tube, J. Mech. Science and Tech.\n23, 3024-3035 (2009).\n[17] G. Donoso, C. Ladera, and P. Martin, Magnet fall inside a condu ctive pipe: motion and the role of the\npipe wall thickness, Eur. J. Phys. 30, 855-869 (2009).\n[18] G. Donoso, C. Ladera, and P. Martin, Damped Fall of Magnets in side a Conducting Pipe, Eur. J. Phys.\n79, 193-200 (2011).\n[19] John D. Jackson, Classical Electrodynamics , (3rd edition), John Wiley & Sons, Inc. See Sec. 5.9C.\n[20] N. Derby and S. Olbert, Cylindrical Magnets and Ideal Solenoids Am. J. of Phys. 78, 229-235, (2010).\n[21] P. J. Salzman, J. R. Burke, and S. M. Lea, The effect of electric fields in a classic introductory physics\ntreatment of eddy current forces, Am. J. Phys. 69, 586-590 (2001).\n[22] K.D. Hahn, E.M. Johnson, A. Brokken, and S. Baldwin, Eddy curr ent damping of a magnet moving\nthrough a pipe, Am. J. Phys. 66, 1066 (1998).\n13"    },    {        "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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Ballester\nDepartamentdeF´ ısica,Universitatdeles IllesBalears,E -07122,PalmadeMallorca,Spain\nroberto.soler@uib.es\nABSTRACT\nWe investigate standing kink magnetohydrodynamic (MHD) os cillations in a\nprominence fine structure modeled as a straight and cylindri cal magnetic tube only\npartiallyfilled withtheprominencematerial,andwithitse ndsfixedattworigidwalls\nrepresenting the solar photosphere. The prominence plasma is partially ionized and\na transverse inhomogeneous transitional layer is included between the prominence\nthread and the coronal medium. Thus, ion-neutral collision s and resonant absorption\nare the considered damping mechanisms. Approximate analyt ical expressions of the\nperiod, the damping time, and their ratio are derived for the fundamental mode in\nthe thin tube and thin boundary approximations. We find that t he dominant damp-\ning mechanism is resonant absorption, which provides dampi ng ratios in agreement\nwith the observations, whereas ion-neutral collisions are irrelevant for the damping.\nThevaluesofthedampingratioareindependentofboththepr ominencethreadlength\nanditspositionwithinthemagnetictube,andcoincidewith thevaluesforatubefully\nfilledwiththeprominenceplasma. Theimplicationsofourre sultsinthecontextofthe\nMHD seismologytechnique are discussed, pointing out that t he reported short-period\n(2 – 10 min) and short-wavelength (700 – 8,000 km) thread osci llations may not be\nconsistent with a standing mode interpretation and could be related to propagating\nwaves. Finally, we show that the inversion of some prominenc e physical parameters,\ne.g., Alfv´ en speed, magnetic field strength, transverse in homogeneity length-scale,\netc., is possible using observationally determined values of the period and damping\ntimeoftheoscillationsalong withtheanalyticalapproxim ationsofthesequantities.\nSubjectheadings: Sun: oscillations— Sun: corona— Sun: magneticfields — waves\n—prominences/filaments– 2 –\n1. INTRODUCTION\nOscillations and propagating waves are commonly reported i n observations of solar promi-\nnences and filaments (see recent reviews by, e.g., Ballester 2006; Engvold 2008; Mackayet al.\n2010). In high-resolution observations, transverse oscil lations of prominence fine structures are\nfrequentlydetected. Thesefinestructures,herecalledthr eads,appearasamyriadoflong(5′′−20′′)\nand thin (0′′.2−0′′.6) dark ribbons in H αimages of filaments on the solar disk (e.g., Linet al.\n2007, 2008, 2009), as well as in observations of prominences in the solar limb from the So-\nlar Optical Telescope (SOT) aboard the Hinode satellite (e. g., Okamotoetal. 2007; Berger et al.\n2008; Chaeet al. 2008; Ninget al. 2009). From the theoretica l point of view, prominence fine\nstructures have been modeled as magnetic flux tubes anchored in the solar photosphere (e.g.,\nBallester&Priest 1989; Rempel et al. 1999), which are piled up to form the prominence body.\nIn this interpretation, only part of the flux tubes would be fil led with the cool (∼104K) filament\nmaterial,whichwouldcorrespondtotheobservedthreads,w hiletherestofthemagnetictube,i.e.,\ntheso-calledevacuatedzone, wouldbeoccupied byhot coron alplasma.\nCommon features of the transverse oscillations of prominen ce fine structures detected in\nDoppler signals and H αsequences are that the reported periods are usually in a narr ow range\nbetween 2 and 10 minutes, that the velocity amplitudes are sm aller than∼3 km s−1, and that the\noscillations seem to be damped after a few periods. Typicall y, the number of oscillatory periods\nobserved before the oscillations disappear is less than 10 ( see, e.g., Molowny-Horaset al. 1999;\nTerradas et al. 2002; Lin 2004; Ning etal. 2009). Theoretica lly, the oscillations have been inter-\npretedintermsofkinkmagnetohydrodynamic(MHD)modessup portedbythefinestructure,mod-\neledasamagneticslab(Joarderet al.1997;D´ ıazet al.2001 ,2003)oracylindricaltube(D´ ıazet al.\n2002,2010;Dymova&Ruderman2005;Terradas et al.2008)par tiallyfilledwiththeprominence\nplasma,whereasseveraldampingmechanismshavebeenpropo sedtoexplainthequickattenuation\n(see, e.g., Oliver2009;Arregui& Ballester2010; Soler201 0).\nBy neglecting the variation of the plasma parameters along t he fine structure and adopting a\nprominence thread model composed of a homogeneous magnetic flux tube with prominence con-\nditions embedded in a coronal environment, Soleret al. (200 8) studied the temporal damping of\npropagating kink MHD waves due to nonadiabatic e ffects (radiative losses, thermal conduction,\nand plasmaheating),whileSoleret al. (2009b) investigate dtheattenuationin thesameconfigura-\ntion but considering ion-neutral collisions as the damping mechanism. These authors concluded\nthat neither nonadiabatic e ffects nor ion-neutral collisions can produce kink mode dampi ng times\ncompatible with those observed. On the other hand, Arreguie t al. (2008b) considered a similar\nmodel but neglected gas pressure (i.e., the β=0 approximation, with βthe ratio of the gas pres-\nsuretothemagneticpressure)andtookintoaccountthepres enceofatransverselyinhomogeneous\ntransitional layer between the thread and the coronal plasm a. In such a configuration, the kink– 3 –\nmode is resonantly coupled to Alfv´ en continuum modes, and s o the kink mode is damped by the\nprocessofresonantabsorption. Arreguiet al.(2008b)nume ricallyobtainedvaluesofthedamping\ntime that are consistent with those reported in the observat ions. Resonant absorption has been\npreviously proposed as an explanation for both the temporal damping of coronal loop transverse\noscillations (e.g., Ruderman &Roberts 2002; Goossenset al . 2002) and the spatial damping of\npropagating kink waves (e.g., Pascoeet al. 2010; Terradas e t al. 2010). Subsequently, Soleret al.\n(2009a) performed a more in-depth analytical and numerical investigationof the damping by res-\nonant absorption in prominence threads by including gas pre ssure, and obtained similar results to\nthose of Arreguiet al. (2008b). More recently, Soleret al. ( 2009c) studied the combined e ffect\nof resonant absorption and the prominence plasma partial io nization on the kink mode damping.\nFor realistic values of the wavelength, Soleret al. (2009c) concluded that partial ionization does\nnot affect the process of resonant absorption, and so the obtained v alues of the damping time are\nthe same as in a fully ionized prominence thread (Arregui eta l. 2008b; Soleret al. 2009a). Ion-\nneutral collisions may become more relevant than resonant a bsorption for the temporal damping\nof propagating kink modes when short wavelengths of the orde r of 103km and smaller values are\nconsidered, while therange of typically observed waveleng ths in prominences is between 5 ×103\n– 105km(Oliver& Ballester2002).\nWhetherthereportedobservationsoftransverselyoscilla tingfilamentandprominencethreads\nare related to propagating waves or standing oscillations i s a subject of debate. For example,\nLinet al.(2009)explainedtheirobservationsintermsofpr opagatingkinkwaves,whereasTerradas et al.\n(2008) proposed standing oscillations as an explanation of the observations by Okamotoet al.\n(2007). It is likelythatpropagatingwaves maybe generated by localizeddisturbances in,e.g., the\nfootpoints of the magnetic tube, while standing oscillatio ns may be related to more global pertur-\nbationsofthewholemagneticstructure. Regardingthedamp ing,alltheworkscitedabovestudied\nthe temporal damping of propagating kink waves, hence the wa velength (or the wavenumber) is a\nfree parameter in their case, but the problem of the damping o f standing oscillations has not been\naddressed yet in the context of prominence fine structures. M oreover, the effect on the damping\nof the longitudinal variation of the plasma parameters alon g the fine structure was not addressed\nin these previous works. Therefore, the aim of the present in vestigation is to broach the problem\nof the damping of standing kink MHD oscillations of longitud inally nonuniform prominence fine\nstructures.\nThemodelconfigurationadoptedhereissimilartothatconsi deredbyD´ ıazetal.(2002,2010),\nDymova&Ruderman (2005), and Terradas et al. (2008), namely a straight and cylindrical mag-\nnetic flux tube only partially filled with prominence plasma. The rest of the magnetic tube, as\nwell as the external medium, has typical coronal properties . Theβ=0 approximation is adopted\nfor the sake of simplicitysince we restrict ourselves to kin k modes, which are correctly described\nby this approximation. Standing oscillations are studied b y imposing the line-tying condition at– 4 –\nthe ends of the cylinder. As in Soler etal. (2009c), we assume that the prominence plasma is\npartially ionized and include a transversely inhomogeneou s transitional layer between the dense\nprominence thread and the external corona. Hence, the mecha nisms of ion-neutral collisions and\nresonant absorption are considered as damping mechanisms. We follow the method introduced\nby Dymova& Ruderman (2005) based on the thin tube limit, and d erive a dispersion relation for\ndamped kink MHD oscillations in the thin boundary approxima tion. Analytical expressions of\nthe period, the damping time, and the ratio of the damping tim e to the period are obtained, while\na general parametric study is performed by numerically solv ing the full dispersion relation. In\na subsequent work, Arreguiet al. (2010) investigate the dam ping of kink oscillations beyond the\nthin tube and thin boundary approximations by numerically s olving the full resistive eigenvalue\nproblemintwo-dimensional,nonuniformthreads.\nThis paper is organized as follows. Section 2 includes a desc ription of the model configura-\ntion and the mathematical method. The dispersion relation o f standing kink MHD oscillations is\nobtained in Section 3, which also contains analytical appro ximations. The results of solving the\ndispersionrelation are given in Section 4, whiletheir impl icationsfor prominence seismologyare\ndiscussedinSection 5. Finally,Section 6 containsourmain conclusions.\n2. MODEL ANDMETHOD\n2.1. Equilibrium Configuration\nFig. 1.—Sketch ofthemodelconfigurationadoptedin thiswor k.\nThe model considered here is schematically plotted in Figur e 1. We consider a straight and\ncylindrical magnetictube of length Land radius a, whose ends are fixed by two rigid walls repre-– 5 –\nsenting the solar photosphere. The magnetic tube is only par tially filled with the cool and dense\nprominence material, and is composed of a dense region of len gthLpwith prominence conditions\nand representing the prominencethread, surrounded by two m uch less dense zones corresponding\nto the evacuated part of the tube. According to the observed t ypical values of thread widths and\nlengths from the high-resolution observations (e.g., Lin 2 004; Lin etal. 2008), the ranges of real-\nistic values of aandLpare 50 km/lessorsimilara/lessorsimilar300 km and 3,000 km /lessorsimilarLp/lessorsimilar28,000 km. On the other\nhand, the total tube length, L, cannot be measured from the observations, but one can relat eLto\nthetypicalspatialscaleinprominencesand filaments,i.e. ,L∼105km.\nFor simplicity, both the prominence and the evacuated (i.e. , coronal) part are taken homoge-\nneous, with densities ρpandρe, respectively. The external, coronal medium has density ρc, which\nisalsohomogeneous. Subscriptsp, e,and cdenotethepromin encepart, theevacuatedregion,and\nthe corona, respectively. In general, the subscript 0 denot es equilibrium quantities without refer-\nring to aparticularregion. Thedensitycontrast oftheprom inencepart with respect to thecoronal\nplasmaisalargeparameter,with ρp/ρc=200avalueusuallyconsidered. Weassumethattheevac-\nuated part ofthetubehas thesamedensityas thecorona. Henc e, for atypical prominencedensity\nofρp=5×10−11kg m−3, the coronal and evacuated densities are ρc=ρe=2.5×10−13kg m−3.\nIntheprominenceregion,weincludeatransverselyinhomog eneoustransitionallayerofthickness\nl, that continuously connects the internal prominence regio n to the external corona. A sinusoidal\nvariation of the density is considered in the transitional l ayer (Ruderman &Roberts 2002). The\nlimitsl/a=0 andl/a=2 correspond to a thread without transverse transitional la yer and a fully\ninhomogeneous thread in the radial direction, respectivel y. The plasma in the prominence region\nisassumedpartiallyionizedwithanarbitraryionizationd egree ˜µp. Boththeevacuatedpartandthe\ncoronaare takenfullyionized,hence ˜ µe=˜µc=0.5.\nWe use cylindrical coordinates, namely r,ϕ, andzfor the radial, azimuthal, and longitudinal\ncoordinates. Themagneticfieldistakenhomogeneousandori entatedalongthe z-direction,namely\nB0=B0ˆez, withB0=5Geverywhere. The z-directionalsocoincideswiththeaxisofthecylinder.\nForl/a=0 and full ionization, the model is equivalent to that assume d by D´ ıazetal. (2002) and\nDymova&Ruderman (2005). In these two works the prominence t hread is located in the center\nof the cylinder. Here, we allow the thread to be displaced fro m the center of the tube. The length\nof the evacuated region on the left-hand side of the thread is L−\ne, whereas the length of the right-\nhand side evacuated region is L+\ne. WhenLandLpare fixed, we can express L+\ne=L−L−\ne−Lp,\nhence it is enough to select a value for L−\nein order to set the length of both evacuated parts.\nThe allowed values of L−\neare in the range 0 ≤L−\ne≤L−Lp. ForL−\ne=0, the thread is totally\ndisplaced to the left-hand side end of the flux tube, while the contrary occurs for L−\ne=L−Lp.\nForL−\ne=L+\ne=1\n2/parenleftig\nL−Lp/parenrightig\nthe prominence thread is located at the center of the tube, i. e., the\nconfiguration studied by D´ ıazet al. (2002) and Dymova&Rude rman (2005). For simplicity, we\nfix the origin of coordinates at the center of the prominence r egion, so that the interfaces between– 6 –\ntheprominence plasmaand the evacuated zones are located at z=±Lp/2. Thephotosphericwalls\nare therefore located at z=z−\nwall=−Lp/2−L−\neandz=z+\nwall=Lp/2+L+\ne, withz−\nwallandz+\nwall\nthe position of the left and right walls, respectively. Thus , forLp=L, i.e, a tube fully filled with\nprominencematerial, themodelreduces to theconfiguration studiedby Soleret al. (2009c).\n2.2. BasicEquations\nThe governing MHD equations for a partially ionized plasma a re derived in Fortezaet al.\n(2007) after the single-fluid treatment of Braginskii (1965 ) (see also details in Pintoet al. 2008;\nSoleret al.2009b;Soler2010). Electrons,ions(i.e.,prot ons),andneutralhydrogenarethespecies\ntaken into account. We assume small-amplitude perturbatio ns over the equilibrium state and\nthe basic equations are linearized (Soler etal. 2009b, Equa tions (4)–(7)). In a partially ionized\nplasma, the induction equation contains di ffusion terms related to the collisions between the dif-\nferent species (see Pinto et al. 2008). For example, Ohm’s di ffusion is governed by electron-ion\ncollisions, whereas Cowling’s di ffusion is dominated by ion-neutral collisions. Here, we negl ect\nOhm’s diffusion from the induction equation (Soler etal. 2009b, Equat ion (7)) because its role in\na partially ionized prominence plasma is much less importan t than that of Cowling’s di ffusion. In\naddition,sinceweadoptthe β=0 approximation,gaspressuree ffects areneglected.\nNext, we follow Fortezaet al. (2008) and Soleret al. (2009b) and take a time dependence of\nthe form exp (−iωt), withωthe oscillatory frequency, so that we can include the e ffect of Cowl-\ning’s diffusion in the definition of a modified Alfv´ en speed squared as Γ2\nA0=v2\nA0−iωηC0, where\nv2\nA0=B2\n0/µρ0is the Alfv´ en speed squared and ηC0is the Cowling’s di ffusion coefficient, with\nµ=4π×10−7N A−2. The expression of Cowling’s di ffusivity,ηC0, in terms of the equilibrium\nproperties is given in, e.g., Soleret al. (2009b) and Soler ( 2010). Thus, the relevant equations for\nourinvestigationare\nρ0∂v1\n∂t=1\nµ(∇×B1)×B0, (1)\n∂B1\n∂t=Γ2\nA0\nv2\nA0∇×(v1×B0), (2)\nwherev1=/parenleftig\nvr,vϕ,vz/parenrightig\nandB1=/parenleftig\nBr,Bϕ,Bz/parenrightig\nare the velocity and the magnetic field perturbations.\nNotethat vz=0 intheβ=0 approximation,and ΓA0/vA0=1 whenCowling’sdi ffusionisabsent.\nEquations(1)and(2)canbecombinedtoarriveatthefollowi ngequationforthetotalpressure\nperturbation, pT=B0Bz/µ, namely\n∂2pT\n∂t2−Γ2\nA0∇2pT=0, (3)– 7 –\nalongwithan equationrelating thetotalpressureand radia l velocityperturbationsas\n∂2vr\n∂t2−Γ2\nA0∂2vr\n∂z2=−1\nρ0∂2pT\n∂t∂r. (4)\nNote that Equation (3) is only valid in the regions with homog eneous densities, hence it cannot\nbe applied in the transversely inhomogeneous transitional layer. Now, we write all perturbations\nproportional to exp (−iωt+imϕ), wheremis an integer representing the azimuthal wavenumber.\nIntheabsenceofmagnetictwist,bothpositiveandnegative valuesof mareequivalent,sohereafter\nwerestrict ourselvesto positivevaluesof m. Forkinkoscillations, m=1. Equation(3)becomes\n∂2pT\n∂z2+1\nr∂\n∂r/parenleftigg\nr∂pT\n∂r/parenrightigg\n+ω2\nΓ2\nA0−m2\nr2pT=0. (5)\nSinceρp,ρe, andρcare uniform in their respective regions, the corresponding Cowling’s\ndiffusivities,ηCp,ηCe, andηCc, respectively, are also uniform. Both the corona and the eva cuated\nregion are fully ionized and much less dense than the promine nce plasma, so we have ηCc≪ηCp\nandηCe≪ηCp. For the sake of simplicity, we set ηCe=ηCc=0, and the effect of Cowling’s\ndiffusion is only considered in the prominence region. If consid ered, Cowling’s diffusion in the\nevacuated and coronal regions would have a very minor influen ce since Cowling’s di ffusivitiesin\ntheseregionsare muchsmallerthan thatin theprominencepl asma.\n2.3. Mathematical Method\nForafullyionizedplasma, thegeneral investigationofthe idealtransverseMHDoscillations\nsupported by our equilibrium was performed by D´ ıazet al. (2 002) in the case l/a=0 and for\nthe prominence thread centered within the magnetic tube. Th ese authors obtained the oscillatory\nfrequencies and eigenfunctions for arbitrary values of L,Lp, anda. Here, we could follow a\ntreatment similar to that of D´ ıazet al. (2002), but this req uires a significant mathematical e ffort\nbeyond the purpose of the present investigation. Instead, w e consider the much simpler approach\nintroduced by Dymova&Ruderman (2005), who studied the same configuration but in the thin\ntube (TT) limit, i.e., for a/L≪1 anda/Lp≪1. To check the validity of this approximation in\nthecontextofprominencethreadoscillations,wetakeinto accountthevaluesof aandLpreported\nfromtheobservations(seeSec.2.1)andassume L∼105km. Weobtain a/Lpanda/Lintheranges\n2×10−3/lessorsimilara/Lp/lessorsimilar0.1 and 5×10−4/lessorsimilara/L/lessorsimilar3×10−3, meaning that the TT approximation is\njustified in prominence fine structures. As shown by Terradas etal. (2008) and D´ ıazet al. (2010),\nthe method of Dymova& Ruderman (2005) shows an excellent agr eement with that of D´ ıazet al.\n(2002) whenrealisticvaluesof a/Lpanda/Laretaken intoaccount.– 8 –\nFollowing Dymova&Ruderman (2005), we can perform a di fferent scaling of Equation (5)\ninside the tube and in the corona. For perturbations inside t he tube, the characteristic scale in the\nr-directionis a,whilethecharacteristicscaleinthe z-directionis L. Sincea/L≪1, theterm with\nthe longitudinalderivativeand the term proportional to ω2are much smaller than the other terms.\nIn sucha case, Equation(5) insidethetubereduces to\n∂\n∂r/parenleftigg\nr∂pTi\n∂r/parenrightigg\n−m2\nr2pTi≈0, (6)\nwithi=p ore. ThesolutionofEquation(6)forregularperturbation satr=0 is\npTi≈Ai(z)/parenleftbiggr\na/parenrightbiggm\n, (7)\nwhereAi(z)isan arbitrary functionof z.\nOntheotherhand,thecharacteristicscaleofperturbation soutsidethetube,i.e.,inthecorona,\nisLinboththe r-andz-directions,sothatnotermscanbeneglectedinEquation(5 ). However,we\ncanexpressthetotalpressureperturbationinthecoronaas pTc=Ac(z)F(r)andusethetechnique\nofseparationofvariablesto obtainthefollowingexpressi ons,\nd2F\ndr2+1\nrdF\ndr−/parenleftigg\nk2\nn+m2\nr2/parenrightigg\nF=0, (8)\nand\nd2Ac\ndz2+ω2\nv2\nAcAc=−k2\nnAc,withAc=0 atz=z±\nwall, (9)\nwhereknis a separation constant, with nan integer accounting for the di fferent radial harmonics.\nIn Equation (9) we have taken into account that Cowling’s di ffusion is neglected in the corona,\nsoΓ2\nAc=v2\nAc. Equation (8) is the modified Bessel Equation. Here, we only c onsider trapped\nmodes and assume k2\nn>0. This last condition may not be satisfied for high harmonics , i.e.,\nlarge values of n, but we need not worry about this issue since here we focus our investigation\non the fundamental mode, which is non-leaky in the present co nfiguration. Then, the solution\nof Equation (8) is F(r)=Km(knr), withKmthe modified Bessel function of the second kind.\nAn asymptotic expansion near the tube boundary (e.g., Abram owitz& Stegun 1972) allows us to\nexpressthetotalpressureperturbationinthecoronaas\npTc≈Ac(z)/parenleftbigga\nr/parenrightbiggm\n. (10)\nNext,ourmethodfollowsclosely thatofDymova&Ruderman (2 005, 2006). Forthesakeof\nsimplicity,weomitherethedetails,whicharegiveninAppe ndixA. Afterconsideringappropriate\nboundary conditions for the solutions of Equations (6) and ( 10), the dispersion relation for kink\noscillationsdampedby resonantabsorptionandCowling’sd iffusionisobtained.– 9 –\n3. DISPERSION RELATIONAND APPROXIMATIONS\nThegeneral dispersionrelationforkinkoscillationsis\n˜ckphcos/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\ncos/parenleftigω\nckeL−\ne/parenrightig\n−ckesin/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\nsin/parenleftigω\nckeL−\ne/parenrightig\n˜ckphsin/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\ncos/parenleftig\nω\nckeL−e/parenrightig\n+ckecos/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\nsin/parenleftig\nω\nckeL−e/parenrightig\n+˜ckphcos/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\ncos/parenleftigω\nckeL+\ne/parenrightig\n−ckesin/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\nsin/parenleftigω\nckeL+\ne/parenrightig\n˜ckphsin/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\ncos/parenleftigω\nckeL+e/parenrightig\n+ckecos/parenleftbigg\nω\n˜ckphLp\n2/parenrightbigg\nsin/parenleftigω\nckeL+e/parenrightig=0,\n(11)\nwith\n˜c2\nkp=ρpΓ2\nAp+ρcv2\nAc\nρp+ρc−iπω/parenleftig\nΓ2\nAp+v2\nAc/parenrightig/parenleftbigg\nρpρc\nρp+ρc/parenrightbigg\nm/a\nωR|∂rρ0|a\n1−iπω/parenleftbigg\nρpρc\nρp+ρc/parenrightbigg\nm/a\nωR|∂rρ0|a, (12)\nb2=−iωπΓ2\nApv2\nAc/parenleftbigg\nρpρc\nρp+ρc/parenrightbigg\nm/a\nωR|∂rρ0|a\nρpΓ2\nAp+ρcv2\nAc\nρp+ρc−iπω/parenleftig\nΓ2\nAp+v2\nAc/parenrightig/parenleftbigg\nρpρc\nρp+ρc/parenrightbigg\nm/a\nωR|∂rρ0|a, (13)\nc2\nkp=ρpv2\nAp+ρcv2\nAc\nρp+ρc,c2\nke=ρev2\nAe+ρcv2\nAc\nρe+ρc, (14)\nandh=/radicalig\n1−b2\nc2\nkp, withωRthe real part of the frequency and |∂rρ0|athe radial derivative of the\ndensity profile evaluated at the resonance position, which h as been approximated by the thread\nmean radius, a. The quantity ˜ ckpis here called the modified kink speed, which takes into accou nt\nboth the effect of Cowling’s di ffusion (throughΓ2\nAp) and the effect of resonant absorption in the\nthin boundary (TB) approach. Extensive details are given in Appendix A. If the terms related to\nresonantabsorptionare omitted,onehas b2=0 and ˜c2\nkpbecomes\n˜c2\nkp=ρpΓ2\nAp+ρcv2\nAc\nρp+ρc, (15)\nwhich reduces to theideal kinkspeed, c2\nkp(Equation(14)), when Cowling’sdi ffusionis neglected,\ni.e.,Γ2\nAp=v2\nAp.\nEquation(11)is atranscendentalequationthathas tobesol vednumerically. Someanalytical\nprogress can be performed if the prominence thread is center ed within the tube, i.e., L−\ne=L+\ne=– 10 –\n1\n2/parenleftig\nL−Lp/parenrightig\n, and we focus on the fundamental kink mode. This solution cor responds to the mode\nwiththelowestfrequency. In such acase, Equation(11)can b esimplifiedto\n1\n˜ckp/radicalig\n1−b2\nc2\nkptanω\n˜ckp/radicalig\n1−b2\nc2\nkpLp\n2−1\nckecot/bracketleftiggω\ncke/parenleftiggL−Lp\n2/parenrightigg/bracketrightigg\n=0. (16)\nThe fundamental kink mode is given by the first root of Equatio n (16). A first-order Taylor ex-\npansion for small arguments of the trigonometric functions of Equation (16) provides us with an\napproximationtothefrequency as\nω2≈4/parenleftig\nL−Lp/parenrightig\nLp˜c2\nkp1−b2\nc2\nkp. (17)\nWe expect Equation (17) to be valid when both ωandLp/Lare small quantities, so that the argu-\nments of the trigonometricfunctions of Equation (16) remai n small. Note that Equation (17) fails\ntorepresentthekinkmodefrequencyinthelimits Lp/L→1andLp/L→0,sooneshouldconsider\nintermediate values of Lp/Lin Equation (17). The correct expressions for the fundament al kink\nmodefrequency intheselimitsare\nω=π\nL˜ckp/radicaligg\n1−b2\nc2\nkp,forLp/L=1, (18)\nand\nω=π\nLcke,forLp/L=0. (19)\nWe can extract two main results from Equation (17). First of a ll, Equation (17) only depends\non the physical properties of the prominence region and the c orona through ˜ ckp,ckp, andb, and\nincludesnocontributionsfromtheevacuatedpart. Andseco nd,theformofEquation(17)issimilar\nto the approximation of the ideal kink mode frequency in a hom ogeneous tube, i.e., ω2≈k2\nzc2\nkp,\nwherekzis the longitudinal wavenumber. Thus, it seems that the main differences between the\nexpression for the homogeneous tube and that for the partial ly filled tube are that 4 /(L−Lp)Lp\nreplacesk2\nz, and that a redefined kink speed has to be taken into account. T his approximation of\nthe frequency is similar to that obtained by Joarder&Robert s (1992) and Oliveret al. (1993) for\nthe string (or hybrid) modes of their slab configuration and t o that obtained by D´ ıazet al. (2010)\nin the context of thread seismology using period ratios. How ever, we must bear in mind that\nEquation(17) isonlyafirst-orderapproximationto thekink modefrequency.\nIn the general case, i.e., when the prominence region is allo wed to be at any position within\nthe tube, one should consider the dispersion relation given by Equation (11). We can follow the\nsame procedure as before and perform a first-order Taylor exp ansion for small arguments of the– 11 –\ntrigonometricfunctions of Equation (11). Then, the follow ing approximation for the frequency is\nobtained,\nω2≈4L/bracketleftig/parenleftig\nL−Lp/parenrightig\nLp+4L−eL+e/bracketrightig\nLp˜c2\nkp1−b2\nc2\nkp. (20)\nNote that the only information from the evacuated zones pres ent in Equation (20) is their lengths\nL−\neandL+\ne, but no additional physical property of these zones contrib utes to Equation (20). In the\ncenteredcase, L−\ne=L+\ne=1\n2/parenleftig\nL−Lp/parenrightig\n,Equation(20)revertstoEquation(17). Now,wecanconside r\nthelimits L−\ne→0 orL+\ne→0,which correspondto theprominencethread totallydispla cedtoward\nan end ofthetube. In suchlimits,Equation(20)becomes\nω2≈4L/parenleftig\nL−Lp/parenrightig\nL2\np˜c2\nkp1−b2\nc2\nkp. (21)\nThe ratio of Equation (17) to Equation (21) estimates the shi ft ofω2when the thread is displaced\nfrom the center toward the end of the tube. Denoting this rati o asδω2, we obtainδω2=Lp/L.\nSinceLp/L<1, weexpectthefrequencyofthekinkmodetoincreaseas thep rominencethread is\ndisplaced form the central position. Obviously, for Lp/L=1 there is no frequency shift because\ntheprominenceplasmaoccupies thewholetube.\n3.1. Periodofthe fundamental kink mode\nFirst, we focus our analytical investigation on the period. Here, we neglect the e ffect of the\ndamping mechanisms since we assume that the kink mode period is only slightly affected by the\npresence of the damping mechanisms. As a first approximation , we consider Equation (20) with\n˜ckp=ckpandb=0, sonowωis areal quantity. Wecomputetheperiodas P=2π/ω, obtaining\nP=π\nvAp/radicaligg\nρp+ρc\n2ρp/radicaligg/bracketleftig/parenleftig\nL−Lp/parenrightig\nLp+4L−\neL+\ne/bracketrightig\nLp\nL. (22)\nSince the expression for the period is known, we could compar e the theoretical periods with\nthose observed and apply the MHD seismology technique to pro minence fine structure oscilla-\ntions. However, in our case Pdepends on many parameters of the model, so Equation (22) alo ne\nis not very useful from a seismological point of view. If we as sume that the prominence thread is\nlocated atthecenterofthemagnetictube,Equation(22)bec omes\nP≈π\nvAp/radicaligg\nρp+ρc\n2ρp/radicalig/parenleftig\nL−Lp/parenrightig\nLp. (23)– 12 –\nIn addition,in thecase ofprominencesonehas ρp≫ρc, soEquation(23)can besimplifiedto\nP≈π√\n2vAp/radicalig/parenleftig\nL−Lp/parenrightig\nLp. (24)\nAlthoughthevalueof LpcanbemeasuredfromH αobservationsoffilaments(e.g.,Lin etal.2007,\n2009),westillhavetwoparameters,i.e.,thetotaltubelen gth,L,andtheprominenceAlfv´ enspeed,\nthatare bothdifficulttodeterminefrom theobservations. Recently, D´ ıazet al.(2010)showedthat\nthe ratio of periods of di fferent overtones is a useful quantity to perform seismology o f promi-\nnencethreads,sinceadditionalparametersas,e.g.,thepr ominenceAlfv´ enspeedaredroppedfrom\nthe expressions (for details about the importance of the per iod ratio for coronal seismology, see\nthe recent review by Andries etal. 2009). Some additional re marks about the MHD seismology\ntechniqueare givenin Section5.\n3.2. Damping by Cowling’sdi ffusion\nHerewestudythekinkmodedamping. Letusconsiderthecasew ithouttransversetransitional\nlayer, i.e., l/a=0, so the damping is exclusively due to Cowling’s di ffusion and the frequency is\ncomplex,ω=ωR+iωI,withωRandωItherealandimaginarypartsofthefrequency,respectively .\nThen, ˜ckpisgivenby Equation(15)and b2=0sincethereisnoresonantdampingfor l/a=0. For\nsimplicity,weconsiderthattheprominencethreadislocat edat thecentralposition. Equation(17)\nallowsustoobtainthereal and imaginaryparts ofthefreque ncy as\nω≈ρpv2\nAp+ρcv2\nAc\nρp+ρc−/parenleftiggρpηCp\nρp+ρc/parenrightigg21/parenleftig\nL−Lp/parenrightig\nLp1/2\n2/radicalig/parenleftig\nL−Lp/parenrightig\nLp\n−i/parenleftiggρpηCp\nρp+ρc/parenrightigg2/parenleftig\nL−Lp/parenrightig\nLp. (25)\nBy setting the real part of Equation (25) equal to zero, we obt ain two critical values of Lp/L,\nnamely\n/parenleftig\nLp/L/parenrightig±\ncrit=1\n2±1\n2/bracketleftigg\n1−/parenleftigg2ρp\nρp+ρc/parenrightigg\n˜η2\nCp/bracketrightigg1/2\n, (26)\nwith ˜ηCp=ηCp/vApL. Hence, thekink modeonlyexistsfor/parenleftig\nLp/L/parenrightig−\ncrit<Lp/L</parenleftig\nLp/L/parenrightig+\ncrit. We cast\nEquation (26) forρp/ρc=200 and the extreme case of an almost neutral plasma with ˜ µp=0.99,\nobtaining/parenleftig\nLp/L/parenrightig−\ncrit≈10−5and/parenleftig\nLp/L/parenrightig+\ncrit≈0.99999. Forsmallervalues of ˜ µp,/parenleftig\nLp/L/parenrightig−\ncritdecreases\nand/parenleftig\nLp/L/parenrightig+\ncritincreases. Hence,thepresenceofthesecriticalvaluesisi rrelevantforrealisticvalues\nofLp/L.– 13 –\nForLp/Lfar from the critical values, one can drop the second term in t he real part of Equa-\ntion(25). Wecomputethedampingtime, τD=1/|ωI|, dueto Cowling’sdi ffusionas\nτD≈1\n2/parenleftiggρp+ρc\nρpηCp/parenrightigg/parenleftig\nL−Lp/parenrightig\nLp, (27)\nwhiletheratioofthedampingtimetotheperiodis\nτD\nP≈1\n2π/parenleftiggρp+ρc\nρp/parenrightigg1/21\n˜ηCp/radicaligg\n2/parenleftigg\n1−Lp\nL/parenrightiggLp\nL. (28)\nForρp/ρc=200,Lp/L=0.1, andL=105km, Equation (28) gives τD/P≈5×103for ˜µp=0.8,\nandτD/P≈150 for ˜µp=0.99. Theseresults indicate that, as in a homogeneousthread, an almost\nneutralprominenceplasmaisneeded,i.e., ˜ µp≈1,forthedampingduetoCowling’sdi ffusiontobe\nefficient. Althoughthe precise ionizationdegree is unknown,s uch large values of ˜ µpare probably\nunrealisticinthecontextofprominences.\nIt is straight-forward to extend Equation (28) to the case in which the prominence region is\nnotat thecenterofthetube,obtaining\nτD\nP≈1\n2π/parenleftiggρp+ρc\nρp/parenrightigg1/21\n˜ηCp/radicaligg\n2/bracketleftigg/parenleftigg\n1−Lp\nL/parenrightiggLp\nL+4L−\neL+\ne\nL2/bracketrightiggLp\nL. (29)\nPerforming the limits L−\ne→0 orL+\ne→0 to this last expression, we can obtain the damping\nratio when the thread is totally displaced toward the ends of the tube, namely (τD/P)end. Then,\nwe compare (τD/P)endwith the damping ratio obtained in the centered case from Equ ation (28),\nnamely(τD/P)center. Thus,(τD/P)center/(τD/P)end=/radicalbigLp/L<1. Therefore, the minimum value\nof the damping ratio by Cowling’s di ffusion takes place when prominence region is located at the\ncenter.\n3.3. Damping by resonant absorption\nHere, we study the general case l/a/nequal0. The full expressions of ˜ ckpandb2given by Equa-\ntions(12)and(13)aretakenintoaccount. Again,weassumet hattheprominencethreadiscentered\nwithinthemagnetictube. We useEquation(17) to providean e xpressionforthe ratio ωI/ωRafter\nneglectingterms of O/parenleftig\nω2\nI/parenrightig\nandO/parenleftig\nωIL−2/parenrightig\n. Thus,weobtain\nωI\nωR≈−π\n8/parenleftig\nρp−ρc/parenrightig2\n/parenleftig\nρp+ρc/parenrightigm/a\n|∂rρ0|a−/parenleftiggρp\nρp+ρc/parenrightigg1/2˜ηCp/radicalig\n2/parenleftig\n1−Lp\nL/parenrightigLp\nL. (30)– 14 –\nThe first term on the right-hand side of Equation (30) is cause d by resonant absorption and the\nsecond term is due to Cowling’sdi ffusion. The term related to Cowling’s di ffusionis also present\nin the case l/a=0. We can compare Equation (30) with Equation (28) of Soler et al. (2009c)\nvalid for a homogeneous tube. We see that both equations coin cide if the replacement of kzby\n2//radicalbig(L−Lp)Lpis done in their expression and ˜ ηCc=0. Therefore, this fact suggests again that\nthe results for the homogeneous tube can be extended to a part ially filled tube by selecting the\nappropriatevalueforthelongitudinalwavenumber.\nNote that the term related to the damping by resonant absorpt ion in Equation (30) takes the\nsame form as in a homogeneous tube and does not depend on Lp/L. This result does not mean\nthat the real and imaginary parts of the frequency do not depe nd onLp/L, but both quantities are\naffected in the same way so that their ratio remains una ffected. This important result is consistent\nwith the conclusions of Andries et al. (2005) and Arreguiet a l. (2005), who found that the kink\nmode frequency of a longitudinallystratified tube is the sam e as that obtained for a homogeneous\ntube with densityρmean, withρmeanthe mean density weighted with the wave energy. Since in\nour equilibriumthe transversely transitionallayer is onl y present in the dense part of the tube and\nboth the evacuated zone and the corona have the same density, resonant absorption only takes\nplace in the dense part of the tube. Therefore, the mean densi ty of the part of the tube where\nresonantabsorptiontakesplaceis,obviously, ρmean=ρp. Hence,accordingtoAndrieset al.(2005)\nand Arregui etal. (2005), the kink mode damping ratio in our c ase must be the same as that of a\nhomogeneoustubewithdensity ρp,asourresultsindicate. Thisconclusionisalsoequivalen ttothat\nobtained by Dymova& Ruderman (2006), who showed that the dam ping ratio in a longitudinally\ninhomogeneous tube in the TT and TB approximations does not d epend on the particular form of\nthelongitudinaldensityprofileif thedensitycontrastbet ween theinternaland externalplasmas is\nconstant.\nBy assuming a sinusoidal density variation in the transitio nal layer, the expression for τD/P\naccording toEquation(30)is\nτD\nP≈2\nπml\naρp−ρc\nρp+ρc+˜ηCp/parenleftiggρp\nρp+ρc/parenrightigg1/24/radicalig\n2/parenleftig\n1−Lp\nL/parenrightigLp\nL−1\n, (31)\nwhichisequivalenttoEquation(29)ofSoleret al.(2009c)f or ˜ηCc=0andkz=2//radicalbig(L−Lp)Lp. To\nperformasimpleapplication,wecompute τD/PfromEquation(31)inthecase m=1,Lp/L=0.1,\nL=107m, andl/a=0.2, resulting inτD/P≈3.18 for a fully ionized thread (˜ µp=0.5), and\nτD/P≈3.16foranalmostneutralthread(˜ µp=0.95). Wenotethattheobtaineddampingtimesare\nconsistentwiththeobservations. Moreover,asobtainedby Soleret al.(2009c),thecontributionof\nresonant absorption to the damping is much more important th an that of Cowling’s di ffusion, so\nthe ratioτD/Pdepends only very slightly on the ionization degree. Then, t he second term on the– 15 –\nright-handsideofEquation(31)can beneglectedand Equati on(31)becomes\nτD\nP≈2\nπ1\nm1\nl/aρp+ρc\nρp−ρc, (32)\nwhichcoincideswiththeexpressionsprovidedbyRuderman & Roberts(2002)andGoossenset al.\n(2002) in the case of longitudinally homogeneous coronal lo ops. Equation (32) can be further\nsimplifiedbyrestrictingourselvestothekinkmode( m=1)and forρp≫ρc,hence\nτD\nP≈2\nπ1\nl/a, (33)\nmeaning that the transverse inhomogeneity spatial-scale c an be estimated if both the period and\ndampingtimearemeasuredfrom theobservations. Bycombini ngEquations(24)and(33), weget\ntheexpressionofthedampingtimeby resonantabsorptionas\nτD≈√\n2\nvAp1\nl/a/radicalig/parenleftig\nL−Lp/parenrightig\nLp. (34)\nFinally,wecancomputethedampingratiowhentheprominenc ethreadisdisplacedfromthe\ncenter of the tube. In such a case, only the term related to Cow ling’s diffusion is modified in the\nwayshowninSection3.2,whilethetermrelatedtotheresona ntdampingisnota ffectedatall. The\ndampingtimebyresonant absorptionin thegeneral case is\nτD≈√\n2\nvAp1\nl/a/radicaligg/bracketleftig/parenleftig\nL−Lp/parenrightig\nLp+4L−\neL+\ne/bracketrightig\nLp\nL, (35)\nwhich has the same dependence on L−\neandL+\neas the period (Equation (22)). Since the resonant\ndampingdominatesoverCowling’sdi ffusion,weanticipatebymeansofthisanalyticalestimation s\nthatthedampingratioisalmostuna ffectedbythepositionoftheprominenceregionwithinthefine\nstructure.\n4. NUMERICALRESULTS\n4.1. Centered prominence thread\nIn this section, we assume that the prominence region is cent ered within the tube, i.e., L−\ne=\nL+\ne=1\n2/parenleftig\nL−Lp/parenrightig\n. We numerically solve the dispersion relation (Equation (1 6)) by means of stan-\ndard methodsand obtainthefrequencyofthefundamentalmod e. Westudythedependenceofthe\nresultson Lp/L.– 16 –\nFig. 2.— A(z)function(inarbitraryunits)correspondingtothefundam entalkinkmode. (a)Results\nforLp/L=0.1 (solid),0.5 (dashed), and 0.9 (dotted)when thepromine ncethread is located at the\ncentralpartofthemagnetictube. (b)Resultswith Lp/L=0.2forL−\ne/L=0.2(dashed),0.4(solid),\nand 0.7 (dotted). The thick part of the lines in panel (b) deno tes the position of the prominence\nthread.\nWe plot in Figure 2a the A(z) function corresponding to the fundamental even kink mode f or\ndifferent values of Lp/L. TheA(z) function gives the dependence of the transverse displacem ent\nin the longitudinal direction at r=a. We see that A(z) is mainly confined within the prominence\npart of the flux tube and satisfies the line-tying condition at z=z±\nwall=±L/2. For a homogeneous\nprominencetube, i.e., Lp/L=1, theA(z)functionbecomesproportionaltocos/parenleftigπ\nLz/parenrightig\n.\n4.1.1. Casewithouttransversetransitionallayer(l /a=0)\nFirst, we take into account the case without transverse tran sitional layer, i.e., l/a=0, and so\nwe study the kink mode damping due to Cowling’s di ffusion exclusively. Figure 3a displays the\nperiod,P, as a function of Lp/Lfor different values of the ionization degree in the prominence\nregion, whereas Figure 3b shows the corresponding values of the damping time, τD. Both values\nare given in dimensionless form with respect to the internal Alfv´ en travel time, τAp=L/vAp. We\nsee thatP/τApincreases as Lp/Lincreases, and tends to the value for a homogeneous prominen ce\ncylinderwhen Lp/L→1. Inaddition,wefindthat Pisindependentoftheionizationdegree. Onthe\ncontrary,τDis stronglydependent on theionizationdegree, as expected .τD/τApslightlyincreases\nasLp/Lbecomes larger. We see that the analytical expressions for t he period (Equation (24)) and\nthedampingtime(Equation(27))areinagreement withtheso lutionofthefulldispersionrelation\nfor realistic, small values of Lp/L, i.e.,Lp/L/lessorsimilar0.4, whereas the approximate expressions diverge\nfrom the actual solution when the prominence region occupie s most of the magnetic tube. As– 17 –\nFig. 3.— Results in the case without transverse transitiona l layer and for the prominence thread\nlocated at the central part of the magnetic tube. (a) Period, P, of the fundamental kink mode in\nunits of the internal Alfv´ en travel time, τAp, as a function of Lp/L. The horizontal dotted line\ncorresponds to the period of the kink mode in a homogeneous pr ominence cylinder. The symbols\naretheapproximationgivenbyEquation(24). (b)Dampingti me,τD,inunitsoftheinternalAlfv´ en\ntravel time,τAp, as a function of Lp/L. The different lines denote ˜µp=0.5 (dotted), 0.6 (dashed),\n0.8 (solid)and 0.95 (dash-dotted). The symbolsare theappr oximation givenby Equation (27) for\n˜µp=0.8. (c)τD/PversusLp/L. The line styles have the same meaning as in panel (b), and the\nsymbolsare theapproximationgivenbyEquation(28).– 18 –\nFig. 4.— Results in the case with a transverse transitional l ayer and for the prominence thread\nlocated at the central part of the magnetic tube. (a) τD, in units of the internal Alfv´ en travel time,\nτAp, and (b)τD/Pas a functions of Lp/L. The different lines in both panels denote l/a=0.05\n(dotted), 0.1 (dashed), 0.2 (solid)and 0.4 (dash-dotted). The symbolsin panels (a) and (b) are the\napproximationsgivenby Equations(34)and (32), respectiv ely,withl/a=0.2.\ncommented before, high-resolution observations suggest t hat the parameter Lp/Lis small in the\nfine structures of prominences. On the other hand, Figure 3c d isplaysτD/PversusLp/L. The\nnumericalsolutionofthedispersionrelationshowslittle dependenceon Lp/L, whiletheanalytical\napproximation(Equation(28))divergesfromthenumerical valueinthelimitoflarge Lp/L. Given\nthe large values of τD/Pobtained, we can conclude that the e fficiency of the damping due to\nCowling’sdiffusioninapartiallyfilledfluxtubedoesnotimprovewithresp ecttothelongitudinally\nhomogeneoustubecaseofSoleret al. (2009b,c).\n4.1.2. Casewith a transversetransitionallayer(l /a/nequal0)\nNow,wetakethecase l/a/nequal0intoaccount. Thekinkmodeisdampedbyresonantabsorptio n\nin the transverse transitional layer. We have computed both the period and the damping time of\nthe fundamental kink mode as a function of the di fferent parameters, namely ˜ µp,l/a, andLp/L.\nRegarding the period, we find that both its value and its depen dence on Lp/Lare the same plotted\ninFigure3abecausetheperiodisalmostindependentof ˜ µpandl/a. Forthesakeofsimplicity,we\ndo not repeat this Figure again and refer to Figure 3a. Theref ore, the presence of the transverse\ntransitional layer in the TB approximation does not modify t he period of kink oscillations with\nrespect to the case l/a=0. The period could be slightly a ffected if thick layers, i.e, l/a>1, are\nconsideredandtheresistiveequationsaresolvednumerica llyinsteadofassumingtheTBapproach– 19 –\n(see, e.g., Van Doorsselaereet al. 2004). This issue will be further addressed by Arreguiet al.\n(2010).\nFigure 4a showsτD/τApversusLp/Lfor different values of l/a. These computations corre-\nspondtoan ionizationdegree ˜ µp=0.8,butequivalentcomputationsforothervaluesof ˜ µpprovide\nalmost identical results because the e ffect of Cowling’s di ffusion is negligible in comparison to\nthat of resonant absorption. As expected, the value of the da mping time decreases with l/a. The\napproximate value of τDgiven by Equation (34) is in good agreement with the full solu tion for\nLp/L/lessorsimilar0.4, as happens for the period. In order to assess the e fficiency of the resonant damp-\ning, Figure 4b displays the corresponding values of τD/P. In comparison to the damping ratio by\nCowling’s diffusion (see Fig. 3c), much smaller values of τD/Pare now obtained. As predicted\nanalytically by Equation (32), τD/Pis almost independent of Lp/L. By comparing Figures 3a and\n4a, we see that both the period and the damping time have a very similar dependence on Lp/L, so\nthedependenceon Lp/Liscanceledwhenthedampingratioiscomputed. Inthiscase, averygood\nagreementbetweenthenumericalresultandtheanalyticala pproximation(Equation(32))isfound\nevenforlargevaluesof Lp/L.\n4.2. Effectofthe positionofthe prominence thread withinthe magne tictube\nInthisSectionwestudythee ffectofthepositionoftheprominenceregionwithinthemagne tic\nflux tube. The results of the previous Section 4.1 correspond to the case in which the thread\nis located at the center of the cylinder. Here, we allow the de nse region to be displaced from the\ncenterofthetube. Hence,wemustconsiderthegeneraldispe rsionrelationgivenbyEquation(11).\nThe dispersion relation is solved numerically for the lowes t frequency solution, equivalent to the\nfundamentalkinkmodeofthecentered case.\nWe display in Figure 2b the A(z) function for different values of L−\ne/LforLp/L=0.2 and\n˜µp=0.8. Since the oscillation is dominated by the prominence phys ical properties, we see that\nthe maximumof A(z) is always in theprominenceregion (denoted by thethick par t of thelines in\nFig. 2b),regardlessofitslocationwithintheflux tube.\nNext, we plot the period (Fig. 5a) and the damping time (Fig. 5 b) as functions of L−\ne/Lfor\nLp/L=0.1, ˜µp=0.8, and different values of l/a. We obtain that the longest period takes place\nwhentheprominencethreadiscenteredwithinthefluxtube,i .e.,when L−\ne/L=1\n2/parenleftig\n1−Lp/L/parenrightig\n=0.45\nfor this particular set of parameters, and Pdecreases symmetrically around L−\ne/L=0.45 when\nL−\ne/Lincreases or decreases. The dependence of τDonL−\ne/Lshows the same behavior as P.\nSuch as happens with the dependence on Lp/L(Fig. 4), the dependence on L−\ne/Lalso cancels\nout when the damping ratio is computed (see Fig. 5c). Hence, i n our model the value of τD/Pis– 20 –\nFig. 5.— (a) Period, P, of the fundamental kink mode in units of the internal Alfv´ e n travel time,\nτAp, as a function of L−\ne/L. The symbols are the approximation given by Equation (22). ( b)\nDamping time,τD, in units of the internal Alfv´ en travel time, τAp, as a function of L−\ne/L. The\ndifferent lines denote l/a=0.05 (dotted), 0.1 (dashed), 0.2 (solid) and 0.4 (dash-dott ed). The\nsymbolsare the approximationgivenby Equation(35) with l/a=0.2. (c)τD/PversusL−\ne/L. The\nlinestyleshavethesamemeaningasinpanel(b), andthesymb olsaretheapproximationgivenby\nEquation(32) with l/a=0.2. In all computations, Lp/L=0.1.– 21 –\nindependentofboth Lp/LandL−\ne/L. WecanalsoseeinFigure5cthattheapproximate τD/Pgiven\nby Equation (32) remains valid even when the thread is not loc ated at the center of the magnetic\ntube.\n5. IMPLICATIONSFOR PROMINENCESEISMOLOGY\nThe obtained results have direct implicationsfor the deter mination of physical parameters in\nprominence fine structures using MHD seismology together wi th observed periods and damping\ntimes. The technique of MHD seismology has been previously a pplied by some authors to ob-\ntaininformationoftheplasmaphysicalconditionsintheco ntextofcoronallooposcillations(e.g.,\nNakariakov& Ofman 2001; Arreguiet al. 2007, 2008a; Goossen set al. 2008), prominenceglobal\noscillations (e.g., Roberts 1991; Regnieretal. 2001; Poug et et al. 2006), and prominence thread\noscillations(e.g.,Terradas etal.2008;Lin et al.2009). A spartialionizationhasanegligiblee ffect\non the damping of oscillations we here concentrate on the inv ersion of parameters using theoret-\nical results for resonantly damped eigenmodes. Following t he analytical and numerical inversion\nschemesbyArreguiet al.(2007)andGoossenset al.(2008)fo rcoronalloops,Arregui& Ballester\n(2010)haverecentlypresentedtheinversionofprominence Alfv´ enspeed,transverseinhomogene-\nity length-scale, and density contrast in oscillating filam ent threads using results for resonantly\ndamped kink oscillations in one-dimensional (1D) filament t hread models. A similar procedure\ncan be followed with two-dimensional (2D) threads by consid ering the impact that the length of\nthethreadhasontheperiodanddampingratioofstandingkin kmodesinpartiallyfilledfinestruc-\ntureoscillations.\nInthefollowingcomputations,weconsiderthatthepromine ncethreadislocatedatthecenter\nofthemagneticfluxtube. Inthe2Dcaseandforsmall Lp/L,i.e,Lp/L/lessorsimilar0.4,theoscillatoryperiod\nisapproximatelygivenbyEquation(23)forarbitrarydensi tycontrast,andbyEquation(24)inthe\nlimitoflargedensitycontrast. In the1Dcase, thekinkmode periodin theTT approximationis\nP≈λ\nvAp/radicaligg\nρp+ρc\n2ρp, (36)\nwhereλis the wavelength. By comparing Equations (23) and (36), we s ee that the expression in\nthe1D case is equivalentto that in the2D case ifa particular or effectivevalueofλis considered,\nnamelyλ≈π/radicalig/parenleftig\nL−Lp/parenrightig\nLp. This effective value ofλallows us to generalize the 1D inversion to\na 2D configuration. Equation (23) is not accurate enough when Lp/L/greaterorsimilar0.4, but we can still use a\nsimilarexpressionfortheperiodforany Lp/Las\nP≈Cπ\nvAp/radicaligg\nρp+ρc\n2ρp/radicalig/parenleftig\nL−Lp/parenrightig\nLp, (37)– 22 –\nFig. 6.—(a)Inversionofphysicalparametersinthe( ρp/ρc,l/a,vAp)spaceforaprominencethread\noscillation with P=20 min andτD/P=3, and for different values of Lp/L(indicated within\nthe Figure). The thin continuous, dotted, and dashed lines c orrespond to the projections of the\nthree-dimensional curves to the ( ρp/ρc,l/a)-, (l/a,vAp)-, and (vAp,ρp/ρc)-planes, respectively. (b)\nInversion of the prominenceAlfv´ en speed, vAp, as a function of the period, P, in the limit of large\ndensity contrast. The di fferent lines correspond to Lp/L=0.2 (solid), 0.4 (dotted), 0.6 (dashed),\nand 1 (dot-dashed). The shaded zone corresponds to the range of typically observed periods in\nthreadoscillations. (c)Magneticfield strength, B0,asafunctionoftheprominencethread density,\nρp, assumingan oscillatoryperiodof P=26 min. Thedifferentlineshavethesamemeaningasin\npanel (b). In allthesecomputations, L=105km.– 23 –\nwithCa correction factor that is computed by comparing the period given by Equation (23) to\nthe general result of Figure 3a (solid line), obtained by sol ving the full dispersion relation (Equa-\ntion(16)). Thus,λ≈Cπ/radicalig/parenleftig\nL−Lp/parenrightig\nLpinthegeneralcase. Forexample,byassuming L=105km,\nweobtain C≈1.05andλ≈9.89×104kmforLp/L=0.1,whereas C≈1.58andλ≈1.98×105km\nforLp/L=0.8. As expected,λ→2L=2×105kminthelimit Lp/L→1.\nOn the other hand, the damping ratio is una ffected by the length of the thread and the same\nexpression for 1D models given by Equation (32) holds. By fol lowing the analytical inversion\nscheme by Goossenset al. (2008) the valid equilibrium model s that explain equally well a given\nset of parameters ( P,τD, andL) are obtained. The resulting one-dimensional curve in the t hree-\ndimensionalparameterspaceisshowninFigure6a,fordi fferentvaluesofthelengthofthethread.\nRegardlessofthevalueof Lp/L,theinversioncurveallowsustoobtainwellconstrainedva luesfor\nvApandl/ain the limit of high density contrast values. Because of the d ecrease inλproduced by\nthedecrease ofthelength ofthethread, theobtainedAlfv´ e n speed intheprominencedecreases as\nLp/Lgets smaller. An accurate estimateofthe lengthofthethrea d, incomparison to thelengthof\nthemagneticfluxtube,isthereforecrucial forthedetermin ationoftheAlfv´ en speedinthethread.\nAspointedoutbyD´ ıazet al.(2010), Lp/Lcanbeobtainedfromtheratioofthefundamentalmode\nperiod to that of the first harmonic, if these values are repor ted from the observations. However,\nbecause of the independence of the damping ratio on Lp/L, the projection of the solution curve\nonto the (ρp/ρc,l/a)-plane remains unaltered regardless of the value of Lp/L. Hence, the inverted\nvalueofthetransverseinhomogeneitylength-scale,which isobtainedtakingthelimit ρp/ρc→∞,\nisnot affected bydifferent valuesofthelengthofthethread.\nThe value of Phas a direct impact on the seismological determination of th e Alfv´ en speed,\nwith the inversion of Figure 6a corresponding to P=20 min. The effect of the period on the\ndetermination of the prominence Alfv´ en speed is indicated in Figure 6b, where we see that the\nAlfv´ en speed decreases as the period grows. Here, it is impo rtant recalling that the usually re-\nported periods of oscillating threads (e.g., Lin etal. 2007 , 2009; Okamotoet al. 2007; Ninget al.\n2009) are in the range 2 – 10 min. Note the very large values of vApobtained in the range of\nobserved periods (shaded zone in Fig. 6b). By assuming ρp=5×10−11kg m−3andB0=5 G\nas typical values of the density and magnetic field strength o f quiescent prominences that can be\nfound in the literature, respectively, the corresponding A lfv´ en speed is vAp≈63 km/s, which for\nL=105kmandLp/L=0.2givesP≈23.5min. Hence,toobtainrealisticvaluesof vAp,wehaveto\nconsider larger periods than those usually observed. Equat ion (22) indicates that, for the same set\nof parameters, the period decreases if the thread is displac ed from the center of the magnetictube\n(seeFig.5a). Themaximumshiftoftheperiodwithrespectto thevalueinthecenteredcaseispro-\nportionalto/radicalbigLp/L. Adoptingthesameparametersasbefore,theperiodvariesf romP≈23.5min\ntoP≈9.4 min when the thread is displaced from the center to the end of the magnetic tube, and\nso the period enters within the observed range. However, the re is no solid basis to assumethat all– 24 –\nshort-period oscillating threads are located at the ends of their magnetic tubes. Alternatively, the\npresence of flows can also shift the oscillatory period. Alth ough mass flow has not been included\nin ourmodel,thetime-dependentsimulationsofflowingthre ads by Terradas et al. (2008) indicate\nthatflowhasaminorinfluenceontheperiodbecausetheflowvel ocitiesaremuchsmallerthanthe\nAlfv´ en speed. In active region prominences, the larger mag netic field strength could cause larger\nAlfv´ en speeds, hence thread standing oscillations in acti ve region prominences may have shorter\nperiodsthan inquiescentprominences.\nMoreover,Lin et al.(2007,2009)andNinget al.(2009)repor tedthatthewavelengthsofthese\nshort-period oscillationsare in the range 700 – 8,000 km. Th e observed wavelengths are between\n1 and 2 orders of magnitude smaller than the wavelengths corr esponding to standing oscillations\ncomputed from Equations (36) and (37). Thus, the reported sh ort wavelengths are impossible to\nreconcile with the fundamental standing mode of the fine stru cture. These results suggest that the\nobserved short-period and short-wavelength oscillations of threads in quiescent prominences may\nnotbeconsistentwithaninterpretationintermsofstandin gkinkoscillations. Amorelikelyexpla-\nnation of these short-period and short-wavelengthoscilla tionsin terms of propagatingkink waves\nhasbeenperformedbyLin et al.(2009). Theseauthorsseismo logicallyinferred realisticvaluesof\nthe Alfv´ en speed and magnetic field strength by assuming a pr opagating wave interpretation (see\ndetailsinLin et al.2009).\nThe limited duration of the currently available Doppler tim e series may prevent the observa-\ntionofstandingmodeswithperiodslargerthan10min(e.g., thetimeserieslastforonly18minin\ntheobservationsofLin etal.2007). However,thereareafew evidencesoflargerperiodsinlonger\ntimeseriesthatcouldbeconsistentwithstandingmodes. Th eselongerperiodshavebeenobtained\nfrom Doppler signals that have been averaged over a large are a. As the spatial scales of standing\nmodesareverylarge,theiroscillatorypatternscouldbefo undinspatiallyaveragedsignals. Onthe\ncontrary, the averaging process could mix signals coming fr om different adjacent threads, and so\nthere is no full confidence that the period corresponds to an i ndividualthread oscillation. Yiet al.\n(1991)detectedthread oscillationswithperiodof16minan dminimumwavelengthof2 ×104km\nin their Doppler observations with low spatial resolution ( 1′′), while Lin (2004) reported 26 min\nand minimumwavelengthof 4 ×104km intheiraveraged Dopplersignals. Althoughnot onlythe\nperiods but also the wavelengths reported in these two works are consistent with a standing oscil-\nlation, we have to be very cautious with this interpretation . Additional information as, e.g., the\npolarisation of the oscillations and the phase di fference between perturbations, would be needed\nforamorerobustanalysisand theunequivocaldeterminatio nofthewavemode.\nTo perform a simple application, let us assume that the perio d of 26 min reported by Lin\n(2004) corresponds to a standing thread oscillation. Then, the estimation of the magnetic field\nstrengthintheprominencethreadispossible. Followingth eanalysispresentedbyLinet al.(2009)– 25 –\nusingtheobservationalperiodandtheseismologicallydet erminedAlfv´ enspeed(asinFig.6a),the\nmagnetic field strength can be computed for a given prominenc e density. This result is plotted in\nFigure 6c. For a typical density of ρp=5×10−11kg m−3, magnetic field strengths in the range\n4 – 7 G are obtained, approximately, when varying Lp/Lbetween 0.2 and 1. These values of the\nmagnetic field strength are in agreement with previous magne tic field measurements in quiescent\nprominences using the Hanle e ffect (e.g., Leroyet al. 1984). Hence, the method can be applie d in\nthefutureusingreal dataifreliableobservationsofstand ingthread oscillationsare reported.\n6. CONCLUSION\nIn this paper, we have investigated standing kink oscillati ons of prominence fine structures.\nThe longitudinal nonuniformity has been taken into account by modeling the fine structure as a\nmagnetic tube only partially filled with the prominence mate rial. We have followed an analytical\nmethod based on the TT approximation and have found a dispers ion relation for kink oscillations\ndamped by Cowling’s di ffusion and resonant absorption in the TB approach. This dispe rsion re-\nlation has been numerically solved and a parametric study of the solution has been performed. In\naddition to the general dispersion relation, we have obtain ed simple analytical approximations to\ntheperiod,thedampingtime,and theirratio.\nBoth approximate and full results conclude that resonant ab sorption is much more e fficient\nthan Cowling’s diffusion for the kink mode damping, with the values of τD/Pin agreement with\nthosereportedintheobservations. Ashappensforlong-wav elengthpropagatingwavesinthedense\npartofthefinestructure(Soler etal.2009c),theprominenc eplasmaionizationdegreeturnsoutto\nbe irrelevant for the resonant damping of the oscillations. In addition, the value of τD/Pis found\nto be independent of both the position of the prominence thre ad within the magnetic tube and the\nlengthoftheprominenceregion,andcoincideswiththevalu eforahomogeneousandfullyionized\nprominencetube(Arreguiet al. 2008a; Soleret al. 2009a).\nFinally, we have discussed the seismological implications of our analytical results, in par-\nticular Equations (24) and (33). With these expressions, it is possible to estimate some relevant\nphysical parameters of oscillating threads if the values of the period and damping time are avail-\nable from the observations. Following this idea, we have per formed a seismological inversion of\nthe prominence thread Alfv´ en speed and the transverse inho mogeneity length-scale by using our\ntheoretical results and adopting ad-hoc values for the peri od and damping time. We have shown\nthat for short-period (2 – 10 min) and short-wavelength (700 – 8,000 km) thread oscillations, the\ndetermined Alfv´ en speeds are much larger than the expected , realistic values, pointing out that\nshort-period and short-wavelength thread oscillations ma y not be consistent with a standing kink\nmodeinterpretationandcouldberelatedtopropagatingwav es. Onthecontrary,threadoscillations– 26 –\nwith periods larger than 10 min and wavelengths larger than 1 04km may be interpreted as stand-\ning oscillations. In this last case, the Alfv´ en speed and ma gnetic field strength estimated by the\nseismological inversion are realistic in the context of pro minences. Thus, the method can be put\ninto practice to extract indirect information of prominenc es when standing thread oscillations are\nunequivocallyobservedandtheoscillationparameters,i. e.,period,dampingtime,andwavelength,\nalongwiththethread lengthareprovidedfrom theobservati ons.\nIn this work, we have assumed that there is an abrupt jump of th e density at the boundary\nbetween the prominence thread and the evacuated part of the m agnetic tube. This simplifica-\ntion has allowed us to proceed analytically. Actually, one s hould expect a continuous variation\nof the plasma properties in the longitudinal direction betw een both regions, which could a ffect\nsomehow our present results. The study of the damping of kink oscillations in fully nonuniform\ntwo-dimensional fine structures is broached numerically by Arreguiet al. (2010) in a following\ninvestigation. Otheradditionalingredientsas,e.g.,mag netictwistorcurvatureandthepresenceof\nflowsmightbeincludedinfutureworks.\nWe acknowledge the unknown referee for valuable comments. R S thanks Marcel Goossens\nfor some useful discussions and for the kind hospitality dur ing his stay in Leuven, where part\nof this work was performed. RS also thanks Jaume Terradas for helpful suggestions. The au-\nthors acknowledge the financial support received from the Sp anish MICINN and FEDER funds\n(AYA2006-07637). The authors also acknowledge discussion within ISSI Team on Solar Promi-\nnenceFormationand Equilibrium: Newdata, newmodels. RS th ankstheCAIB fora fellowship.\nA. DERIVATIONOFTHEDISPERSION RELATION\nHere,wegiveextensivedetailsaboutthemethodthatleadsu stothedispersionrelation(Equa-\ntion(11)).\nA.1. Boundary conditions at r=a\nFirst, we must consider appropriate boundary conditions fo r the solutions of Equations (6)\nand (10) at the cylinder edge, i.e., r=a. In the evacuated part of the tube, we assume ρe=ρc\nand there is no transverse transitional layer. Hence, the bo undary conditions are those given by\nDymova&Ruderman (2005)intheirEquation(4),namely\n/bracketleftbig/bracketleftbigpT/bracketrightbig/bracketrightbig=0,[[vr]]=0,atr=afor|z|>Lp/2, (A1)– 27 –\nwhere[[X]]standsforthejumpofthequantity X.\nOn the other hand, in the prominence part of the tube we consid er the effect of resonant ab-\nsorptioninthetransitionallayer. Wefollowthetreatment byAndrieset al.(2005),whogeneralize\nthe concept of the jump conditions at the resonance position of Sakurai et al. (1991) to the case\nofalongitudinallyinhomogeneoustube. Andrieset al. (200 5)combinedthejumpconditionswith\nthe TB approximation,i.e, l/a≪1, to obtain analytical expressionsof the dispersionrelat ion and\nthe frequency for longitudinally stratified tubes. The accu racy of this analytical method was nu-\nmerically verified by Arreguiet al. (2005), who found a good a greement between the expressions\nof Andrieset al. (2005) and their numerical computations. I n the thin tube approximation, i.e.,\na/L≪1,Dymova& Ruderman(2006)followasimilarformalismandal soprovideequivalentex-\npressions for the jump conditions that can be applied to our p erturbations. Hence, it is convenient\ntoexpress pTandvras\npT=∞/summationdisplay\nn=1pTnGn,vr=∞/summationdisplay\nn=1vrnGn, (A2)\nwherepTnandvrnare the coefficients of the series expansions of pTandvr, respectively, with\nrespect tothefunctions Gndeterminedby theSturm-Liouvilleproblem\nv2\nA(r)d2Gn\ndz2=−λ2\nn(r)Gn, (A3)\nwithappropriateboundary conditionsfor Gnatz=±Lp/2, withλ2\nnthecorrespondingeigenvalues.\nEquation (A3) describe the spectrum of Alfv´ en modes, with λn(r)the corresponding frequencies\nof the Alfv´ en continuum. In general, it is not straight-for ward to deduce the boundary conditions\nforGnatz=±Lp/2 because they are givenby the continuityof Gnatz=±Lp/2, and thevalue of\nGnatz=±Lp/2 is also determined by the properties of the evacuated regio n. In a longitudinally\nhomogeneous tube, i.e., for Lp=Land with the Alfv´ en speed depending on the radial direction\nonly,we simplyhavethat the boundaryconditionsare Gn(±L/2)=0 and obtainλn(r)≡ωA(r)=\nnπ\nLvA(r), withn=1,2, ...\nInournotation,thejumpconditionsfor pTnandvrnprovidedbyDymova& Ruderman(2006)\nare\n/bracketleftbig/bracketleftbigpTn/bracketrightbig/bracketrightbig=0,[[vrn]]=−πωRm2/a2\n|ρ0∆n|rAnpTn,atr=rAnfor|z|<Lp/2,(A4)\nwhererAnis the Alfv´ en resonance position for the nth mode and∆n=d\ndr/parenleftig\nω2\nR−λ2\nn/parenrightig\n, withωRthe\nreal part of the frequency. According to Equation (A2), the c ondition for pTnin Equation (A4)\nleads to/bracketleftbig/bracketleftbigpT/bracketrightbig/bracketrightbig=0 atr=rAnfor|z|<Lp/2. On the contrary, the condition for vrndepends on\n|ρ0∆n|rAn, and amoredetailedanalysisisneeded.\nForoursubsequentanalysis,wedonot need theprecisevalue ofλn(r)but onlyitsfunctional\ndependenceontheradialdirection. Forthegivensinusoida ldensityprofileinthetransitionallayer,– 28 –\nwecan expresstheAlfv´ en speedsquared in thetransitional layer asv2\nA(r)=v2\nAp/f(r), with\nf(r)=1\n2/braceleftigg/parenleftigg\n1+ρc\nρp/parenrightigg\n−/parenleftigg\n1−ρc\nρp/parenrightigg\nsin/bracketleftbiggπ\nl(r−a)/bracketrightbigg/bracerightigg\n. (A5)\nHence, Equation(A3)isrewrittenas\nv2\nApd2Gn\ndz2=−λ2\nn(r)f(r)Gn. (A6)\nWithnolossofgenerality,wecanassumethat Gnisonlyafunctionof z,i.e.,thedifferentmagnetic\nsurfaces are not coupled to each other. So, according to Equa tion (A6), the quantity λ2\nn(r)f(r)\ncorresponds to the Alfv´ en eigenvalue squared in the promin ence part of the tube. Since ρpis\nhomogeneous, its corresponding Alfv´ en eigenvalue does no t depend on r, meaning that the radial\ncontribution ofλ2\nn(r)andf(r)cancel out. Thus, we define λ2\npn≡λ2\nn(r)f(r), withλpna constant\ncorresponding to the Alfv´ en eigenvalue in the homogeneous prominence thread. Therefore, we\nhave\nλ2\nn(r)=λ2\npn\nf(r), (A7)\nwhere all the radial dependence of λ2\nn(r)comes from the function f(r). With the help of Equa-\ntion (A7), we obtain that ∆n=λ2\nn(r)f′(r)/f(r), where the prime denotes the radial derivative.\nFinally,weusetheresonant condition,namely λ2\nn(rAn)=ω2\nR, and write\n|ρ0∆n|rAn=ω2\nR|∂rρ0|rAn. (A8)\nThus,theconditionfor vrninEquation(A4)becomes\n[[vrn]]=−πm2/a2\nωR|∂rρ0|rAnpTn,atr=rAnfor|z|<Lp/2. (A9)\nThe value of rAnis in principle different for each value of nand could be determined from the\nresonant condition λ2\nn(rAn)=ω2\nRif the eigenvaluesλ2\nn(r)werea prioriknown and the number of\nAlfv´ en eigenmodes that are resonant to the kink modeis also known. Hence, the derivativeof the\ndensity profile at each resonant position could be computed. A reasonable assumption in the TT\nandTBlimitsistoconsider rAn≈aforalln,sothat|∂rρ0|aisaconstantindependentof n,meaning\nthatweareassumingthatallresonancestakeplaceatthesam eposition. Foroursinusoidalprofile,\n|∂rρ0|a≈π/parenleftig\nρp−ρc/parenrightig\n/2l. Note that this is not a very strong restriction for the funda mental kink\nmode, sinceit is likely that the resonant conditionis satis fied forn=1 only, becauseλ2\nn(r)grows\nasnincreasesandsoonlyoneresonancetakesplace. Theapproxi mationrAn≈amightnotbevalid\nforthekinkmodeovertones,butherewerestictourselvesto thefundamentalmode. Alternatively,\na simplerlinear density profile could be adopted (Goossense t al. 2002) in which the derivativeof– 29 –\nthe densityprofile does not depend on theposition. Therefor e and using Equation (A2), we arrive\nat\n[[vr]]=−πm2/a2\nωR|∂rρ0|apT,atr=afor|z|<Lp/2. (A10)\nForLp=LthejumpconditionofEquation(A10)consistinglyreducest othatprovidedbySakurai et al.\n(1991).\nA.2. Solutioninthe evacuatedregions\nLet us consider first the boundary conditions for the evacuat ed parts (Equation (A1)). The\nanalysis here is identical to that of Dymova& Ruderman (2005 ). For the condition on the total\npressureperturbation,weobtain Ae(z)=Ac(z)=A(z). Next,werewriteEquation(4)as\nv2\nA0∂2vr\n∂z2+ω2vr=−iω\nρ0∂pT\n∂r. (A11)\nWe evaluate Equation (A11) for r≈aon both sides of the tube boundary. Thus, in the evacuated\npart,\nv2\nAe∂2vre\n∂z2+ω2vre=−iω\nρem\naA(z),forr/lessorapproxeqla, (A12)\nwhereas in thecorona,\nv2\nAc∂2vrc\n∂z2+ω2vrc=iω\nρcm\naA(z),forr/greaterorapproxeqla. (A13)\nAccording to the boundary condition for vrgiven by Equation (A1), vre=vrc. Thus, we combine\nEquations(A12)and (A13)tofind thefollowingtwoexpressio ns\n∂2vr\n∂z2=iωm\naµ\nB2\n0ρe+ρc\nρe−ρcA(z), (A14)\nvr=−im\na1\nω2\nρe−ρcA(z), (A15)\nwhere we have considered that the magnetic field is homogeneo us. Now, we differentiate Equa-\ntion (A15) with respect to ztwice and compare the resulting expression with Equation (A 14). We\nobtain\nd2A(z)\ndz2+ω2\nc2\nkeA(z)=0. (A16)\nThe quantity ckecorresponds to the kink speed in the evacuated region (Equat ion (14)). Note that\ninourparticularapplication ρe=ρc,socke=vAe. However,wekeepthegeneralnotation ckeinthe\nfollowingexpressions.– 30 –\nTosolveEquation(A16),weconsidertheline-tyingconditi onatthephotosphere,i.e., A/parenleftig\nz±\nwall/parenrightig\n=\n0. Therefore, thesolutioninthetwoevacuatedzones is\nA(z)=C1sin/bracketleftig\nω\ncke/parenleftig\nz−Lp/2−L+\ne/parenrightig/bracketrightig\n,forz>Lp/2,\nC2sin/bracketleftigω\ncke/parenleftig\nz+Lp/2+L−\ne/parenrightig/bracketrightig\n,forz<−Lp/2,(A17)\nwhereC1andC2are constants.\nA.3. Solutionintheprominence thread\nIn the prominence thread, we adopt the TB approach and use the jump conditions given by\nEquation (A4) as our boundary conditions. The combination o f both formalisms to investigate\nresonant waves in coronal flux tubes has been reviewed by Goos senset al. (2006). Again, the\ncondition over the total pressure perturbation gives Ap(z)=Ac(z)=A(z). Near the boundary\nwe express vrc=vrp+δvr, whereδvris the jump of the radial velocity perturbation provided by\nEquation(A10), namely\nδvr=−πm2/a2\nωR|∂rρ0|apT. (A18)\nAs before, we evaluate Equation (4) on both sides of the tube b oundary and, after some algebra,\nwearriveat thefollowingexpressions\n∂2vrp\n∂z2=m\naiω\nv2\nAc−Γ2\nApρp+ρc\nρpρcA(z)\n+m2/a2\nωR|∂rρ0|aπ\nv2\nAc−Γ2\nAp/bracketleftigg\nv2\nAcd2A(z)\ndz2+ω2A(z)/bracketrightigg\n, (A19)\nvrp=−m\nai\nωρpΓ2\nAp+ρcv2\nAc\nρpρc/parenleftig\nv2\nAc−Γ2\nAp/parenrightigA(z)\n−π\nω2Γ2\nAp\nv2\nAc−Γ2\nApm2/a2\nωR|∂rρ0|a/bracketleftigg\nv2\nAcd2A(z)\ndz2+ω2A(z)/bracketrightigg\n. (A20)\nNow,wedifferentiateEquation(A20)withrespectto ztwiceandcomparetheresultingexpression\nwithEquation(A19), obtaining\nd4A(z)\ndz4+ω2Γ2\nAp+v2\nAc\nΓ2\nApv2\nAc+iω\nπωR|∂rρ0|a\nm/aρpΓ2\nAp+ρcv2\nAc\nρpρcΓ2\nApv2\nAcd2A(z)\ndz2\n+ω2ω2\nΓ2\nApv2\nAc+iω\nπωR|∂rρ0|a\nm/aρp+ρc\nρpρcΓ2\nApv2\nAcA(z)=0. (A21)– 31 –\nThegeneralEquation(65)ofDymova&Ruderman(2006)andour Equation(A21)areequivalent\nif a constant piecewise density is assumed in the former and C owling’s diffusion is omitted in the\nlatter. Equation (A21) can be solved by taking a solution of t he form exp (ikzz)and obtaining the\nsubsequent forth-order polynomial for kz. Two independent values of kzare possible, namely kz1\nandkz2. Thus,thegeneral solutionofEquation(A21)is\nA(z)=D1exp(ikz1z)+D2exp(−ikz1z)+D3exp(ikz2z)+D4exp(−ikz2z),(A22)\nwithD1,D2,D3, andD4constants that are determined by the boundary conditions at z=±Lp/2.\nHowever,to keep thisgeneral analysis impliesthat thefoll owingexpressionsare complicatedand\nrequire an additional mathematical e ffort. Instead, we choose a more restrictive way to simplify\nmatters.\nForournextanalysis,Equation(A21)isrewrittenin aconve nientform as\nb2\nω2d4A(z)\ndz4+d2A(z)\ndz2+ω2\n˜c2\nkpA(z)=0, (A23)\nwhere ˜c2\nkpandb2are defined inEquations(12)and(13).\nIn thecaseofthefundamentalmode,onecouldassumethat,wh en thetermsrelatedtoCowl-\ning’sdiffusionandresonantabsorptionarepresent,thecharacteris ticscaleforthevariationsofthe\neigenfunctions in the z-direction is only slightly modified with respect to the idea l case without\ntransitional layer. Therefore, a reasonable approximatio n is to relate the forth-order derivative of\nA(z)inEquation(A23)withthesecond-orderderivativeas follo ws\nd4A(z)\ndz4∼−K2d2A(z)\ndz2, (A24)\nwhere the quantity Kplays the role of the longitudinal wavenumber. We can approx imateKby\nitsexpressionintheidealcase, namely\nK2≈ω2\nc2\nkp, (A25)\nwithc2\nkptheidealkinkspeed. Hence, Equation(A23)becomes\nd2A(z)\ndz2+ω2\n˜c2\nkp/parenleftbigg\n1−b2\nc2\nkp/parenrightbiggA(z)≈0, (A26)\nwhich is formally identical to Equation (A16). It is importa nt recalling that the approximation\nof the forth-order z-derivative of A(z)may introduce some uncertainty in the solutions of Equa-\ntion (A26) in comparison with the solutions of the full Equat ion (A21). However, we expect a– 32 –\nminor discrepancy in the case of the fundamental mode becaus e its characteristic scale in the z-\ndirection should not be essentially modified when the terms r elated to Cowling’s di ffusion and\nresonantabsorptionare takenintoaccount intheequations .\nThesolutionofEquation(A26)is\nA(z)=E1cosω\n˜ckp/radicalig\n1−b2\nc2\nkpz+E2sinω\n˜ckp/radicalig\n1−b2\nc2\nkpz,if|z|≤Lp/2,(A27)\nwithE1andE2constants. 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Phys.,132,63\nThispreprintwaspreparedwiththeAAS L ATEXmacrosv5.2."    },    {        "title": "2308.01711v1.Flavor_wave_theory_with_quasiparticle_damping_at_finite_temperatures__Application_to_chiral_edge_modes_in_the_Kitaev_model.pdf",        "content": "Flavor-wave theory with quasiparticle damping at finite temperatures: Application to chiral edge\nmodes in the Kitaev model\nShinnosuke Koyama1and Joji Nasu1\n1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan\n(Dated: August 4, 2023)\nWe propose a theoretical framework to investigate elementary excitations at finite temperatures within a lo-\ncalized electron model that describes the interactions between multiple degrees of freedom, such as quantum\nspin models and Kugel-Khomskii models. Thus far, their excitation structures have been mainly examined us-\ning the linear flavor-wave theory, an SU( N) generalization of the linear spin-wave theory. These techniques\nintroduce noninteracting bosonic quasiparticles as elementary excitations from the ground state, thereby elu-\ncidating numerous physical phenomena, including excitation spectra and transport properties characterized by\ntopologically nontrivial band structures. Nevertheless, the interactions between quasiparticles cannot be ignored\nin systems exemplified by S=1/2 quantum spin models, where strong quantum fluctuations are present. Re-\ncent studies have investigated the e ffects of quasiparticle damping at zero temperature in such models. In our\nstudy, extending this approach to the flavor-wave theory for general localized electron models, we construct a\ncomprehensive method to calculate excitation spectra with the quasiparticle damping at finite temperatures. We\napply our method to the Kitaev model under magnetic fields, a typical example of models with topologically\nnontrivial magnon bands. Our calculations reveal that chiral edge modes undergo significant damping in weak\nmagnetic fields, amplifying the damping rate by the temperature increase. This e ffect is caused by collisions\nwith thermally excited quasiparticles. Since our approach starts from a general Hamiltonian, it will be widely\napplicable to other localized systems, such as spin-orbital coupled systems derived from multi-orbital Hubbard\nmodels in the strong correlation limit.\nI. INTRODUCTION\nIn modern condensed matter physics, the topological nature\nof electronic systems is one of the crucial ingredients enrich-\ning physical phenomena. For example, in quantum Hall sys-\ntems, the Chern number, a topological invariant in electronic\nband dispersions, is closely related to a quantized value of the\nHall conductance and the number of chiral edge modes [1, 2].\nThis concept has been applied to magnons, which are collec-\ntive spin-wave excitations from a magnetically ordered state in\ninsulating magnets [3–5]; the Chern number of magnon bands\npossibly becomes nonzero, leading to the presence of chiral\nedge modes in the gap between the bands. Since magnons are\ncharge-neutral bosonic excitations, quantum Hall e ffects do\nnot occur. Instead, one expects the emergence of thermal Hall\neffects [3–5], which is a phenomenon exhibiting a thermal\ncurrent induced by a temperature gradient along the perpen-\ndicular direction when magnon bands possess nonzero Berry\ncurvature [6]. Such magnons are called topological magnons,\nusually caused by anisotropic spin interactions beyond the\nHeisenberg coupling or non-collinear magnetic orders. Thus\nfar, the emergence of topological magnons has been theo-\nretically proposed in spin systems with the Dzyaloshinskii-\nMoriya or Kitaev-type interactions [5, 7–12], and the ther-\nmal Hall e ffect has been observed the pyrochlore materials,\nLu2V2O7, Ho 2V2O7, In 2Mn 2O7and Tb 2Ti2O7[4, 13, 14],\nthe layered honeycomb material VI 3[15, 16], and the lay-\nered Kagome materials, Cu(1,3-benzenedicarboxylate) and\nCd-kapellasite [17, 18].\nSince magnons are collective excitations from a magneti-\ncally ordered state, they behave as bosons with zero chemical\npotential and possess positive energies. These characteristics\nlead to crucially di fferent behavior from fermionic systems.\nIn quantum Hall systems composed of electrons, the thermalHall coe fficient divided by temperature takes a nonzero quan-\ntized value determined by the Chern number of each band in\nthe zero-temperature limit. On the other hand, in its magnonic\ncounterpart, the thermal Hall coe fficient divided by tempera-\nture must vanish at zero temperature because chiral magnon\nedge modes appear not across zero energy but in between\npositive-energy bands [19, 20]. This consideration implies\nthat thermally excited magnons play a crucial role in the trans-\nport phenomena originating from the chiral edge modes.\nThe topological nature of collective excitations has been\ndiscussed not only on magnons but also on other bosonic\nquasiparticles, such as triplons from a singlet-dimer covered\nground state [21–23]. A comprehensive approach to include\nthese cases has been developed as the flavor-wave theory, an\nextension of the spin-wave theory. This method was origi-\nnally proposed for localized SU( N) systems [24–27], but it\ncan be applied to other systems with interacting local degrees\nof freedom. By using the generalized theory, it has been pro-\nposed that the Shastry–Sutherland model possesses the band\nstructures of triplon excitations with nonzero Chern numbers\ndue to Dzyaloshinskii–Moriya interactions [21].\nWhile the topological character of collective excitations has\nbeen examined under the assumption of noninteracting quasi-\nparticles within a linear flavor-wave approximation, nonlinear\nterms beyond this approximation are inevitably present, which\nappear due to quantum fluctuations intrinsic to localized spins.\nThis e ffect gives rise to interactions between collective ex-\ncitations and should be addressed appropriately, particularly\nforS=1/2 quantum spin systems. In magnetically ordered\nsystems, the interactions cause a finite lifetime to magnons,\nnamely the damping of magnons, at zero temperature [28–\n31] and finite temperatures [32]. The e ffects of the magnon\ndamping have been introduced to understand the broad spec-\ntrum observed in inelastic neutron scattering measurements,arXiv:2308.01711v1  [cond-mat.str-el]  3 Aug 20232\nand calculations incorporating the magnon damping have suc-\ncessfully explained experimental results. Nevertheless, it is\noften challenging to address the impact of the magnon damp-\ning on the topological nature of magnons and related phenom-\nena, such as the thermal Hall e ffect [5, 33].\nRecently, it has been reported that the thermal Hall conduc-\ntivity originating from collective excitations is much smaller\nthan the theoretically predicted value obtained within the free-\nmagnon picture in the quasi-two-dimensional quantum mag-\nnets SrCu 2(BO 3)2with triplon excitations [34] and Cr 2Ge2Te6\nwith magnon excitations [35]. These results suggest that the\ninteractions between quasiparticles are crucial in topological\ntransport phenomena. Therefore, clarifying the damping ef-\nfects of topological bosonic quasiparticles at zero and nonzero\ntemperatures is highly desired.\nOther well-known simple examples exhibiting topologi-\ncal excitations are the Kitaev-related systems under magnetic\nfields. In the pure Kitaev model [36], magnetic fields along\nthe [111] direction yield topologically nontrivial magnon\nbands [10, 37]. Moreover, recent theoretical studies based on\nthe linear spin-wave theory have reported that the Chern num-\nber of each magnon band changes by introducing other in-\nteractions, such as the Heisenberg one and varying magnetic-\nfield directions [10, 12, 37–45]. Correspondingly, the thermal\nHall e ffect has been observed in the Kitaev candidate mate-\nrialα-RuCl 3[15, 46–56]. Although the experimental results\nimply the presence of topological quasiparticles in this com-\npound, there is a debate about which quasiparticles are re-\nsponsible for the thermal Hall e ffect; several works have pro-\nposed that only magnons are responsible [15, 52], while oth-\ners have suggested Majorana fermions or phonons as possible\norigins [49, 50, 53, 55, 56].\nMeanwhile, it has been pointed out that strong magnon\ndamping is crucial in understanding spin dynamics [57, 58],\nsuggesting that nonlinear terms beyond the linear spin-wave\napproximations are also crucial for the topological nature\nof the magnons in the Kitaev-related model under magnetic\nfields. Topologically nontrivial magnon bands accompany\nchiral edge modes, and their stability has been examined at\nzero temperature [10]. On the other hand, it has been known\nthat possible magnon-damping processes at finite tempera-\ntures are entirely di fferent from that at zero temperature [29];\nthe latter only originates from spontaneous decay by splitting\ninto multiple magnons, but collisions with other thermally ex-\ncited magnons also contribute to the finite-temperature pro-\ncesses [32]. Therefore, examining the finite-temperature ef-\nfects of magnon damping on chiral edge modes is needed to\nclarify the role of topological magnons in thermal transport.\nIn this paper, we construct a calculation framework ca-\npable of handling nonlinear terms beyond the linear flavor-\nwave theory for generalized many-body models with interact-\ning local degrees of freedom at finite temperatures. We apply\nthis approach to the Kitaev model under magnetic fields. In\nthe present scheme, we start from the Hamiltonian consist-\ning of one-body and two-body terms for local degrees of free-\ndom and apply the mean-field (MF) theory with the arbitrary\nnumber of sublattices. Utilizing the generalized Holstein-\nPrimako fftransformation, we rewrite the model Hamiltonianinto a bosonic representation. This bosonic Hamiltonian con-\nsists of bilinear terms of bosons and higher-order terms be-\nyond these contributions. The former describes noninteract-\ning bosons. We treat the latter contributions corresponding\nto interactions between bosons by the self-consistent imag-\ninary Dyson equation (iDE) approach, which enables us to\navoid the appearance of unphysical divergences in excitation\nspectra. Here, we apply the present scheme to the Kitaev\nmodel under magnetic fields on a honeycomb lattice, which\npossesses magnon bands with nonzero Chern numbers. In\nthis model, the MF solution is a forced ferromagnetic state\nregardless of the magnetic-field intensity. To examine the\neffect of the magnon damping on edge states, we calculate\nmagnon damping rates and magnon spectra on clusters with\nopen boundaries. The chiral edge modes are strongly damped\nat zero temperature in weak magnetic fields due to a magnon\ncollapsing into two. The damping rate decreases with increas-\ning the intensity of an applied magnetic field and vanishes\nabove a certain field intensity. The disappearance of overlap\nbetween the magnon dispersions and the two-magnon contin-\nuum determines the critical value of the field intensity. With\nincreasing temperature, the damping rate of the magnon edge\nmodes increases due to another magnon damping process, col-\nlisions with thermally excited magnons. The damping rate is\nnonzero, even above the critical magnetic field. We demon-\nstrate that the nonlinear terms giving rise to the magnon damp-\ning strongly a ffect not only bulk spectra but also edge modes\nat finite temperatures. Our results suggest that the e ffect of the\nmagnon damping is relevant to finite-temperature topological\ntransport phenomena such as the thermal Hall e ffect.\nThis paper is organized as follows. In the next section,\nwe present the method used in the present study. The MF\ntheory and generalized Holstein-Primako fftransformation are\ndescribed in Sections II A and II B, respectively. We intro-\nduce the linear flavor-wave theory in Sec. II C. The extensions\nconsidering the nonlinear parts of flavor-waves are given in\nSec. II D. Section II E provides the method we have developed\nto evaluate e ffects of the damping of collective modes in the\nnonlinear flavor-wave theory at finite temperatures. In Sec. III,\nwe introduce the S=1/2 Kitaev honeycomb model to which\nour method is applied in the present study. The results are\ngiven in Sec. IV. First, we show the MF results for the two\nsystems with di fferent boundary conditions in Sec. IV A. In\nSec. IV B, we show the magnon spectra at low magnetic fields\nat zero temperatures. The results for higher magnetic fields\nat finite temperatures are given in Sec. IV C. In Sec. V, we\ndiscuss the magnetic-field dependence and temperature evo-\nlution of the magnon damping of chiral edge modes. We also\nmention the relevance to real materials and observables such\nas the thermal Hall e ffect. Finally, Sec. VI is devoted to the\nsummary.3\nII. METHOD\nA. Mean-field theory\nBefore showing the details of the flavor-wave theory, we\nfirst explain the MF approximation. We start from a general\nlocalized model, which is given by\nH=1\n2X\ni,jX\nγγ′Jγγ′\ni jOγ\niOγ′\nj−X\niX\nγhγ\niOγ\ni, (1)\nwhereOγ\nirepresents the γcomponent of the local operator\ndefined at site iwith the local dimension N,Jγγ′\ni jstands for\nthe interaction between the operators Oγ\niandOγ′\nj. The last\nterm of Eq. (1) is the one-body term with the local field hγ\ni\nfor the operatorOγ\ni. In the MF theory, each local operator is\ndecomposed by the local average and the deviation from it as,\nOγ\ni=δOγ\ni+⟨Oγ⟩l (2)\nwhere we assume that the local average ⟨Oγ⟩ldepends on sub-\nlattice lto which the site ibelongs. Here, we prepare Msub-\nlattices to realize a stable MF solution.\nThe original Hamiltonian given in Eq. (1) is decomposed to\nthe MF Hamiltonian HMFand the deviation from it, H′, as\nH=HMF+H′, (3)\nwhereHMFis given by the sum of the local Hamiltonians as\nHMF=X\niHMF\ni+const. (4)\nThe local MF Hamiltonian HMFat site iis represented as\nHMF\ni=X\nγX\nl′X\nj∈l′X\nγ′Jγγ′\ni j⟨Oγ′⟩l′−hγ\niOγ\ni. (5)\nHere,⟨Oγ⟩l=⟨0;i|Oγ|0;i⟩denotes the expectation value for\nthe ground state |0;i⟩of the local Hamiltonian HMF\niwith\nsiteibelonging to sublattice l. To obtain the ground state,\nwe diagonalize the N×N-matrix representation of HMF\ni\nand obtain the ground state |0;i⟩with the eigenenergy El\n0\nandm-th excited states |m;i⟩with the eigenenergy El\nmfor\nm=1,2,···,N−1. Note that the Hilbert space of the to-\ntal Hamiltonian is spanned by the direct product of the local\neigenstates|m;i⟩ofHMF\ni, but the eigenenergies and eigen-\nstates at sites belonging to the same sublattice are equivalent,\nrespectively. Namely, the eigenenergy and eigenstate are la-\nbeled not by the site index but by the sublattice index l.\nB. Generalized Holstein-Primako fftransformation\nThe excitation structure is described by the contribution be-\nyond the MF Hamiltonian, H′, which is given by\nH′=1\n2X\ni,jX\nγγ′Jγγ′\ni jδOγ\niδOγ′\nj. (6)We rewrite the above Hamiltonian using bosons to evaluate\nthe elementary excitations from the MF ground state. Here,\nwe expand Eq. (2) using the eigenstates of the local Hamilto-\nnian at site iin sublattice las\nδOγ\ni=N−1X\nm,m′=0Xmm′\niδOγ\nmm′;l, (7)\nwhere Xmm′\ni=|m;i⟩⟨m′;i|is the local projection operator at\nsiteiandδOγ\nmm′;l=⟨m;i|δOγ|m′;i⟩, which depends only on\nthe sublattice index lto which site ibelongs. The projec-\ntion operator is represented by bosons using the generalized\nHolstein-Primako fftransformation [24, 39, 59, 60]. We intro-\nduceN−1 bosons described by the creation (annihilation)\noperators a†\nmi(ami) with m=1,2,···,N−1. For m≥1,X0m\ni\nandXm0\niare written as\nXm0\ni=a†\nmiS−N−1X\nn=1a†\nniani1/2\nX0m\ni=\u0010\nXm0\ni\u0011†, (8)\nand, for 1≤m,m′,Xmm′\niis given by\nXmm′\ni=a†\nmiam′i. (9)\nNote thatSis introduced as\nS=X00\ni+N−1X\nn=1a†\nniani, (10)\nand it should be unity because of the local constraint intrin-\nsic to the projection operator,PN−1\nm=0Xmm\ni=1. The bosonic\nexpressions in Eqs. (8)–(10) reproduce the commutation rela-\ntions that the projection operators should satisfy. The above\nbosonic representation for the local operators has been used\nperticularly for SU( N) systems. The approaches applied to\nthese systems is known as the flavor-wave theory [24–27].\nAlthough the above bosonic representation using the gener-\nalized Holstein-Primako fftransformation is exact, the pres-\nence of the square root in Eq. (8) complicates further cal-\nculations. When the number of excited bosons is small\nenough, the square root can be expanded with respect to 1 /S\nas [24, 39, 59–61]\nXm0\ni=√\nSa†\nmi1−1\n2SN−1X\nn=1a†\nniani+O(S−3/2). (11)\nUsing this expression, H′in Eq. (6) is represented by the\nbosons and expanded for 1 /Sas\nH′=SH′\n2+√\nSH′\n3+H′\n4+O(S−1/2), (12)\nwhereH′\n2,H′\n3, andH′\n4are the terms composed of the products\nof two, three, and four boson creation /annihilation operators\ninH′.\nOn the other hand, the local MF Hamiltonian is given by\nHMF\ni=N−1X\nm=0El\nmXmm\ni=SEl\n0+N−1X\nm=1∆El\nma†\nmiami (13)4\nwhere ∆El\nm=El\nm−El\n0is the energy di fference between the\nexcited state and ground state of the local MF Hamiltonian at\nsiteibelonging to sublattice l. Note that the average ⟨Oγ⟩l\nfor the MF ground state is of O(S), which is understood from\nEq. (10). If one assumes that hγ\niis the order ofS, the local MF\nforOγ\niin Eq. (5) should be the order of O(S), and thereby, the\nenergies El\nm(m=0,1,···,N−1) are in this order. Thus, the\nfirst and second terms of Eq. (13) are in the orders of O(S2)\nandO(S), respectively. Here, we introduce HLFW, which is\ngiven by the sum of the second term of Eq. (13) and SH′\n2in\nEq. (12). Since these give contributions with O(S), the total\nHamiltonian is represented as\nH=S \nHLFW+1√\nSH′\n3+1\nSH′\n4+O(S−3/2)!\n+const.(14)\nNote that, while a†\nmiandamido not appear alone because of\nthe stable condition of the MF solution, other odd-order terms\nare allowed to be present in the Hamiltonian [29].\nC. Linear flavor-wave theory\nFirst, we consider HLFW, which is given by the bilinear\nfrom of amianda†\nmi. The approximation considering only\nHLFWis called the linear flavor-wave theory. By introducing\nthe Fourier transformation of amiwith respect to i, the Hamil-\ntonianHLFWis formally written as\nHLFW=1\n2B.Z.X\nkA†\nkMkAk, (15)\nwhereMkis a 2 N×2NHermitian matrix and N=(N−1)M\nis the number of collective mode branches (see Appendix A).\nThe 2 N-dimensional vector A†\nkis given by\nA†\nk=\u0010\na†\n1,ka†\n2,k···a†\nN,ka1,−ka2,−k···aN,−k\u0011\n, (16)\nwhere aι,kwithι=(ml) being the composite index of local\nexcited state mand sublattice lis the Fourier transformation\nofami, which is represented by\naι,k=r\nM\nNtX\ni∈lamie−ik·ri. (17)\nHere, riis the position of site i, and Ntis the number\nof sites. We diagonalize Mkby applying the Bogoliubov\ntransformation as Ek=T†\nkMkTk, where Tkis a parauni-\ntary matrix andEkis the diagonal matrix given by Ek=\ndiag{ε1,k,ε2,k,···,εN,k,ε1,−k,ε2,−k,···,εN,−k}[62]. Using\nthis transformation, the Hamiltonian is rewritten as the fol-\nlowing diagonalized form:\nHLFW=1\n2B.Z.X\nkB†\nkEkBk, (18)Here, we introduce the set of bosonic operators Bk=T−1\nkAk,\nwhich is given by\nB†\nk=\u0010\nb†\n1,kb†\n2,k···b†\nN,kb1,−kb2,−k···bN,−k\u0011\n, (19)\nwhere b†\nη,kis regarded as the creation operator of a quantized\nflavor-wave excitation with the energy εη,k. The sum of kis\ntaken in the first Brillouin zone.\nD. Nonlinear flavor-wave theory\nIn the previous section, we consider only the bilinear terms\nof bosonic operators. In this case, the Hamiltonian is writ-\nten as a free boson system without interactions, as shown\nin Eq. (18). Here, we address e ffects of the higher-order\nterms beyond the linear flavor-wave Hamiltonian HLFW in\nEq. (14). These contributions are treated as perturbation\nterms, andHLFW is regarded as an unperturbed term. As\ndiscussed in Sec. II B, H′is expanded with respect to 1 /S,\nand we take account of O(1/S) corrections from the bilinear\ntermHLFW [28, 30, 63, 64]. In this sense, we need to deal\nwithH′\n3/√\nSup to second-order perturbations and H′\n4/Sup\nto first-order perturbations [see Eq. (B15)].\nIn the present study, we focus on damping e ffects on the\nbosonic quasiparticles, and hence, we examine the imaginary\npart of the self-energy of the bosonic quasiparticles. Note\nthat the first-order perturbations contribute only to the real\npart of the self-energy [30–32, 64–67]. Thus, we concen-\ntrate second-order perturbations for the cubic term of bosons,\nH′\n3/√\nS. The cubic term can be decomposed into two terms,\nH′\n3/√\nS=H(d)\n3/√\nS+H(s)\n3/√\nS. The first term involves the\nprocess with a quasiparticle splitting into two particles and\nsecond term stands for the process of creating (annihilating)\nthree quasiparticles simultaneously. The details are given in\nAppendix B, and only the results are presented here as\n1√\nSH(d)\n3=1\n2!r\nM\nNtSNX\nηη′η′′k+q=pX\nkqp \n¯Vη,η′←η′′\nk,q←pb†\nη,kb†\nη′,qbη′′,p+H.c.!\n,\n(20)\n1√\nSH(s)\n3=1\n3!r\nM\nNtSNX\nηη′η′′k+q=−pX\nkqp \n¯Wη,η′,η′′\nk,q,pb†\nη,kb†\nη′,qb†\nη′′,p+H.c.!\n,\n(21)\nwhere ¯Vη,η′←η′′\nk,q←pand ¯Wη,η′,η′′\nk,q,pare the bare vertex functions,\nwhich is given explicitly in Eqs. (B10) and (B11), respec-\ntively. Figures 1(a) and 1(b) represent the schematic diagrams\nof the processes with ¯Vη,η′←η′′\nk,q←pand ¯Wη,η′,η′′\nk,q,p, respectively.\nHere, we treat Eqs. (20) and (21) as perturbation terms,\nwhere Eq. (18) is regarded as a unperturbed Hamiltonian. We\ncalculate the self-energy of bosonic quasiparticles, Σk(ω,T),\nwhich is defined in Eq. (B16) with Tbeing temperature. In the\npresent calculations, we focus on the quasiparticle damping\ncaused within each collective mode branch. To this end, we\nconsider only the imaginary part of the diagonal components5\nextracted from the self-energy Σk(ω,T) up to 1/Srelative to\nHLFW. We denote the self-energy to which the above simpli-\nfication is applied as ˜Ση,k(ω,T). This self-energy is split into\ntwo contributions:\n˜Ση,k(ω,T)= Σ(c)\nη,k(ω,T)+ Σ(d)\nη,k(ω,T), (22)\nwhere Σ(c)\nη,k(ω,T) and Σ(d)\nη,k(ω,T) are the contributions with\nO(1/S) from the diagrams shown in Figs. 1(c) and 1(d),\nrespectively. The explicit representations are given in\nEqs. (B22) and (B23). As shown in Fig. 1(c), Σ(c)\nη,k(ω,T)\ncomes from a collision with excited quasiparticles. At zero\ntemperature, there are no thermally excited quasiparticles, and\nhence, Σ(c)\nη,k(ω,T) is nonzero only at finite temperatures. On\nthe other hand, Σ(d)\nη,k(ω,T) originates from a decay process\nto multiple magnons. This process is not involved with ex-\ncited quasiparticles and contributes to the self-energy even\nat zero temperature. Note that there is another process in-\nvolved with ¯Wη,η′,η′′\nk,q,p[Fig. 1(b)]. This contribution, defined as\nΣ(s)\nη,k(ω,T), is represented by the diagram that the arrows of\nintermediate states in Fig. 1(d) are reversed [see Appendix B\nand Eq. (B24)]. Indeed, the imaginary part of Σ(s)\nη,k(ω,T) van-\nishes due to the energy conservation law.\nWe comment on the first-order perturbation for the cu-\nbic termH′\n3, which does not contribute to the self-energy in\nthe present scheme, as mentioned before. This contribution\ncan be considered by applying the Hartree-Fock (HF) decou-\npling toH′\n3. It has been reported that the correction originat-\ning from this e ffect modifies the MF solution in the ground\nstate [10, 30]. Namely, an MF value ⟨δOγ⟩l, which vanishes\nwithout the first-order perturbation of H′\n3, becomes nonzero\ndue to contributions from HF decouplings like ⟨a†a⟩a. Such\nterms destabilize the existing MF ground state of HMFand\ncompel us to find a new MF solution. After solving the self-\nconsistent equation determining MFs, we reconstruct a flavor-\nwave Hamiltonian with the newly found MFs. Together with\nthe HF decoupling for H4, we can take into account the HF\ncorrections up to the order of O(1/S). If we consider the real\npart of the self-energy at zero temperature, we need to address\nthem and Σ(s)\nη,k(ω,T) properly [30, 64].\nE. Imaginary Dyson Equation approach\nIn this section, we show the details of the iDE approach\nused in the present calculations. This method was developed\nin Refs. [64, 68] and has been widely applied to magnon sys-\ntems at zero temperature [32, 33, 57, 58, 69–71]. The details\nare shown in Appendix B. We start from the Dyson equation\nshown in Eq. (B18). The pole of a bosonic Green’s fuction is\ndetermined by\ndeth\nωσ 3−Ek−Σk(ω,T)i\n=0, (23)\nwhereσ3=\u00101N×N\n−1N×N\u0011\n. We introduce the damping rate as\nthe imaginary part of the self-energy as follows:\nΓk(ω,T)≡−ImΣk(ω,T). (24)\nFIG. 1. (a),(b) Schematic pictures of the three-quasiparticle in-\nteractions corresponding to Eqs. (20) and (21), respectively. (c),(d)\nLowest-order contributions of H3to the self-energy ˜Σin Eq. (22).\nAt zero temperature, only the diagram shown in (d) contributes to\nthe self-energy.\nAs mentioned in Sec II D, we consider the diagonal part of\nthe self-energy and neglect its real part, and such a self-energy\nhas been introduced as ˜Ση,k(ω,T) in the previous section. In\nthe following, we restrict the range of ηto 1,2,···,N. In this\ntreatment, Eqs. (23) and (24) are simplified as\nω=εη,k+˜Ση,k(ω,T), (25)\nΓη,k(ω,T)=−Im˜Ση,k(ω,T). (26)\nThe simplest approximation to eliminate the ωdependence\nis the on-shell approximation, where the argument ωof˜Ση,k\nis replaced to the one-particle energy εη,k. By applying the\napproximation, the damping rate up to 1 /Scorrections is ex-\npressed as Γη,k(T)≃−Im˜Ση,k(εη,k,T) (see Appendix C for\nmore details). This approximation in the 1 /Scorrection cor-\nresponds to the Born approximation that considers only one\nloop in Figs. 1(c) and 1(d). Note that the imaginary part\nof˜Ση,k(εη,k,T) exhibits unphysical divergent behavior due to\ntreating initial particle and two-particle states with di fferent\naccuracy in the on-shell approximation [29, 32, 64]. In gen-\neral, the issue may be alleviated by higher order 1 /Scorrec-\ntions, but such calculations are highly complicated, and per-\nforming them would involve significant computational costs.\nInstead, to suppress singularities in the self-energy encoun-\ntered within the 1 /Scorrection, the iDE approach has been\nproposed [64, 68]. In this approach, a finite lifetime is in-\ntroduced in the one-particle energy of the corresponding self-\nenergy. By considering the finite lifetime, one can relax the\nenergy conservation law in the self-energy, which allows us\nto reduce the artificial singularity and enables regularizing\nthe quasiparticle spectrum. To determine the self-energy, we\nsolve the following equation:\nω=εη,k+iIm˜Ση,k(ω∗,T), (27)\nwhere the complex conjugate ω∗inΣη,koriginates from\ncausality [64]. Note that, in this approach, the real part of ω\nremains unchanged, and the imaginary part is determined iter-\natively. The obtained complex number value ωgives the pole6\nof the bosonic Green’s function into the upper half-plane, and\nthe imaginary part corresponds to the damping rate Γη,k(T),\nwhich is expressed as Γη,k(T)=−Im˜Ση,k(ω∗,T). This pro-\ncedure is equivalent to changing the delta function in the on-\nshell approximation [see Eq. (C1)] to the Lorentzian-type [see\nEqs. (B22) and (B23)]. It has been shown that the iDE ap-\nproach can regularize the singularity of the damping rate ap-\npearing in the on-shell approximation [57, 64, 68]. We nu-\nmerically confirmed that the damping rate calculated using\nour scheme with the iDE approach is consistent with previ-\nous studies at zero temperature [33].\nIn the present study, we extend the iDE method to finite-\ntemperature calculations. To demonstrate the validity of this\nmethod at finite temperatures, we calculate the dynamical spin\nstructure factor in the Kitaev model, which is introduced in\nthe next section. The results are presented in Appendix D. The\npresent iDE results agree with those obtained by a continuous-\ntime quantum Monte Carlo (CTQMC) method [72] at the\nsame temperature. Therefore, we conclude that the iDE ap-\nproach is valid, at least for the temperature region where the\nmagnon picture is justified.\nHere, we introduce the spectral function for bosonic excita-\ntions. We approximate the retarded Green’s function defined\nin Eq. (B19) as\nGR\nη,k(ω,T)≃1\nω−εη,k+iΓη,k. (28)\nBy using Eq. (28), the spectral function, Ak, can be written as\nAk(ω,T)=−1\nπ1\nNNX\nηImGR\nη,k(ω,T)\n≃1\nNNX\nηΓη,k/π\n(ω−εη,k)2+ Γ2\nη,k. (29)\nFIG. 2. Schematic picture of the honeycomb lattice on which the\nS=1/2 Kitaev model is defined. The red, blue, and green lines stand\nfor the x,y, and zbonds, respectively. The black arrows represent\nthe two primitive translation vectors. The inset shows the relation\nbetween the coordinates of the spin space spanned by ( Sx,Sy,Sz)\nand the real space by ( a,b,c).III. MODEL\nWe apply the iDE method explained in the previous section\nto the S=1/2 Kitaev model on a honeycomb lattice [10, 36,\n73–75]. The Hamiltonian of this model is given by\nH=2KX\nγ=x,y,zX\n⟨i,j⟩γSγ\niSγ\nj−X\nih·Si, (30)\nwhere Sγ\ni(=x,y,z) represents the S=1/2 spin at site i, and\nK(<0) is the exchange constant of the ferromagnetic Kitaev\ninteraction between spins on the nearest neighbor (NN) sites.\nThe Kitaev interaction is bond-dependent, and ⟨i,j⟩γdenotes\nthe NNγbond on the honeycomb lattice (see Fig. 2). The last\nterm of Eq. (30) is the Zeeman term with the magnetic field\nh. The Kitaev model is believed to be realized in compounds\nwith 4 dor 5dtransition metal ions [76, 77]. Considering the\nconnection to real materials, we introduce the spin coordinate\nsuch that the [111] direction in the spin space is parallel to\nthecaxis in the real space and the Szdirection is on the c-a\nplane (see the inset of Fig. 2). Hereafter, we set the Kitaev\ninteraction to K=−1 and apply the magnetic field perpendic-\nular to the honeycomb plane, h∥c.\nIn our calculations, we introduce two clusters characterized\nby distinct boundary conditions: One is a cluster with peri-\nFIG. 3. (a) Schematic picture of the 12 zigzag chains on the honey-\ncomb cluster with zigzag edges where a periodic boundary condition\nis imposed along the adirection. (b) Spatial distribution of the direc-\ntion of spin moment at several magnetic fields. The inset shows the\ndefinition of θ, which is the angle of the spin moment from the caxis\non the a-cplane.7\n012ka/⇡0.01.02.03.0\n0.01.53.0\n\u0000KM\u00000.01.02.03.0\n0.01.53.0\n012ka/⇡0.01.02.03.0!\u0000KM\u00000.01.02.03.0!\n1612zigzag chain010203040✓[\u0000]h=0.01h=0.05h=0.10h=0.50h=0.60h=1.00\ntickslabel= 26label=28(a)\n(c)(b)\n(d)\n𝑎𝑐𝜃<latexit sha1_base64=\"hC1DSgZBpmVOZ2Q09cEAQ0nsyRI=\">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</latexit>hSii\n⨀!\t#\t$\t135791124681012(e)\n(f)ΓKM\n0.01.02.03.0\nFIG. 4. (a) Magnon dispersion presented along high-symmetry lines in the kspace for h=0.1 in the system with the periodic boundary\nconditions imposed. (b) Magnon spectral functions at h=0.1 and T=0, where the magnon damping calculated by the iDE approach. In (b),\nthe dashed lines represent the bare magnon dispersions shown in (a). (c),(d) Corresponding plots for the system with open boundaries where\nthe lattice terminates with zigzag edges [see Fig. 3(a)].\nodic boundary conditions along the two primitive translation\nvectors for the honeycomb lattice [see Fig. 2(a)], and the other\npossesses the open boundary with zigzag edges and the peri-\nodic condition imposed along the direction of the zigzag chain\n[see Fig. 3(a)]. For the latter, we consider the cluster including\n12 zigzag chains as shown in Fig. 3(a).\nIV . RESULT\nA. Mean-field solution\nFirst, we mention the MF ground states of the Kitaev model\nunder magnetic fields on the two clusters. In the cluster with\nthe periodic boundary conditions, the MF ground state is the\nspin-polarized state parallel to the magnetic field direction re-\ngardless of the magnetic-field intensity. On the other hand, in\nthe case of the cluster with zigzag edges, the spin direction de-\npends on the distance from the edge. Spins near the center of\nthe cluster are almost parallel to the magnetic field direction.\nMeanwhile, in the vicinity of the edges, the spin direction is\ntilted from the magnetic field direction ( h∥c) to the aaxis, as\nshown in Fig. 3(b). This is due to the lack of zbonds for edge\nsites; spins on edge sites tilt toward the Sx-Syplane to gain\nthe exchange energies on xandybonds. The deviation from\nthe applied field direction becomes small with increasing the\nfield intensity.B. Magnon spectra for low-field regime at zero temperature\nIn this section, we show the results for the magnon spec-\ntra in the Kitaev model under the magnetic field with h=0.1\nat zero temperature. Figure 4(a) displays the magnon disper-\nsions obtained by the linear flavor-wave theory on the clus-\nter with the periodic boundary conditions. The dispersions\nare plotted along the lines shown in Fig. 4(a) in the first\nBrillouin zone. There are two branches because there are\ntwo spins in a unit cell of the honeycomb lattice. The re-\nsults are consistent with the magnon dispersions obtained by\nthe previous studies [10, 39]. We also calculate the magnon\ndispersions on the cluster with zigzag edges. As shown in\nFig. 4(c), there are two modes connected between two bulk\nbands corresponding to the two magnon dispersions shown in\nFig. 4(a). These two ingap modes are chiral edge modes along\nthe two zigzag edges, which result from topologically nontriv-\nial magnon bands caused by an applied magnetic field.\nHere, we introduce the damping e ffect of magnons. Fig-\nures 4(b) and 4(d) show the magnon spectral functions de-\nfined in Eq. (29), where the imaginary part of the self-energy\nis evaluated by the iDE method, in the cluster with the periodic\nboundary conditions and that with zigzag edges, respectively.\nIn the former case shown in Fig. 4(b), magnons around the Γ\npoint in the low-energy branch survive even in the presence of\nthe magnon-magnon interactions. Away from the Γpoint, the\ndamping e ffect of magnons becomes more significant, which\nis reflected by the enhancement of the imaginary part of the\nself-energy. For the higher-energy magnon branch, magnons\nappear to decay throughout the Brillouin zone. In the case of\nthe cluster with zigzag edges [Fig. 4(d)], low-energy magnons8\nFIG. 5. Two-magnon DOS at h=0.1 in (a) the system with the pe-\nriodic boundary conditions imposed and (b) that with zigzag edges.\nThe black lines represent the one-particle magnon dispersion.\naround ka=0 are stable against introducing magnon-magnon\ninteractions similar to the results in the cluster with the pe-\nriodic boundaries. On the other hand, low-energy magnons\naway from the ka=0 point are strongly damped, as well as\nhigh-energy magnons. It is worth noting that the chiral edge\nmodes also exhibit strong damping at zero temperature.\nTo clarify the kand energy dependence of the magnon\ndamping, we focus on the self-energy of magnons. There are\ntwo contributions in the self-energy as shown in Eq. (22), but\nthe second term Σ(d)\nη,k(ω,T) corresponding to the decay process\n[Fig. 1(d)] only contributes at zero temperature. From the ex-\nplicit representation of Σ(d)\nη,k(ω,T) presented in Eq. (B23), it\nis naively expected that the imaginary part is zero when the\nconditionω−εη′,q−εη′′,k−q=0 is satisfied for ω=εη,k.\nThis consideration is correct for the on-shell Born approxi-\nmation. We apply it to the present results obtained by the\niDE approach. Here, we introduce two-magnon density of\nstates (DOS), which is defined as\nD(d)\nk(ω)≡1\nN2NX\nη′,η′′M\nNtB.Z.X\nqδ(ω−εη′,q−εη′′,k−q). (31)\nWhen the dispersion relation ω=εη,koverlaps with nonzero\nD(d)\nk(ω), such magnons possibly collapse due to the decay pro-\ncess inherent in the self-energy Σ(d)\nη,k(ω,T).\nThe two-magnon DOS at h=0.1 is shown in Fig. 5, where\nFigs. 5(a) and 5(b) correspond to the results in the system\nunder the periodic boundary conditions and that with zigzagedges, respectively. In both cases, the upper one-magnon\nbranches overlap with the two-magnon continuum. In the\non-shell Born approximation, a necessary condition for the\nsingle-magnon with εη,ksplitting into a two-magnon contin-\nuum is represented as\nεη,k=εη′,q+εη′′,k−q. (32)\nThis condition indicates that the high-energy magnons decay\nas long as ¯Vη,η′←η′′\nq,k−q←k,0, and the strong magnon decay ob-\ntained by the iDE approach [Figs. 4(b) and 4(d)] is consistent\nwith this consideration. On the other hand, the chiral edge\nmodes also decay strongly, as shown in Fig. 4(d), although\nthe intensity of D(d)\nk(ω) is weak compared to the high-energy\nregion. Indeed, we have numerically confirmed that the chiral\nedge modes rarely decay in the on-shell Born approximation.\nContrary to this approximation, the criteria for magnon decay\nshown in Eq (32) do not necessarily have to hold strictly in\nthe iDE approach because this approach takes account of the\nenergy fluctuations of magnons.\nC. Temperature dependence of magnon spectra with higher\nmagnetic fields\nIn this section, we focus on the results for higher magnetic\nfields in the cluster with zigzag edges. Figure 6(a) shows the\nmagnon spectrum for h=0.6 at zero temperature calculated\nby the iDE approach. As shown in this figure, the chiral edge\nmodes appear to survive particularly below ω≲1.5 in con-\ntrast to the case with h=0.1. Moreover, we also find that\nmagnons in the low-energy band survive, although those in\nthe high-energy band collapse due to the nonzero lifetime of\nthe magnons. These results are understood from the overlap\nbetween the single-magnon dispersion relation εη,kand two-\nmagnon DOS D(d)\nk(ω); the lower-energy magnons and chiral\nedge modes do not overlap with nonzero D(d)\nk(ω) but this DOS\ntakes larger values in the energy window of the higher-energy\nmagnons, as shown in Fig. 6(c). Further increase of the mag-\nnetic field leads to stable chiral edge modes in the wider en-\nergy region, as shown in Fig. 6(e). This is understood from\nthe high-energy shift of two-magnon DOS [Fig. 6(g)].\nHere, we examine the e ffect of thermal fluctuations on the\nmagnon damping. Figure 6(b) shows the magnon spectrum\nforh=0.6 atT=0.6. As shown in this figure, the chiral edge\nmodes are smeared strongly by thermal fluctuations. To clar-\nify the origin of this e ffect, we discuss the contribution from\nΣ(c)\nη,k(ω,T) in Eq. (22), which is nonzero at finite temperatures.\nThis contribution originates from the collision with thermally\nexcited magnons. Here, we introduce the two-magnon colli-\nsion DOS as\nD(c)\nk(ω)≡1\nN2NX\nη′,η′′M\nNtB.Z.X\nqδ(ω+εη′,q−εη′′,k+q). (33)\nSimilar to the case of D(d)\nk(ω), the overlap between the single-\nmagnon dispersion and the two-magnon collision DOS gives9\nFIG. 6. (a),(b) Magnon spectral function given in Eq. (29) in the system with zigzag edges under the magnetic field with h=0.6 at (a) T=0\nand (b) T=0.6. The dashed lines represent the one-particle magnon dispersions. (c) Two-magnon DOS and (d) two-magnon collision DOS\nath=0.6. The black lines represent the one-particle magnon dispersions. (e)–(h) Corresponding plots for h=1.0.\nthe necessity condition of nonzero Im Σ(c)\nη,k(ω,T) within the on-\nshell Born approximation [see Eq. (B22)]. Figure 6(d) shows\nthe two-magnon collision DOS D(c)\nk(ω). Unlike the case of\nD(d)\nk(ω) shown in Fig. 6(c), the chiral edge modes overlap with\nthe two-magnon collision DOS. This result indicates that the\ndecay of the chiral edge modes at h=0.6 and T=0.6 is\ndue to the collision process with thermally excited magnons\nin the bulk bands. The present consideration is also applied to\nthe case with higher magnetic fields. Figure 6(f) displays the\nmagnon spectrum for h=1.0 atT=0.6. In this case, the chi-\nral edge modes appear to survive even at finite temperatures.\nThe existence of the stable edge modes against thermal fluc-\ntuations is understood from the two-magnon collision DOS as\nshown in Fig. 6(h); the edge modes do not overlap the two-\nmagnon collision DOS. Therefore, in the high-field case, thechiral edge magnon is robust in the presence of the magnon-\nmagnon interactions at both zero and finite temperatures, and\nthe topological magnon picture in terms of the noninteracting\nlimit is well established.\nV . DISCUSSION\nIn this section, we discuss the magnetic-field dependence\nof the magnon damping for the chiral edge mode. As shown\nin Fig. 4(c), the two magnon branches along the zigzag edges\nare present between two bulk bands. These chiral edge modes\ncross at ka=π. We focus on the magnon damping of the\nedge modes and introduce the damping rate for each edge\nmode at this point as Γedge. We have confirmed that the val-\nues of the damping rate for the two magnon edge modes are10\n0.0 0.2 0.4 0.6 0.8 1.0\nh0.01.02.03.04.0ΓedgeT=0\nT=0.2\nT=0.4T=0.6\nT=0.8\nT=1.0\nFIG. 7. Magnetic-field dependence of the edge-mode damping rate\nΓedge, which is defined in the main text, at several temperatures.\n0.0 0.2 0.4 0.6 0.8 1.0\nh0.01.02.03.0ωεedge\nE(d)\nmin\nE(c)\nmaxhc1 hc2\n0.01.02.03.0\nFIG. 8. Magnetic field dependence of the energy of the edge-mode\nεedge, the lowest energy of the two-magnon DOS E(d)\nmin, and the highest\nenergy of the two-magnon collision DOS E(c)\nmaxatka=π.\nthe same at ka=π. Figure 7 shows the magnetic field de-\npendence of Γedgeat several temperatures. At T=0,Γedge\nmonotonically decreases with increasing hand vanishes above\nh≃0.48. The condition for vanishing Γedgecan be under-\nstood from the overlap between the chiral edge mode and two\nmagnon DOS defined in Eq. (31). As shown in Fig. 5(b), the\nchiral edge modes overlap with the two magnon continuum,\nindicating the finite lifetime of chiral edge magnons. With\nincreasing the magnetic field, the continuum is shifted to the\nhigher-energy side and does not overlap with the edge modes,\nas shown in Figs. 6(c) and 6(g). Thus, we calculate the lowest\nenergy of the continuum D(d)\nk(ω) atka=π. Figure 8 shows\nthe magnetic field dependence of the lowest energy defined as\nE(d)\nmin. We find that E(d)\nminis larger than the edge mode energy\nεedge=εka=πabove hc1≃0.52. This value is close to the\ncritical field where Γedgevanishes at T=0, suggesting that\nthe damping e ffect at zero temperature basically is understood\nfrom the decay to two magnons. Meanwhile, the monotonic\ndecrease of Γedgewith increasing hcannot be understood only\nfrom two magnon DOS. This behavior originates from the ma-\ntrix elements of the vertex ¯Vη,η′←η′′\nk,q←pand the smearing e ffects\ninΣ(c)\nη,k(ω,T) taken by the iDE calculations.Next, we discuss the magnetic-field e ffect on the magnon\ndamping of the edge modes at finite temperatures. As shown\nin Fig. 7, Γedgeincreases with increasing temperature, and\nthermal fluctuations result in the shift to the high-field side\nwithout changing the overall hdependence. Accordingly, the\ncritical field where Γedgevanishes shifts to the high-field side.\nSince the magnon-damping e ffect originating from the self-\nenergy Σ(d)\nη,k(ω,T) contributes only in h≲hc1, the shift is due\nto the collision process coming from Σ(c)\nη,k(ω,T). This process\nis brought about when magnon energy overlaps with the two-\nmagnon collision DOS presented in Eq. (33). As shown in\nFigs. 6(d) and 6(h), the two-magnon collision DOS has a dis-\ntribution centered at the zero energy. To discuss the overlap\nwith the chiral edge modes, we focus on the highest energy\nof two-magnon collision DOS at ka=π, which we define\nasE(c)\nmax. The magnetic field dependence of E(c)\nmaxis shown\nin Fig. 8. We find that E(c)\nmaxandεedgecross at hc2≃0.86,\nwhich is larger than hc1. This result clearly indicates that ther-\nmal fluctuations give rise to the magnon damping of the chiral\nedge modes even when the noninteracting magnon picture for\nthe edge modes is justified at zero temperature.\nIn the Kitaev candidate material α-RuCl 3, the strength of\nthe Kitaev interaction is considered to be |2K|=100 K [78–\n82]. In this case, damping e ffects on the chiral edge magnons\nare inevitably present as long as we apply the magnetic fields\nwith a Zeeman energy comparable to the exchange coupling,\nwhose intensity is approximated as ∼100 T. Therefore, we\nconclude that, within the magnon picture, chiral edge modes\npossess a finite lifetime, and the damping e ffect is relevant in\nthe pure Kitaev model under magnetic fields. When we con-\nsider additional interactions such as Heisenberg and Γterms,\nthe energy structures of two-magnon DOS and two-magnon\ncollision DOS can be changed. The additional interactions\ncould stabilize the chiral edge modes if these two kinds of\ncontinuum do not overlap with chiral mode branches. The ef-\nfects of the additional terms remain as future work.\nFinally, we note that topological thermal transport phenom-\nena may occur even if the chiral edge mode possesses a finite\nlifetime. The thermal Hall e ffect at low temperatures is mainly\ncaused by the finite Berry curvatures defined at the low energy\nbranches [20, 83, 84]. At h=0.1, Figs. 4(b) and 4(d) show\nthat the low energy magnons are stable in the presence of the\nmagnon-magnon interactions despite the strong damping of\nthe chiral edge magnon mode. Therefore, if the low-energy\nmagnons survive, the thermal Hall e ffect may be detectable\nregardless of the stability of chiral edge magnons. To answer\nthis question, one desires the formulation of the thermal Hall\nconductivity involved in the magnon damping in bulk systems.\nOur calculation framework starts from a general Hamilto-\nnian describing interactions between local degrees of free-\ndom with any number of states and the MF ground state with\nany number of sublattices. Thus, it is widely applicable to\nother systems with local degrees of freedom such as spin-\norbital entangled systems with multipolar interactions [59–61]\nand molecular-orbital crystals including spin-dimerized sys-\ntems [21, 85–87], and also to systems with the magnetic or-\ndering characterized by long-period waves such as skyrmion11\ncrystals [88]. Moreover, since our formalism is mapped onto\nbosonic Hamiltonian with no restriction of the number of lo-\ncal bosons, it can be applied to electronic models coupled with\nphonons, such as dynamical Jahn-Teller systems [89]. We\nalso expect that our method is applicable to anticipating ex-\ncitation spectra of collective modes in real materials. It has\nbeen proposed that the general Hamiltonian in Eq. (1) based\non the present study can be derived as a Kugel-Khomskii-type\neffective model for a Mott insulator from a realistic multior-\nbital Hubbard model [90]. Combining this approach with our\nmethod could systematically demonstrate excitation spectra in\nMott insulators based on first-principles calculations.\nVI. SUMMARY\nIn summary, we have proposed a framework that takes\naccount of damping e ffects on collective excitations from a\nmean-field ground state in a general interacting model be-\ntween local degrees of freedom. We have introduced bosonic\nquasiparticles based on the flavor-wave theory. We evaluate\ndamping e ffects on the quasiparticles at finite temperatures\nby extending the imaginary Dyson equation method, which\nhas been widely employed in analyzing various magnon sys-\ntems. We have found two contributions to quasiparticle damp-\ning processes in the self-energy: One occurs due to collisions\nwith excited quasiparticles, and the other is a decay process\ninto two quasiparticles. The latter occurs even at zero tem-\nperature, but the former appears only at finite temperatures.\nWe have applied our method to the Kitaev model on a hon-\neycomb lattice under a magnetic field, a well-known system\nwith topologically nontrivial magnon bands. We have val-\nidated the method by comparing the dynamical spin struc-\nture factor with that obtained by the continuous-time quan-\ntum Monte Carlo simulations. We focus on chiral edge modes\nin a cluster with zigzag edges to examine damping e ffects on\ntopological magnons. We have demonstrated that the chiral\nedge modes are strongly damped in weak magnetic fields at\nzero temperature. With increasing the intensity of the applied\nmagnetic field, the lifetime of the chiral edge modes becomes\nlonger and well-defined quasiparticle excitations. We have\nalso found that thermal fluctuations give rise to an increase\nin the damping rate of the chiral edge modes. This e ffect\noriginates from collisions with thermally excited magnons in\nthe bulk. Since the present approach starts from a general\nHamiltonian, it can widely apply to other localized systems,\nsuch as spin-orbital and electron-phonon coupled systems. We\nalso expect an application to real materials by integrating with\nfirst-principles calculations giving a localized model.ACKNOWLEDGMENTS\nThe authors thank S. Murakami, P. McClarty, S. Hoshino,\nA. Ono, and R. Iwazaki for fruitful discussions. Parts of\nthe numerical calculations were performed in the supercom-\nputing systems in ISSP, the University of Tokyo. This\nwork was supported by Grant-in-Aid for Scientific Research\nfrom JSPS, KAKENHI Grant No. JP19K03742, JP20H00122,\nJP22H01175, JP23H01129, and JP23H04865, and by JST,\nthe establishment of university fellowships towards the cre-\nation of science technology innovation, Grant Number JP-\nMJFS2102.\nAppendix A: Linear flavor-wave theory and Bogoliubov\ntransformation\nIn this section, we show the details of the linear flavor-\nwave theory and Bogoliubov transformation given in Sec. II C.\nThe 2 N×2NHermitian matrixMkin the linear flavor-wave\nHamiltonianHLFWin Eq. (15) is given by\nMk= ¯Jk+¯J†\nk+ ∆Ediag Jk+JT\n−k\nJ∗\n−k+J†\nk¯J∗\n−k+¯JT\n−k+ ∆Ediag!\n(A1)\nwhere ∆Ediag=diag{∆E1,∆E2,···,∆EN}with ∆Eι= ∆El\nm\nis aN×Ndiagonal matrix, and Jkand ¯Jkare defined as the\nfollowing N×Ndiagonal matrices:\nJk;ιι′=Jk;(ml)(m′l′)=1\n2X\ni∈l,j∈l′X\nγγ′Jγγ′\ni jeik·(rj−ri)¯Ol\nγm¯Ol′\nγ′,m′,\n(A2)\n¯Jk;ιι′=¯Jk;(ml)(m′l′)=1\n2X\ni∈l,j∈l′X\nγγ′Jγγ′\ni jeik·(rj−ri)¯Ol\nγm¯Ol′∗\nγ′,m′.\n(A3)\nTo diagonalize this Hamiltonian into Eq. (18), we introduce\nthe paraunitary matrix Tkas,\nAk=TkBk= \nUkV∗\n−k\nVkU∗\n−k!\nBk, (A4)\nor equivalently,\naι,k=NX\nη\u0010\nUk,ιηbη,k+V∗\n−k,ιηb†\nη,−k\u0011\n, (A5)\na†\nι,k=NX\nη\u0010\nU∗\nk,ιηb†\nη,k+V−k,ιηbη,−k\u0011\n. (A6)\nHere, UkandVkareN×Nunitary matrices. We can find\nthe paraunitary matrix Tkby combining with the Cholesky\ndecomposition and exact diagonalization [62].12\nAppendix B: Nonlinear flavor-wave theory\nIn this section, we present theoretical treatments for nonlinear terms in the flavor-wave theory. Here, we focus on the cubic\nterm,H′\n3/√\nSin Eq. (14), which is neglected in the linear flavor-wave theory. By performing straightforward calculations, one\ncan represent the cubic term as\n1√\nSH′\n3=1\n2√\nSMX\nll′X\ni∈l,j∈l′X\nγγ′N−1X\nmm′m′′Jγγ′\ni j\u0010¯Ol\nγmδ¯Ol′\nγ′,m′m′′a†\nmia†\nm′jam′′j+δ¯Ol\nγ,m′m′′¯Ol′\nγ′ma†\nm′iam′′ia†\nm j+H.c.\u0011\n, (B1)\nwhere ¯Ol\nγ,m=⟨m;i|Oγ|0;i⟩andδ¯Ol\nγ,m′m′′=⟨m′;i|δOγ|m′′;i⟩. By using Eq. (17), H′\n3is expressed by\n1√\nSH′\n3=r\nM\nNtSMX\nll′X\nkk′N−1X\nmm′m′′\u0010\nJlδl′\nk;m,m′m′′a†\nml,ka†\nm′l′,k′am′′l′,k+k′+J¯lδ¯l′\nk;m,m′m′′aml,−ka†\nm′l′,k′am′′l′,k+k′\n+Jδll′\nk;m′m′′,ma†\nm′l,k′am′′l,k′−ka†\nml′,−k+Jδ¯l¯l′\nk;m′m′′,ma†\nm′′l,k′am′l,k′−kaml′,k\u0011\n, (B2)\nwhere the coe fficients are given as follows:\nJlδl′\nk;m,m′m′′=1\n2X\ni∈l,j∈l′X\nγγ′Jγγ′\ni jeik·(rj−ri)¯Ol\nγmδ¯Ol′\nγ′,m′m′′,Jl′δl\nk;m,m′m′′=1\n2X\ni∈l,j∈l′X\nγγ′Jγ′γ\njieik·(ri−rj)¯Ol′\nγ′mδ¯Ol\nγ,m′m′′,\nJ¯lδl′\nk;m,m′m′′=1\n2X\ni∈l,j∈l′X\nγγ′Jγγ′\ni jeik·(rj−ri)¯Ol∗\nγmδ¯Ol′\nγ′,m′m′′,J¯l′δl\nk;m,m′m′′=1\n2X\ni∈l,j∈l′X\nγγ′Jγ′γ\njieik·(ri−rj)¯Ol′∗\nγ′mδ¯Ol\nγ,m′m′′,\nJδll′\nk;m′m′′,m=1\n2X\ni∈l,j∈l′X\nγγ′Jγγ′\ni jeik·(rj−ri)δ¯Ol\nγ,m′m′′¯Ol′\nγ′m,Jδl′l\nk;m′m′′,m=1\n2X\ni∈l,j∈l′X\nγγ′Jγ′γ\njieik·(ri−rj)δ¯Ol′\nγ′,m′m′′¯Ol\nγm,\nJδl¯l′\nk;m′m′′,m=1\n2X\ni∈l,j∈l′X\nγγ′Jγγ′\ni jeik·(rj−ri)δ¯Ol\nγ,m′m′′¯Ol′∗\nγ′m,Jδl′¯l\nk;m′m′′,m=1\n2X\ni∈l,j∈l′X\nγγ′Jγ′γ\njieik·(ri−rj)δ¯Ol′\nγ′,m′m′′¯Ol∗\nγm,\nJ¯lδ¯l′\nk;m,m′m′′=1\n2X\ni∈l,j∈l′X\nγγ′Jγγ′\ni jeik·(rj−ri)¯Ol∗\nγmδ¯Ol′∗\nγ′,m′m′′,J¯l′δ¯l\nk;m,m′m′′=1\n2X\ni∈l,j∈l′X\nγγ′Jγ′γ\njieik·(ri−rj)¯Ol′∗\nγ′mδ¯Ol∗\nγ,m′m′′.(B3)\nUsing the relations between the above coe fficients such as J¯lδ¯l′\nk;m,m′m′′=Jlδl′∗\n−k;m,m′m′′=Jδl′l∗\nk;m′m′′,m, one can represent Eq. (B2) as\n1√\nSH′\n3=r\nM\nNtSMX\nll′X\nkk′N−1X\nmm′m′′\u0010\nJlδl′\nk;m,m′m′′a†\nml,ka†\nm′l′,k′am′′l′,k+k′+Jδll′\n−k′;m′m′′,ma†\nm′l,ka†\nml′,k′am′′l,k+k′+H.c.\u0011\n. (B4)\nWe introduce the following interaction vertex:\nVι,ι′←ι′′\nk,q←p≡Jlδl′\nk;m,m′m′′\u0002(δl′,l′′−δl,l′′)+2δl,l′′δl′,l′′\u0003+Jl′δl\nq;m′,mm′′\u0002(δll′′−δl′,l′′)+2δl,l′′δl′,l′′\u0003. (B5)\nThe above cubic term is simplified as\n1√\nSH′\n3=1\n2r\nM\nNtSNX\nιι′ι′′k+q=pX\nkq\u0010\nVι,ι′←ι′′\nk,q←pa†\nι,ka†\nι′,qaι′′,p+H.c.\u0011\n, (B6)\nwhereι=(ml),ι′=(m′l′), andι′′=(m′′l′′). Similar expressions are derived for spin models in previous studies [91, 92].\nBy applying the Bogoliubov transformation, Eqs. (A5) and (A6), H′\n3is represented by\n1√\nSH′\n3=1√\nSH(d)\n3+1√\nSH(s)\n3, (B7)\nwhere\n1√\nSH(d)\n3=1\n2!r\nM\nNtSNX\nηη′η′′k+q=pX\nkqp \n¯Vη,η′←η′′\nk,q←pb†\nη,kb†\nη′,qbη′′,p+H.c.!\n, (B8)\n1√\nSH(s)\n3=1\n3!r\nM\nNtSNX\nηη′η′′k+q=−pX\nkqp \n¯Wη,η′,η′′\nk,q,pb†\nη,kb†\nη′,qb†\nη′′,p+H.c.!\n. (B9)13\nThe interaction vertices in the above expressions are given by\n¯Vη,η′←η′′\nk,q←p=NX\nιι′ι′′(\nVι,ι′←ι′′\nk,q←pU∗\nk,ιηU∗\nq,ι′η′Up,ι′′η′′+\u0010\nVι,ι′←ι′′\n−k,−q←−p\u0011∗V∗\nk,ιηV∗\nq,ι′η′Vp,ι′′η′′\n+Vι,ι′←ι′′\nq,−p←−kU∗\nq,ιη′Vp,ι′η′′V∗\nk,ι′′η+\u0010\nVι,ι′←ι′′\n−q,p←k\u0011∗V∗\nq,ιη′Up,ι′η′′U∗\nk,ι′′η\n+Vι,ι′←ι′′\n−p,k←−qVp,ι,η′′U∗\nk,ι′ηV∗\nq,ι′′η′+\u0010\nVι,ι′←ι′′\np,−k←q\u0011∗Up,ιη′′V∗\nk,ι′ηU∗\nq,ι′′η′)\n, (B10)\n¯Wη,η′,η′′\nk,q,p=NX\nιι′ι′′(\nVι,ι′←ι′′\nk,q←−pU∗\nk,ιηU∗\nq,ι′η′V∗\np,ι′′η′′+\u0010\nVι,ι′←ι′′\n−k,−q←p\u0011∗V∗\nk,ιηV∗\nq,ι′η′U∗\np,ι′′η′′\n+Vι,ι′←ι′′\nq,p←−kU∗\nq,ιη′U∗\np,ι′η′′V∗\nk,ι′′η+\u0010\nVι,ι′←ι′′\n−q,−p←k\u0011∗V∗\nq,ιη′V∗\np,ι′η′′U∗\nk,ι′′η\n+Vι,ι′←ι′′\np,k←−qU∗\np,ιη′′U∗\nk,ι′ηV∗\nq,ι′′η′+\u0010\nVι,ι′←ι′′\n−p,−k←q\u0011∗V∗\np,ιη′′V∗\nk,ι′ηU∗\nq,ι′′η′′)\n. (B11)\nThese expressions are the same as those in previous studies [91, 93]. Figures 1(a) and 1(b) illustrate these vertices, ¯Vη,η′←η′′\nk,q←pand\n¯Wη,η′,η′′\nk,q,p, respectively.\nTo treat the anharmonic terms as perturbations up to order 1 /S, we utilize the standard Green’s function approach [65]. In\nterms of the bosons bη,k, we define the temperature Green’s function as follows:\nGηη′,k(τ)≡−\nTτBη,k(τ)B†\nη′,k\u000b \u0000η,η′=1,2,···,2N\u0001(B12)\nGηη′,k(iωn)≡Zβ\n0dτeiωnτGηη′k(τ), (B13)\nwhere Tτis the time-ordering operator in the imaginary time, ωn=2nπ/βis the Matsubara frequency with nbeing integer,\nand⟨·⟩stands for the thermal average. β=(kBT)−1is the inverse temperature, where kBis the Boltzmann constant. The bare\ntemperature Green’s function is represented by\nG(0)\nk(iωn)=1\niωnσ3−Ek. (B14)\nThe temperature Green’s function up to the 1 /Scorrection with respect to the HLFW[see Eq. (14)] can be written as [65]\nGηη′,k(τ)≃G(0)\nηη′,k(τ)+1\nSZβ\n0dτ1D\nTτH′\n4(τ1)Bη,k(τ)B†\nη′,kE\n0−1\nS1\n2!Zβ\n0dτ1Zβ\n0dτ2D\nTτH′\n3(τ1)H′\n3(τ2)Bη,k(τ)B†\nη′,kE\n0,(B15)\nwhereH′\n4is the four-quasiparticle interaction, and ⟨·⟩ 0represents the thermal average in HLFW. We also introduce the self-\nenergy, which is defined as\nΣk(ω,T)≡Σk(iωn→ω+i0+) (B16)\nΣk(iωn)≡h\nG(0)\nk(iωn)i−1−h\nGk(iωn)i−1. (B17)\nBy using the self-energy, the temperature and retarded Green’s function can be written as [65]\nGk(iωn)=h\n1−G(0)\nk(iωn)Σk(iωn)i−1G(0)\nk(iωn)=h\niωnσ3−Ek−Σk(iωn)i−1, (B18)\nGR\nk(ω,T)≡Gk(iωn→ω+i0+)=h\n(ω+i0+)σ3−Ek−Σk(ω,T)i−1. (B19)\nSince we focus on the damping phenomena of the bosonic quasiparticles, we consider only the imaginary part of the self-\nenergy within the 1 /Scorrection relative to HLFW. The lowest-order diagrams with the imaginary part are Figs. 1(c) and 1(d),\nwhich represent the spontaneous decays and collisions with thermally excited bosons, respectively. Since multiple thermally\nexcited bosons are required for the collision process, only contributions from Fig. 1(d) are nonzero at zero temperature. As in\nRefs. [57, 91], we treat only diagonal terms of the self-energy, defined as ˜Ση,k(iωn). In the following, we restrict the range of η\nto 1,2,···,N. Then, the temperature Grees’s function are expressed by\nGη,k(iωn)=1\niωn−εη,k−˜Ση,k(iωn). (B20)14\nFIG. 9. The lowest-order diagram involved with H(s)\n3defined in Eq. (21) [see also Fig. 1(b)].\nFrom Figs. 1(c) and 1(d), ˜Ση,k(ω,T) can be written as\n˜Ση,k(ω,T)=˜Ση,k(iωn→ω+i0+)= Σ(c)\nη,k(ω,T)+ Σ(d)\nη,k(ω,T), (B21)\nwhere\nΣ(c)\nη,k(ω,T)=M\nNtSNX\nη′,η′′B.Z.X\nq\f\f\f\f¯Vη,η′←η′′\nk,q←k+q\f\f\f\f2\nω+εη′,q−εη′′,k+q+i0+h\ng(εη′,q)−g(εη′′,k+q)i\nforω≥0, (B22)\nΣ(d)\nη,k(ω,T)=1\n2M\nNtSNX\nη′,η′′B.Z.X\nq\f\f\f\f¯Vη′,η′′←η\nq,k−q←k\f\f\f\f2\nω−εη′,q−εη′′,k−q+i0+h\ng(εη′,q)+g(εη′′,k−q)+1i\n. (B23)\nThe above quantities Σ(c)\nη,kandΣ(d)\nη,kcorrespond to the self-energies depicted by Figs. 1(c) and 1(d), respectively. The analytic\ncontinuation is performed under the condition that sgn( ω)ImΣk(ω)≤0 is satisfied. Note that we also consider the process\nΣ(s)\nη,k(ω,T) shown in Figure 9, which is involved with the magnon interaction presented in Fig. 1(b). This is given by\nΣ(s)\nη,k(ω,T)=−1\n2M\nNtSNX\nη′,η′′B.Z.X\nq\f\f\f\f¯Wη′,η′′,η\nq,k−q,k\f\f\f\f2\nω+εη′,q+εη′′,−k−q−i0+h\ng(εη′,q)+g(εη′′,−k−q)+1i\n. (B24)\nSinceεη′,qandεη′′,−k−qare positive, the imaginary part of this quantity is zero. We do not take it into account in the present\nscheme.\nAppendix C: On-shell approximation\nIn the on-shell approximation, ωin the self-energy ˜Ση,k(ω,T) is replaced to the one-particle energy εη,k. The on-shell ap-\nproximation up to the 1 /Scorrection corresponds to the Born approximation. In this approximation, the damping rate Γη,kis\nevaluated as\nΓη,k(T)≃−Im˜Ση,k(ω=εη,k,T)\n=M\nNtSNX\nη′η′′B.Z.X\nq(1\n2\f\f\f\f¯Vη′,η′′←η\nq,k−q←k\f\f\f\f2h\ng(εη′,q)+g(εη′′,k−q)+1i\nδ(εη,k−εη′,q−εη′′,k−q)\n+\f\f\f\f¯Vη,η′←η′′\nk,q←k+q\f\f\f\f2h\ng(εη′,q)−g(εη′′,k+q)i\nδ(εη,k+εη′,q−εη′′,k+q))\n. (C1)\nWe numerically confirmed that calculation results of the damping rate in the antiferromagnetic Heisenberg model on the square\nand tetragonal lattice are the same as those in the previous studies at T=0 [30, 31]. We note that since this approxima-\ntion completely neglects the magnon-magnon interactions in the intermediate states of the self-energy, it shows non-analytic\ndivergences [29].15\nAppendix D: Dynamical structure factor\nIn this appendix, we verify the validity of our method incorporated with the iDE approach at finite temperatures. We calculate\nthe dynamical structure factor in Eq. (30) and compare with the previous results computed by a continuous-time quantum Monte\nCarlo (CTQMC) method in Ref. [72].\nThe dynamical structure factor is defined as\nSγγ′(k,ω)≡Z∞\n−∞dt\n2πeiωt\nδOγ\nk(t)δOγ′\n−k\u000b(D1)\n=1\nMMX\nll′Z∞\n−∞dt\n2πeiωt\nδOγ\nl,k(t)δOγ′\nl′,−k\u000b, (D2)\nwhere landl′are the sublattice indexes. The Fourier transformation of the operator δOγ\niintroduced in Eq. (2) is defined as\nδOγ\nk≡r\n1\nNtX\niδOγ\nie−ik·ri=r\n1\nMMX\nlδOγ\nl,k, (D3)\nwith\nδOγ\nl,k≡r\nM\nNtX\ni∈lδOγ\nie−ik·ri. (D4)\nTo calculate Eq. (D2), we introduce the following time-ordered correlation function:\nΦγγ′(k,τ)≡1\nMMX\nll′\nTτδOγ\nl,k(τ)δOγ′\nl′,−k\u000b. (D5)\nThe Fourier transformation with respect to the imaginary time is given by\nΦγγ′(k,iΩ)=Zβ\n0dτΦγγ′(k,τ)eiΩτ. (D6)\nHere, we introduce the dynamical susceptibility as an analytic continuation from Φγγ′(k,iΩ) as follows:\nχγγ′(k,ω)≡Φγγ′(k,iΩ→ω+i0+). (D7)\nForω> 0, the diagonal components of Eq. (D2) is expressed as\nSγγ(k,ω)=1\nπImχγγ(k,ω) (D8)\nIn the present study, we only take into account the lowest-order contributions of δOγ\niand approximate the retarded Green’s\nfunction for bosonic quasiparticles as Eq. (28). By using this representation of the Green’s function, Sγγ(k,ω) is written as\nSγγ(k,ω)=1\nMNX\nη\f\f\f˜Wγkη\f\f\f2 Γη,k/π\n(ω−εη,k)2+ Γ2\nη,k, (D9)\nwhere\n˜Wγkη≡NX\nι\u0010¯Ol\nγmVk,ιη+¯Ol∗\nγmUk,ιη\u0011\n. (D10)\nWe calculate the trace of the dynamical structure factor, S(k,ω)=P\nγSγγ(k,ω). Figure 10 shows S(k,ω) at\n(h/|2K|,T/|2K|)=(0.15,0.05) and (0.12,0.15). We choose these parameters to compare the results with the previous ones\nin Ref. [72]. Figures 10(b), and 10(e) show S(k,ω) obtained by the on-shell Born approximation. We find that the spectral\nweight around the Γ′point is smeared due to the magnon damping compared to the results calculated by the linear spin-wave\ntheory [Figs 10(a), and 10(d)]. In addition, unnatural suppression in the high-energy region is observed. This behavior is due\nto the artifact intrinsic to the on-shell Born approximation as mentioned in Sec. C. On the other hand, in the results by the iDE16\nFIG. 10. Dynamical structure factor S(k,ω) ath/|2K|=0.15 and T/|2K|=0.05 obtained by (a) the linear spin-wave theory and nonlinear\ntheories with (b) the on-shell approximation and (c) the iDE approach. (d)–(f) Corresponding plots for h/|2K|=0.12 and T/|2K|=0.15.\nThe smearing factor is introduced as 0.1. The parameters for upper and lower sides correspond to those for Figs. 3(h) and 3(e) in Ref. [72],\nrespectively. (g) First Brillouin zones centered at the Γpoint and adjoined one centered at the Γ′point. The red lines represent the lines\nconnected with the high-symmetry points used in (a)–(f).\napproach, such unnatural behavior is not observed, as shown in Figs. 10(c), and 10(f). 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Azevedo 1 \n1Departamento de Física, Universidade Federal de Per nambuco, 50670-901, Recife, PE, Brasil. \n2Facultad de Física, Pontificia Universidad Católica  de Chile, Casilla 306, Santiago, Chile. \n \nWe show that in ferromagnetic (FM)/normal metal (NM ) bilayers the dynamic coupling at the \ninterface transfers an additional magnetic relaxati on from the heavily damped motion of the \nconduction electron spins in the NM layer to the FM  spins. While the FM relaxation rates due \nto two-magnon scattering and spin pumping decrease rapidly with increasing FM film \nthickness, the damping due to the dynamic coupling does not depend on the FM film thickness. \nThe proposed mechanism explains the very large broa dening of ferromagnetic resonance lines \nin thick films of yttrium iron garnet after deposit ion of a Pt layer.  \n \nPACS: 76.20.+q, 76.50.+g, 75.70.Cn, 72.25.Mk \n*Corresponding author: E-mail: rezende@df.ufpe.br  \n \nOne of the fundamental properties of a magnetic sys tem is \nthe manner by which its magnetization relaxes towar ds \nequilibrium. This is governed by the spin interacti ons and \nthe structure of the magnetic system and its detail ed \nunderstanding is important from the point of view o f basic \nphysics and for technological applications. For sev eral \ndecades the magnetic relaxation has been investigat ed \nexperimentally in bulk and thin film materials main ly by \nmeasuring the linewidth of the ferromagnetic resona nce \n(FMR) at microwave frequencies. In bulk magnetic \ninsulators the relaxation occurs through intrinsic \nmechanisms involving magnon-magnon and magnon-\nphonon processes as well as extrinsic mechanisms su ch as \nscattering by impurities. 1,2  In bulk metallic materials the \nrelaxation is dominated by processes involving the \nconduction electrons. 3 In very thin films and multilayers \nnew physical relaxation processes have been discove red in \nthe last fifteen years, the most important ones bei ng two-\nmagnon scattering from the irregularities at the su rfaces or \ninterfaces 4,5  and the spin pumping mechanism.6,7  These \nprocesses contribute with additional relaxation rat es that \nincrease as the magnetic film thickness decreases a nd thus \nbecome very important in ultra-thin films. 5,8 \nIn recent years structures made of bilayers of \nferromagnetic metal (FM) / normal metal (NM) films have \nbeen attracting considerable interest due to the di scoveries \nof the spin Hall effect 9,10  and the inverse spin Hall effect \n(ISHE). 11,12  In a FM/NM bilayer undergoing ferromagnetic \nresonance (FMR) it has been found 11-14  that the precessing \nspins in the FM inject spins into the adjacent NM l ayer \ngenerating a spin-pumping dc voltage by means of th e \nISHE opening immense possibilities in the field of \nspintronics. 15  A very important recent development in this \nfield was the demonstration that the ferrimagnetic insulator \nyttrium iron garnet (YIG) can be used in FM/NM stru ctures to convert charge current into spin current and vic e-versa.16  \nDue to its small magnetic damping, YIG films can be  used \nto transport spin information over much larger dist ances \nthan in FM metals so that YIG/Pt structures have at tracted \nincreasing scientific attention.17-26 However it has been \nobserved that the deposition of a Pt layer on thick  YIG \nfilms produces an unusually large broadening of the  \nmicrowave absorption lines,17,18  which is quite surprising \nbecause one expects the spin pumping mechanism to b e \neffective only at ultra-thin films.  \nIn this paper we show that when a NM layer is \ndeposited on a FM film, in addition to the spin pum ping \nprocess there is another mechanism for magnetic rel axation \nwhich is effective in thick FM films. The mechanism  relies \non the transferred relaxation due to the dynamic co upling \nof the precessing magnetization in the FM with the heavily \ndamped precession of the conduction electron spins in the \nNM layer. This process is effective in FM metallic or \ninsulating films and is independent of the spin-pum ping \nmechanism, although both originate in the spin coup ling at \nthe interface. While the spin-pumping mechanism is due to \nthe flow of angular momentum out of the FM layer in to the \nNM layer and relaxes the longitudinal component of the \nmagnetization, the new mechanism relaxes the transv erse \ncomponents of the magnetization. We show that the \nrelaxation due to dynamic coupling at the interface  explains \nthe observed broadening of the FMR lines in thick Y IG \nfilms with deposition of a Pt layer.  \nWe consider a bilayer of a ferromagnetic material w ith \na nonmagnetic metal in which the precessing magneti c \nmoments of the FM layer interact with the heavily d amped \nspins of the conduction-electron spins in the NM la yer \nthrough the dynamic exchange coupling at the interf ace. In \norder to treat the coupled mode problem we follow t he \nmacroscopic approach of Ref. [16] and consider that  at the  2interface sites i the spins isr of the conduction electrons in \nthe NM layer interact with the spins iSr\n in the FM side \nthrough the s-d exchange interaction, i ii sd sd sSJHrr\n⋅ −=∑, \nwhere Jsd  is the exchange coupling constant. Writing the \nrelation between the magnetization and the spins as  \n)( )(i ii B rrSgrMrrr rr\n− =∑δ µ , where g is the Landé factor and \nBµ is the Bohr magneton, the summation on the interfa ce \nsites i can be written as a surface integral along the \ninterface and the coupling between the magnetizatio n \n),(trMrr\n in the FM side and the magnetization ),(trmNrr of \nthe conduction electrons in the NM side can be writ ten as, \n      ∫∫⋅ −= ),()(),( )/( trmyatrMdy dz dx AJHN eff ex sd rr rr\nδ ,        (1) \nwhere y is the direction perpendicular to the interface pl ane \nx-z with area A at y = 0, )/( MSJJe sd ex γh=  is the \ndimensionless exchange coupling constant, S is an effective \nblock spin per unit cell and M is the magnetization of the \nFM, eγ is the gyromagnetic ratio of the conduction \nelectrons in the NM, 2/se eff ava= is the effective interaction \nrange, ev is the volume per conduction electron, and sa is \nthe lattice constant of the localized spins at the interface on \nthe FM side. In order to make the interface couplin g \ntractable we follow Ref. [16] and consider that the  \nmagnetizations do not vary along the interface plan e and \n),(),( tymtrmN Nrrr=, so that from Eq. (1) we obtain,  \n         ) , 0()( tymatMJHN eff ex sd =⋅ −=rr\nζ ,                             (2) \n where ζ is a factor that accounts for the surface integral : \n1=ζ corresponds to an atomically flat surface and unif orm \nmagnetizations at the interface plane; 1<ζ accounts for \nirregularities at the interface, such as roughness,  or to a \nspatially varying ),(trMvr\n as in a spin wave. Equation (2) \nrepresents the interface energy per unit area and s ince the \nmagnetization is distributed over the whole FM volu me, the \nenergy per unit volume on the FM side is \nFM N eff ex FM \nv dtmatMJ E / ), 0 ( )(rr\n⋅ −=ζ , where FM d is the FM \nlayer thickness. One can write the effective field acting on \nthe FM magnetization due to the interface exchange as, \n      FM Neff ex FM \nv FM \nE dtmaJ\nMEH / ), 0 (rrr\nζ=\n∂∂−= .                           (3) \n Thus one can write the Landau \n     MdamMJHMdt Md\nFM \nFM eff \nN ex r vr rrr\nη ζγ γ − × −×−= )] 0 ( [ ,          (4) \nwhere γ  is the gyromagnetic ratio ( ×=πµ2/hBg 2.8 \nGHz/kOe for YIG) and FM η is the relaxation rate of the FM \nmagnetization in the absence of coupling at the int erface. \nThe equation of motion for the magnetization ),(tymNr of \nthe conduction electrons must take into account the  spin \ndiffusion into the NM side and the exchange couplin g at the \ninterface. The magnetization in the NM can be writt en 16  as )()( )(0 ymyamymN eff Nr rrδδ+ =  where MJ mex Nr rζχ=0  is the \nequilibrium magnetization, Nχ is the paramagnetic \nsusceptibility of the conduction electrons and ),(tymNrδ  \nrepresents the spin accumulation. The equation of m otion \nthen becomes,16 \n     N N Nsf eff N ex e NeN\nmDmyaMmJHmtm\nv vrv rvv\nδ ηδ ζγ γ\n2)()(\n∇+−× −×−=∂∂\n ,          (5) \nwhere sf η is the relaxation rate of the spin accumulation, \nrelated to the electron spin-flip time sf τ by sf sf τη/ 1=, and \nND is the spin diffusion constant. We consider that t he \nstatic field is in a direction parallel to the inte rface plane, \ndesignated the z-direction of a coordinate system that has \nthe y-direction perpendicular to the interface and write  the \ntwo magnetizations as z y x MzmymxM ˆˆˆ ++=r and \nz\nNy\nNx\nN N mzmymxm δδδδ ˆˆˆ ++=r. Equations (4) and (5) \ndescribing the coupled motion of the magnetizations  can be \nsolved for all magnetization components. This has b een \ndone in Ref. [16] for the longitudinal spin accumul ation \n),(tymz\nNδ  from which one can calculate the spin current \ndensity in the NM using z\nN yNBz\ns mDeJ δ µ∇ −= )/( , which \nleads to, in units of angular momentum/(area.time),  \n         )()1 (2) 0 (2 2 22\nzyx\nNBNN z\nSMmmDJΓ+=λµχωh,                               (6) \nwhere 2 / 1)(sf N NDτλ=  is the spin diffusion length, Γ is the \nparameter defined in Ref. [16], with 1=ζ, \n) ( /eff sf sd N aJSτ λh=Γ  and we have assumed circular \nprecession. Comparison of Eq. (6) with the well kno wn \nexpression for the current density in terms of the transverse \ncomponents of the magnetization 6,7  leads to a convenient \nrelation between the exchange coupling parameter an d the \nspin-mixing conductance ↑↓ g, \n            )1 (2\n2 2Γ+=↑↓ \nNBNNDgλµχπh.                                               (7)  \nNotice that this relation differs by a factor of 2 from the one \nin Ref. [16] because we have considered \nyx y x mmmmmm 22 2≈+=−+. The spin current in Eq. (6) \nrepresents a flow of spin angular momentum out of t he FM \nlayer resulting in the relaxation of the FM magneti zation \nthrough the spin pumping mechanism giving rise to a  FMR \nlinewidth (half width at half maximum) given by  6,7     \n                  \nFM SP dMgHπω\n4↑↓ =Δh.                                                  (8) \nwhere Mπ4is the saturation magnetization. Note that the \nspin pumping process relaxes the zMcomponent of the \nmagnetization so that it produces a relaxation time  of the \ntype T1 in the Bloch-Bloembergen formulation 1,2  of the \nLandau-Lifshitz equation. In the derivation of Eq. (6) one  3neglects 16 the reaction of the spin accumulation Nmrδ on \nthe FM magnetization Mr. The full solution of Eqs. (4) and \n(5) for the coupled transverse components of the \nmagnetizations reveals another independent contribu tion to \nthe FM relaxation of the type T2 arising in the dynamic \ncoupling between the spins at the interface. Using for the \ntransverse variables y\nNx\nN N mimm δδδ +=+ and y xmimm+=+ \nwe can solve the equation for the spatial dependenc e of the \ntransverse spin accumulation ),(tymN+δ  obtained from Eq. \n(5). Considering for simplicity a NM layer thicknes s much \nlarger than the spin diffusion length we find \n)/exp( ), 0 ( ),(N N N ytmtym λ δ δ − =+ + so that from Eq. (5) we \nobtain an equation relating )(tm+ and ), 0 (tmN+δ. This \nequation together with the one obtained from Eq. (4 ) form a \nset of two equations for the transverse variables. \nConsidering the time dependence )exp( tiω one obtains the \ncoupled equations,  \n   ) 0 ()/( ])( [+ +−= −−N FM eff ex z FM FM mdaJMmi δ ζγ ηωω  ,       (9) \n   ++\n−−= − −−\nm Mmi\nsf Nz H ex N sf ex H\nηχγωωλδ ηλωω\n]/ ) [( ) 0 (] )1 ( 2 / ) ( [,   (10) \nwhere HeHγω= is the conduction electron spin resonance \nfrequency, )4(MHHFM π γω + = is the FMR frequency and \nΓ=/ζλex  is a dimensionless coupling parameter. From Eqs. \n(9) and (10) one obtains, \n   \n0]) [( 2)] (2 ) [( ]) [( \n=Δ+−−+−Δ+− −−\nCi i\nFM FM ex sf FM FM FM FM \nωωωληωωωηωω\n         (11) \nwhere H FM FM ωωω −=Δ  and )/( 0 FM ex sf eff dM amC λη= . Eq. \n(11) leads to a quadratic equation with complex \ncoefficients, the roots of which are the two comple x \neigenmode frequencies,  \n                    sf Hiηωω 21+≈ ,                                           (12) \n     ] )( )(4 [0 4 4\n2\nFM eff \nsf ex sf ex FM FM da\nMmi ηληληωω +++≈  .             (13) \nThe real and imaginary parts of Eqs. (12) and (13) \ncorrespond respectively to the eigenmode oscillatio n \nfrequencies and relaxation rates. Clearly 1ω is associated \nwith the motion dominated by the conduction electro n spins \nin the NM layer, whereas 2ω is associated with the spin \nprecession in the FM layer. Since 10 10 ~Hωs-1 and \n12 10 ~sf ηs-1 the motion of the spins in Pt is heavily \noverdamped. The important result revealed in Eq. (1 3) is \nthat the relaxation rate of the FM layer has, in ad dition to \nthe intrinsic term FM η, two contributions proportional to the \nfourth power of the exchange coupling parameter and  to the \nconduction electron spin relaxation sf η. Note also that since \n1/0<Mm  and 1/<< FM eff da , for YIG films with µm \nthickness, the third term in the imaginary part of Eq. (13) is \nnegligible compared to the second. As will be shown  in the \nfollowing, the damping transferred from the conduct ion electron spins in the NM layer to the FM magnetizat ion \nrepresented by the second term in the imaginary par t of Eq. \n(13) explains the large broadening of the FMR lines  \nobserved in YIG/Pt. \nThe experiments were carried out at room temperatur e \nwith several samples consisting of a single-crystal  YIG film \nwith thickness in the range 8 to 28 µm, grown by li quid-\nphase epitaxy (LPE) on 0.5 mm thick (111) gadoliniu m \ngallium garnet substrates and cut in an approximate  square \nshape with lateral dimensions of 2 to 4 mm. Measure ments \nwere made with the bare YIG films and with a Pt lay er with \nthickness 6 nm or 20 nm deposited by magnetron \nsputtering. The samples are mounted at the tip of a  plastic \nrod to allow measurements as a function of the angl e and \nplaced in the middle of a shorted rectangular waveg uide \nbetween the poles of an electromagnet. The static m agnetic \nfield with intensity H and the microwave magnetic field are \nkept perpendicular to each other. Field sweep \nmeasurements were made to obtain spectra of the \nmicrowave absorption derivative at several values o f the \nangle Hθ between the field Hr\n the film plane to investigate \nthe out-of-plane angle dependence of the spectra. \nAll samples investigated exhibited similar behavior . \nThe FMR linewidths in the bare YIG film are less th an 1 \nOe with the applied field parallel or perpendicular  to the \nfilm plane. After deposition of a 6 nm or 20 nm Pt layer the \nlinewidths increase to several Oe. We present here only \nsome representative results obtained with microwave  \nfrequency 9.4 GHz and input power less than 10 mW. Fig. \n1 shows the field derivative d P/d H of the microwave \nabsorption spectra for a 28  µm thick YIG film for two \ndirections of the field, parallel and perpendicular  to the film \nplane. The spectra in Figs. 1(a) and 1(b) obtained before \ndeposition of the Pt layer allow a clear identifica tion of the \nabsorption lines. They correspond to standing spin- wave \nmodes that have quantized in-plane wave numbers due  to \nthe boundary conditions at the edges of the film. 17,18,27   The \nstrongest line corresponds to the FMR mode that has  \nfrequency given by 2 / 1 2 / 1\n0 )4(MHH π γω + ≈ . In Fig. 1(a) \nwith the field in the plane, the lines to the left of the FMR \nmode correspond to hybridized surface modes whereas  \nthose to the right are volume magnetostatic modes.  17,18  As \nshown in Figs. 1(c) and (d) the deposition of a 6 n m thick \nPt layer on the YIG film produces a pronounced \nbroadening of the lines for all modes. They have ne arly the \nsame peak-to-peak linewidth pp HΔ of about 6 Oe, which is \nnearly ten times larger than in bare YIG. Similar r esults \nwere obtained with YIG film thickness 8 and 15 µm. In \nboth samples the linewidths in YIG/Pt have nearly t he same \nvalue for θΗ = 0 and 90 o, which are respectively 2.5 and 3.0 \nOe larger than in the corresponding bare YIG film.   \nThe first mechanism one would consider to explain t he \nincreased damping in YIG caused by deposition of Pt  is the \nspin pumping process.  6,7,23 Using for the effective spin-  4 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1 . Field scan microwave field derivative absorption \nspectra at a frequency 9.4 GHz of a 28 µm thick YIG  film with \nlateral dimensions 2 x 3 mm with the magnetic field  applied \nparallel or perpendicular to the film plane as indi cated. In (a) and \n(b) the YIG film is bare while in (c) and (d) it is  covered with a 6 \nnm Pt layer. \n \nmixing conductance the value 13 10 5 . 1×≈↑↓ g cm -2 determined \nin Ref. [18] from measurements of VSP  using the spin-Hall \nangle 08 . 0=Hγ\n and spin-diffusion length 7 . 3=Nλnm, 14,18  \nand Mπ4= 1.76 kG for YIG, we obtain with Eq. (8) for a \n28 µm thick YIG/Pt bilayer a linewidth of 410 4−×=ΔSP H Oe. \nThis is definitely too small compared to the observ ed line \nbroadening ruling out the spin pumping as the sourc e of the \nadditional relaxation. Another possible origin of t he line \nbroadening with Pt deposition is the enhancement in  the \ninterface two-magnon scattering relaxation similar to the \none observed when a FM layer is in contact with an \nantiferromagnetic material. 5 However, the two-magnon \nlinewidth 4 varies with 2/ 1FM d and like the spin pumping it is \nimportant only in films with thickness of a few nan ometers. \nMoreover, with the field normal to the plane the FM R \nfrequency is at the bottom of the spin-wave manifol d and \nthere are relatively few degenerate states into whi ch the \nFMR mode can decay into, so that the relaxation due  to \ntwo-magnon scattering should be small as demonstrat ed \ntheoretically and observed experimentally. 29,30  As it is clear \nin Fig. 1, the linewidths in YIG/Pt at =Hθ0 and 90 o are \nvery similar, ruling out the two-magnon scattering as the \nmechanism responsible for the line broadening. \nWe will show now that the transferred relaxation \nmechanism proposed here accounts comfortably for th e \nobserved line broadening in YIG/Pt. For FM films wi th \nthickness in the µm range the additional damping in  Eq. \n(13) is dominated by the second term in the imagina ry part \nso one can write the contribution of the transferre d \nrelaxation to the FMR linewidth as,  \n           γηλ/4\nsf ex trans H=Δ ,                                          (14) where the 1/<< Γ=ζλex  exchange parameter is related to \nthe spin mixing conductance by,  \n              \nNNNB\nex Dgχπλµζλh222\n2 ↑↓ = .                                  (15) \nAtomic force microscopy analysis has revealed that \nthe surfaces of the LPE grown YIG films are flat in  the \natomic level, so we consider 1=ζ. Using the spin mixing \nconductance 18  for YIG/Pt, -2 13 cm  10 1.5 ×=↑↓ g , and the \nfollowing parameters for Pt, nm  7 . 3=Nλ (Ref.14), \n12 10 =sf ηs-1 (Ref.16), 1 . 02==sf N NDηλ cm 2s-1, 510 1 . 2−×=Nχ \n(paramagnetic susceptibility of bulk Pt, Ref.31), w e obtain  \n024 . 02=ex λ . This gives for the theoretical additional FMR \nlinewidth due to the transferred relaxation a value  \n32 ≈Δtrans H Oe. This value is six to ten times larger than the \nmeasured ones, revealing that the dynamic interacti on at \nthe interface indeed transfers a considerable dampi ng from \nthe overdamped motion of the conduction electron sp ins in \nPt to the magnetization in YIG. The discrepancy bet ween \ntheory and experiment can be attributed to several factors. \nFirst we note that there is a large discrepancy in the \nparameters for Pt reported by different authors. 32  Second, \nand perhaps more important, we have used in the \ncalculation of ex λ the value of the paramagnetic \nsusceptibility for bulk Pt, which is the only one a vailable in \nthe literature. 31  However, there are experimental evidences \nthat when Pt is deposited on a FM metal, such as Ni  and \nPy, the atomic layers close to interface become spi n \npolarized due to the proximity effect. 33,34  This effect most \ncertainly occurs also in YIG/Pt, producing an enhan cement \nof the Pt susceptibility near the interface, which decreases \nthe value of ex λ and consequently lowers the theoretical \nprediction for trans HΔ. Note that an enhancement by a factor \nof 3 in the Pt susceptibility, which is well within  the \nincrease in the magnetic moment near the interface \nmeasured is Refs. [33,34], is sufficient to produce  a nice \nagreement between theory and experiments. \nIn conclusion, we have demonstrated that in FM/NM \nbilayers, the coupling of the overdamped motion of the \nconduction electron spins in the NM layer to the sp ins in \nthe FM layer gives rise to a transferred damping \nmechanism. This additional damping has an origin th at is \nindependent of the well known spin-pumping \nmechanism.6,7  Both relaxation processes originate in the \ninterface coupling but they are due to quite differ ent \nmechanisms. The conventional spin-pumping is due to  the \nflow of angular momentum out of the FM layer into t he \nNM layer and relaxes the longitudinal component of the \nmagnetization. It does not depend on the conduction  \nelectron spin-flip relaxation rate sf η and varies inversely \nwith the thickness of the FM layer. On the other ha nd, the \nmechanism proposed here relaxes the transverse \ncomponents of the magnetization, resulting in a dam ping 2.50 2.55 2.60 2.65 Signal dP/dH  (arb. units) θH = 0° \nHH(a) \n5.07 5.12 5.17 5.22 (b) θH = 90° \nH\nH\n2.50 2.55 2.60 2.65 (c) Signal dP/dH  (arb. units) \nMagnetic Field H (kOe) θH = 0° HH\n5.07 5.12 5.17 5.22 (d) \nMagnetic Field H (kOe) θH = 90° \n  5rate proportional to sf η and independent of the thickness of \nthe FM layer. The existence of this novel relaxatio n \nmechanism is clearly observed in the broadening of the \nferromagnetic resonance lines in relatively thick Y IG films \nafter deposition of a Pt layer.  \nThis work was supported in Brazil by the agencies \nCNPq, CAPES, FINEP and FACEPE and in Chile by the \nMillennium Science Nucleus \"Basic and Applied \nMagnetism\" No. P10-061-F. \n \nReferences \n1 M. Sparks, Ferromagnetic Relaxation  (Mc Graw-Hill, New \nYork, 1964). \n2 R.M. 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B 83 , 075407 (2011). \n \n \n \n \n \n "    },    {        "title": "2204.09776v1.Ferrimagnet_GdFeCo_characterization_for_spin_orbitronics__large_field_like_and_damping_like_torques.pdf",        "content": " \n \n1 \n Ferrimagnet GdFeCo  characterization for spin-orbitronics:  large field -like and \ndamping -like torques  \n \nHéloïse Damas1*, Alberto Anadon1, David Céspedes -Berrocal1,2, Junior Alegre -Saenz1,2, Jean -\nLoïs Bello1, Aldo Arriola -Córdova1,2, Sylvie Migot1, Jaafar Ghanbaja1, Olivier Copie1, Michel \nHehn1, Vincent Cros3, Sébastien Petit -Watelot1* and Juan -Carlos Rojas -Sánchez1* \n \n1Université de Lorraine,  CNRS, Institute Jean Lamour, F -54000 Nancy, France  \n2Universidad Nacional de Ingeniería, Rímac 15333, Peru  \n3Unité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767 Palaiseau, France  \n \nCorresponding authors:  heloise.damas@univ -lorraine.fr, sebastien.petit@univ -lorraine.fr,  \nJuan-Carlos.ROJAS -SANCHEZ@univ -lorraine.fr  \n \nH. Damas, Dr. A. Anadon, D. Céspedes -Berrocal, A.Y. Arriola -Córdova, J -L Bello, S. Migot, \nJ. Ghanbaja, Dr. O. Copie, Prof. M. Hehn, Dr. S. Petit -Watelot,  Dr. J. -C. Rojas -Sánchez.  \nUniversité de Lorraine,CNRS, Inst itute Jean Lamour, F -54000 Nancy, France  \nE-mail:  heloise.damas@univ -lorraine.fr , sebastien.petit@univ -lorraine.fr  , Juan-\nCarlos.ROJAS -SANCHEZ@univ -lorraine.fr   \n \nD. Céspedes -Berrocal, A.Y. Arriola -Córdova , J. Alegre -Saenz  \nUniversidad Nacional de Ingeniería, Rímac 15333, Peru  \n \nDr. V. Cros,  \nUnité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767 Palaiseau, France  \n \nKeywords:  \nFerrimagnet GdFe Co; spin -orbit torque; spin -torque ferromagnetic resonance; spin anomalous \nHall effect; spin Hall effect  \n \nSpintronics is showing promising results in the search for new materials and effects to reduce \nenergy consumption in information technology. Among these materials, ferrimagnets are of \nspecial interest, since they can produce large spin currents that trigge r the magnetization \ndynamics of adjacent layers or even their own magnetization. Here, we present a study of the \ngeneration of spin current by GdFeCo in a GdFeCo/Cu/NiFe trilayer where the  FeCo  sublattice \nmagnetization is domina nt at room temperature. Magn etic properties such as the saturation \nmagnetization are deduced from magnetometry measurements while damping constant is  \n \n2 \n estimated from spin -torque ferromagnetic resonance (ST -FMR). We show that the overall \ndamping -like (DL)  and field -like (FL) effective fields as well as the associated spin Hall angles \ncan be reliably obtained by performing the dependence o f ST-FMR by an added dc current. The \nsum of the spin Hall angles  for both  the spin Hall effect (SHE) and the spin anomalous Hall \neffect ( SAHE ) symmetries  are: 𝜃𝐷𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐷𝐿𝑆𝐻𝐸=−0.15±0.05 and 𝜃𝐹𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐹𝐿𝑆𝐻𝐸=\n0.026±0.005. From the symmetry of ST -FMR signals we find that 𝜃𝐷𝐿𝑆𝐻𝐸 is positive and \ndominated  by the negative 𝜃𝐷𝐿𝑆𝐴𝐻𝐸. The present study paves the way f or tuning the different \nsymmetries in spin conversion in highly efficient ferrimagnetic systems.  \n \n1. Introduction  \n \nIn the last years, ferrimagnets have attracted growing interest for their potential utility in \nspintronic devices  [1]. In particular, GdFeCo ferrimagnetic alloy is extensively studied  as it \nexhibits  a wide diversity of phenomena arising from the specific properties of rare earth -\ntransition metal (RE -TM) ferrimagnets. Furthermore , the two antiferromagnetically coupled \nsublattices have a different response to external stimuli and the spin -orbit couplin g (SOC) of \nthe Gd 5d state allows the interplay between charge, spin , and orbital transport. The different \nrelaxation times of these two coupled sublattices are thought to be responsible for the all -optical \nhelicity -independent switching (AO -HIS) in GdFeCo  demonstrated for almost a decade  [2,3] . \nAO-HIS has also been recently observed in TbCo  [4]. Nowadays, GdFeCo is used to perform \nthe AO -HIS of Co/Pt  [5–7] or CoNi/Pt  [8] ferromagnetic multilayers. Moreover, it is possible \nto tune the Dzyaloshinskii -Moriya interaction  in thin GdFeCo ferrimagnetic alloys  [9], a \nrelevant property for skyrmions formation.  It has been shown that GdFeCo ferrimagnet  also \nhosts  large self-induced spin -orbit torque, or self -torque  [10,11] , with recent theoretical \nadvances  [12,13] . Ferro and ferrimagnetic materials  are the source of spin currents with  \ndifferent symmetries  [10,12]  coming from the s pin anomalous Hall effect (SAHE)  [14–16] and \nthe spin Hall effect (SHE)  [17]. In the SAHE, the spin polarization of the spin current 𝐽sSAHE is  \n \n3 \n parallel to the magnetization while in the SHE it is perpendicular to both the injected charge \ncurrent and the produced spin current 𝐽sSHE. A giant overall spin Hall angle  for SAHE -like and \nSHE -like symmetries  has been reported in a Gd -rich GdFeCo/Cu at r oom temperature  [10]. \nSizable i nterconversion efficiencies ha ve also been reported for  other magnetic materials such \nas NiFe  [18–20] and CoFeB [15,21] . In the present work, we study  room temperature  FeCo -\nrich GdFeCo in a //Gd 25Fe65.6Co9.4(8 nm)/Cu(4 or 6 nm)/Ni 81Fe19(4 nm) trilayer by structural, \nmagnetic and spintronics characterization. W e use two complementary ST -FMR techniques to \nreveal the signs and ma gnitudes of the contributions coming from the different spin current \nsymmetries in GdFeCo. Namely, the modulation of the damping along with the  shift of \nresonance field to extract the overall parameters (sum of the SHE -like and SAHE -like \ncontributions) and  the symmetry of the ST -FMR signal which is sensitive only to the SHE -like \nparameters. We found that damping -like (DL) SAHE spin Hall angle, 𝜃𝐷𝐿𝑆𝐴𝐻𝐸, is negative  for \nFeCo -rich GdFeCo . In contrast, the DL SHE -like symmetry , is positive.   \n \n2. Structur al and chemical characterization  \n \nSamples were grown on thermally oxidized Si wafers using dc magnetron sputtering at room \ntemperature with  an Ar gas pressure of 3 mTorr and base pressure of 1x10-7 Torr. GdFeCo \n(Gd 25Fe65.6Co9.4) was co -deposited using separate Gd, Co , and Fe targets. All the samples in the \npresent study were capped with 3 nm of naturally oxidized Al. The c omposition was controlled \nby varying the sputter gun power on each target. The d eposition rate was calibrate d by X -ray \nreflectivity and lift -off and profilometer measurements of the thickness.  In order to perform \nstructural characterization, a thin lamella was extracted by focused ion beam (FIB) milling \nusing an FEI Helios Nanolab dual -beam 600i.  \nTransmission e lectron microscopy (TEM) investigations were carried out using a JEM - ARM \n200F Cold FEG TEM/STEM  (Scanning TEM)  operating at 200 kV, coupled with a GIF \nQuantum 965 ER and equipped with a spherical aberration (Cs) probe and image correctors  \n \n4 \n (point resoluti on 0.12 nm in TEM mode and 0.078 nm in Scanning TEM  (STEM)  mode) . High -\nResolution TEM (HRTEM) micrographs were performed to study the atomic structure of the \ndeposit layers  as shown in  Figure 1 a. The Fast Fourier Transformation (FFT) patterns in Figure \n1b,c confirm that the Cu/NiFe layers are [111] textured along the growth direction while \nGdFeCo is amorphous as evidenced by the diffuse rings.  Electron Energy Loss Spectroscopy \n(EELS) maps were carried out systematically on the different samples and confirm the nominal \ncomposition and thickness of the different materials  (Figure 1 d). We also evidence a slight  \nGdFeCo composition variation along the growing direction as usually observed in RE -TM \nferrimagnets  [10,11,22] . The EELS maps displayed in Figure 1 d were performed with \n1ev/channel and  a step of 0.3 nm.   \n \n \nFigure 1.  TEM/STEM characterization of  Gd 25Fe65.6Co9.4(8)/Cu(6)/Ni 81Fe19(4)/AlOx . (a) \nHRTEM micrograph of the deposit ed layers. The yellow (blue) square shows where the FFT \nanalysis have been performed on Cu/NiFe (GdFeCo). The FFT patterns ( b) and ( c) indicate the \n[111] growth direction of textured Cu/NiFe and that GdFeCo is amorphous. ( d) High Angle \nAnnular Dark Field (H AADF) -STEM micrograph and the corresponding individual EELS \nelemental maps obtained from the green rectangle area in the HAADF micrograph. Co (yell ow), \nFe (gr een), Gd (orange), O (red), Ni (cyan), Cu (pink).  \n \n \n5 \n  \n \n \n \n3. Magnetic characterization: magnetic anisotropies in GdFeCo  \n \n3. 1. SQUID magnetometry  \n \nMagnetization loops were performed at room temperature on a //GdFeCo (8)/Cu(6)/NiFe (4) \nstack with the applied field  parallel and perpendicular to the field plane. Both 𝑀(𝐻) \nmeasurements  are display ed in Figure 2a show ing an open hysteresis loop . This confirm s that \nthe NiFe magnetization  direction  𝒎̂NiFe lies in the plane of the sample while that of GdFeCo , \n𝒎̂GdFeCo ,  is spontaneously perpendicular to the film plane  as shown in the inset . We assume  \nthat the 6  nm thick Cu layer decouples the two magnetic layers to extract their distinct saturation \nmagnetization and saturation magnetic field . For NiFe, the saturation magnetization 𝑀sNiFe is \n625 kA/m  and the saturation field 𝜇0𝐻sat−zNiFe to place 𝒎̂NiFe out of the plane of the film is 0.85 \nT. In the case of GdFeCo, the  saturation magnetization 𝑀sGdFeCo is 115 kA/m  and the saturation \nfield 𝜇0𝐻sat−xyGdFeCo to align 𝒎̂GdFeCo along the plane is about  0.13 T which are typical values  for \nboth NiFe and Gd25Fe65.6Co9.4 at room temperature   [23–25]. From the  saturation field  and \nmagnetizations , we can also estimate the effective saturation magnetization for both magnetic \nmaterials, it results  𝑀effNiFe=676  kA/m  and 𝑀effGdFeCo=103  kA/m. The relatively  low \nperpendicular magnetic anisotropy of GdFeCo allows its magnetization to be easily placed \nalong t he plane  of the film  which is useful for ST-FMR measurements.  The trilayer used in the \nnext section has a 4 nm Cu spacer and displays a lower saturation field 𝜇0𝐻sat−xyGdFeCo to align \n𝒎̂GdFeCo along the plane,  which is about  ~0.047 T.  \n \n \n  \n \n6 \n  \nFigure 2. Bulk magnetization data using a SQUID magnetometer. (a) Magnetic \nhysteresis loop of the Gd25Fe65.6Co9.4(8)/Cu(6)/Ni 81Fe19(4) trilayer. Magnetization \nvalues are normalized by the surface sample. To identify the different saturation fields \nand effective saturation magnetization, we consider that the magnetic layers are \ndecoupled  by the 6 nm of Cu . The inset shows a s chematic of the sample with the \nspontaneous magnetization alignment of GdFeCo (out -of-plane) and NiFe (in -plane) \naccording to SQUID results.  \n \n \n \n3. 2. Spin -torque FMR study  \nWe perform ST -FMR measurements  [26–31] on a GdFeCo (8)/Cu(4)/ NiFe (4) trilayer  to extract \nproperties such as the damping constant 𝛼 and the Landé g -factor of the magnetic layers. From \nMagneto optic Kerr effect measurements we have verified that this GdFeCo (8) is FeCo -rich at \nroom temperature.  The experimental setup is described in Figure 3a. A radiofrequency (rf) \ncharge current , 𝑖rf,  is applied  along the 𝒙̂ direction and generates an oscillating Oersted field \nwhich  triggers the magnetization precession  at the resonance condition . A sweeping dc \nmagnetic field 𝐻dc is applied in the xy plane of the device, at an angle of 𝜑𝐻 with respect to  the \ncurrent line. At the resonance field 𝐻res, a dc voltage 𝑉mix composed of a mix ing of a \nsymmetric and antisymm etric Lorentzians of amplitude Vsym and Vanti respectively can be \nmeasured using a bias tee. The measured mixed voltage  display ed in Figure 3b can be fitted \nwith the following general expression:  \n𝑉mix=𝑉offset+𝑉symΔ𝐻2\nΔ𝐻2+(𝐻−𝐻res)2+𝑉anti(𝐻−𝐻𝑟𝑒𝑠)Δ𝐻\nΔ𝐻2+(𝐻−𝐻𝑟𝑒𝑠)2, (1) \n \n \n7 \n where  we consider an additional offset 𝑉offset  and where Δ𝐻 is the linewidth. In Figure 3b, we \nobserve the two resonance lines corresponding to the NiFe resonance  (lower resonance field)  \nand the GdFeCo resonance  (higher resonance field ). For the sake of clarity,  it is only show n at \n8, 12 and 14  GHz . Then, from  broadband frequency dependence  ST-FMR  we can extract the \neffective saturation magnetization  Meff (it results negative f or system s where  perpendicular \nmagne tic anisotropy dominate s over shape anisotropy ), and the Landé g -factor considering the  \nfollowing  expression : \n𝑓=𝛾\n2π√(𝐻+𝐻uni)(𝑀eff+𝐻+𝐻uni) ,          (2) \nwhere 𝛾=𝑔μB\nℏ is the gyromagnetic ratio and where  Huni stands for a small in -plane uniaxial \nmagnetic anisotrop y. Equation 2  applies for a thin film ferromagnetic  layer with  a magnetic \nfield applied in the plane . We fix the NiFe  Landé g -factor to 2.10 . We  determine the effective \nsaturation magnetization of NiFe , 𝑀effNiFe=569±1kA/m . The difference with previous \nSQUID results  comes from  the difference in Cu thickness which affect s the NiFe anisotropy . \nWe also evaluate  a rather small  𝐻uni=−7±1 Oe. We exploit the same Equation 2 for \nGdFeCo resonance condition to determine the GdFeCo  Landé g -factor  and its effective \nsaturation magnetization 𝑀effGdFeCo. We obtain  g=2.87±0.04, and 𝑀effGdFeCo=−37±4 kA/\nm (−46.5 mT). The fitted experimental data is shown in Figure 3c. Finally, from  the frequency \ndependence  of the linewidth  H, we calculate the Gilbert -type magnetic damping constant  𝛼: \n𝛥𝐻=𝛥𝐻0+2π𝑓\n𝛾𝛼 , (3) \nwhere H0 is the f-independent contribution due to inhomogeneity . We have fixed g -Landé \nfactor for both NiFe and GdFeCo. The fits on the measurements are shown in Figure 3d.  The \ndamping of in-plane NiFe is estimated  as 𝛼𝑁𝑖𝐹𝑒=0.012±0.001 which  is about 8  times \nsmaller than the damping of our out -of-plane Gd25Fe65.6Co9.4 𝛼GdFeCo=0.085±0.006 but \ncomparable with in -plane Gd12.5Fe76.1Co11.4 [32]. Recently, it has been point ed out that actual   \n \n8 \n or intrinsic damping in ferrimagnet s is lower than th at measured  directly  due to different spin \ndensity for each magnetic sublattice in GdFeCo and  determine d by domain wall mobility  [33].  \nIn the next section , we show how we c an estimate the effective field s that drive the spin -orbit \ntorque from Gd25Fe65.6Co9.4(8)/Cu(4)  to Ni81Fe19(4). \n \n \n \n \n \nFigure 3. Determination of Meff and damping using broadband ST -FMR in \nGd 25Fe65.6Co9.4(8)/Cu(4)/Ni 81Fe19(4), and g -Landé factor for GdFeCo . (a) Illustration of a \ntypical ST -FMR device along with  the dc magnetic field applied at 𝜑𝐻 to the trilayer slab  which  \nis along 𝑥̂. (b) Typical ST -FMR spectr a at 8, 12 and 14  GHz. At a higher field s, the GdFeCo \nresonance line  is observed . The symmetrical (orange) and antisymmetrical (green) voltage \ncontribution s are shown  for NiFe at 8 GHz. Broadband frequency  dependence of Hres (c) and \nlinewidth H (d) are used to determine Meff and , respectively. Equation 2 is used for NiFe \nand GdFeCo layer  in (c). For Gd 25Fe65.6Co9.4 (8 nm), we estimate   𝑔=2.87±0.04 and 𝛼=\n0.085±0.006. Black (green ) experimental data are obtained for the resonance of NiFe \n(GdFeCo) as depicted in (b). \n \n \n \n \n9 \n  \n4. Damping -like and field -like efficiencies  determination  by ST -FMR techniques  \n4. 1. Spin torque  symmetries  and ST -FMR signal   \nAs discussed, there are two symmetries for spin current generation in magnetic materials, \nSAHE -like and SHE -like. When these spin currents are absorbed by a nother  magnetic layer, \nthey contribute to the total torque on the magnetization :  \n𝚪tot=𝚪SHE+𝚪SAHE . (4)  \nIn the geometry of our ST -FMR measurements , the GdFeCo and NiFe magnetizations are both \naligned with an angle of  𝜑𝐻 with respect to the 𝑥̂ axis. The sp in polarization corresponding to \nthe SHE -like symmetry, 𝜎̂SHE, lies along the  𝑦̂ axis regardless of the direction  of both \nmagnetizations . The spin polarization direction related to the SAHE-like spin current , 𝜎̂SAHE , \nlies along the direction of 𝑚̂GdFeCo  which in turn is aligned with the equilibrium direction of \nthe NiFe magnetization , 𝑚̂NiFe . The different contributions to the total torque  can further be \ndivided into two contributions, coming from the  damping -like (ℎDL) and the field-like (ℎFL) \neffective  fields:  \n𝚪SHE\n𝛾𝑀SNiFe= ℎDLSHE𝒎̂NiFe×(𝝈̂SHE⏟\n𝒚̂×𝒎̂NiFe)+ℎFLSHE 𝒎̂NiFe×𝝈̂SHE⏟\n𝑦̂,   (5𝑎) \n𝚪SAHE\n𝛾𝑀SNiFe= ℎDLSAHE  𝒎̂NiFe ×( 𝝈̂SAHE⏟  \n𝒎̂GdFeCo×𝒎̂NiFe)+ℎFLSAHE 𝒎̂NiFe×𝝈̂SAHE⏟  \n𝒎̂GdFeCo.(5𝑏) \nThe efficiency of the charge -to-spin current  conversion is described  by the spin Hall angles \n𝜃DL(FL)SAHE and 𝜃DL(FL)SHE. They are related to the SAHE and SHE -like effective fields generated by \nGdFeCo and acting on NiFe layer as follow s  [14,15,34,35] : \nℎDL(FL)SHE=ℏ\n2|𝑒|𝑗cGdFeCo\n𝜇0𝑀sNiFe𝑡NiFe 𝜃DL(FL) SHE, (6𝑎) \nℎDL(FL)SAHE=ℏ\n2|𝑒|(𝒎̂GdFeCo×𝑱cGdFeCo).𝒛̂\n𝜇0𝑀sNiFe𝑡NiFe  𝜃DL(FL)SAHE , (6𝑏) \nwith (𝒎̂GdFeCo×𝑱cGdFeCo).𝒛̂=sin(𝜑𝐻) 𝐽cGdFeCo in our geometry , as depicted in Figure 3a .  \n \n10 \n  \nWe show in the following subs ections that ST -FMR techniques can be  useful tool s to further \nstudy the sign and quantification of the different contributions.  Indeed, t he analytical expression \nfor the lon gitudinal voltage obtained by ST-FMR measurements reads   [15,27,36,37] :  \nVdc=−∆𝑅AMRNiFe\n2sin(2𝜑𝐻)𝐼rf(𝜒𝜑𝜃′𝛿ℎ𝜃+𝜒𝜑𝜑′𝛿ℎ𝜑), (7) \nwhere Δ𝑅AMRNiFe is the anisotropic magnetoresistance amplitude , 𝜒𝜑𝜃′ and 𝜒𝜑𝜑′ are respectively \nthe real part of the 𝜑𝜃 and the 𝜑𝜑 components of the susceptibility matrix  of NiFe . And, 𝛿ℎ𝜃 \nand 𝛿ℎ𝜑 are respectively the polar and azimuthal component of the exciting fi eld 𝛿ℎ (whose \nexpression is discussed in the next subsection). We can see that only the transverse components \nof the excitation fields contribute to the ST -FMR voltage . We will discuss  the different \ncontributions to the total torque: i) first considering the symmetries  of Equation 7 , and  ii) \nadding a dc current which will modify the susceptibility  components.  \nWe highlight here that all the equations and signs that our model describe s have been verified \nby considering the results obtained in a //Pt(5)/NiFe (4) reference system. Namely, in this system , \n𝜃𝐷𝐿=𝜃𝐷𝐿𝑆𝐻𝐸>0 and 𝜃𝐹𝐿=𝜃𝐹𝐿𝑆𝐻𝐸>0 (with a negative Oersted field).    \n \n4. 2. Symmetry of the ST -FMR signal  \nThe NiFe magnetization resonance is triggered by the rf current induced Oersted field,  and the \nspin torques described in Equation 5a  and Equation 5b . We gather the different contributions  \nunder the general term of the exciting field, 𝛿ℎ. Here, the delta means  that the excitation is \nweak.  The dynamics around the equilibrium position, which takes place in the (𝑒̂𝜃,𝑒̂𝜑) plane \nin spherical coordinates, is only sensitive to the polar and azimuthal components of the exciting \nfield 𝛿ℎ𝜃 and 𝛿ℎ𝜑. Since 𝝈̂SAHE lies along the NiFe magnetizatio n equilibrium position \n𝒎̂𝑁𝑖𝐹𝑒=𝒆̂r, the associated SAHE effective fields do not contribute to the magnetization \ndynamics. On the contrary, the effective fields associated to the SHE -like symmetry contribute  \n \n11 \n to the dynamics since 𝝈̂𝑆𝐻𝐸∥ 𝒚̂ and 𝛿ℎ𝜃=ℎDLSHEcos (𝜑𝐻) and 𝛿ℎ𝜑=cos (𝜑𝐻)(ℎOe−\nℎFLSHE)  [36,37] . Since at the resonance  𝜒𝜑𝜑′ is an antisymmetric function of the app lied \nmagnetic field and 𝜒𝜑𝜃′ is a symmetric function, we can express th e symmetrical voltage \n𝑉𝑠𝑦𝑚 amplitude  and the antisymmetrical amplitude  𝑉𝑎𝑛𝑡𝑖 introduced in Equation 1  by replacing \nthe suitable expressions  in Equation 7 :    \n𝑉sym= −sin(𝜑𝐻)1\n4𝐼𝑟𝑓 Δ𝑅AMRNiFe\n𝜇0(2𝐻+𝑀effNiFe) 2π𝑓\n𝛾 ℎDLSHE\nΔ𝐻 , (8) \n𝑉anti= −sin(𝜑𝐻)1\n4𝐼rf Δ𝑅AMRNiFe\n𝜇0(2𝐻+𝑀effNiFe) 2π𝑓\n𝛾 [1+𝑀eff\n𝐻res]1\n2\n ℎOe−ℎFLSHE\nΔ𝐻 .(9) \n \n𝑉𝑠𝑦𝑚 (𝑉𝑎𝑛𝑡𝑖) only depends on ℎ𝐷𝐿𝑆𝐻𝐸(ℎ𝑂𝑒−ℎ𝐹𝐿𝑆𝐻𝐸) but the extraction of the effective fields using \nEquation 8 and Equation 9 is not trivial since  the rf current has to be evaluate d. Nevertheless, \nwe can discuss the signs of the SHE effective fields. As depicted in Figure 3b , 𝑉𝑠𝑦𝑚 is positive  \nwhich me ans that ℎ𝐷𝐿𝑆𝐻𝐸>0. 𝑉𝑎𝑛𝑡𝑖 is negative , and thus  ℎ𝐹𝐿𝑆𝐻𝐸>0 assuming that the Oersted \nfield is lower than the FL effective field.  \n \n4. 3. Adding  a dc bias in ST -FMR: damping modulation and shift of 𝑯𝐫𝐞𝐬 \n \n \nWhen adding a dc bias to the previous ST -FMR measurement, a constant torque is applied on \nthe oscillating magnetization which results in a change in the expression of its dynamical \nsusceptibility matri x. This change induces a modulation of the linewidth and a shift in the \nresonant field, which can be both probed by the ST -FMR technique with an added dc bias. \nBecause the susceptibility is related to the effective field along which the magnetization lies, \nonly the spin polarizations with a projection along this effective field induce a change in the \nsusceptibility. The modulation of damping technique is thus sensitive to both the SHE and \nSAHE -like symmetries and allows to extract overall parameters.   \n \n12 \n In the limit of low current densities where we can neglect strong heating contribution that \ndeformed the linear behavior, we can modify the expressions developed for magnetic tunnel \njunctions  [38,39] , to apply it in our system  [10,15,26] . For the modulation of the NiFe linewidth , \nit reads : \n𝜕𝛥𝐻NiFe\n𝜕𝑖dc=−𝑓\n𝛾𝑁𝑖𝐹𝑒2\n(2𝐻𝑟𝑒𝑠𝑁𝑖𝐹𝑒+𝑀𝑒𝑓𝑓𝑁𝑖𝐹𝑒)𝑆GdFeCo\n𝑊𝑡GdFeCo   (𝝈̂SAHE.𝒎̂NiFe ⏟        \n1𝜕ℎ𝐷𝐿𝑆𝐴𝐻𝐸\n𝜕𝐽𝑐𝐺𝑑𝐹𝑒𝐶𝑜+ 𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟        \nsin(𝜑𝐻)𝜕ℎ𝐷𝐿𝑆𝐻𝐸\n𝜕𝐽𝑐𝐺𝑑𝐹𝑒𝐶𝑜), (10) \nwhere the left -hand term in the equation is the slope of the modulation of NiFe linewidth,  \n𝛾NiFe=𝑔𝑁𝑖𝐹𝑒𝜇𝐵\nℏ . 𝑆GdFeCo  accounts for the shunting of the  GdFeCo layer  by the other conductive \nlayer s, i.e., the current density flowing in GdFeCo layer is 𝐽𝑐GdFeCo=𝑆GdFeCo\n𝑊𝑡GdFeCo𝑖dc with 𝑊 the \nwidth of the slab (10 m). For simplicity, Equation 10 can also be written in terms of the Hall \nangles  using Equation 6a,b  in the following way : \n𝜕𝛥𝐻NiFe\n𝜕𝑖dc=−𝑓\n𝛾NiFe2\n(2𝐻resNiFe+𝑀effNiFe)𝑆GdFeCo\n𝑊𝑡GdFeCo ℏ\n2|𝑒|sin(𝜑𝐻)  𝜃DLSAHE+ 𝜃DLSHE\n𝜇0𝑀𝑠NiFe𝑡NiFe,(11) \nThe slope s 𝜕𝛥𝐻NiFe\n𝜕𝑖dc that account for the linewidth modulation at 8 GHz are displayed in Figure \n4b for 𝜑𝐻=135°  and 𝜑𝐻=−45°. The resistivities were determined independently through \nthe dependence of the GdFeCo and Cu thicknesse s for the different layers  obtaining  𝜌𝐶𝑢=15 \ncm, 𝜌𝐺𝑑𝐹𝑒𝐶𝑜=175 cm, and 𝜌𝑁𝑖𝐹𝑒=40 cm. It follows 𝑆GdFeCo=0.11. We note \nthat the slope s obtained when 𝜑𝐻=135° , for all the different frequencies measured, are \nopposite  than the one s measured for  the //Pt/NiFe reference sample  (not shown) . It indicates \nthat the DL overall spin Hall angle,   𝜃DLSAHE+ 𝜃DLSHE,  is negative and opposite  to the one  of Pt  \nwhere only the SHE is present . From the average of positive and negative dc fields , or 135° and \n-45°, and  for 8, 12 and 14 GHz , we evaluate the overall DL efficiency  𝜃DL𝑆𝐴𝐻𝐸+𝜃𝐷𝐿𝑆𝐻𝐸=\n−0.15±0.05 for the FeCo -rich GdFeCo interfaced with Cu .   \n \n13 \n Furthermore, the same experiment also allows  us to obtain the corresponding field -like \nvalue s, ℎFL and 𝜃FL. Based on the work of ref . [38,39] , we also obtain the following expression  \nthat account s for the linear displacement of the resonance f ield with an added dc current :  \n𝜕𝐻resNiFe\n𝜕𝑖dc=𝑆GdFeCo\n𝑊𝑡GdFeCo  (𝝈̂SAHE.𝒎̂NiFe ⏟        \n1𝜕ℎFLSAHE\n𝜕𝐽𝑐GdFeCo+ 𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟        \nsin(𝜑𝐻)𝜕ℎFLSHE\n𝜕𝐽𝑐GdFeCo )−𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟        \nsin(𝜑𝐻)𝜕ℎOe\n𝜕𝑖dc, (12) \nwhere ℎ𝑂𝑒 is the Oersted field which lies along the −𝒚̂ direction in the geometry of our system. \nIts amplitude can be approximated with ℎOe=−1\n2(𝑗cGdFeCo𝑡GdFeCo+𝑗cCu𝑡Cu). Equation 12 \nreads in terms of the FL Hall angles  (Equation 6a,b ): \n𝜕𝐻resNiFe\n𝜕𝑖dc=sin(𝜑𝐻)[𝑆GdFeCo\n𝑊𝑡GdFeCo (ℏ\n2|𝑒|𝜃FLSAHE+ 𝜃FLSHE\n𝜇0𝑀𝑠NiFe𝑡NiFe)− 𝜕ℎOe\n𝜕𝑖dc] , (13) \nThe slope  obtain ed from the shift of the resonance field  vs. 𝑖dc is displayed in Figure 4c for \ndifferent frequencies . We observe that the slope is frequency -independent in agreement with \nEquation 1 3. Moreover, the slope has the same sign  as the one in  the //Pt/NiFe reference system . \nThat implies that if there is any FL contribution on  the GdFeCo/Cu/NiFe system studied here it \nhas the same sign as for  the //Pt/NiFe. The slope is  evaluate d as 𝜕𝐻resNiFe\n𝜕𝑖dc=0.037 T/A. The \nOersted field is approximated  as  𝜕ℎOe\n𝜕𝑖dc=−0.0476  T/A. Finally, considering  Equation 13, the \noverall FL efficiency is assessed a s 𝜃FLSAHE+𝜃FLSHE=0.026±0.005. This value  has the same \nsign and is comparable  to the one measure d in NiFe/Pt  [37,40] . We have independently \nmeasured a control //Cu/NiF e sample with out a  sizable effect. We can therefore exclude the \nCu/NiFe interface  as the origin behind the FL measured in GdFeCo/Cu/NiFe . The sizable  \noverall FL value would indicate that even though  GdFeCo is not in contact with NiFe, a \nsignificant  FL contribution can  still be detected. The origin of the FL effect in the trilayer is not \nclear at this sta ge. \n  \n \n14 \n  \nFigure 4. Damping modulation and Resonance field shift . (a) Schematic of the NiFe \nresonance condition  with additional 𝑖dc current injected . (b) 𝑖dc dependence of the NiFe \nlinewidth  for a rf frequency of 8 GHz . (c) Resonance field shift vs. 𝑖dc for three frequencies. \nUnlike the damping or linewidth modulation, we can see that the resonance field shift is \nfrequency independent.  \n \n5. Discussion and conclusions  \nThe overall efficiencies for FeCo -rich GdFeCo/Cu/NiFe are evaluated 𝜃DL𝑆𝐴𝐻𝐸+𝜃𝐷𝐿𝑆𝐻𝐸=\n−0.15±0.05 and 𝜃FLSAHE+𝜃FLSHE=0.026±0.005. For sake of comparison, the SAHE \nefficiency of a ferromagnet such as CoFeB is 𝜃𝑆𝐴𝐻𝐸𝐶𝑜𝐹𝑒𝐵=−0.14  [15], and the SHE efficiency of \nPt heavy metal is  𝜃𝑆𝐻𝐸𝑃𝑡=0.056−0.076   [29,41,42] . Seki et al . show in FePt that DL \n𝜃𝑆𝐴𝐻𝐸+𝑆𝐻𝐸𝐹𝑒𝑃𝑡=0.25 from the linewidth modulation  [43].  \nThe damping -like SAHE contribution dominates over the SHE one: |𝜃𝐷𝐿𝑆𝐴𝐻𝐸|>|𝜃𝐷𝐿𝑆𝐻𝐸| with a \nnegative SAHE contribution  for FeCo -rich GdFeCo , and a positive SHE contribution . We also \nshow that the field -like SHE contribution is positive. However, we cannot  estimate the \nindividual value of each contribution.  We perform the same experiments at 15 K where our \nferrimagnet is Gd -rich and its magnetization aligns in -plane with a field above 0.4 T.  From the \nsign of the symmetric contribution we confirm that SHE remains positive when crossing the \nmagnetic compensation temperature. This is consistent with the fact that the SHE does not \ndepend on the GdFeCo magnetic propert ies.  In contrast, we cannot conclude of any 𝜃𝐷𝐿𝑆𝐴𝐻𝐸 sign \nchange  because the modulation of linewidth experiments at 15 K is hidden by others effect that \n \n \n15 \n are out of the scope of this study.  However, the large variation in absolute value between these \nresults and the one previously reported,  for a Gd-rich GdFeCo at room temperature,  |𝜃𝐷𝐿𝑆𝐴𝐻𝐸+\n𝜃𝐷𝐿𝑆𝐻𝐸|=0.80±0.05  [10], suggest that the si gn of 𝜃𝐷𝐿𝑆𝐴𝐻𝐸 changes between FeCo -rich and Gd -\nrich samples . If so, t he opposite DL -SAHE sign for  FeCo -rich GdFeCo might indicate that the \nSAHE spin polarization comes always from the same magnetic sublattice . Despite that , further \nstudies could  be carried out to confirm that.     \nGdFeCo can thus generate efficient spin currents and the different symmetries allow this \nmaterial to be used in a wide variety of devices for spintronics. For instance, the SHE spin \ncurrent can generate self -torque  [10] and can be used for the electrical switching of the \nmagnetization, as shown in epitaxial FePt   [44] or CoTb  [45]. Also, the total spin current \n(SAHE+SHE) can be used to induce a torque on another magnetic layer or for the manipulation \nof skyrmions.  \n \nIn summary, w e have studied  FeCo -rich GdFeCo/Cu/NiFe  heterostructure  at room t emperature . \nFirst, structural,  and chemical analyses were performed by HRTEM and EELS. T hen, the \nmagnetic properties and the relevant spin -orbitronics  parameters  were determined by \ncombining magnetometry, spin-torque ferromagnetic resonance and additional dc current \ndependence . The overall damping -like and field -like efficienc ies, which include the SHE -like \nand the SAHE-like symmetries, are 𝜃𝐷𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐷𝐿𝑆𝐻𝐸=−0.15±0.05 and 𝜃𝐹𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐹𝐿𝑆𝐻𝐸=\n0.026±0.005 at room temperature . We show that  SAHE  dominates over SHE contribution on \nthe DL torque. Furthermore , this study show s that the SHE contribution does not change sign \nwhen crossing the magnetic compensation temperature while SAHE may change sign \ndepending on the dominant  sublattice of the ferrimagnet . All this underlines the importance of \nGdFeCo, and RE -TM ferrimagnets in general, as promising materials in spintronics for the \nexploitation of their strong spin-orbit  torque .  \n  \n \n16 \n  \nData availability  \nThe data that support the fi ndings of this study are available from the corresponding author on \nreasonable request.  \n \nAcknowledgements  \nWe acknowledge A. Fert for fruitful discussions. This  work was supported from Agence \nNationale  de la Recherche (France) under contract ANR -19-CE24 -0016 -01 (TOPTRONIC) , \nANR -20-CE24 -0023 (CONTRABASS),  and ANR -17-CE24 -0025 (TOPSKY), from the \nFrench PIA project “Lorraine Universit é d’Excellence ”, reference ANR -15IDEX -04-LUE  and \nby the « SONOMA» projec t co-funded by FEDER -FSE Lorraine et Massif des Vosges 2014 -\n2020, a European Union Program . DCB and JAS also thanks 2019 and 2021 Master -LUE \nprogram internship. Devices in the present study were patterned at MiNaLor  clean -room \nplatform which is partially s upported by FEDER and Grand Est Region through the RaNGE \nproject.  \n \n \nReferences  \n[1] S. K. Kim, G. S. D. Beach, K. -J. Lee, T. Ono, T. Rasing, and H. Yang, “Ferrimagnetic \nspintronics” Nat. Mater.  21, 24 (2022).  \n[2] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo -\nFesenko, S. El Moussaoui, L. Le Guyader, E. Mengotti, L. J. Heyderman, F. Nolting, \nA. 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Commun.  12, 4555 (2021).  \n \n \n \n \n "    },    {        "title": "0809.4644v2.Damping_and_magnetic_anisotropy_of_ferromagnetic_GaMnAs_thin_films.pdf",        "content": "Anisotropic Magnetization Relaxa tion in Ferromagnetic GaMnAs \nThin Films \n \nKh.Khazen, H.J.von Bardeleben, M.Cubukcu, J.L.Cantin \nInstitut des Nanosciences de Paris,  \nUniversité Paris 6, UMR 7588 au CNRS \n140, rue de Lourmel, 75015 Paris, France \n \nV.Novak, K.Olejnik, M.Cukr \nInstitut of Physics, Academy of Sciences,  \nCukrovarnicka 10, 16253 Praha, Czech Republic \n \nL.Thevenard, A. Lemaître \nLaboratoire de Photonique et  des Nanostructures, CNRS \nRoute de Nozay, 91460 Marcoussis, France \n \n \nAbstract: \n \n The magnetic properties of annealed, epitaxial Ga 0.93Mn 0.07As layers under tensile and \ncompressive stress have been investigat ed by X-band (9GHz) and Q-band (35GHz) \nferromagnetic resonance (FMR) spectroscopy. From  the analysis of the linewidths of the \nuniform mode spectra the FMR Gilbert damping factor α has been determined. At T=4K we \nobtain a minimum damping factor of α = 0.003 for the compressively stressed layer. Its value \nis not isotropic. It has a minimum value for th e easy axes orientations of the magnetic field \nand increases with the measuring temperature. It s average value is for both type of films of \nthe order of 0.01 in spite of strong differences  in the inhomogeneous linewidth which vary \nbetween 20 Oe and 600 Oe for the layers grown on GaAs and GaInAs substrates respectively .   \n \n \nPACS numbers: 75.50.Pp, 76.50.+g, 71.55.Eq  \nIntroduction: \n \nThe magnetic properties of ferromagnetic Ga 1-xMn xAs thin films with Mn \nconcentrations between x=0.03 and 0.08 have been studied in great detail in the recent years \nboth theoretically and experimentally. For recent reviews see references [1, 2].  A \nparticularity of GaMnAs ferro magnetic thin films as comp ared to conventional metal \nferromagnetic thin films is the predominance of  the magnetocrystalline anisotropy fields over \nthe demagnetization fields. The strong anisotropy fields are not directly related to the crystal \nstructure of GaMnAs but are induced by the la ttice mismatch between the GaMnAs layers and \nthe substrate material on which they are grow n. When grown on (100) GaAs substrates the \ndifference in the lattice constants induces biaxial strains  of ≈ 0.2% which give rise to \nanisotropy fields of several 103 Oe. The   low value of the de magnetization fields (~300Oe) is \nthe direct consequence of the small spin conc entration in diluted magnetic semiconductors \n(DMS) which for a 5% Mn doping leads to a saturation magnetization of only  40 emu/cm3. \nAs the strain is related to the lattice mismat ch it can be engineer ed by choosing different \nsubstrate materials. The two systems which have been investigated most often are (100) GaAs \nsubstrates and (100)GaInAs pa rtially relaxed buffer layers. These two cases correspond to \ncompressive and tensile strained Ga MnAs layers respectively [3].  \nThe static micro-magnetic pr operties of GaMnAs layers can be determined by \nmagnetization, transport, magneto-optical and ferromagnetic resonance techniques. For the \ninvestigation of the ma gnetocrystalline anisotropies the ferromagnetic resonance spectroscopy \n(FMR) technique has been shown to be partic ularly well adapted [2, 4]. The dynamics and \nrelaxation processes of the magnetization of such layers have hardly been investigated up to \nnow [5-7]. The previous FMR studies on this  subject concerned either unusually low doped \nGaMnAs layers [5, 7] or employed a single microwave frequency [6] which leads to an \noverestimation of the damping factor. The knowle dge and control of the relaxation processes \nis in particular important for device applications as they de termine for example the critical \ncurrents necessary for current induced magne tization switching. It is thus important to \ndetermine the damping factor for state of the art samples with high Curie temperatures of T C ≈ \n150K, such as those used in this work.  Anothe r motivation of this work is the search for a \npotential anisotropy of the ma gnetization relaxation in a dilu ted ferromagnetic semiconductor in which the m agnetocrystalline anisotro pies are strong and dom inant over the \ndemagnetization contribution.  \nThe intr insic sm all angle m agnetization re laxation is generally described by one \nparam eter, the Gilbert d amping coefficient α, which is defined by the Landau Lifshitz Gilbert \n(LLG) equation of m otion for the m agnetization: \n \n⎥⎦⎤\n⎢⎣⎡× +⎥⎦⎤\n⎢⎣⎡× −= ⋅dtsdMeffH MdtMdrr rrr\nγα\nγ1   eq.1 \nwith M the m agnetization, H eff the effective m agnetic field, α the dam ping fa ctor, γ the \ngyrom agnetic ratio and s the uni t vector parallel to M.   \nThe dam ping factor α is generally assum ed to be a scal ar quantity [8, 9] . It is defined \nfor sm all angle precess ion relaxatio n   which  is the case of FMR experim ents. This param eter \ncan be experim entally determ ined by FMR spectr oscopy either from  the angular variation of \nthe linewidth or from  the variati on of the uniform  mode linewidth ∆Hpp with th e microwave \nfrequency. In this second case the linewidth is given  by: \n \nω\nγω ⋅\n⋅⋅ + ∆= ∆+ ∆= ∆\nMGH H H Hin inpp\n2 hom hom hom32)(  eq.2 \nWith ∆Hpp the first derivative p eak-to -peak  linewid th of  the uniform mode of \nLorentzian lineshape, ω the angular m icrowave frequency an d G the Gil bert dam ping factor \nfrom which the m agnetization independent damping factor α can be deduced as α=G/γM. In  \neq. 2 it is assum ed that the m agnetiz ation and th e applied magnetic f ield are collin ear which is \nfulfilled for high symm etry direction s in GaMnAs  such as [001], [110] and [100]. Ot herwise a \n1/cos ( θ-θH) term  has to be added to eq.2 [8]. \n∆Hinhom is the inhom ogeneous, frequency indepe ndent linewidth; it can be further \ndecom posed in three con tributions, re lated to the crysta lline imperf ection of  the f ilm [10]: \n \nint\ninthom HHH H HHr\nH\nHr\nH\nHr\nin ∆⋅ + ∆⋅ + ∆⋅∂= ∆δδφδφδθθδ  eq.3 \n \nThese three term s were introduced to take in to account a slight m osaic structure of the \nmetallic thin f ilms def ined by the polar angles  (θ, φ) and their distributions ( ∆θ, ∆φ)  - \nexpressed in the first two term s in eq.3- and a distribution of  the internal anisotropy fields H int – the last term of eq.3. In the case of  homo epitaxial III-V films obtained by MBE growth like \nGaMnAs on GaAs, films of high crystalline qu ality are obtained [LPN] and only the third \ncomponent ( ∆Hint) is expected to play an important role.  \n Practically, the variation of the FMR linewid th with the microwave frequency can be   \nmeasured with resonant cavity systems at different frequencies between 9GHz and 35GHz; \nthe minimum requirement -used also in this work - is the use of two frequencies. We disposed \nin this work of 9GHz and 35GHz spectrometers. The linewidth is decomposed in a frequency independent inhomogeneous part and a linear fr equency dependent homogeneously broadened \npart. For most materials the inhomogeneous fr action of the linewidth is strongly sample \ndependent and depends further on the interface quali ty and the presence of cap layers. It can \nbe smaller but also much larger than the intrinsic linewidth. In Ga\n0.95Mn 0.05As single films \ntotal X-band linewidths be tween 100Oe and 1000Oe have been encountered. These \nobservations indicate already the impor tance of inhomogeneous broadening. The \nhomogeneous linewidth will depend on the intrinsic sample properties. This approach supposes that the inhomogeneous  linewidth is frequency independent and the homogenous \nlinewidth linear dependent on the frequency, two assumptions generally valid for high \nsymmetry orientations of the a pplied field for which the magne tization is parallel to the \nmagnetic field.  \nIt should be underlined that  in diluted magnetic semiconductor (DMS) materials like \nGaMnAs the damping parameter is not only determined by the sample composition x Mn [5]. It \nis expected to depend as we ll on (i) the magnetic compensati on which will vary with the \ngrowth conditions, (ii) the (hol e) carrier concentration respon sible for the ferromagnetic Mn-\nMn interaction which is influenced by the presence of native donor defects like arsenic \nantisite defects or Mn interstitial ions [11] and (iii) the valence bandstructure, sensitive to the \nstrain in the film. Due to the high out-of –plane and in-plane anisotropy of the magnetic \nparameters [12] which further vary with the applied field and the temperature a rather \ncomplex situation with an anisotropic and te mperature dependent da mping factor can be \nexpected in GaMnAs.  \nWhereas the FMR Gilbert damping factor has been determined for many metallic \nferromagnetic thin films [8] only three experimental FMR studies have been published for GaMnAs thin films up to now [5-7]. In ref.[ 5,7] low doped GaMnAs laye rs with a critical \ntemperature of 80K which do not correspond to the high quality, standa rd layers available \ntoday were studied. In the ot her work [6] higher doped layers  were investigated but the \nexperiments were limited to a single microw ave frequency (9GHz) and thus no frequency dependence could be studied.  In this work we present the results of FMR studies at 9GHz and \n35 GHz on two high quality GaMnAs layers with  optimum critical temperatures: one is a \ncompressively strained layer grown on a GaAs buffer layer and the othe r a tensile strained \nlayer grown on a (Ga,In)As buffer layer. Due to  the opposite sign of the strains the easy axis \nof magnetization is in-plane [ 100] in the first case and out-o f-plane [001] in the second. The \nGaMnAs layers have been annealed ex-situ after their growth in order to  reduce the electrical \nand magnetic compensation, to homogenize the laye rs and to increase the Curie temperature \nto ≈ 130K. Such annealings have become a st andard procedure for improving the magnetic \nproperties of low temperature molecular b eam epitaxy (LTMBE) grown GaMnAs films. \nIndeed, the low growth temperature required to incorporate the high Mn concentration \nwithout the formation of precipitates gives rise to native defect the conc entration of which can \nbe strongly reduced by the annealing.  \n \nExperimental details \nA first sample consisting of a Ga 0.93Mn 0.07As layer of 50nm thickness has been grown \nat 250° C by low temperature molecular beam epitaxy on a semi-insulating (100) oriented \nGaAs substrate. A thin GaAs buffer layer has been grown before the deposition of the \nmagnetic layer. The second sample, a 50 nm thick Ga 0.93Mn 0.07As layer have been grown \nunder very similar conditions on a partially relaxed (100) Ga 0.902In0.098As buffer layer; for \nmore details see ref. [13]. After the growth the structure was thermally annealed at 250° C for \n1h under air or nitrogen gas fl ow. The Curie temperatures were 157K and 130K respectively. \nBased on conductivity measuremen ts the hole concentratio n is estimated in the 1020cm-3 \nrange. \nThe FMR measurements were performed with Bruker X-band and Q-band \nspectrometers under standard conditions: mW microwave power and 100 KHz field \nmodulation. The samples were measured at te mperatures between 4K and 170K. The angular \nvariation of the FMR spectra was measured in  the two rotation planes (110) and (001). The \npeak-to peak linewidth of the first derivati ve spectra were obtained from a lineshape \nsimulation. The value of the st atic magnetization M(T) had been determined by a commercial \nsuperconducting quantum interference device (SQUID) magnetometer. A typical hysteresis \ncurve is shown in the inset of fig.8. \n \nExperimental results: The saturation magnetizations of the two laye rs and the magneto crystalline anisotropy \nconstants which had been previously de termined by SQUID and FMR measurements \nrespectively are given in table I. The anisotropy constants had been determined in the whole \ntemperature range but for clarity only its values  at T=55K and T=80K are given in table I. We \nsee that the dominant anisotropy constant K 2⊥ are of different sign with -55000 erg/cm3 to \n+91070 erg /cm3 and that the other three constants ha ve equally opposite signs in the two \ntypes of layers. The easy axes of magnetization are the in-pla ne [100] and the out-of-plane \n[001] direction respectively. Howe ver the absolute values of th e total effective perpendicular \nanisotropy constant Ku=K 2⊥ +K 4⊥ are less different for the two samples: -46517erg/cm3 and \n+57020erg/cm3 respectively. More detailed inform ation on the measurements of these \nmicromagnetic parameters will be published elsewhere.  \nFor the GaMnAs/GaAs layers the peak-to-peak linewidth of the first derivative \nuniform mode spectra has been strongly re duced by the thermal annealing; in the non \nannealed sample the X-band linewidth was highl y anisotropic with va lues between 150Oe and \n500Oe  at T=4K. After annealing it is reduced to  an quasi isotropic average value of 70Oe at \nX-band. Quite differently, for the GaMnAs/GaInA s system the annealing process decreases \nthe linewidth of the GaMnAs layers only marginally. Although full angular dependencies \nhave been measured by FMR we will analyze only the linewidth of the four high symmetry field orientations H//[001], H //[100], H//[1-10], H//[ 110] corresponding to the hard and easy \naxes of magnetization. As will be shown below, in spite of rather similar high critical temperatures (157K/130K) the linewidth are drastically di fferent for the two cases.  \n \n1. GaMnAs on GaAs \n In fig. 1a and 1b we show typical low te mperature FMR spectra at X-band and Q-band \nfrequencies for the hard [001] /intermediate  [100] axis orientation of the applied magnetic \nfield. The spectra are characterized by excelle nt signal to noise ra tios and well defined \nlineshapes.  We see that at both frequencies the lineshape is close to a Lorentzian. In addition \nto the main mode one low intensity spin wave  resonance is observed at both frequencies at \nlower fields (not shown). \n The linewidth at X-band (fig.2) is of  the order of 50Oe to 75Oe with a weak \norientation and temperature dependence. Above T>130K, close to the criti cal temperature, the \nlinewidth increase strongly. At Q-band we observe a systematic increase by a factor of two of \nthe total linewidth (fig.3) with an increase d temperature and orient ation dependence. As generally observed in GaMnAs, the easy axis orie ntation gives rise to th e lowest linewidth. At \nQ-band the lineshape is perfectly Lorentzian (f ig.1b). These linewidth are among the smallest \never reported for GaMnAs thin films, which re flects the high crystal line and magnetic quality \nof the film. \n To determine the damping factor α we have plotted the frequency dependence of the \nlinewidth for the different orientations and at various temperatures. An example is given in \nfig. 4 for T=80K; this allows us to determin e the inhomogeneous linewidth obtained from a \nlinear extrapolation to zero frequency and the damping factor from the slope. The \ninhomogeneous linewidth at T=80K is of the order of 30 Oe, i.e. 50% of the total linewidth at \nX-band. This shows that the approximation ∆Hinhom<< ∆Hhomo which had been previously \nused [5] to deduce the damping factor from a single (X-band) frequenc y measurement is not \nfulfilled here.  \n The temperature dependence of the inhomogeneous linewidth is shown in fig.5. \nSimilar trends as for the total linewidth in the non annealed films are observed: the linewidth \nis high at the lowest temperatures, decreases with increasing temperat ures up to 120K   and \nincreases again close to T C.   \n From the slope of the linewidth variati on with microwave frequency we obtain the \ndamping factor α (fig.6). Its high temperature value is of the order of 0.010 but we observe a \nsystematic, linear variation with the temperatur e and a factor two difference between the easy \naxis orientation [100] and the hard axis orientation [001].  \n \n \n2. GaMnAs on GaInAs \n Similar measurements have been performed on the annealed tensile strained layer. In \ntensile strained GaMnAs films the easy axis  of magnetization ([001]) coincides with the \nstrong uniaxial second order anisotropy directio n.  For that reason no FMR resonance can be \nobserved at temperatures below T=80K for the easy axis orientation H// [001] at X-band. For \nthe other three orientations the resonances  can be observed at X-band in the whole \ntemperature range 4K to T C. Due to the strong temperature dependence of the anisotropy \nconstants and the parallel decr ease of the internal anisotropy fields the easy axis resonance \nbecomes observable at X-band for temperatures above 80K. In the films on GaInAs much \nhigher linewidth are encountered th an in films on GaAs, the values are up to ten times higher \nindicating a strong inhomogeneity in this film. A second low fiel d resonance is systematically \nobserved at X-band and Q-band; it is equally attributed to a spin wave resonance.   Figures 7a and 7b show typical FMR spectra  at X- and Q-band re spectively. At both \nfrequencies the lineshape can no longer be simu lated by a Lorentzian but has changed into a \nGaussian lineshape. \n Contrary to the first cas e of GaMnAs/GaAs the X-band linewidth varies monotonously \nin the whole temperature region (fig.8). We observe a linewidth of ~600Oe at T=4K, which \ndecreases only slowly with temperature; the linewidth becomes minimal in the 100 K to 140K \nrange. The Curie temperature “s een” by the FMR spectroscopy is s lightly higher as compared \nto the one measured by SQUID due to th e presence of the applied magnetic field. \n At low temperature the Q-ba nd linewidth vary strongly w ith the orientation of the \napplied field with values be tween 500Oe and 700Oe. The lowest  value is observed for the \neasy axis orientation. They decrease as at X-band only slowly with increasing temperature \nand increase once again when approaching the Curie temperature. At Q-band the easy axis \nFMR spectrum, which is also accompanied by a str ong spin wave spectrum at lower fields, is \nobservable in the whole temperature range. \n For this sample we observe especially at Q-band a systematic difference between the cubic axes [100], [001] linewidth and the one for the in-plane [110] and [1-10] field \norientations (fig.8). The most surprising observation is that for temperatures below T<100K \nthe linewidth for H//[100] and H//[110] are co mparable at X-band and Q-band and thus an \nanalysis in the simple model discussed above is not possible. We attribute this to much higher \ncrystallographic/magnetic inhomogeneities, which mask the homogenous linewidth. The \norigin of the strong inhomogeneity is still unclear.  The only orientation for which in the whole temperature range a systematic increas e in the linewidth between X-and Q-band is \nobserved is the H//[1-10] orienta tion. We have thus analyzed th is variation (fig.10) according \nto eq.1.  \n In spite of important differences in the lin ewidth the slope varies only weakly which \nindicates that the inhomogeneous linewidth is  very temperature dependent and decreases \nmonotonously with increasing temperature from 570Oe to 350Oe.   \nIn the high temperature range (T ≥100K) the easy axis orientation could also be \nanalyzed (fig.11). The inhomogeneous  linewidth are lower than for the hard axis orientation at \nthe same temperatures and are in the 300Oe range (fig.12). The homogenous linewidth at \n9Ghz is in the 50Oe range which is close to the values determined in the first case of \nGaMnAs/GaAs.  \n From the slope (fig.13) we obtain the da mping factor which for the hard axis \norientation is α=0.010 in the whole temperature range. Th is value is comparable to the one measured for the GaMnAs/GaAs film for H//[110]. The damping factor for the easy axis \norientation is lower but   increases close to T C as in the previous case.  \n \nDiscussion: \n  An estimation of the FMR intrinsic damping factor in a ferromagnetic GaMnAs thin \nfilm has been made within a model of localiz ed Mn spins coupled by p-d kinetic exchange \nwith the itinerant-spin of holes treated by the 6-band Kohn-Luttinger Hamiltonian [5]. Note, that these authors take for the effective kinetic exchange field the value in the mean-\nfield approximation, i.e. H\neff=JN<S>, so that their calculation are made within the random \nphase approximation (RPA). RPA calculations of α have been made by Heinrich et al. [14] \nand have recently been used by Tserkovnyak et al .[15] for numerical app lications to the case \nof Ga 0.95Mn 0.05As.  Both models however, are phenomenological and include an \nadjustable parameter: the quasiparticle lifetime Γ for the holes in [5] and the spin-flip \nrelaxation T 2 in [15]. These models do not take into account neither multi-magnon \nscattering nor any damping beyond the RPA. It has been argued elsewhere [16], that in diluted \nmagnetic semiconductors such affects are only impor tant at high temperatur e (i.e. at T>Tc). In \nparticular, the increase of α in the vicinity of Tc may be attributed to such effects that are \nbeyond the scope of the models of references  [5] and [14]. At low temperatures T<<Tc \nhowever, where the corrections to the RPA are expected to be negligib le, the models of [5, \n14,15] provide us with a numerical value of α in agreement with our experiments if we \nintroduce reasonable values of   these parameters. For GaMnAs films with metallic \nconductivity, Mn concentrations of x= 0.05 and hole concentr ations of 0.5 nm-3 (5x1020cm-3) \nSinova et al [5] predict an isotropi c  low temperature damping factor α between 0.02 and 0.03 \ndepending on the quasiparticle life time broa dening. Tserkovnyak et al [15] found a similar \nvalue of α ≈ 0.01 for the isotropic damping factor fo r a typical GaMnAs film with 5% Mn \ndoping and full hole polarization. \n Both predicted values are of the same orde r of magnitude as the experimental values \ndetermined in this study. Our results for GaMn As/GaAs show further that the damping factor \nis not isotropic as generally a ssumed but is anisotropic with a lowest value for the in-plane  \neasy axis orientations of the applied magnetic field H//[100], H//[1-10] and an increase of up \nto a factor of two for the hardest axis orientat ion H// [001]. Intrinsic anisotropic damping is \nrelated to the fact that the free energy density depends on the orientation of the magnetization which in the case of GaMnAs is related to th e anisotropy of the p-hole Fermi surface. We have shown (table I) that the anisotropy of the magnetocristalline constants and the related \nfields are important in these strained layers and it is thus not surprisi ng to find also anisotropy \nof the damping factor. For further discussions on this subject see reference [9]. The system \nmight also contain extrinsic an isotropies related to the pres ence of lattice defects. Their \ninfluence can be deduced from the value and anisotropy of the inhomogeneous linewidth. In \nthe case of the compressively strained layers (G aMnAs/GaAs) we see that their value is small \nand rather isotropic quite differen tly from the tensile strained film. It is in the first case that \nour measurements show a factor of two anisot ropy of the Gilbert damping factors. A further \nindication for the intrinsic charac ter of the anisotropy is the fact that the damping factor for \nthe perpendicular orientation has the highest value. In this case any contribution from two \nmagnon scattering will be minimized. Anyway, such contributions are generally only \nimportant at low frequency measurements in the 2-6GHz range but even there they were \nfound to be negligible [7]. \n Additional material related parameters are expected to further influence the damping \nfactor. As the spin flip relaxation times will depend on the sample properties and in particular \nthe presence of scattering centers we will not expect to find a unique damping factor even for GaMnAs/GaAs samples with the same Mn composition x. More likely, different damping \nfactors are expected to be found in  real films and their values might be used to assess the film \nquality. In this sense the GaMnAs/GaAs film studi ed here is of course “better” than the one \non GaInAs in line with the strong difference in the sample inhomogeneities.  \n The inhomogeneous linewidth originates from spatial inhomogeneities in the local \nmagnetic anisotropy fields and inhomogeneities in the local exchange interactions. Given the particular growth conditions of these films, low temper ature molecular beam epitaxy, \ninhomogeneities can not be expected to be neglig ible in these materials.  If we had analyzed \nour X-band results of the GaMnAs/GaAs film in the spirit of ref. [5], i.e. assuming a \nnegligible inhomoge neous linewidth - ( ∆H\ninhom=0) -, we would have obtained artificially \nincreased damping factors. A further contribution might be expected from the intrinsic \ndisorder in these films: as GaMnAs is a diluted magnetic semiconductor with random \ndistribution of the Mn ions, this  disorder will even for crysta llographically perfect crystals \ngive some importance to this term.  \n In the previous studies of  the FMR damping factor in GaMnAs/GaAs single films \nhigher values have been reported. Matsuda et  al [6] found damping factors between 0.02 and \n0.06 in the T=10K to T=20K temperature range. They  observed the same tend encies as in this \nwork concerning the anisotropy and temperature dependence of α: the lowest damping factor is seen for the easy axis orientation and its va lues increases with increasing temperatures. The \nfilms of their study were however significantly different: (i) the Mn doping concentration was \nlower, x=0.03 and thus the hole concentratio n was equally lower and (ii) the critical \ntemperature of the annealed film was only T=80K. The anisotropy in the inhomogeneous \nlinewidth at T=20K was equally much higher, varying between 30Oe for the easy axis to \n250Oe for the hardest axis [110]. In a second study Sinova et al [5] have measured an \nannealed GaMnAs/GaAs sample with a similar composition (x= 0.08) and critical temperature \n(TC=130K) as the one studied here. They deduced a damping factor of the order of α=0.025 \nwith only a slight temperature dependence betw een 4K and 80K and an increase close to T C. \nHowever, these measurements were done at one  (X-band) microwave frequency only and the \nnumerical value of α was obtained by assuming a neglig ible inhomogeneous linewidth. As \nexplained above, the value  of α can be expected to be overestimated in this case.In the \nphotovoltage measurements of ref.7 the dampi ng factor of a low (x=0.02) doped GaMnAs \nlayer has been determined with microwave fr equencies from 2 to 19.6GHz but for one field \norientation H//[001] (h ard axis) and one temperature (T =9K) only. Interestingly, their \nmeasurements show a linear behavior even in  the low frequency range down to 4GHz which \ndemonstrates the negligible contribution from two magnon scattering in this case. \n The intrinsic damping factor α plays also an important role in the critical currents \nrequired to switch the magnetization in FM/NM/ FM trilayers [5]. However, in trilayer \nstructures interface and spin pumping effects wi ll add to the intrinsic damping factor of the \nferromagnetic material and give rise to an incr eased effective damping fa ctor. Sinova et al [5] \nhave estimated the critical current for real istic GaMnAs layers: based on a value of α= 0.02, \nthey estimated the critica l current density to J C=105A/cm2. It should be noted that the damping \nfactor involved in the domain wall motion [ 17, 18] is by definition different form the FMR \ndamping factor. Both are however linearly related with αFMR<αDW [17, 18]. First observations \nof current induced magne tization switching in Ga 0.956Mn 0.044As/GaAs/Ga 0.967Mn 0.033As \ntrilayers confirm these theoretical predictions [19]. These authors observed a critical current \ndensity of ≈  105A/cm2 which would have been predicted from Slonczewski’s formula [20] for \na  domain wall damping factor of αDW=0.002. \n \nConclusion:  \n  We have determined the intrinsic FM R Gilbert damping factor for annealed \nGa0.93Mn 0.07As thin films with high critical temperatures. To evaluate the influence of the \nstrain the two prototype cases of  compressive and tensile strained  layers were studied.   In \nboth cases we find an average damping factor of the order of 0.01. We thus see that the sign \nof the strain does not seem to  influence the damping factor strongly.  The homogeneity of the \nfilms as judged from the inhomogeneous linewid th is much higher in the case of GaAs \nsubstrates than for GaInAs substrates. This must  be attributed to the high dislocation density \nin the GaInAs layer [13]. In the case of the GaMnAs/GaAs layers, where the small linewidth \nallows a finer analysis of the data, we observe  an anisotropy of the da mping factor, which has \nthe lowest value for the easy axis orientation. This value of α[1−10]=0.003 is an order of \nmagnitude lower than the previous reported va lues. The few experiment al results available \nseem to indicate that the damping factor decr eases with increasing Mn concentration. This \ncorresponds well to the th eoretical predictions by Sinova et al [5] for the case of small quasi-\nparticle lifetime broadening. It wi ll be interesting to test this behavior further in more highly \ndoped layers (x ≈0.15), which become now available.  \n \nAcknowledgment: We thank Alain Mauger of the IMPMC laboratory of the University Paris \n6 and Bret Heinrich from the Simon Frazer University in Vancouver for many helpful \ndiscussions. \n \nReferences : \n \n[1] T. Jungwirth, J. Sinova, J. Masek, J. Kucera, and A.H. MacDonald, Rev.Mod.Phys. 78, \n809 (2006). \n[2] X. Liu, and J. K. Furdyna , J. Phys.: Condens. Matter 18, R245 (2006). \n[3] X. Liu, Y. Sasaki and J. K. Furdyna, Phys.Rev.B 67 , 205204 (2003). \n[4] K.Khazen, H.J.von Bardeleben, J.L.Cantin, L.  Thevenard, L. Largeau, O. Mauguin, and A. \nLemaître,  Phys.Rev. B77, 165204 (2008) . \n[5] J.Sinova, T.Jungwirth, X.Liu, Y.Sa saki, J.K.Furdyna, W.A.Atkinson, and \nA.H.MacDonald, Phys.Rev. B69, 085209 (2004). \n [6] Y.H.Matsuda, A.Oiwa, K.Ta naka, and H.Munekata, Physica B 376-377,668 (2006) \n[7] A.Wirthmann, X.Hui, N.Mecking, Y.S.Gu i, T.Chakraborty, C.M.Hu, M.Reinwald, \nC.Schüller, and W.Wegschneider, Appl.Phys.Lett.92, 232106 (2008) \n[8] M. Farle, Rep.Prog.Phys. 61, 755 (1998). [9] B.Heinrich, « Spin relaxation in magnetic me tallic layers and multilayers », in Ultrathin \nMagnetic Structures III, ed. by J.A.C.Bland, and B.Heinrich, Springe r Verlag (Berlin 2005), \np.143 \n[10] C.Chappert,K.LeDang,P.Beauvilain,H .Hurdequint, and D.Renard, Phys.Rev.B34 , 3192 \n(1986) \n[11] F. Glas, G. Patriarche, L. Largeau, A. Lemaître, Phys. Rev. Lett., 93, 086107 (2004) \n[12] L.Thevenard, L.Largeau,, O.Mauguin, A.Lemaitre, K.Khazen and H.J.von Bardeleben, \nPhys. Rev.B75 ,195218 (2006) \n[13] L. Thevenard, L. Largeau, O. Mauguin, G. Patriarche, A. Lemaître, N. Vernier and J. Ferré, Phys. Rev. B 73, 195331 (2006)  \n[14] B.Heinrich, D.Fraitova, V.Kambersky , Phys. Stat. Solidi 23, 501 (1967). \n[15] T.Y.Tserkovnyak, G.A.Fiete, B.I.Halperin, Appl.Phys.Lett. 84, 5234 (2004) \n[16] A.Mauger, D.L. Mills, Phys.Rev. B29, 3815 (1984) \n[17] S.C.Chen and HL.Huang, I EEE Transactions on Magnetics 33, 3978 (1997) \n[18] H.L.Huang, V.L.Sobolev, and S.C.Chen,  J.Appl.Phys. 81, 4066(1997) \n[19] D.Chiba, Y.Sato, T.Kita, F. Matsukura, and H.Ohno, Phys.Rev.Lett. 93, 216602(2004) \n[20] J.C.Slonczewski, J.Magn.Magn.Mater. 159, L1(1996) \n \n \nFigure Captions: \n \nFigure 1a: X-band FMR spectrum of the GaMnAs /GaAs film taken at T=20K and for H// \n[001]; the peak-to-peak linewidth is 60Oe. The experimental  spectrum is shown by circles and \nthe Lorentzian lineshape simulation by a line. \n \n \nFigure 1b: Q-band FMR spectrum of the GaMnAs /GaInAs film taken at T=80K and for H// \n[100]; the peak to peak  linewidth is 120Oe. The experimental spectrum is shown by circles \nand the Lorentzian lineshape simulation by a line \n \nFigure 2 (color on line): X-band peak-to-peak li ne widths for the GaMnAs/GaAs film for the \nfour main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] \nred circles, H//[100] blue upper triangles.   \nFigure 3 color on line): Q-band peak-to-peak line widths for the GaMnAs/GaAs film for the \nfour main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] \nred circles, H//[100] blue upper triangles. \n \nFigure 4 (color on line): Peak-t o-peak linewidth at 9GHz and 35GHz for the GaMnAs/GaAs \nfilm; T=80K and H//[001] black squares, H//[1-10] olive lowe r triangles, H//[110] red circles, \nH//[100] blue upper triangles \n \nFigure 5 (color on line): GaMnAs/GaAs inhomogeneous linewidth as a function of temperature for four orientations of the applie d field: H//[001] black squares, H//[1-10] olive \nlower triangles, H//[110]  red circles, H//[100] blue upper triangles \n \nFigure 6 (color on line): damping factor α as a function of temperature and magnetic field \norientation; H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] blue upper \ntriangles, H//[100] red circles; the maximum erro r in the determination of the linewidth is \nestimated to 10G which co rresponds to an error in α of  0.001 \n \nFigure 7a: X-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard \naxis); the low field spin wave resonance (SW) is of high intensity in this case; circles: \nexperimental points, line: simulation with Gaussian lineshape \n \nFigure 7b: Q-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard \naxis); the low field spin wave resonance (SW) is of high intensity in this case; circles: experimental points, line: simulation with Gaussian lineshape \n \n \nFigure 8 (color on line): GaMnAs/GaInAs:  X-band FMR linewidth as a function of \ntemperature for the four orientations of the applied magnetic field: H//[001] black squares, \nH//[1-10] olive lower triangles, H//[110] red circ les, H//[100] blue upper triangles; the easy \naxis FMR spectrum is not observable below T=100K ; a typical hysteresis  curve as measured \nby SQUID is shown in the inset. \n Figure 9 (color on line): GaMnAs/GaInAs : Q-band FMR linewidth as a function of \ntemperature for the four orientations of the applied magnetic field : H//[001] black squares, \nH//[1-10] olive lower triangl es, H//[110] red circles, H//[100] blue upper triangles \n \nFigure 10 (color on line): GaMnAs/GaInAs : FM R linewidth as a function of microwave \nfrequency for H//[1-10] at different temper atures; T=10K (black squares), T=25K (red \ncircles), T=55K(black spades), T=80K( blue stars), T=115K (red triangles) \n \nFigure 11 (color on line): GaMnAs/GaInAs: FMR linewidth as a function of microwave \nfrequency for H//[001] (easy ax is)  at different temperatur es; T=100K (red circles), T=120K \n(blue spades),T=130K(black squares) \n \nFigure 12: GaMnAs/GaInAs inhomogeneous linewi dth as a function of temperature for two \norientations of the applied field: H//[001]  squares, H//[1-10]  lower triangles,  \n \nFigure 13 : GaMnAs/GaInAs damping factor  α as a function of temperature and magnetic \nfield orientation; H//[001]  squa res, H//[1-10]  lower triangles.  \n \nTables \n \n \nGa 0.93Mn 0.07As/GaAs G a0.93Mn 0.07As/Ga 0.902In0.098As \nMs(T=4K)= 47e mu/cm3 Ms(T=4K)= 38 e mu/cm3 \nTC =157K  (SQUID) TC=130K  (SQUID) \nAnisotropy constants (T=80K) Anisotropy constants (T=55K) \nK2⊥ = - 55000 e rg/cm3 K 2⊥ = +91070 e rg/cm3 \nK2// =2617 er g/cm3 K 2// = -2464 erg/cm3 \nK4⊥ = 8483  erg/cm3 K 4⊥ = -34050  erg/cm3 \nK2// =2590 er g/cm3 K 2// = -1873er g/cm3 \nEasy axis of magnetization Easy axis of magnet ization \n[100] 4K<T<T C [001] 4K<T< T C \n \n \nTable I \n \nTable caption \n \nTable I: M icrom agnetic  param eters of the two sam ples studied in this work: s aturation \nmagnetization M s at T=4K, critical tem perature T C, magneto crystalline anisotropy constants \nof second and fourth order K 2⊥ , K 2//, K 4⊥, K 4// at T=55K and T=80K respectively and the \norientation of the easy axis for m agnetization. \n \n \n \n \n \n \n \n            \n \n7500 7600 7700 7800 7900 8000SW\n  FMR Signal (arb.u.)\nMagnetic Field H ( Oe)\n \n \nFigure 1a \n \n10000 1020 0 10400 1060 0 10800 1100 0\n  FMR Signal (arb.u.)\nMagnetic Field H (Oe)\n \nFigure 1b \n 0 20 40 60 80 100 120140 160050100150200250300Linew idth ∆H (Oe)\nTempera ture (K) 001\n 110\n 100 1-10\n \nFigure 2 \n \n0 20 40 60 80 100 120 140160050100150200250300Linew idth ∆H (Oe)\nTemperature ( K) 001\n 110\n 100 1-10\n \nFigure 3 \n \n 0 5 10 15 20 25 30 35 4004080120160200Linewi dth ( G)\nFreq uency  (GHz)\n  \n \n \nFigure 4 \n \n20 40 60 80 100 120 140 160020406080100120∆Hinhom (Oe)\nTemperature (K) 001\n 110\n 100\n 1-10\n \n \nFigure 5 \n \n 0 20 40 60 80 100 120140 1600.0040.0060.0080.0100.0120.0140.0160.0180.020Damping factor α\nTemperature (K ) 001\n 110\n 100\n 1-10\n \nFigure 6 \n \n4000 5000 6000 7000 8000SW\n  FMR S ignal (arb. u.)\nMagnetic Field (Oe )\n \nFigure 7a \n 13000 14000 15000 16000 17000SW\n  FMR Signal (arb. u.)\nMagnet ic Field  (Oe)\n \n \nFigure 7b \n \n0 20 406 08 0 100120140300400500600700800\n  Linewidth ∆H (Oe)\nTemperature (K) [110]\n [100]\n [1-10]\n [001]-500 0 500-40040 M (emu/cm2)\n Magnetic F ield (O e)\n \nFigure 8 \n 0 204 0 608 0 100 120 140300400500600700800\n  Linewidth ∆H (Oe)\nTemperat ure (K ) [110]\n [100]\n [1-10]\n [001]\n \n \nFigure 9 \n \n0 5 10 15 20 25 30 35 40350400450500550600650700750\n115K80K55K25K10K\n  Linewidth ∆H (Oe)\nFreque ncy (GHz)\n \n \nFigure 10 \n 0 5 10 15 20 25 30 35 40200250300350400450500\n100K120K130K\n  Linewidth ∆H (Oe)\nFreque ncy (GHz)\n \n \nFigure 11 \n \n0 204 0 608 0 100120140200300400500600\n  ∆Hinhom (Oe)\nTemperat ure (K ) [001]\n [1-10]\n \n \n \nFigure 12 \n 0 204 06 0 80 100 120 1400.0040.0060.0080.0100.0120.0140.0160.0180.020Dam ping factor α\n  \nTemperature (K)\n \n \nFigure 13 \n \n \n \n "    },    {        "title": "1205.4895v3.Heavy_quark_damping_rate_in_hot_viscous_QCD_plasma.pdf",        "content": "arXiv:1205.4895v3  [hep-ph]  22 Aug 2013Heavy quark damping rate in hot viscous QCD plasma\nSreemoyee Sarkar∗and Abhee K. Dutt-Mazumder†\nHigh Energy Nuclear and Particle Physics Division,\nSaha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkat a-700 064, INDIA\nWe derive an expression for the heavy quark damping rate in ho t quark gluon\nplasma in presence of flow. Here all the bath particles are out of equilibrium due\nto the existence of non-zero velocity gradient. The magneti c sector shows similar\ninfrared divergences even after hard thermal loop correcti ons as one encounters in\ncase of non-viscous plasma. We estimate the first order corre ction in ( η/s) for heavy\nquark damping rate due to the non-zero viscosity of the QCD pl asma.\nI. INTRODUCTION\nIt is now generally accepted that the matter produced in Relativistic Heavy Ion Collider\nexperiments behave like near ideal fluid. In such collisions the ratio of the shear viscosity\n(η) and the entropy density ( s) is estimated to be not more than few times the lower bound\nofη/s= 1/4π[1]. This conclusion has been drawn from the success of ideal hydro dynam-\nics particularly in explaining both hadron transverse momentum spec tra and elliptic flow\n(v2(pT)) in the low pTregion.\nThev2(pT), isthesecond harmonic oftheazimuthal distributionof theprodu cedparticles\nwith respect to the reaction plane which is measured as a function of transverse momenta\nof various particle types and for different impact parameters. The ideal hydrodynamic\ndescription however breaks down for pT>2GeVbeyond which v2(pT) does not rise as\npredicted by non-viscous hydrodynamics[2, 3].\nIn ideal hydrodynamics v2(pT) shows a rising trend with increasing pT[3]. Recently\nseveral investigations have been performed to address this issue and it has been shown that\nthefallingtrendof v2(pT)inthehigher pTregioncannaturallybeexplained byinvoking non-\n∗Electronic address: sreemoyee.sarkar@saha.ac.in\n†Electronic address: abhee.dm@saha.ac.in2\nideal(viscous) hydrodynamics. Itistobenotedthatinnon-idealfl uidtheenergymomentum\ntensor (Tµν) apart from the ideal (non-interacting) part will also receive a cor rection term\nδTµνinvolving both the coefficients of shear ( η) and bulk viscosity ( ζ). Stating differently in\nnon-ideal fluid, the sound attenuation length Γ s= (4η)/(3(ǫ+p)) is non-zero. However, the\nattenuation length is small compared to the expansion rate τ, whereǫis the energy density\nandpis the pressure of the fluid [2].\nIt is to be noted that a strong hydrodynamic response is possible wh en Γs< τ. The\nnon-zero η(or Γs) modifies both the equation of motion and also the particle distributio n\nfunctions. The latter now will have a viscous correction term δfi.efor the fluid particle dis-\ntribution function and we shall write f=f0+δf, wheref0is the local thermal distribution\nfunction [3]. δfinvolves the sound attenuation length which however can be determ ined by\nsolving the linearized Boltzmann equation. In general δfcontains both the shear and the\nbulk viscosity coefficients, here, we restrict ourselves only to the m odification of δfdue to\nη. A brief discussion on this will be presented in section IIA.\nOne of the major activities in the area of high energy heavy ion exper iments have been\nto understand how the dissipative effects modify various experimen tal observables like par-\nticle spectra, HBT radii or elliptic flow due to the modified energy mome ntum tensor or\nthe viscous corrected distribution function [4–6]. Recently the effe ct of non-zero ηon the\nphoton and the dilepton production rates have been estimated [7–9 ]. In case of photon the\neffect of non-zero viscosity leads to a larger thermalization time . Ac cording to the authors\nof [9] the role of non-ideal effects is to increase the net photon yield due to slowing down\nof the hydrodynamic expansion. For dilepton the space-time integr ated transverse momen-\ntum spectra shows a hardening where the magnitude of the correc tion increases with the\nincreasing invariant mass. Here the authors argue that the therm al description is reliable\nfor an invariant mass <(2τ0T2\n0)/(η/s) [7], where, τ0is the thermalization time and T0is the\ninitial temperature. The effect of the viscous correction to the glu on dielectric function has\nalso been studied [10] with a specific choice of the direction of the inte rmediate momentum\nexchange for the gluons. Attempt has also been made to calculate t he drag and diffusion\nco-efficient including the viscous corrections numerically [11].\nIn the present work we intend to calculate the heavy quark damping rate in presence\nof longitudinal flow with viscous corrections upto O(η/s) which allows us to obtain closed\nform analytical results. The calculation of quark damping rate in equ ilibrium QED or QCD3\nplasma has already been studied in past few years [12–16]. From the s tudy it has been\nestablished that in non-viscous medium the quark damping rate is plag ued with infrared\ndivergences. In the non-relativistic plasma, where one considers o nly the coulomb or electric\ninteraction such divergences are removed by the Debye screening effects. The problem be-\ncomes non-trivial in dealing with the relativistic plasma where one has t o worry about both\nthe electric and the magnetic interactions. This additional complicat ion actually arises due\nto the absence of the static magnetic screening. In case of QED, in [14, 15] the authors have\nshown that the electric contribution to the quasiparticle damping ra te with plasma screen-\ning effects is finite and is of the order Γ long∼g2T(gis the coupling constant) whereas the\ntransverse part remains divergent even after inclusion of the plas ma corrections. This is be-\ncause the latter is only dynamically screened. To obtain a finite result another resummation\nscheme has to be developed by using Bloch-Nordsieck propagator [ 14, 15]. In case of QCD\none also encounters similar problem in hot plasma where either one can use the magnetic\nmass for the gluons or adopt similar resummation scheme as develope d in [14, 15]. The same\nproblem was latter addressed in [17] by adopting renormalization gro up formalism. Both\nthe formalisms however yield the same final result with non-perturb ative corrections for the\ndamping rate showing non-exponential time dependence. It would b e worthwhile to note\nthat in degenerate plasma one obtains finite results with magnetic int eractions without fur-\nther resummation unlike its high temperature counterpart. For de tailed discussions about\nthese issues we refer the readers to [18–20].\nThe viscous part is operative only when there exists momentum aniso tropicity i.ewe\nconsider there exists a non-zero velocity gradient in such plasma. N aturally this is a major\ndeparture from the above cited calculations where always the bath particles are assumed to\nbe in equilibrium. This is true only when there exists no velocity or tempe rature gradient\nin the plasma and there is no external force.\nII. FORMALISM\nIn order to calculate the heavy quark damping rate in a viscous plasm a, we first recall\nwhat is done in case of non-viscous medium. There one starts with th e Boltzmann kinetic4\nequation given by\n/parenleftbigg∂\n∂t+vp.∇r+F.∇p/parenrightbigg\nfp=−C[fp], (1)\nhere, the right hand side represents the collision integral which can be evaluated once the\ninteractions are known. vp=p/Epis the particle velocity.\nIn absence of the external force and gradients of temperature , velocity or density Eq.(1)\ntakes very simple form,\n∂fp\n∂t=−C[fp]. (2)\nThe collision integral can be written as sum of multiple terms represen ting various scattering\nprocesses,\nC[fp] =−(C[fp]1→2+C[fp]2→2+C[fp]2→3+···). (3)\nIn the present work we are interested only in the 2 →2 processes and calculate only the\ndamping rate leaving collisional energy loss calculation for future wor k [21]. For two body\ninteraction ( P+K→P′+K′), the explicit form of the collision integral is the following,\nC[fp] =1\n2Ep/integraldisplayd3k\n(2π)32Ekd3p′\n(2π)32E′\npd3k′\n(2π)32E′\nk\n×fpfk(1±fp′)(1±fk′)−fp′fk′(1±fp)(1±fk)\n×(2π)4δ4(P+K−P′−K′)1\n2/summationdisplay\nspin|M|2. (4)\nIt is to be noted that in Eq.(4) all the distribution functions designat ed byfifori=k,k′,p′\nare either the Fermi or Bose distribution functions for the quarks or gluons respectively.\nThe±signs include both stimulated emission or the Pauli blocking respective ly.\nInthepresent scenario we areconcerned withthe heavy quark da mping ratescattering off\nfrom quarks and gluons in the medium. The injected heavy quark now has both equilibrium\nand a fluctuating part ( fp=f0\np+δfp) and all the bath particles are in equilibrium. f0\ni\ndenotes the equilibrium fermion or boson distribution function. Inse rtingfp=f0\np+δfpin\nEq.(2) one can write,\n∂δfp\n∂t=1\n2Ep/integraldisplayd3k\n(2π)32kd3p′\n(2π)32E′pd3k′\n(2π)32k′fEpf0\nk(1±f0\nk′)\n×(2π)4δ4(P+K−P′−K′)1\n2/summationdisplay\nspin|M|2. (5)5\nNote the difference of the thermal phase space here with that of t he light quarks in [22].\nWhile writing the above equation for high energetic parton the possib ility of back scattering\nhas been excluded and the approximation (1 ±f0\nE′p)≃1 has also been incorporated in the\nthermal phase space since Ep′>> T. In the relaxation time approximation one can write,\n∂δfp\n∂t=−C[fp] =−δfpΓ0\np. (6)\nWe can identify Γ0\npas the particle damping rate given by,\nΓ0\np=1\n2Ep/integraldisplayd3k\n(2π)32kd3p′\n(2π)32E′pd3k′\n(2π)32k′f0\nk(1±f0\nk′)(2π)4δ4(P+K−P′−K′)1\n2/summationdisplay\nspin|M|2.(7)\nNote that Γ0\npis independent of non equilibrium part of fpand depends only on the\ndistribution of the bath particles.\nA. Viscosity corrected distribution function\nTo incorporate the effect of flow of the medium, we take viscous cor rected distribution\nfunction as fi=f0\ni+δfη\nifor the bath particles, where, i=p′,k,k′.δfη\niis the first order\ncorrection to the thermal distribution function which actually is con strained by the viscosity\nor energy momentum stress tensor and also related to Γ s. The form of δfη\nidepends on the\nvarious ansatz [3, 23–25]\nδfη\ni=χ(k)f0\nk(1±f0\nk)\nTˆkiˆkj∇iuj. (8)\nIn principle χ(k) can be determined from various microscopic theories as discussed in [3].\nIn most of the hydrodynamic calculations it is assumed that δfk∝k2f0\nkand the propor-\ntionality constant is independent of the particle type. This is known a s quadratic ansatz\nwhich recently has been called into question [3]. For QCD it has however been shown that\nχ(q)/χ(g)≈1.70. For the present case we assume these to be equal.\nFor a boost invariant expansion without transverse flow one can inc orporate the viscous\ncorrection to the distribution function in the following way, [2, 9, 11],\nδfη\ni(k) =f0\ni(1±f0\ni)Φi(k), (9)6\nwhere,\nΦi(k) =1\n2T3τη\ns/parenleftbiggk2\n3−k2\nz/parenrightbigg\n. (10)\nThe correction term given above is based upon the ”first approxima tion” described in [26]\nand ”one-parameter ansatz” for a variational solution of [25]. The formal procedure for\ndetermining the viscous correction using variational principle has be en discussed in [25] in\ngreat detail. The viscous modification holds true only in the local rest frame of the fluid and\nit contains the first order correction in the expansion of shear par t of the stress tensor. τis\nthe thermalization time of the quark-gluon plasma (QGP) and the flow is alongzaxis. From\nthe above expression this is also evident that the non-equilibrium par t of the distribution\nfunction becomes operative only when there is a momentum anisotro py in the system.\nIn a medium with non-zero flow gradient with the distribution function s of the form\nmentioned in Eqs.(9) and (10) the collision integral can be expressed as,\nC[fp] =1\n2Ep/integraldisplayd3k\n(2π)32kd3p′\n(2π)32E′\npd3k′\n(2π)32k′/summationdisplay\ni=1,2αi(2π)4δ4(P+K−P′−K′)1\n2/summationdisplay\nspin|M|2,(11)\nwhere,αi’s represent the viscous modified phase-space distribution functio ns.α1contains\nthe equilibrium part of the distribution functions, this gives us the us ual interaction rate\nmentioned in Eq.(7) where all the bath particles are in thermal equilibr ium,\nα1=δfpf0\nk(1±f0\nk′). (12)\nα2involves terms due to the viscous modifications to the light quark dist ribution functions\nfor the bath constituents,\nα2≃δfp/bracketleftbig\nΦkf0\nk(1±f0\nk′)±Φk′f0\nk′f0\nk/bracketrightbig\n. (13)\nIn absence of viscosity Φ i’s are zero and we get back our result for usual quasiparticle\ndamping rate which is given by Eq.(7). In the relaxation time approxima tion using Eqs.(12)\nand (13) the collision integral can be written as,\nC[fp]≃δfp/parenleftbig\nΓ0(p)+δΓη(p)/parenrightbig\n(14)\nwhereδΓηreceives contribution from α2.7\nP P\nK K ''\n(a) (b) (c) (d)\nFIG. 1: Amplitudes for heavy quark elastic scattering in a QC D plasma. A curly line denotes a\ngluon (QCD). The amplitude (d) is specific to the QCD case. The blob in (a) and (d) denotes\nthe resummed hard thermal loop boson propagator, which is ne cessary to screen the t-channel\ncontribution in the infrared domain.\nB. Damping rate in presence of flow\nIt has already been indicated in the previous section that in presenc e of non-zero flow\ngradientparticledistributionfunctiongetsmodifiedwithaterminvolv ingviscouscorrections\nas given by Eqs. (8), (9) and (10). For the heavy quark or the tes t particle the distribution\nfunction also has a non-equilibrium fluctuating component δfpin additionto the equilibrium\npartf0\np(δfp<< f0\np). It is to be noted that the heavy quark distribution function is\nindependent of η.\nIn Eq.(4) we now insert the above mentioned viscous corrected dist ribution function to\nobtain,\nδΓη\nq(p) =1\n2Ep/integraldisplay\nk,p′,k′/bracketleftbig\nΦkf0\nk(1±f0\nk′)±Φk′f0\nk′f0\nk/bracketrightbig\n(2π)4δ4(P+K−P′−K′)1\n2/summationdisplay\nspin|M|2,(15)\nin the above equation/integraltext\nkis shorthand for/integraltext\nd3k/((2π)32k). The above expression has been\narrived at by neglecting terms O((η/s)2) andO(f3\ni). To proceed further we have to know\nthe interaction. In case of quark-quark (q-q) scattering of diffe rent flavours in the tchannel\nand the quark-gluon (q-g) scattering in the same channel the mat rix amplitudes squared are\ngiven by (see Fig.(1)) [27],\n1\n2/summationdisplay\nspin|M|2\nq1q2,t=8g4d2\nFC2\nF\nCA/parenleftbiggs2+u2\nt2/parenrightbigg\n,\n1\n2/summationdisplay\nspin|M|2\nqg,t= 8g4dFCFCA/parenleftbiggs2+u2\nt2+s2+t2\nu2/parenrightbigg\n. (16)\nThe above matrix elements are singular because of the t−2= (ω2−q2)−2dependence, where\nωandqare the energy and momentum transfer. There are now two singula rities because8\nof the Mandelstam variables, uandt. For small momentum transfers the singular behavior\ncan be cured with the help of the plasma screening which we discuss lat er in the present\nsection.\nFor the fermion exchange diagrams the matrix element is given by [27 ],\n1\n2/summationdisplay\nspin|M|2\nqg,s+u=−8g4dFC2\nF/parenleftBigu\ns+s\nu/parenrightBig\n. (17)\nFrom the above matrix elements we take only the terms which give lead ing contributions to\nthe heavy quark damping rate, hence, we approximately write,\n1\n2/summationdisplay\nspin|M|2\nt,qq=Ab,q˜s2\nt2\n1\n2/summationdisplay\nspin|M|2\ns+u,qg=−Af/parenleftbigg˜s\n˜u+˜u\n˜s/parenrightbigg\n, (18)\nfor heavy quark ˜ s=s−m2\nq, ˜u=u−m2\nq. The group factor for tchannel diagrams are\ngiven by Ab,q=d2\nFC2\nF\nCAandAb,g=dFCFCA, where,Ab,qis relevant for quark-quark and Ab,g\nfor quark-gluon scattering. Afforsanduchannel diagrams is Af=dFC2\nF. The divergences\nin thetchannel diagrams are usually removed with the help of the plasma scr eening. The\nusual way to include this medium modification is to use the Hard Therma l Loop (HTL)\ndressed propagator instead of bare one [28–30]. With the dressed gluon propagator the\nquark-quark matrix amplitude squared looks like,\n1\n2/summationdisplay\nspin|M|2\nqq= 8Ab,qg4/bracketleftbigg1\n(q2+ΠL)−vp,T.vk,T\n(q2−ω2+ΠT)/bracketrightbigg2\n. (19)\nIn theabove equation the medium modified gluonpropagatorcontain s thepolarizationfunc-\ntionsΠ LandΠ T, which describe plasma screening ofinterparticle interaction bylong itudinal\nand transverse plasma perturbations, respectively. In the large wavelength limit q << T\n[23],\nΠL(q,ω) =m2\nDχL,ΠT(q,ω) =m2\nDχT, (20)\nwhere,\nχL=/bracketleftbigg\n1−x\n2ln/parenleftbiggx+1\nx−1/parenrightbigg/bracketrightbigg\n,\nχT=/bracketleftbiggx2\n2+x(1−x2)\n4ln/parenleftbiggx+1\nx−1/parenrightbigg/bracketrightbigg\n, (21)9\nandmDis the gluon Debye mass. In the static limit ( x=ω/q∼0) above mentioned\npolarization functions can be expanded to give rise to χL∼1 andχT∼iπx\n4when|x|<<1.\nIn this region the squared matrix element becomes,\n1\n2/summationdisplay\nspin|M|2\nqq= 8Ab,qg4\n1\n(q2+mD)2+/parenleftbig\nv2\np−x2/parenrightbig\nq2cos2φ\n(1−x2)/parenleftBig\nq6+π2ω2m4\nD\n16/parenrightBig\n, (22)\nwhere, we have omitted the cross term of longitudinal and transve rse interaction because\nazimuthal angle φintegration gives zero contribution for this term. With the above ma trix\namplitude squared we now compute viscous corrected quark dampin g rate for the quark-\nquark scattering δΓη\nqq,\nδΓη\nqq≃1\n2Ep/integraldisplay\nk,p,p′/bracketleftBig\nΦkf0\nk(1−2f0\nk+2ωf′0\nk)+ωΦ′\nkf02\nk+O(ω2)/bracketrightBig\n×1\n2/summationdisplay\nspin|M|2(2π)4δ4(P+K−P′−K′) (23)\nwhile writing the above equation we have used the following expansion,\nf0(k′) =f0(k−ω)≃f0\nk−ωf′0\nk,\nΦk′= Φ(k−ω)≃Φk−ωΦ′\nk. (24)\nTo simplify δΓη\nqqwe neglect higher order terms in ω. With the bare interaction one observes\nthat the above mentioned damping rate has infrared divergences a nd also the order of di-\nvergences for different terms are different. Both/integraltext\ndq/q3and/integraltext\ndq/qdivergences are present.\nAs mentioned earlier in this section the usual way to handle these dive rgences is to incor-\nporate the effects of plasma screening. The method of calculating t he effects of screening\ndeveloped by Braaten and Yuan [31] involves introducing an arbitra ry momentum scale q∗\nto separate the region of hard momentum transfer q∼Tfrom the soft region q∼gT. The\narbitrary momentum scale is chosen in the way so that gT << q∗<< T, which is possible\nin the weak-coupling limit g→0. The contribution from the hard momentum region q > q∗\nis calculated using tree-level scattering diagrams where the lower lim itq∗acts as infrared\ncutoff.\nThe detailed calculations of all the terms in Eq.(15) have been presen ted in the Appendix\nA and B. Here, we only quote the final results. The final expression s for both the electric10\nand the magnetic sectors in case of q-q scattering in the tchannel take the following forms,\nδΓη,L\nqq=/parenleftBigη\ns/parenrightBigg4Ab,q\n4(2π)3τvp/bracketleftbigg\nπ2/parenleftbigg\n−vp\n6−v3\np\n9+v5\np\n10/parenrightbigg\n+ζ(3)/parenleftbigg3\n4vp+3\n2v3\np−21\n20v5\np/parenrightbigg/bracketrightbigg\n×/parenleftbigg\n−1\n2+ln/parenleftbiggqmax\nmD/parenrightbigg/parenrightbigg\n,\nδΓη,T\nqq=/parenleftBigη\ns/parenrightBigAb,qg4πT2\n16(2π)3τvpm2\nD/parenleftbigg7π4\n15−45ζ(5)\n2/parenrightbigg/integraldisplayq∗\n0dq\nq\n+/parenleftBigη\ns/parenrightBigAb,qg4\n8(2π)3τvp/bracketleftBigg/parenleftbiggπ2\n3−3ζ(3)/parenrightbigg/parenleftbigg\n−16v5\np\n225−4v5\np\n15ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2qmax\nmD√πv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg/bracketrightBigg\n.(25)\nIn the high energy limit qmax∼/radicalbig\nEpT[32]. From the above equations it is evident that\nthe transverse sector contains both the finite and the infrared d ivergent terms. It is to be\nnoted that presence of ωin the numerator in Eq.(23) makes some of the divergent terms\nfinite once the integrations are performed. Still however/integraltext\ndq/qdivergence remains for the\nfirst two terms in Eq.(23). This is reminiscent of what happens for th e case of quasiparticle\ndamping rate in non-viscous plasma which requires further resumma tion as discussed in\n[14, 15].\nThe physical processes responsible for these divergences are th e collisions involving the\nexchange oflong wavelength, quasistatic, magnetic gluons, which a renot screened by plasma\neffects and show logarithmic divergence. The leading divergences ca n be resummed using a\nnonperturbative treatment based on a generalization of the Bloch -Nordsieck model at finite\ntemperature [14, 15]. The resulting expression of the fermion prop agator is free of infrared\ndivergences, and exhibits a nonexponential damping at large times [1 4, 15].\nLater Boyanovsky et alhave shown that result obtaind in [14, 15] can be reproduced\nby invoking the renormalization group method [17]. This allows a consist ent resummation\nof infrared effects associated with the exchange of quasistatic tr ansverse gluons leading to\nanomalous logarithmic relaxation of the form e−αTtln[ωpt]for hard momentum excitations\nwhereαis the fine structure constant and ωpis the plasma frequency.\nIn this context we recall the result of the heavy quark damping rat e in non-viscous\nmedium. It has been calculated long ago by Thoma and Gyulassy [33] in hot QCD plasma.\nHere, we quote the final result,\nΓ(p)0,L\nqq=g4AbT3\n96π/parenleftbigg1\nm2\nD−1\nq2max/parenrightbigg\nΓ(p)0,T\nqq=g4AbT3vp\n48πm2\nD/integraldisplayq∗\n0dq\nq. (26)11\nThus we see that both the viscous corrected and non-viscous dist ribution functions for the\nbath particles give similar divergent result without any additional dive rgence.\nWe now compute the contribution ( δΓη\nqg) to the heavy quark damping ratefrom Compton\nscattering. We start with Eq.(15), where, the matrix element for q -g scattering including all\nthree channels M=Ms+Mt+Muhas to be inserted. First we estimate the contributions\nfrom the sanduchannel diagrams,\nδΓη\nqg=/parenleftBigη\ns/parenrightBigAfg4\n32T3τπ3Ep/bracketleftbigg2π4T4\n135−T3m2\nq\n3Ep/parenleftbigg\n2ζ(3)ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle4EpT\nm2q/vextendsingle/vextendsingle/vextendsingle/vextendsingle+3ζ(3)−2γζ(3)+2ζ′(3)/parenrightbigg/bracketrightbigg\n.(27)\nThe contribution of quark-gluon scattering in heavy quark damping rate in a non-viscous\nmedium has the following form,\nΓ(p)0\nqg=1\n16πEp/integraldisplayd3kfk\n(2π)32k/integraldisplay\ndt|M|2\ns−m2\nq\n=g4T2Af\n48πEp/bracketleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingle4EpT\nm2q/vextendsingle/vextendsingle/vextendsingle/vextendsingle+O(1)/bracketrightbigg\n. (28)\nThe final expression for the heavy quark damping rate can be obta ined by adding the\ncontributions from q-q (Eq.(25) and q-g (Eq.(27)) scatterings,\nδΓη\nq=δΓη\nqq+δΓη\nqg\n=/parenleftBigη\ns/parenrightBigg4Ab\n4(2π)3τvp/bracketleftbigg\nπ2/parenleftbigg\n−vp\n6−v3\np\n9+v5\np\n10/parenrightbigg\n+ζ(3)/parenleftbigg3\n4vp+3\n2v3\np−21\n20v5\np/parenrightbigg/bracketrightbigg/parenleftbigg\n−1\n2+ln/parenleftbiggqmax\nmD/parenrightbigg/parenrightbigg\n+/parenleftBigη\ns/parenrightBigAbg4πT2\n16(2π)3τvpm2\nD/parenleftbigg7π4\n15−45ζ(5)\n2/parenrightbigg/integraldisplayq∗\n0dq\nq\n+/parenleftBigη\ns/parenrightBigAbg4\n8(2π)3τvp/bracketleftBigg/parenleftbiggπ2\n3−3ζ(3)/parenrightbigg/parenleftbigg\n−16v5\np\n225−4v5\np\n15ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2qmax\nmD√πvp/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg/bracketrightBigg\n+/parenleftBigη\ns/parenrightBigAfg4\n32T3τπ3Ep/bracketleftbigg2π4T4\n135−T3m2\nq\n3Ep/parenleftbigg\n2ζ(3)ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle4EpT\nm2\nq/vextendsingle/vextendsingle/vextendsingle/vextendsingle+3ζ(3)−2γζ(3)+2ζ′(3)/parenrightbigg/bracketrightbigg\n.(29)\nwhere, we have defined Ab=Ab,q+Ab,g.\nAn interesting implication of the characteristic behaviour of the dam ping rate as ob-\ntained above is closely related to the heavy ion experiments where no n-equilibrated heavy\nquark passes through the longitudinally expanding QGP medium. Damp ing rate eventually\ndetermines how rapidly non-equilibrated heavy quark approaches it s equilibrium state.12\nIII. SUMMARY AND CONCLUSION\nIn the present work we have calculated heavy quark damping rate in a viscous medium\nrestricting ourselves only to two body scattering processes i.ewe consider only the quark-\nquark and quark-gluon scatterings. It has been shown in the text , how does the viscosity\nenter into the calculation viathe viscous corrected distribution function in the phase-space\nfactor in presence of a flow gradient. We have restricted ourselve s only to the leading order\ncontributions in η/sby dropping all the higher order terms. To further simplify the collisio n\nintegral the powers beyond the quadratic terms of the distributio n functions have been\ndropped. Furthermore, for the gluon exchange only the soft fre quencies have been retained\nfollowing the standard techniques what one adopts to calculate the damping rate formalism\nin non-viscous plasma. These approximations, in effect, allow us to pr esent closed form\nanalytical results. The final result is based upon the ansatz mentio ned in Eqs.(9) and (10).\nIt is to be noted for the flow we consider only the longitudinal gradien t by assuming that\nthere is no transverse expansion. One of the interesting findings o f the present work has\nbeen the infrared behaviour of the transverse damping rate which has the same/integraltext\ndq/qform\nboth for the viscous and the non-viscous part. To cure this diverg ence, which remains even\nafter using finite temperature HTL propagator due to the non-ex istence of screening for the\nstatic gluons one may perform further resummation by using Bloch- Nordsieck propagator\nor renormalization group method [14, 15, 17] as mentioned in the tex t.\nAppendix: Damping rate calculation\n1. Quark-quark scattering\nIn this section we explicitly show the computation of the q-q and q-g s cattering rates\nin thetchannel in a viscous medium. First we evaluate the tchannel diagram for q-q\nscatterings. For this we start from the expression Eq.(15),\nδΓη\nqq≃1\n2Ep/integraldisplay\nk,p,p′/bracketleftBig\nΦkf0\nk(1−2f0\nk+2ωf′0\nk)+ωΦ′\nkf02\nk+O(ω2)/bracketrightBig\n×1\n2/summationdisplay\nspin|M|2(2π)4δ4(P+K−P′−K′)\n=δΓ1+δΓ2+δΓ3+δΓ4. (A.1)13\nConsidering the first two terms of the coefficient Φ k,\nδΓ1+2(p) =/parenleftBigη\ns/parenrightBig1\n24(2π)52T3τ/integraldisplayd3p′d3k\nEpEp′kk′f0\nk/parenleftbig\n1−2f0\nk/parenrightbig/parenleftbiggk2\n3−k2\nz/parenrightbigg\nδ(Ep+k−Ep′−k′)1\n2/summationdisplay\nspin|M|2. (A.2)\nIn the collision integral we use the spatial delta function to perform thek′integration and\nto shift the p′integration into an integration over q, where,q=p′−p. It is convenient to\nintroduce a dummy integration variable ω,\nδ(Ep+k−Ep′−k′) =/integraldisplay∞\n−∞dωδ(ω+Ep−Ep′)δ(ω−k+k′). (A.3)\nEvaluating q=p′−pin terms of p, q and cos θpqand defining t=ω2−q2we find,\nδ(ω+Ep−Ep′) =1\nvpqδ/parenleftbigg\ncosθpq−ω\nvpq−t\n2pq/parenrightbigg\n,\nδ(ω−k+k′) =1\nqδ/parenleftbigg\ncosθkq−ω\nq+t\n2kq/parenrightbigg\n. (A.4)\nUsing the above delta functions one can arrive at the following equat ion,\nδΓ1+2(p) =/parenleftBigη\ns/parenrightBig1\n(2π)4242T3τvp/integraldisplaydqdωdcosθqk2dkdcosθkdφk\nEpEp+qk(k−q)f0\nk/parenleftbig\n1−2f0\nk/parenrightbig/parenleftbiggk2\n3−k2\nz/parenrightbigg\n×δ/parenleftbigg\ncosθpq−ω\nvpq−t\n2pq/parenrightbigg\nδ/parenleftbigg\ncosθkq−ω\nq+t\n2kq/parenrightbigg1\n2/summationdisplay\nspin|M|2. (A.5)\nTo perform the phase space integration we choose qalong the zaxis,pin thex−zplane\nandkremains arbitrary,\nq= (0,0,1)q,\np= (sinθp,0,cosθp)p,\nk= (sinθkcosφk,sinθksinφk,cosθk)k. (A.6)\nUsing the first delta function integration over dcosθqcan be done. The delta function\nimposes the following condition on the angle θpq,\nˆp.ˆq= cosθpq=ω\nvpq+t\n2pq. (A.7)\nSecond delta function yields,\nˆk.ˆq= cosθkq=ω\nq−t\n2kq. (A.8)14\nWith the help of the first delta functions the bounds on ωcan be fixed as follows,\nω±=Ep−/radicalBig\nE2\np+q2∓2Epvpq. (A.9)\nIn case of a high energetic quark i.ewhenEp>> q, the above limits can be approximated\nasω≈ ±vpq. Using the second delta function we explicitly write the term coming fr om the\nviscous corrected distribution function as follows,\n/parenleftbigg1\n3−cos2θkq/parenrightbigg\n=/parenleftbigg1\n3−χ/parenrightbigg\n. (A.10)\nFirst we consider the soft sector of Eq.(A.1),\nδΓsoft\n1+2+3+4=/parenleftBigη\ns/parenrightBigAb,qg4\n(2π)34T3τvp/integraldisplay∞\n0k4dk/integraldisplayq∗\n0dq/integraldisplayvpq\n−vpqdω/bracketleftBigg\n/parenleftbig\nf0\nk−2f02\nk+2ωf0\nkf0\nk′/parenrightbig/parenleftbigg1\n3−χ/parenrightbigg\n+ Φ′\nkωf02\nk/bracketrightBigg\n1\n(q2+m2\nD)2+/parenleftBig\nv2\np−ω2\nq2/parenrightBig\n2/parenleftBig\n1−ω2\nq2/parenrightBig1/parenleftBig\nq4+π2m4ω2\n4q2/parenrightBig\n. (A.11)\nWe use following results for the ωintegration to proceed further,\n/integraldisplayvpq\n−vpq/parenleftBigg\n1\n3−/parenleftbiggω\nq−ω2−q2\n2kq/parenrightbigg2/parenrightBigg\ndω\n=q3\nk2/parenleftbigg−vp\n2+v3\np\n3−v5\np\n10/parenrightbigg\n+2qvp\n3/parenleftbig\n1−v2\np/parenrightbig\n,\n/integraldisplayvpq\n−vpq/parenleftBigg\nω\n3−ω/parenleftbiggω\nq−ω2−q2\n2kq/parenrightbigg2/parenrightBigg\ndω\n=q3\nk/parenleftbigg\n−2v3\np\n3+2v5\np\n5/parenrightbigg\n. (A.12)\nHence, for the first two terms in Eq.(A.11) in the electric sector we g et,\nδΓsoft\n1+2,l=/parenleftBigη\ns/parenrightBigAb,qg4\n(2π)34T3τvp/bracketleftBigg/parenleftbigg−vp\n2+v3\np\n3−v5\np\n10/parenrightbigg/integraldisplay∞\n0k2(f0\nk−2f02\nk)dk/integraldisplayq∗\n0q3dq\n(q2+m2\nD)2\n+/parenleftbigg−2v3\np\n3+2v5\np\n5/parenrightbigg/integraldisplay∞\n0k4(f0\nk−2f02\nk)dk/integraldisplayq∗\n0qdq\n(q2+m2\nD)2/bracketrightBigg\n=/parenleftBigη\ns/parenrightBig/bracketleftBigg\nAb,qg4\n(2π)34τvp/parenleftbigg−vp\n2+v3\np\n3−v5\np\n10/parenrightbigg/parenleftbiggπ2\n3−3ζ(3)\n2/parenrightbigg/parenleftbigg\n−1\n2−ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglemD\nq∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n+Ab,qg4T2\n(2π)34τvp/parenleftbigg−v3\np\n3+v5\np\n5/parenrightbigg/parenleftbigg7π4\n15−45ζ(5)\n2/parenrightbigg/parenleftbigg1\nm2\nD−1\nq∗2/parenrightbigg/bracketrightBigg\n. (A.13)15\nHard gluon exchange gives,\nδΓhard\n1+2,l=/parenleftBigη\ns/parenrightBig/bracketleftBigg\nAb,qg4\n(2π)34τvp/parenleftbigg−vp\n2+v3\np\n3−v5\np\n10/parenrightbigg/parenleftbiggπ2\n3−3ζ(3)\n2/parenrightbigg/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingleqmax\nq∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n+Ab,qg4T2\n(2π)34τvp/parenleftbigg−v3\np\n3+v5\np\n5/parenrightbigg/parenleftbigg7π4\n15−45ζ(5)\n2/parenrightbigg/parenleftbigg1\nq∗2−1\nq2\nmax/parenrightbigg/bracketrightBigg\n.(A.14)\nIn the high energy limit from the kinematics qmaxcan be taken as/radicalbig\nEpT[32], hence,\nδΓ1+2,l=/parenleftBigη\ns/parenrightBig/bracketleftBigg\nAb,qg4\n(2π)34τvp/parenleftbigg−vp\n2+v3\np\n3−v5\np\n10/parenrightbigg/parenleftbiggπ2\n3−3ζ(3)\n2/parenrightbigg/parenleftbigg\n−1\n2+ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleqmax\nmD/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n+Ab,qg4T2\n(2π)34τvp/parenleftbigg−v3\np\n3+v5\np\n5/parenrightbigg/parenleftbigg7π4\n15−45ζ(5)\n2/parenrightbigg/parenleftbigg1\nm2\nD−1\nq2\nmax/parenrightbigg/bracketrightBigg\n. (A.15)\nThe magnetic interaction on the other hand gives rise to,\nδΓsoft\n1+2,t=/parenleftBigη\ns/parenrightBigηAb,qg4\n(2π)38T3τvp/integraldisplay∞\n0k4(f0\nk−2f02\nk)dk/integraldisplayq∗\n0dq/integraldisplayvpq\n−vpqdω/parenleftbigg1\n3−χ/parenrightbigg/parenleftBig\nv2\np−ω2\nq2/parenrightBig\n2/parenleftBig\n1−ω2\nq2/parenrightBig/parenleftBig\nq4+π2m4ω2\n4q2/parenrightBig\n≃/parenleftBigη\ns/parenrightBigηAb,qg4T2π\n(2π)316τvpm2\nD/parenleftbigg7π4\n15−45ζ(5)\n2/parenrightbigg/integraldisplayq∗\n0dq\nq. (A.16)\nThe transverse part is infrared divergent even after using the HT L resummation and the\nform of divergence is same as obtained in the non-viscous medium [14 , 15].\nComputation of the last two terms of the damping rate is presented below,\nδΓsoft\n3+4,l=/parenleftBigη\ns/parenrightBigηAb,qg4\n(2π)32T3τvp/parenleftbigg\n−v3\np\n3+v5\np\n5/parenrightbigg/integraldisplay∞\n0(2k3f0\nkf′0\nk−k2f02\nk)dk/integraldisplayq∗\n0q3dq\n(q2+m2\nD)2\n=/parenleftBigη\ns/parenrightBigAb,qg4\n(2π)32τvp/parenleftbigg\n−v3\np\n3+v5\np\n5/parenrightbigg/parenleftbiggπ2\n3−3ζ(3)/parenrightbigg/parenleftbigg\n−1\n2+ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleq∗\nmD/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\nδΓ3+4,l=/parenleftBigη\ns/parenrightBigAb,qg4\n(2π)32τvp/parenleftbigg\n−v3\np\n3+v5\np\n5/parenrightbigg/parenleftbiggπ2\n3−3ζ(3)/parenrightbigg/parenleftbigg\n−1\n2+ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleqmax\nmD/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n.(A.17)\nThe transverse interaction on the other hand gives,\nδΓsoft\n3+4,t=/parenleftBigη\ns/parenrightBigAb,qg4\n8(2π)3τvp/bracketleftBigg/parenleftbiggπ2\n3−3ζ(3)/parenrightbigg/parenleftbigg\n−16v5\np\n225−4v5\np\n15ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2qmax\nmD√πv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg/bracketrightBigg\n.(A.18)\nUnlike the previous two terms here in this case we obtain finite interac tion rate in the\nmagnetic sector using the resummed propagator. This is because o f the fact that one extra\nωin the numerator coming from the phase-space distribution functio n (ω∂fk/∂k) cures the\nlogarithmic divergence.\nOne obtains same result both for the q-q and q-g scattering tchannel diagrams except\nthe multiplicative group factor. Ab,gdiffers from the quark case according to Eq.(16).16\n2. Quark-gluon scattering\nIn this section we present the detailed calculation of the contributio n of quark-gluon\nscattering in the sanduchannels to the total heavy quark damping rate. We start with\nthe following expression,\nδΓη\nqg=/parenleftBigη\ns/parenrightBig8Afg4\n2T3τ/integraldisplayd3kd3k′d3p′\n(2π)52Ep2Ep′2k2k′/bracketleftbig\nΦkf0\nk(1+f0\nk′)+Φk′f0\nk′f0\nk/bracketrightbig\n×δ4(P+K−P′−K′)/bracketleftbigg−˜u\n˜s+−˜s\n˜u/bracketrightbigg\n. (A.19)\nIn short notation we write the above equation as,\nδΓη\nqg=/parenleftBigη\ns/parenrightBig8Afg4\n2T3τ/integraldisplayd3kd3k′d3p′\n(2π)52Ep2Ep′2k2k′/bracketleftbig\nΦkf0\nk(1+f0\nk′)+Φk′f0\nk′f0\nk/bracketrightbig\n×δ4(P+K−P′−K′)g(s,t,ω), (A.20)\nwhere,g(s,t,ω) depends on the Mandelstam variables and exchanged energy. The k′inte-\ngration can be expressed as\n/integraldisplay\nk′Φkf0\nk(1+f0\nk′)+Φk′f0\nk′f0\nk\n2k′(2π)4δ(4)(P+K−P′−K′)\n= 2πΦkf0\nk(1+f0\nk−ω)+Φk−ωf0\nk−ωf0\nkΘ(k−ω)δ((K−Q)2), (A.21)\nwith the help of the following expression\n/integraldisplayd3k′\n(2π)31\n2k′= 2π/integraldisplayd4k′\n(2π)4Θ(k−ω)δ((K−Q)2).\nTo perform the integration over p′,pis chosen along the zaxis and kin they−zplane,\nhence,\np= (0,0,1)p,\nk= (0,sinθk,cosθk)k,\np′= (sinθp′sinφp′,sinθp′cosφp′,cosθp′)p′. (A.22)\nThe integration over φcan be done with the help of the delta function as shown below,\n/integraldisplay2π\n0dφδ((K−Q)2) =2√fΘ(f), (A.23)\nwhere,f=B2−A2.BandAcan be expressed in terms of the Mandelstam invariants\n[34, 35],\nA=s−m2\nq+t−2kEp′+2kp′cosθkcosθp′,\nB= 2kp′sinθksinθp′. (A.24)17\nWe now change the variables from p′and cosθp′totandωrespectively by the following\ntransformation,\nt= 2(m2\nq−EpEp′+pp′cosθ),\nω=Ep−Ep′. (A.25)\nWith this Eq.(A.20) now becomes,\nδΓη\nqg=/parenleftBigη\ns/parenrightBigAfg4\n4T3τπ2pEp/integraldisplay\nk1\n2k/integraldisplay0\n−∞dt/integraldisplay∞\n−∞dω/radicalbig\nf(ω)/parenleftbig\nΦkf0\nk(1+f0\nk−ω)+Φk−ωf0\nk−ωf0\nk/parenrightbig\ng(s,t,ω). (A.26)\nBounds on the integrals ωandtarise from the condition f=B2−A2≥0.f(ω) can now\nbe written as follows [34, 35],\nf(ω) =−a2ω2+bω+c, (A.27)\nthe coefficients of the above equation are [34, 35],\na=s−m2\nq\np,\nb=−2t\np2(Ep(s−m2\nq)−k(s+m2\nq)),\nc=−t\np2[t((Ep+k)2−s)+4p2k2−(s−m2\nq−2Epk)2]. (A.28)\nf(ω) is positive only in the domain ωmin<< ω << ω max, where the discriminant D=\n4a2c+b2is positive. Thus we have [34, 35],\nωmax\nmin=b±√\nD\n2a2,\nD=−t/parenleftbig\nst+(s−m2\nq)2/parenrightbig/parenleftbigg4ksinθk\np/parenrightbigg2\n. (A.29)\nThe condition D≥0 leads to the 2 →2 scattering processes with one massless and one\nmassive particle in the limit tmin≤t≤0 with [34, 35],\ntmin=−(s−m2\nq)2\ns. (A.30)\nWeshowlatter inthissectionthat thefirst terminEq.(A.26)gives ust hefinitecontribution,\nthe other terms are either quadratic in fkor higher oreder in ω. Hence, for the present\npurpose evaluation of the first term is sufficient. Considering only th e first term we obtain,\nδΓη\nqg=/parenleftBigη\ns/parenrightBigAfg4\n4T3τπ2pEp/integraldisplay\nk/parenleftbigg1\n3−cos2θkz/parenrightbiggkf0\nk\n2/integraldisplay0\ntmindt/integraldisplayωmax\nωmindω/radicalbig\nf(ω)/bracketleftbigg−˜u\n˜s+−˜s\n˜u/bracketrightbigg\n.(A.31)18\nEvaluation of the ωintegral gives,\nIω=/integraldisplayωmax\nωmindω1/radicalbig\nf(ω)= Re/integraldisplay∞\n−∞dω1/radicalbig\nf(ω)=π\na. (A.32)\nEq.(A.31) now becomes,\nδΓη\nqg=/parenleftBigη\ns/parenrightBigAfg4\n4T3τπEp/integraldisplay\nk/parenleftbigg1\n3−cos2θkz/parenrightbiggkf0\nk\n2/integraldisplay0\ntmindt1\ns−m2\nq/bracketleftbigg−˜u\n˜s+−˜s\n˜u/bracketrightbigg\n.(A.33)\nDominating logarithmic contribution from q-g scattering to the heav y quark damping rate\ncomes from the domain ˜ umin<<˜u <<˜umaxgiving rise to,\n/integraldisplay˜umax\n˜umind˜u\n˜s/bracketleftbigg−˜u\n˜s+−˜s\n˜u/bracketrightbigg\n= ln/vextendsingle/vextendsingle/vextendsingle/vextendsingles\nm2q/vextendsingle/vextendsingle/vextendsingle/vextendsingle+1\n2−m4\nq\n2s2. (A.34)\nWe now focus in the limit Ep>> m2\nq/T, which implies s=m2\nq+2PK∼ O(EpT)>> m2\nq.\nIn this limit we can consider only the logarithmic term ln/vextendsingle/vextendsingle/vextendsingles\nm2q/vextendsingle/vextendsingle/vextendsingle. The remaining kintegral\ncan be evaluated as follows,\n1\n8π2/integraldisplay∞\n0k3dkf0\nk/integraldisplay1\n−1dcosθk/parenleftbigg1\n3−cos2θkz/parenrightbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2Epk(1−cosθpk)+m2\nq\nm2q/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=1\n8π2/integraldisplay∞\n0k3dkf0\nk/parenleftbigg2\n9−m2\nq\n3kEpln/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+4Epk\nm2\nq/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n=1\n8π2/parenleftbigg2π4T4\n135−T3m2\nq\n3Ep/parenleftbigg\n2ζ(3)ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle4EpT\nm2q/vextendsingle/vextendsingle/vextendsingle/vextendsingle+3ζ(3)−2γζ(3)+2ζ′(3)/parenrightbigg/parenrightbigg\n.(A.35)\nThe final expression for the q-g scattering in the sand theuis now given by,\nδΓη\nqg=/parenleftBigη\ns/parenrightBigAfg4\n32T3τπ3Ep/parenleftBigg\n2π4T4\n135\n−T3m2\nq\n3Ep/parenleftbigg\n2ζ(3)ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle4EpT\nm2q/vextendsingle/vextendsingle/vextendsingle/vextendsingle+3ζ(3)−2γζ(3)+2ζ′(3)/parenrightbigg\n. 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Nembach,6, 7Jürgen Fassbender,1, 3\nand Helmut Schultheiss1, 3\n1)Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden,\nGermany\n2)Institut für Physik, Technische Universität Chemnitz, 09107 Chemnitz, Germany\n3)Fakultät Physik, Technische Universität Dresden, 01062 Dresden, Germany\n4)Walther-Meißner Institute, Bayerische Akademie der Wissenschaften, 85748 Garching,\nGermany\n5)Physik-Department, TU München, 80799 Munich, Germany\n6)Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado 80305,\nUSA\n7)JILA, University of Colorado, Boulder, Colorado 80309, USA\n(Dated: 15 July 2020)\nWe report on the impact of nonlinear four-magnon scattering on magnon transport in microstructured\nCo25Fe75waveguides with low magnetic damping. We determine the magnon propagation length with mi-\ncrofocused Brillouin light scattering over a broad range of excitation powers and detect a decrease of the\nattenuation length at high powers. This is consistent with the onset of nonlinear four-magnon scattering.\nHence, it is critical to stay in the linear regime, when deriving damping parameters from the magnon propaga-\ntion length. Otherwise, the intrinsic nonlinearity of magnetization dynamics may lead to a misinterpretation\nof magnon propagation lengths and, thus, to incorrect values of the magnetic damping of the system.\nIn the growing field of magnonics,1–4one aims at the\nuse of magnons, the excitation quanta in magnetically\nordered systems, to transport and process information.\nIn order to allow for coherent long-distance transport of\nsignals in complex magnonic networks, the search for ma-\nterials with low magnetic damping was reinitiated. In\n2016, ferromagnetic resonance (FMR) measurements of\ncontinuous films revealed intrinsic damping values as low\nas(5\u00061:8)\u000210\u00004fortheconductorCo 25Fe75,5approach-\ning values found for the ferrimagnetic insulator yttrium\niron garnet6, with the added benefit of seminconductor\ncompatibility. As a consequence, studies of the propaga-\ntion characteristics in Co 25Fe75microstructures followed,\ncomparing damping values obtained from FMR to those\nderived from magnon propagation lengths.7–9In Ref. 7, a\n2.5 times higher damping is reported for magnon trans-\nport measurements than for FMR analysis of extended\nfilms, which is attributed to significant extrinsic contri-\nbutions to the magnetic damping in the microstructured\nsample, such as local inhomogeneities and two-magnon\nscattering.\nHowever, onemustconsiderthateveninaperfectcrys-\ntal, higher-order nonlinear scattering can lead to a signif-\nicant increase of losses. The efficiency of such nonlinear\nprocesses depends on the population of the initial and\nthe final magnon states. Hence, if fewer final states are\navailable, e.g., by patterning the continuous film to a\nmicrostructured waveguide, a higher population of final\nstates is achieved more easily, and the threshold for non-\nlinearities is reduced strongly.10This is even more the\na)Electronic mail: k.schultheiss@hzdr.decase in materials with low intrinsic damping. As a di-\nrect consequence, the efficiency of magnon transport cer-\ntainly depends on the excitation amplitude, i.e., whether\nthe system is operated in the linear or nonlinear regime.\nOne may think that in order to obtain most effective\nmagnon transport, it is necessary to drive the excita-\ntion of magnons with the maximum power available. But\ncounterintuitively, this may lead to increased losses due\nto nonlinear magnon scattering.\nIn this Letter, we demonstrate that the application\nof high excitation powers above the threshold for non-\nlinear scattering may easily result in an underestima-\ntion of the magnon propagation lengths. Using spatially\nresolved Brillouin light scattering (BLS) microscopy of\nmagnon transport in a Co 25Fe75waveguide, we prove\nthat magnon propagation lengths are significantly larger\nif excited in the linear compared to the nonlinear regime.\nAs illustrated in Fig. 1a, we study a 5:25µmwide\nand 60µmlong magnon waveguide that was patterned\nfrom a Pt(3)/Cu(3)/Co 25Fe75(26)/Cu(3)/Ta(3) multi-\nlayer (all thicknesses in nanometers) using electron beam\nlithography, sputter deposition, and subsequent lift-off.\nIn a previous study, we reported an intrinsic damping\nfor this metallic thin film of \u000b0\u00143:18\u0002104in out-\nof-plane geometry.9In a second step, we patterned a\nCr(5)/Au(70) film to a coplanar waveguide (CPW) with\na2:1µmwide center conductor, separated by 960 nm\nfrom the 1:5µmwide ground planes. The magnon waveg-\nuide extends up to 38µmon one side of the CPW. Mi-\ncrowave (rf) currents running through the CPW allow\nfor the excitation of magnons with well-defined frequen-\ncies and high amplitudes, easily reaching the nonlinear\nregime.\nIn order to achieve magnon propagation in Damon-arXiv:2005.12113v2  [cond-mat.mtrl-sci]  14 Jul 20202\nEshbach configuration,11an external magnetic field of\n\u00160Hext= 46 mT was applied perpendicularly to the\nlong axis of the waveguide. At this field, the mag-\nnetic moments align parallel to the external field over\na4:5µmwide center region, whereas the demagneti-\nzation field causes canting of the magnetization at the\nedges. This is confirmed by micromagnetic simulations\nusing MuMax312(black solid line in Fig. 1b). Further-\nmore, laser scanning magneto-optic Kerr effect measure-\nments confirm that the magnetization in the center of\nthe magnon waveguide is saturated for \u00160Hext>20 mT\n(Fig. 1c). Hence, magnon propagation along the waveg-\nuide is governed by the characteristic dispersion with ~k?\n~Mas plotted by the black solid line in Fig. 1d, together\nwith the corresponding group velocities vg= 2\u0019\u000efsw\n\u000ek(red\ndashed line). The calculation follows the formalism of\nKalinikos and Slavin13assuming a quantization of the\nwavevector kyacross thestripe withan effective widthof\n4:1µmand an effective magnetic field of \u00160He\u000b= 40 mT ,\nboth obtained from micromagnetic simulations (Fig. 1b).\nFor our calculations and micromagnetic simulations, we\nassume material parameters as in Ref. 9: Aex= 26 pJ=m,\ng= 2:051, thickness t= 26 nm, andMs= 1700 kA=m.\nMsis reduced compared to 1870 kA=mreported in Ref. 9\nto account for heating due to the scanning BLS laser.14\nIn order to analyze the influence of nonlinear scat-\ntering on magnon propagation, we employed Brillouin\nlight scattering microscopy ( µBLS).15,16This technique\nrelies on the inelastic scattering of light and magnons\nfollowed by spectral analysis of the scattered light via\na high-resolution interferometer. The incident light, fo-\ncusedwithamicroscopeobjectiveontothesample, yields\na spatial resolution of about 350 nm. The advantage of\nthis technique compared to, e.g., time-resolved magneto-\noptical Kerr effect or ferromagnetic resonance, lies in its\ncapability to detect not only coherent but also incoher-\nent magnons. This means we do not only have access\nto magnons that are directly excited by microwave fields\nbut also to magnons created by nonlinear processes.\nSuch nonlinear interactions occur above a critical\nmagnon amplitude. In our experiments, we determine\nthis threshold in terms of a critical microwave power di-\nrectly supplied by a signal generator without using any\namplifiers. To this end, we apply a fixed microwave fre-\nquency offrf= 9:5 GHzto the CPW and gradually in-\ncreasethemicrowavepowerfrom \u00005 dBmto25 dBm. We\nmeasure the magnon intensity integrated in a 600 MHz\nwindow around the excitation frequency in 1µmdistance\nfrom the antenna (red triangles in Fig. 1e).\nIn the range between \u00005 dBmand 15 dBm, the re-\nsponse of the magnetic system linearly follows the in-\ncreasing excitation power (dashed black line). Above\n15 dBm, however, the measured intensities deviate from\nthe extrapolated linear response. In this regime, the in-\ntensity is lower than would be expected from the increas-\ning input power, indicating the onset of nonlinear scat-\ntering processes, which we will elucidate in more detail\nbelow.\n00.511.527891011121314\n024681012141618y=8,4 (0-20,101)dispersionv_g-2-1012Position y across width (µm)05001000150001020304050\n-50510152025Excitation power (dBm)BLS intensity (a.u.)linear response9.5 GHz-60-3003060-101\nfminMy (kA/m)\nMy /Ms µ0Hext (mT)\nkx (rad/µm)fsw (GHz)\nBeff, y (mT)CPWCo25Fe75(a)(b)(c)\n(d)(e)vg (km/s)Position y across width (µm)FIG. 1. (a) Schematic of the investigated microstructure.\nBlue depicts the 5:25µmwide Co 25Fe75waveguide, yellow\nshows the coplanar waveguide (CPW). (b) Micromagnetic\nsimulation of the y-component of the magnetization (black\nsolidline)andoftheeffectivemagneticfield Be\u000b;y(reddashed\nline), both extracted along the width of the waveguide for\n\u00160Hext= 46 mT . (c) Hysteresis loop of the waveguide in hard\naxis direction measured by laser scanning MOKE in the cen-\nterofthewaveguide. (d)Calculateddispersionrelation(black\nsolid line) and group velocity (red dashed line). (e) BLS in-\ntensity measured in a 600 MHz window around the excitation\nfrequencyfrf= 9:5 GHzas a function of the excitation power\nin1µmdistance to the CPW.\nNonlinear processes have extensively been studied\nin a variety of publications, both experimentally and\ntheoretically.10,17–22In general, several nonlinear multi-\nmagnon scattering processes, often referred to as Suhl\ninstabilities17, can cause a reduction of the population of\ndirectly excited magnons, all of which naturally obey en-\nergy and momentum conservation. Considering the dis-\npersion relation in Fig. 1d, the lowest order process of\nthree-magnon scattering (one magnon with frfsplits in\ntwo magnons with f1+f2=frf) can be excluded in our\nsystem since no states exist to satisfy this condition for\nfrf= 9:5 GHz.\nThe next higher order nonlinear process is given by\nfour-magnon scattering: two primary magnons, that are\nexcited atfrf, scatter and form two secondary magnons\nwithfrf\u0006\u000ef, whichiseasilypossibleinoursystemdueto\nthe quasi-linear magnon dispersion over a wide range of\nwave vectors. To demonstrate that the intensity reduc-\ntion in Fig. 1e is indeed in accordance with four-magnon\nscattering, we measure BLS spectra at three different mi-\ncrowave powers ( 15 dBm,20 dBm, and 25 dBm) and in\ntwo distances Lfrom the antenna ( 0:5µmand20µm).\nThe results are summarized in Fig. 2. Each column in3\n89101112\n891011 \n891011121314BLS frequency (GHz)\n891011 891011121314BLS frequency (GHz)\nmax\nminlog. BLS intensity (a.u.)2frf - fmin\nExcitation frequency frf (GHz)L = 0.5 µm15 dBm\nL = 20 µm15 dBm(a)(b)(c)\n(d)(e)(f)L = 0.5 µm20 dBm\nL = 20 µm20 dBmL = 0.5 µm25 dBm\nL = 20 µm25 dBmfmin\nFIG. 2. µBLS spectra measured in 0:5µm(a-c) and 20µm\n(d-f) distance to the antenna and at 15 dBm(a,d), 20 dBm\n(b,e), and 25 dBm(c,f) microwave power. At low powers, the\nlinear response of the magnetization dynamics to the external\ninput dominates (white dotted lines). Especially for the high-\nest power and close to the antenna, four-magnon scattering\nis apparent, which causes a significant broadening of the ex-\ncitation spectra. All BLS intensities are plotted on the same\nlogarithmic color scale.\nthe color maps shows a BLS spectrum that was recorded\nfor microwave frequencies ranging from 8 GHzto12 GHz.\nThemeasuredintensitiesarecolorcodedonthesamelog-\narithmic scale.\nClose to the antenna, at 15 dBm(Fig. 2a), the mea-\nsured magnon frequencies ( y-axis) linearly follow the ex-\ncitation frequency frf(x-axis), as indicated by the white\ndotted line. For increasing powers (Fig. 2b,c), however,\nwe see a significant broadening around frf. The limits\nof this broadening are indicated by black dashed lines\nand match the limits set by the bottom of the magnon\nmanifold (fmin) and energy conservation ( 2frf\u0000fmin).\nThis broadening is in agreement with previous studies\ndriven at high excitation powers close to the exciting\nantenna and is a well known evidence for four-magnon\nscattering.10,21\nNote that the intensities of nonlinearly excited\nmagnons above frf(white dashed line in Fig. 2b,c) are\nlower than in the range from fmintofrf. This can\nbe related to the decreasing detection sensitivity of the\nBLS microscope with increasing wave vector, i.e., with\nincreasing frequency15. Additionally, in first order ap-\nproximation, the lifetime of magnons scales inversely\nwith their frequency which reduces the overall popula-\n0510152068101214\n05101520\n05101520BLS frequency (GHz)11.6 dBmDistance x to antenna (µm)(a)17.3 dBm23.0 dBm(b)(c)\nmax\nminlog. BLS intensity (a.u.)FIG. 3. µBLS spectra measured along the waveguide for\ndifferent excitation powers at a fixed excitation frequency\nfrf= 9:5 GHz(white dotted line). The grey dashed lines\nmark the distribution of magnons that are parametrically ex-\ncited by four-magnon scattering.\ntion of states with higher frequencies23and thus leads to\na smaller signal.\nAs a next step, we study the influence of four-magnon\nscattering on the propagation characteristics in the sys-\ntem. At 20µmdistance to the antenna, the detected\nmagnon intensities are attributed only to propagation,\nwhich was confirmed by phase-resolved BLS measure-\nments in a previous investigation.9Nonetheless, Fig. 2f\nreveals that four-magnon scattering can still be evident\nat the highest excitation power, revealing the significant\nimpact of four-magnon scattering on magnon propaga-\ntion at high power levels.\nTofurtherelaborateonthisphenomenon,wefixtheex-\ncitation frequency at frf= 9:5 GHzand measure magnon\nintensities as a function of distance xto the antenna for\nthree different excitation powers (Fig. 3). At 11:6 dBm,\nwe detect only magnons that are directly excited at\n9:5 GHz(white dotted line) and continuosuly decay along\nthewaveguide. Withincreasingpower(Fig.3b,c), thede-\ntected magnon spectrum broadens again, which is most\npronounced close to the antenna and gets more and more\nnarrow with increasing distance (indicated by dashed\nlines in Fig. 3b,c). One can assume that this spatial\nprofile is defined by several contributions: First, pri-\nmary magnons excited at frfby the antenna propagate\nalong the waveguide with their amplitude decaying ac-\ncording to their damping. The same is true for secondary\nmagnons that are excited by nonlinear scattering above\nthe nonlinear threshold. In the same manner, they prop-\nagate along the waveguide and suffer from damping. Sec-\nond, four-magnon scattering causes energy flux from the\nprimary to the secondary magnons. This energy flux pro-\nvides an additional damping channel for the directly ex-\ncited magnons at frfbut also compensates losses of the\nsecondary magnons. The coupling is based on dipole-\ndipole interaction and inversely proportional to the wave\nvector mismatch of the contributing states19. Therefore,\nscattering into states iwith small \u000eki=jk(!rf)\u0000k(!i)j\nis expected to have lower threshold amplitudes, and to\nbe more efficient. In the case of high excitation powers,\nthis translates into the presence of these scattered states\nat larger distances from the antenna.\nFor a more detailed analysis of the nonlinear propa-4\n00.10.2Transmission812162024L    (µm)decay\n05101520Distance to antenna (µm)Integrated BLS intensity (a.u.)23 dBm20.1 dBm11.6 dBm0 dBm\n-1001020Excitation power (dBm)BLS int. (a.u.)magnons excited by 4-magnon scatteringrange for fit\n(a)(b)(c)Distance x to antenna (µm)\nLatt (µm)\nFIG. 4. a) Integrated BLS intensities of magnons directly\nexcited atfrf= 9:5 GHzas a function of the distance xto the\nantenna for various RF powers. Solid lines show the results of\nexponentialfitsintherangebetween x= 1µmandx= 12 µm.\nMeasurements at different powers are shifted vertically for\nclarity. b) Attenuation lengths (red triangles) that were de-\ntermined for different powers from the exponential fits in a.\nRelative transmission through the waveguide (black squares)\ngiven by the ratio of the integrated intensities measured in\n1µmand22:7µmdistance to the antenna. c) BLS intensi-\nties only of magnons excited by four-magnon scattering (in-\ntegrated in the ranges fBLS<9:2 GHzandfBLS>9:8 GHz)\nand integrated over all measurement positions.\ngation characteristics, we repeat the same spatial mea-\nsurements as in Fig. 3 in a broader range of excitation\npowers. We integrate the BLS intensities over the entire\nfrequency range of primary and secondary magnons and\nplot this as a function of the distance xto the antenna.\nFigure 4a shows the integrated intensities exemplarily for\n0, 11.6, 20.1, and 23 dBm. Thedataupto x= 12 µmsug-\ngest magnon propagation following a simple exponential\ndecay of the intensity I(x) = exp(\u00002x=L att) +cwith\nLattgiving the attenuation length. The constant offset c\nincludes the thermal magnon background as well as ther-\nmal noise of the used photon counter. The factor two in\nthe exponent accounts for the measurement of magnon\nintensities via BLS, which are proportional to the square\nof the magnon amplitudes. For fitting the data, we con-\nsider only the range 1µm\u0014x\u001412µmand plot the\nresults as solid lines in Fig. 4a.\nAt0 dBmand11:6 dBm, the fits agree well with the\nmeasured data for the entire range of distances from\nthe antenna. However, at 20:1 dBmand 23 dBm, the\nfits strongly deviate from the experimental results for\nx >12µmand the measured intensities exceed the pre-\ndictions of the exponential fits. This already indicates\nthat nonlinear processes have a crucial impact on the\npropagation of magnons when driven above the critical\nthreshold. Fitting data in a limited range close to the\nantenna at high excitation powers may lead to an under-\nestimation of the actual decay lengths.\nRed triangles in Fig. 4b summarize the attenuation\nlengthsLattthat we obtained from the exponential fits\nin the range 1µm\u0014x\u001412µmfor all excitation powers.\nFrom\u000010 dBmup to 10 dBm, the attenuation lengthsremain on a constant level around 19µm. For higher\nexcitation powers, a pronounced reduction of the atten-\nuation lengths occurs, reaching a minimum of 8:5µmat\n23 dBm. However, if we fit the BLS intensities for further\ndistances from the antenna, i.e.,x>12µm, all fits over\nthe entire power range yield propagation lengths in the\nrange of 20µm.\nAs a different means to describe the total signal losses\nover the propagation length of 22:7µm, we calculated a\ntransmission factor T=If=Ii, whereIi(If) is the mea-\nsured intensity at x= 1µm(x= 22:7µm), respectively\n(black squares in Fig. 4b). The power dependence of this\nparameter is consistent with the determined attenuation\nlengths. In the investigated range of microwave powers,\nthetransmissionisreducedbyalmostoneorderofmagni-\ntude, showing the significance of four-magnon scattering\non the effective losses.\nTo demonstrate the influence of four-magnon scatter-\ning more directly, we integrated only the contributions of\nthe secondary magnons, i.e., we integrated the BLS sig-\nnal in the ranges fBLS<9:2 GHzandfBLS>9:8 GHz.\nFigure 4c displays the intensity of those magnons, that\nare excited via four-magnon scattering, as a function of\nthe excitation power. At lower powers, this signal stays\nwithin the noise level of the measurement. Only above\n12 dBm, contributions from four-magnon scattering dras-\ntically increase, which is consistent with the decrease of\nthe measured attenuation lengths and transmission in\nFig. 4b and the measurement of the threshold for nonlin-\near four-magnon scattering in Fig. 1e.\nIn conclusion, we quantitatively studied the impact of\nnonlinear four-magnon scattering on long range magnon\ntransport in a Co 25Fe75waveguide. The increase of\npropagation losses coincides with the onset of nonlinear\nfour-magnon scattering. Measurements of the full\nmagnon spectrum along the waveguide showed the pres-\nence of nonlinearities for propagation lengths exceeding\n10µm. Our studies show that it is crucial to stay in the\nlinear regime in order to quantify propagation lengths in\nsystems with reduced dimensionality. If one is not aware\nthat the excitation powers already reach the nonlinear\nregime, one may underestimate the performance of their\nstructure, especially when comparing the results to fer-\nromagnetic resonance measurements on continuous films.\nFinancial support within DFG programme SCHU\n2922/1-1 is acknowledged. K.S. acknowledges funding\nwithin the Helmholtz Postdoc Programme. LL, LF\nand MW acknowledge funding by the DFG via projects\nWE5386/4-1 and WE5386/5-1. The samples were par-\ntially fabricated at the Nanofabrication Facilities (Nano-\nFaRo) at the Institute of Ion Beam Research and Materi-\nals Research at HZDR. We thank B. Scheumann for the\ndeposition of the Cr/Au film.5\nDATA AVAILABILITY\nThe data that support the findings of this study are\navailable from the corresponding author upon reasonable\nrequest.\nREFERENCES\n1S. Neusser and D. Grundler. Adv. Mater. 21, 2927-2932 (2009).\n2V. V. Kruglyak, S. O. Demokritov, and B. Hillebrands. J. 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Chris Hammel, and Fengyuan Yang \nDepartment of Physics, The Ohio State University, Columbus, OH 43210 \n \nAbstract \nY3Fe5O12 is arguably the best magnetic materi al for magnonic quantum information science \n(QIS) because of its extremely low damping. We  report ultralow damping at 2 K in epitaxial \nY3Fe5O12 thin films grown on a diamagnetic Y 3Sc2Ga3O12 substrate that contains no rare-earth \nelements. Using these ultralow damping YIG film s, we demonstrate for the first time strong \ncoupling between magnons in patterned YIG thin films and microwave photons in a \nsuperconducting Nb resonator. This result pave s the road towards scalable hybrid quantum \nsystems that integrate superconducting microw ave resonators, YIG film magnon conduits, and \nsuperconducting qubits into on-chip QIS devices.   2\nY3Fe5O12 (YIG) is a well-known ferrimagnetic in sulator with extremely low magnetic \ndamping, which makes it one of the best ma terials for fundamental studies and potential \napplications in magnonics, spintronics, and QIS.1-8 To date, mm-scale YIG single-crystal \nspheres have been the material of choice fo r coherently coupling magnons to superconducting \nqubits4 as well as for quantum sensing.4, 7, 9, 10 For QIS applications that require scalable on-\nchip integration, ultralow-dam ping magnetic films, which can be  patterned and integrated in \nhybrid quantum systems, are highl y desired. However, there is a major barrier for using YIG \nthin films in QIS because epitaxial  YIG films are mostly grown on Gd 3Ga5O12 (GGG) or other \ngarnet substrates containing ra re-earth elements, which induce excessive damping loss in YIG \nfilms across the interface at low temperatures.11 If this substrate-induced damping enhancement \ncan be eliminated while maintaining high YIG film  quality, it will enable scalable on-chip QIS \ndevices based on ultralow-d amping YIG films operati ng at mK temperatures. \nPreviously, we have shown that a diam agnetic epitaxial buffer layer of Y 3Sc2.5Al2.5O12 \ncan effectively separate the magnetic coupli ng between the YIG film and GGG substrate, \nresulting in much improved damping at low temperatures.12 Here, we demons trate the growth \nof YIG epitaxial films on a new diamagnetic Y 3Sc2Ga3O12 (YSGG) single-crystal substrate \ncontaining no rare-earth elemen ts, which exhibit extremely low damping that decreases rapidly \nwith temperature below 5 K. By integrating patterned YIG epitaxial films on YSGG with \nsuperconducting Nb coplanar waveguide (CPW) microwave resonators, we observe strong \ncoupling between microwave phot ons and magnons at 2 K. \nYIG thin films are epitaxially grown on YSGG (111) substrates using off-axis \nsputtering13 at a substrate temperature of 675 ℃. The bulk lattice constant of YSGG is a = \n12.466 Å, which causes a 0.72% tensil e strain in the YIG film (bulk a = 12.376 Å). The \ncrystalline quality of the YIG/ YSGG films is characterized by X-ray diffraction (XRD) and X-\nray reflectivity (XRR), as shown in Figs. 1a and 1b, respectivel y. The clear Laue oscillations 3\nof the YIG (444) peak indicate highly coherent crystalline orde ring. The XRR peaks reveal the \nfilm thickness (30 nm) and low interfacial and surface roughness.  \nBroadband ferromagnetic resonance (FMR) measurement is performed on a YIG(30 \nnm)/YSGG(111) film at va rious temperatures ( T) between 2 and 300 K in  a Physical Property \nMeasurement System (PPMS) from Quantu m Design, as described previously.12 We obtain \nFMR absorption spectra as a function of an in -plane magnetic field at various microwave \nfrequencies ( f) and temperatures. Figure 2a shows a re presentative derivative FMR spectrum \nat T = 2 K and f = 4.3 GHz. The resonance field 𝐻୰ and linewidth Δ𝐻 are extracted from fitting \nthe FMR spectrum using 𝑦ൌሾ𝐴െ𝐵ሺ𝐻െ𝐻୰ሻሿ ሾሺ𝐻െ𝐻୰ሻଶሺΔ𝐻/2ሻଶሿଶ⁄ , where 𝐴and 𝐵 are \nthe symmetric and antisymmetric amplitudes of th e lineshape, respectively. From Fig. 2a, we \nobtain 𝐻୰ = 710 Oe and Δ𝐻 = 13.2 Oe ( √3ൈ of peak-to-peak linewidth of 7.6 Oe). \nWe also measure the frequency-dependent FMR absorption at 2 K with a fixed in-plane \nfield of 710 Oe using a vector network analyzer  (VNA), as shown in Fig. 2b. The resonance \nfrequency 𝑓୰ and linewidth ∆𝑓 are extracted from fitting the spectrum with a Lorentzian \nfunction 𝑦ൌ𝐴ሾሺ𝑓െ𝑓୰ሻଶሺΔ𝑓/2ሻଶሿ ⁄ . The obtained 𝑓୰ = 4.307 GHz and ∆𝑓 = 36.3 MHz \nagree well with the field linewidth Δ𝐻 = 13.2 Oe as related by th e gyromagnetic ratio of free \nelectron, 𝛾2π⁄  2.8 MHz/G, where 36.3 MHz corresponds to 13.0 Oe. \nFigure 2c shows the extracted field linewidths of the YIG(30 nm)/YSGG film as a \nfunction of frequency at T = 2 to 300 K. As temperature decreases from 300 K, the linewidth \nand the slope of their frequency dependence in crease and peak at 20-30 K, after which both \nΔ𝐻 and the slope decrease quickly down to 2 K ( limited by our PPMS). This behavior agrees \nwith the slow-relaxation theory in YIG,14-16 where the existence of rare-earth impurities in YIG \ninduces a significant enhancement of relaxation at temperatures of 10’s K. At temperatures \nbelow this regime, the linewidth an d the slope decrease significantly.  \nAccording to the Landau-Lifs hitz-Gilbert (LLG) equation,17 the FMR linewidth is 4\nlinearly dependent on the microwave fre quency with the slope determined by the \nphenomenological Gilbert damping coefficient ( 𝛼): Δ𝐻 ൌ 4π𝛼𝑓 𝛾⁄Δ𝐻, where Δ𝐻 is the \ninhomogeneous broadening whic h is generally attributed  to magnetic nonuniformity18 and \nsurface defects.19 We note that the LLG equation is a simplified theory and does not explain \nthe nonlinearity observed in the frequency dependence of Δ𝐻, particularly at low temperatures, \nwhich requires additional contri butions such as the slow-relax ation mechanism. Nonetheless, \nthe phenomenological damping coefficient α from the LLG equation can serve as a figure of \nmerit for a magnetic material. \nWe fit the linewidth vs. frequency data in Fig. 2c using the LLG equation and extract \ndamping 𝛼and inhomogeneous broadening Δ𝐻 for each temperature, as shown in Fig. 2d. The \ndamping starts at a very low value of 𝛼 = 3.3  10-4 at room temperature and increases as \ntemperature drops. After reaching a peak of 𝛼 = 1.3  10-3 at 45 K, the damping decreases \nquickly at lower temperatures. Remarkably, the FMR linewidths ar e essentially frequency \nindependent at 5 and 2 K, indi cating an essentially zero dampin g according to the LLG equation. \nThe rapid decrease of linewidth below 10 K implies that Δ𝐻 will likely decrease even further \nat sub-Kelvin temperatures.  \nThe extremely low damping of the YI G epitaxial films on diamagnetic YSGG \nsubstrates at very low temperatures is highly pr omising for QIS studies. As an initial step in \nthis regard, we integrate the YIG/YSGG film s with superconducting resonators for the study \nof coupling between magnons in a YIG film and microwave photons emitted by a \nsuperconducting CPW resonator. First, we pattern a YIG(30 nm)/YSGG film into 10- μm wide \nstrips using photolithography a nd Ar ion milling. Then, 300-nm thick Nb resonators are \ndeposited by sputtering on the YS GG substrate with photolithogra phy patterns followed by lift-\noff such that the YIG strips lie within the gaps of the CPW resonators.  \nThe microwave resonator design includes a main bus channel which is capacitively 5\ncoupled to two resonators, as shown in Fig. 3a . The two resonators are 13- and 13.5-mm long \nwith both ends open-circuit, so they resona te with half-wavelengt h standing waves, as \ndetermined by 𝑓ൌ𝑐2𝑙ඥ𝜖ୣ ⁄ , where 𝑐 is the speed of light, 𝑙 is the length of the resonator, \nand 𝜖ୣ is the average dielectric constant of vacuum and YSGG substrate. The center conductor \nof the CPW resonators is 20- μm wide with a spacing gap of 15- μm from the group conductor \non either side. A YIG strip is located within a gap of each resonator near the middle of the \nresonator length, as shown in the insets of Fig.  3a, which enables the ma gnetic field component \nof the resonating microwave to drive the FMR of the YIG strip. We fabricate two such devices \nwith a total of four Nb resonators coupled to four YIG strips with 10- μm width and lengths of \n300, 600, 900, and 1200 μm. In addition, we fabricate an other such device with two Nb \nresonators on YSGG without YI G strips as a reference sample . The devices are mounted on a \nhome-made sample holder and wire -bonded as shown in the insets of Fig. 3a, which is loaded \ninto the PPMS and cooled down to 2 K.  \nWe measure microwave transmission spectr a of these devices using a VNA with an \noutput power of -40 dBm, where each spectrum is averaged over 40 scans. Figure 3b shows \nthe transmission spectrum (S21) of  a device with two Nb resonato rs and no YIG, which exhibits \ntwo sharp dips at f = 4.203 and 4.364 GHz, corresponding to the resonances of the 13.5- and \n13-mm resonators, with high quality factors (Q) of  56,800 and 49,000, respectively. It is noted \nthat the measured resonance frequency ratio of (4.364 GHz)/(4.203 GHz) = 1.0383 is almost \nidentical to the length ratio of (13.5 mm)/( 13 mm) = 1.0385, validating the resonator design \nand fabrication. \nFigure 3c compares the transmission spectra of  a bare Nb resonator (no YIG) and a Nb \nresonator coupled to a 10  1200 μm2 YIG strip at zero field or in the presence of a magnetic \nfield of 550 Oe, while Fig. 3d shows similar sp ectra for a Nb resonator coupled to a 10  600 \nμm2 YIG strip. All Nb resonators with or wit hout coupling to YIG strips exhibit high quality 6\nfactors between 44,000 and 72,700 at zero magnetic field. To evaluate the performance of the \nNb resonators in a magnetic field needed for strong microwave phot on-magnon coupling, we \napply an in-plane field of 550 Oe, wh ich lowers Q by a factor of ~10 , although the quality \nfactor remains quite high (>4,000). This behavi or together with shift of resonance frequency \ndue to the applied field and coupling to YIG are commonly seen in other microwave \nphotonmagnon coupling reports.20, 21 \nSuch resonator-ferromagnet hybr id systems can be modeled as a macrospin coupled to \nan LC resonator by the radio-frequenc y (rf) magnetic field component brf.20 The \neigenfrequencies of this syst em can be calculated as, \n𝜔േൌ൫𝜔𝜔ሺ𝐻ሻ൯2⁄േඥሺ𝜔ሺ𝐻ሻെ𝜔ሻଶ4𝑔ଶ2⁄,   (1) \nwhere 𝜔୰ is the resonance frequency of a standalone microwave resonator, 𝜔ሺ𝐻ሻ is the \nstandalone ferromagnet’s  FMR frequency (depende nt on the applied field 𝐻), and 𝑔 is the total \nphoton-magnon coupling strength. Th e total coupling strength scales with the number of spins \n𝑁 in the ferromagnet as 𝑔ൌ𝑔௦√𝑁, where 𝑔௦ is the coupling strength between microwave \nphotons and individual spins in the ferromagne t, which depends on the device design and \nmicrowave magnetic field strength brf at the ferromagnet. Note that the number of spins here \nis a net value due to YIG’s ferrimagnetism.  \nWe perform field-dependent transmission measurement on the 13- and 13.5-mm Nb \nresonators to quantify their coupling strength to the 10  300 μm2, 10  600 μm2, 10  900 μm2, \nand 10  1200 μm2 YIG strips. Figures 4a-4d show the microwave transmission of the four \nresonator-YIG hybrid devi ces as a function of field and frequency measured at 2 K. The \nanticrossing features in the plots are the signatures of mi crowave photon-magnon coupling, \nwhere YIG’s FMR frequency 𝜔ሺ𝐻ሻ meets the microwave resonator’s resonance frequency \n𝜔 and the degeneracy is lifted. By comb ining Eq. (1) and the Kittel equation, 𝜔୫ሺHሻൌ7\n𝛾ඥ𝐻ሺ𝐻4π𝑀ୣሻ, where 𝑀ୣ is the effective saturation magnetization of the YIG film, we \nobtain the angular frequencies of the hybrid systems’ modes, \n𝜔േൌ൫𝜔୰𝛾ඥ𝐻ሺ𝐻4π𝑀ୣሻ൯2⁄േට൫𝛾ඥ𝐻ሺ𝐻4π𝑀ୣሻെ𝜔୰൯ଶ\n4𝑔ଶ2ൗ (2) \nWe point out that different aspect ratios of  the YIG strips can give rise to different \ndemagnetizing factors, which slightly shift 𝜔୫ሺ𝐻ሻ. However, this shift is smaller than 1 Oe \nover the field and frequency range of interest and thus is ignored here. We  fit the anticrossing \nfeatures with Eq. (2) and extract the coupling strength 𝑔 which is dependent on the magnetic \nvolume and the number of spins 𝑁. For the fitting, a background linearly dependent on the field \nis added to Eq. (2) to account for the fiel d-induced microwave resonator frequency shift. \nTo determine the number of spins in a YIG st rip, we measure magnet ic hysteresis loops \nfor the YIG(30 nm)/YSGG film using a superc onducting quantum interference device (SQUID) \nmagnetometer at 2 and 300 K, as shown in Fig.  4e. The saturation magne tization is determined \nto be 𝑀ௌ = 180 emu/cm3 at 2 K and 131 emu/cm3 at 300 K. The hysteresis loops remain largely \nsquare with a small coercivity of 1.1 and 3.1 Oe at 300 and 2 K, respectively, corroborating the \nhigh magnetic uniformity at both ro om and cryogenic temperatures. \nFigure 4f shows the linear de pendence of coupling strength 𝑔2𝜋⁄ on the square root of \nboth the YIG volume and the number of sp ins. From Fig. 2b, the decay rate 𝜅୫2π⁄ of YIG’s \nresonance mode near 4.3 GHz at 2 K and H = 710 Oe is determined to be 36.3 MHz. Next, the \ndecay rate 𝜅୰2π⁄of the Nb resonator is estimated to be 1.037 MHz based on the transmission \nlinewidth of the resonator coupled to the 10  600 μm2 YIG strip at H = 550 Oe as shown in \nFig. 3d. Thus, the cooperativity for the 10  600 μm2 YIG is calculated to be 𝑔ଶሺ𝜅𝜅ሻ ⁄   \n57.9, showing that the coupling strength is much  stronger than the decay rates of individual \ncomponents, which is required for coherent coupling . Using 𝑀ௌ = 180 emu/cm3 at 2 K for the \nYIG/YSGG film, we determine the average couplin g strength to an individual spin to be 𝑔௦/2𝜋 8\n= 25.1 Hz. This coupling strength 𝑔௦/2𝜋is comparable to other reports on microwave photon-\nmagnon coupling using ferromagnetic metals20-22 and similar CPW re sonator-ferromagnet \nstructures. Considering that the microwave magne tic field in the gap of a CPW (where our YIG \nstrip is located) is weaker than that for a ferromagnet directly on top (or underneath) the \nsuperconducting center conductor channel, on e can achieve stronger microwave photon-\nmagnon coupling by positioning the YIG strip near  a stronger microwave magnetic field or \nutilizing resonator designs such as lumped-element LC resonator, which can provide far \nstronger coupling strength.20  \nThis work provides the first demonstration that ultralow-damping YIG epitaxial films \non YSGG can be integrated with superconducto r resonators to achieve strong microwave \nphoton-magnon coupling at few Kelvin temperatur es. Such ultralow-damping YIG films offer \nclear advantages (in terms of decay rate) over metallic ferromagnets for on-chip hybrid \nquantum systems that incorporate magnonic c onduits, microwave superconductor resonators, \nand superconductor qubits for QIS applica tions that operate in the mK regime. \nIn summary, we demonstrate the growth of  high-quality epitaxial YIG thin films on \ndiamagnetic YSGG substrate, which exhibi t narrow FMR linewidt h and extremely low \ndamping at 2 K. We couple these YIG films to superconducting Nb resonators to create hybrid \nstructures that achieve strong microwave photon-magnon coupling. This  work demonstrates \nthe potential power of ultralow-damping YIG films for scalable, integrated QIS applications at \nlow temperatures.  \nThis work was primarily supported by the Center for Emergent Materials: an NSF \nMRSEC under award number DMR-2011876. D.R.  acknowledges partial support from the \nNational Science Foundation under award numbe r DMR-2225646 (YIG film growth and X-\nray characterizations).  9\nFigure Captions: \nFigure 1 . X-ray diffraction results of YIG films.  (a) 2𝜃/𝜔XRD scan and (b) XRR scan of \na YIG(30 nm) epitaxial film grown on YSGG (111)  substrate, indicati ng the high crystalline \nquality of the YIG film. \nFigure 2 . FMR measurements of a YIG(30 nm)/YSGG film.  (a) Field-dependent derivative \nFMR spectrum at 2 K driven by a microwave frequency of 4.3 GHz. (b) Frequency-dependent \nFMR spectrum at 2 K with an in-plane fiel d of 710 Oe. (c) Frequency dependence of FMR \nlinewidth at different temperatures, from wh ich the damping constant and inhomogeneous \nbroadening are obtained by linear fitting. (d ) Temperature dependence of damping (red) and \ninhomogeneous broadening ∆𝐻 (blue). The error bars are from the linear fitting in (c). \nFigure 3 . Microwave transmission of Nb CPW resona tors with or without YIG strips and \nmagnetic field at 2 K.  (a) Schematic of Nb resonator devi ce with YIG strips placed within \ntheir gaps. The size of the whole device is 3.5  4.4 mm2. The lengths of the two Nb resonators \nare 13 mm and 13.5 mm. Insets: optical microscope  images of selected areas with the same \nmagnification. The YIG strips shown (color contrast augmented) are 10   900 μm2 (top) and \n10  300 μm2 (bottom). (b) Microwave transmission (S21) spectrum of two Nb resonators \nwithout a YIG strip in their gaps. The two sharp dips (resonances) at 4.364 and 4.203 GHz \ncorrespond to the resonance frequencies of th e 13 mm and 13.5 mm resonators, respectively. \n(c) Microwave transmission spectra of the 13.5 mm resonators without YIG at zero field (blue), \nwith a 10  1200 μm2 YIG strip at zero field (orange), and with a 10  1200 μm2 YIG strip in \nthe presence of a 550 Oe field (green). (d ) Microwave transmission spectra of the 13 mm \nresonators without YIG at zero field (blue), with a 10  600 μm2 YIG strip at zero field (orange), \nand with a 10  600 μm2 YIG strip at 550 Oe (green). 10\nFigure 4. Microwave photon-magnon coup ling in Nb resonator-YIG(30 nm) hybrid \nstructures on YSGG.  Microwave transmission in Nb resona tors coupled to YIG strips of (a) \n10  300 μm2, (b) 10  600 μm2, (c) 10  900 μm2, and (d) 10  1200 μm2, as a function of \nmicrowave frequency and in-plane  magnetic field at 2 K. (e) Ma gnetic hysteresis loops of a \nYIG(30 nm)/YSGG film at 2 and 300 K. (f ) Coupling strength between microwave photons \nand magnons in the YIG films as a function of th e square root of the magnetic volume as well \nas the number of spins in YIG.   11\nReferences: \n1. A. A. Serga, A. V. Chumak and B. Hillebrands, \"YIG magnonics,\" J. Phys. D-Appl. \nPhys.  43, 264002 (2010). \n2. A. V. Chumak, V. I. Vasyuchka, A.  A. Serga and B. 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Rev. B  95, 174411 (2017). \n12. S. D. Guo, B. McCullian, P. C. Hamm el and F. Y. Yang, \"L ow damping at few-K \ntemperatures in Y 3Fe5O12 epitaxial films isolated from Gd 3Ga5O12 substrate using a \ndiamagnetic Y 3Sc2.5Al2.5O12 spacer,\" J. Magn. Magn. Mater.  562, 169795 (2022). \n13. F. Y. Yang and P. C. Hammel, \"Topica l review: FMR-Driven Spin Pumping in \nY3Fe5O12-Based Structures,\" J. Phys. D: Appl. Phys.  51, 253001 (2018). \n14. H. Maier-Flaig, S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. Huebl and \nS. T. B. Goennenwein, \"Temperature-dep endent magnetic damping of yttrium iron \ngarnet spheres,\" Phys. Rev. B  95, 214423 (2017). \n15. P. E. Seiden, \"Ferrimagnetic Resonanc e Relaxation in Rare-Earth Iron Garnets,\" Phys. \nRev. 133, A728-A736 (1964). \n16. E. G. Spencer, R. C. LeCraw and A. M. Clogston, \"Low-Temperature Line-Width \nMaximum in Yttrium Iron Garnet,\" Phys. Rev. Lett.  3, 32-33 (1959). \n17. S. V. Vonsovskii, Ferromagnetic Resonance: The Phen omenon of Resonant Absorption \nof a High-Frequency Magnetic Fi eld in Ferromagnetic Substances . (Pergamon, 1966). \n18. E. Schlömann, \"Inhomogeneous Broade ning of Ferromagnetic Resonance Lines,\" Phys. \nRev. 182, 632-645 (1969). \n19. S. Klingler, H. Maier-Flaig, C. Dubs, O. Surzhenko, R. Gross, H. Huebl, S. T. B. \nGoennenwein and M. Weiler, \"Gilbert damp ing of magnetostatic modes in a yttrium \niron garnet sphere,\" Appl. Phys. Lett.  110, 092409 (2017). 12\n20. J. T. Hou and L. Q. Liu, \"Strong Coupling between Microwave Photons and \nNanomagnet Magnons,\" Phys. Rev. Lett.  123, 107702 (2019). \n21. Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Z. Zhang, J. J. Ding, T. Khaire, \nH. Saglam, R. Divan, J. Pearson, W.-K. Kwok, Z. L. Xiao, V. Novosad, A. Hoffmann \nand W. Zhang, \"Strong Coupling between Magnons and Microwave Photons in On-\nChip Ferromagnet-Superconduc tor Thin-Film Devices,\" Phys. Rev. Lett.  123, 107701 \n(2019). \n22. I. W. Haygood, M. R. Pufall, E. R. J. Edwards, J. M. Shaw and W. H. Rippard, \"Strong \nCoupling of an FeCo All oy with Ultralow Damping to Superconducting Co-planar \nWaveguide Resonators,\" Phys. Rev. Appl.  15, 054021 (2021). \n \n  13\n \n \nFigure 1. \n  101103105\n1234\n2 (deg)XRR Intensity (c/s)YIG(30 nm)/YSGG(b)100102104106\n47 49 51 53\n2 (deg)XRD Intensity (c/s)(a) YIG(30 nm)/YSGG14\n \n \nFigure 2.   680 700 720 740FMR Int. (arb. unit)H (Oe)(a)\nT = 2 K\nf = 4.3 GHzH = 13.2 Oe\n4.2 4.3 4.4VNA Int. (arb. unit)f (GHz)T = 2 K\nH = 710 Oef = 36.3 MHz (b)\n0102030\n0 5 10 15 20FMR Linewidth (Oe)\nf (GHz)T = 300 K150 K45 K 75 K30 K\n20 K\n10 K\n5 K\n2 K(c)\n00.00050.001\n01020\n0 100 200 300Damping H0 (Oe)\nT (K)YIG(30 nm)/YSGG\n(d)15\n \n \nFigure 3.  \n-40-30-20-10\n4.2 4.21 4.22 4.23S21 (dB)\nf (GHz)Resonator 1\nH = 0\nNo YIG\nQ = 56,800H = 550 Oe\n10x1200 m YIG\nQ = 5,054\nH = 0 Oe\n10x1200 m YIG\nQ = 44,900(c)\n-40-30-20-10\n4.36 4.37 4.38 4.39S21 (dB)\nf (GHz)Resonator 2\nH = 0\nNo YIG\nQ = 49,000H = 550 Oe\n10x600 m YIG\nQ = 4,230\nH = 0\n10x600 m YIG\nQ = 72,700(d)-30-20-10\n4.2 4.25 4.3 4.35S21 (dB)\nf (GHz)(b)\nResonator 1\n(13.5 mm)\nResonator 2\n(13 mm)\nfRes1 = 4.203 GHz fRes2 = 4.364 GHzT = 2 K16\n \n \n \nFigure 4. \n020406080\n0 5 10 15 20Coupling g/2 (MHz)\nYIG Volume1/2 (m3/2)Number of spins (1012)\n0 14 9\n(f)\n-200-1000100200\n-20 -10 0 10 20Ms (emu/cm3)\nH (Oe)T = 2 K300 KYIG\nYSGG\n(e)"    },    {        "title": "2401.16697v1.The_Velocity_Space_Signature_of_Transit_Time_Damping.pdf",        "content": "Under consideration for publication in J. Plasma Phys. 1\nThe Velocity-Space Signature of\nTransit-Time Damping\nRui Huang †1, Gregory G. Howes1, and Andrew J. McCubbin2\n1Department of Physics and Astronomy, University of Iowa, IA 52242, USA\n2Applied Physics Laboratory, Johns Hopkins University, MD 20723, USA\n(Received ?; revised ?; accepted ?. - To be entered by editorial office)\nTransit-time damping (TTD) is a process in which the magnetic mirror force—induced\nby the parallel gradient of magnetic field strength—interacts with resonant plasma par-\nticles, leading to the collisionless damping of electromagnetic waves and the resulting\nenergization of those particles through the perpendicular component of the electric field,\nE⊥. In this study, we utilize the recently developed field-particle correlation technique\nto analyze gyrokinetic simulation data. This method enables the identification of the\nvelocity-space structure of the TTD energy transfer rate between waves and particles\nduring the damping of plasma turbulence. Our analysis reveals a unique bipolar pattern\nof energy transfer in velocity space characteristic of TTD. By identifying this pattern,\nwe provide clear evidence of TTD’s significant role in the damping of strong plasma\nturbulence. Additionally, we compare the TTD signature with that of Landau damping\n(LD). Although they both produce a bipolar pattern of phase-space energy density loss\nand gain about the parallel resonant velocity of the Alfv´ enic waves, they are mediated\nby different forces and exhibit different behaviors as v⊥→0. We also explore how the\ndominant damping mechanism varies with ion plasma beta βi, showing that TTD domi-\nnates over LD for βi>1. This work deepens our understanding of the role of TTD in the\ndamping of weakly collisional plasma turbulence and paves the way to seek the signature\nof TTD using in situ spacecraft observations of turbulence in space plasmas.\nPACS codes:\n1. Introduction\nA key area of research in the study of turbulence in weakly collisional plasmas is un-\nderstanding how energy from the fluctuating plasma flows and electromagnetic fields\nis converted into plasma particle energy. This phenomenon is especially relevant in he-\nliospheric plasmas like the solar wind, where the characteristic low density and high\ntemperature lead to weakly collisional plasma dynamics. The dissipation of turbulence\nin such space and astrophysical plasmas is likely mediated by three categories of mech-\nanisms: (i) resonant wave-particle interactions, such as Landau damping (Landau 1946;\nChen et al. 2019), transit-time damping (Stix 1992; Barnes 1966), and cyclotron damp-\ning (Isenberg & Hollweg 1983; Isenberg & Vasquez 2019); (ii) non-resonant wave-particle\ninteractions, including stochastic heating (Chandran et al. 2010, 2013; Martinovi´ c et al.\n2020; Cerri et al. 2021), magnetic pumping (Lichko & Egedal 2020; Montag & Howes\n2022), and “viscous” damping mediated by kinetic temperature anisotropy instabilities\n†Email address for correspondence: rui-huang@uiowa.eduarXiv:2401.16697v1  [physics.plasm-ph]  30 Jan 20242\n(Arzamasskiy et al. 2023); and (iii) dissipation within coherent structures, in particular\ncollisionless magnetic reconnection that may occur in current sheets that are found to\narise naturally in plasma turbulence (Osman et al. 2011; Zhdankin et al. 2015; Mallet\net al. 2017; Loureiro & Boldyrev 2017).\nGiven the low collisionality of these plasma environments, the six-dimensional (3D-3V)\nkinetic plasma theory is essential for analyzing the evolution of the turbulence and its\ndissipation through collisionless interactions between electromagnetic fields and plasma\nparticles (Howes 2017). Although in situ spacecraft measurements in the solar wind\nprovide invaluable data, they are often limited to a single point, or a few points, in\nspace, which presents a significant challenge for the investigation of the physical mech-\nanisms that remove energy from the turbulent fluctuations and consequently energize\nthe plasma particles. The recently developed field–particle correlation technique (Klein\n& Howes 2016; Howes et al. 2017; Klein et al. 2017) enables direct measurements of the\nelectromagnetic fields and particle velocity distributions at a single point in space to be\ncombined to create a velocity-space signature of particle energization that can be used to\nidentify the physical mechanisms responsible for damping the turbulence and to estimate\nthe resulting rate of the change of particle energy density. Consequently, this technique\nprovides an innovative means to utilize in situ spacecraft observations to identify specific\ncollisionless damping mechanisms and determine particle heating rates.\nThis technique has shown success in identifying several damping mechanisms in weakly\ncollisional turbulent plasmas, such as ion Landau damping (Klein et al. 2017; Li et al.\n2019), ion cyclotron damping (Klein et al. 2020; Afshari et al. 2023), electron Landau\ndamping (Chen et al. 2019; Li et al. 2019; Afshari et al. 2021; Conley et al. 2023), and\nmagnetic pumping (Montag & Howes 2022). However, the role of transit-time damping,\na resonant wave-particle interaction, in the damping of plasma turbulence remains un-\nconfirmed. The focus of this paper is to employ the field-particle correlation technique\nto identify the velocity-space signature of ion energization through transit-time damping\nand to recover this signature from simulations of strong plasma turbulence.\nThe structure of this paper is laid out as follows. We derive the specific form of the\nfield-particle correlation for transit-time damping in §2.1. This is followed by an explo-\nration of the expected transit-time damping signature in §2.2. In §3, we conduct single\nkinetic Alfv´ en wave simulations to investigate the velocity-space signature characteris-\ntic of transit-time damping. Subsequently, in §4, we delve into turbulence simulations,\npresenting details for distinguishing transit-time damping from turbulence damping pro-\ncess. §5 summarizes our findings and outlines potential future applications for further\nresearch.\n2. Transit-Time Damping\nThe idea of transit-time damping (TTD) had its origins in mid-20th-century plasma\nphysics when transit-time magnetic pumping was proposed as a means to heat confined\nplasma (Spitzer Jr & Witten 1951). This method is characterized by a modulation of\nthe magnetic field magnitude at a frequency considerably lower than the ion cyclotron\nfrequency; the double adiabatic evolution of the parallel and perpendicular temperatures,\ncombined with a weak collisionality, leads to a net transfer of energy to the plasma\nparticles. The term “transit-time” refers to the duration necessary for an ion to traverse\nfrom one side to the other across the confined region.\nThe magnetic mirror force plays a key role in the dynamics of TTD. In a static mag-\nnetic field with a spatial variation of the magnetic field magnitude along the direction\nparallel to the field, the mirror force accelerates charged particles in the direction of de-Transit-Time Damping 3\nzxoB∇BFFzFFryFrFz\nFigure 1. Diagram of the radial component Frand axial component Fzof the Lorentz force\nof the magnetic field (red) on a positively charged particle (red +) in a converging magnetic\nfield (green) with increasing magnitude in the + zdirection. Averaged over the Larmor orbit of\nthe particle (blue), the net magnetic mirror force is in the direction of decreasing magnetic field\nmagnitude, here the −zdirection.\ncreasing field magnitude. In a cylindrical coordinate system aligned with the magnetic\nfield direction, the condition ∇·B= 0 implies that an increase of the magnetic field along\nthe axial direction must be accompanied by the convergence of the field in the radial di-\nrection, as shown in Figure 1. For a particle with a guiding center on the axis, the particle\nwill experience an inward radial field throughout its Larmor orbit. The Lorentz force,\nwhich acts perpendicularly to the magnetic field direction at the particle position, will\ntherefore have both a large radial and a small axial component. Averaged over the full\nLarmor orbit, the net nonzero axial component accelerates the particle in the direction\nof the decreasing magnetic field magnitude. Because the magnetic field can do no work,\nthe total energy of the particle remains constant—the change in the parallel velocity is\naccompanied by a small change in the perpendicular velocity governed by the average of\nthe radial component of the Lorentz force. The net effect of the magnetic mirror force is\nthat, as a particle moves in the direction of the increasing magnetic field, the mirror force\nreduces the velocity v∥parallel to the mean magnetic field over the Larmor orbit and\nincreases perpendicular velocity v⊥to maintain a constant total velocity v= (v2\n⊥+v2\n∥)1/2.\nAlthough the mirror force in a static magnetic field cannot change the energy of par-\nticles, any changes of the magnetic field in time will induce an electric field according\nto Faraday’s Law. Work done by that induced electric field can do work on the parti-\ncles, providing the key element underlying the physics of TTD. In a collisionless plasma,\ncollisionless wave-particle interactions are governed by the resonance condition, given\nbyω−k∥v∥=nΩs, where nΩsforn= 0,±1,±2, . . .incorporates the cyclotron har-\nmonics of the particle motion in a magnetic field (Melrose 1980). The n= 0 resonance,\nknown as the Landau resonance, describes resonant interactions with particles that have\nparallel velocities near the phase velocity of the wave, v∥≃ω/k∥, enabling energy ex-\nchange between the particles and the wave through two mechanisms: (i) the electrostatic\nforce due to parallel component of the electric field governs the phenomenon of Landau\ndamping (LD) (Landau 1946; Villani 2014); (ii) the magnetic mirror force governs the4\nphenomenon of TTD (Stix 1992), also known as Barnes damping (Barnes 1966). In the\ncase of TTD, the perpendicular component of the electric field, induced by the change\nin the magnetic field magnitude along the parallel direction, accelerates the particle by\nchanging the perpendicular velocity v⊥; the mirror force effectively converts this perpen-\ndicular velocity into parallel velocity, leading to a net acceleration of the particle along\nthe axial direction parallel to the magnetic field (Howes et al. 2024). For a Maxwellian\ndistribution of particles, there are more particles with parallel velocities v∥< ω/k ∥than\nparticles with v∥> ω/k ∥, so the net effect on the distribution is an increase of the particle\nenergy, leading to damping of the wave. A detailed demonstration of this phenomenon\nfor a model moving magnetic mirror field is presented in §2.2.\n2.1.Field-Particle Correlation for Transit-Time Damping\nTo determine the appropriate form of the field-particle correlation to diagnose TTD via\nthe magnetic mirror force, we start with the Vlasov equation for a species sbeing acted\nupon by a general force Fs,\n∂fs\n∂t+v· ∇fs+Fs\nms·∂fs\n∂v= 0. (2.1)\nMultiplying the Vlasov equation by msv2/2, we obtain an expression for the rate of\nchange of the phase-space energy density ,ws(r,v, t)≡msv2fs(r,v, t)/2,\n∂ws(r,v, t)\n∂t=−v· ∇ws−v2\n2Fs·∂fs\n∂v. (2.2)\nPrevious analysis of this equation (Klein & Howes 2016; Howes et al. 2017) has shown\nthat, if integrated over space (with appropriate infinite or periodic boundary conditions),\nthe change in the total kinetic energy of the particles Ws(t) =R\nd3rR\nd3vws(r,v, t) is\ndue to work done on the particle species by the force Fs. Therefore, the field-particle\ncorrelation due to a general force Fsat spatial position r0is defined as a time-average\nover a correlation interval τof the last term on the right hand side of (2.2),\nCFs(r0,v, t;τ)≡1\nτZt+τ/2\nt−τ/2−v2\n2Fs(r0,v, t′)·∂fs(r0,v, t′)\n∂vdt′. (2.3)\nNote here that the correlation interval τis a parameter of the field-particle correlation\nanalysis, and so is included as a secondary argument, separated by a semicolon from\nthe primary arguments that define the dimensions of the 3D-3V phase space in position,\nvelocity, and time.\nIf we consider the force due to an electric field Fs=qsE, we obtain the established\nfield-particle correlation due to the electric field (Klein & Howes 2016; Howes et al. 2017;\nKlein et al. 2017),\nCE,s(r0,v, t;τ) =1\nτZt+τ/2\nt−τ/2−qsv2\n2E(r0, t′)·∂fs(r0,v, t′)\n∂vdt′. (2.4)\nThe collisionless transfer of energy between electromagnetic waves and particles in\nTTD is mediated by the magnetic mirror force, Fs=−µsˆb· ∇B, where the magnetic\nmoment for a particle of species sis given by µs=msv2\n⊥/(2B), the unit vector in the\ndirection of the magnetic field is given by ˆb≡B/B, and the magnitude of the magnetic\nfield is B=|B|. Substituting the magnetic mirror force into (2.3), we obtain\nCB,s(r0,v, t;τ) =1\nτZt+τ/2\nt−τ/2msv2v2\n⊥\n4B(ˆb· ∇)B(r0, t′)·∂fs(r0,v, t′)\n∂vdt′. (2.5)Transit-Time Damping 5\nA few modifications of the form of the field-particle correlation for TTD given in (2.5)\nare helpful for its application to the gyrokinetic simulations presented here. First, we\nexploit two important characteristics of TTD and turbulence: (i) TTD is most effective\nfor electromagnetic waves with wavelengths at the ion scales, kρi∼1; and (ii) for most\nturbulent space and astrophysical plasmas of interest, the amplitude of the magnetic\nfluctuations δBat ion scales kρi∼1 is much smaller than the magnitude of the mean\nmagnetic field B0. Therefore, if we separate the magnetic field into its mean plus the\nfluctuations, B=B0+δB, where |δB| ≪ |B0|, the change in the magnetic field magni-\ntude δ|B|(which is the key ingredient for the magnetic mirror force) can be expressed\nasδB∥by recognizing\nδ|B|=|B| − |B0|=p\n(B0+δB)2−B0=q\nB2\n0+ 2δB·B0+|δB|2−B0≃δB∥(2.6)\nwhere we use a binomial expansion to eliminate the square root, neglect the small |δB|2\nterm, and write δB∥=δB·(B0/B0) as the variation in the component of the perturbed\nmagnetic field parallel to the mean magnetic field. Furthermore, separating term v2=\nv2\n⊥+v2\n∥in the correlation (2.5), it is easy to show that the v2\n⊥contribution yields a\nperfect differential in fswhen integrated over all parallel velocity (Howes et al. 2017),\nso we choose to omit this term since it leads to zero net change in the particle energy.\nFinally, we write the gradient along the magnetic field direction by ∇∥≡ˆb· ∇, leading\nto our preferred form of the field-particle correlation for TTD,\nCδB∥,s(r0,v, t;τ) =1\nτZt+τ/2\nt−τ/2msv2\n∥v2\n⊥\n4B∇∥δB∥∂fs(r0,v, t′)\n∂v∥dt′. (2.7)\nIn the gyrokinetic system of equations (Antonsen Jr & Lane 1980; Frieman & Chen\n1982; Howes et al. 2006; Schekochihin et al. 2009), the gyro-averaged effect of parallel\nmagnetic field gradients leads to the magnetic mirror force, which can accelerate particles\nin the direction parallel to the magnetic field via the Landau resonance, and therefore\ncan lead to collisionless TTD of electromagnetic fluctuations. A rigorous derivation of the\ngyrokinetic equation in Appendix A shows explicitly the two collisionless wave-particle\ninteractions via the Landau resonance—specifically, LD and TTD. The natural form of\nthe field-particle correlation arising from the gyrokinetic version of the generalized energy\ndensity equation is slightly different from the field-particle correlation for TTD given in\n(2.7), but the gyrokinetic form requires the gyroaveraged distribution function, which is\nnot accessible through single-point spacecraft measurements and is not a natural quantity\nthat can easily be derived from other kinetic simulation approaches, such as particle-in-\ncell or Vlasov simulations. Therefore, we choose here to use the perturbed distribution\nfunctions and electromagnetic fields generated by our gyrokinetic simulations, but we\nanalyze them using (2.7) which is more directly applicable to spacecraft measurements\nor alternative kinetic simulation approaches, such as particle-in-cell codes.\nUtilizing the gyroaveraged distribution function and corresponding fields, the quantity\nCδB∥,s(r0,v, t;τ) reveals the velocity-space signature of TTD on the gyrotropic velocity\nspace ( v⊥, v∥). To simplify the notation, henceforth we shall employ CδB∥,s(v∥, v⊥) to\nsignify the gyrotropic correlation, explicitly noting the associated spatial position r0,\ntime t, and the correlation interval τonly when necessary. The resonant structure of\nthe velocity-space signature of TTD is primarily a function of v∥, so it is often useful to\ndefine the reduced parallel field-particle correlation by integrating CδB∥,s(v∥, v⊥, t) over\nv⊥, given by CδB∥,s(v∥, t)≡2πR\nCδB∥,s(v∥, v⊥, t)v⊥dv⊥, where the extra 2 πv⊥factor\narises from the integration over the gyrophase in 3V phase space. A timestack plot of the6\nreduced parallel correlation CδB∥,s(v∥, t) reveals the persistence in time of any resonant\nvelocity-space signatures in v∥. We can also consider the rate of change of the kinetic\nenergy density of species sdue to TTD by integrating the gyrotropic correlation over all\nvelocity space: ( ∂Ws/∂t)TTD=R\nCδB∥,s(v∥, v⊥)d3v.\nIn closing, note that the gyroaveraging procedure employed in the derivation of the\nsystem of gyrokinetics enables the variations in the perpendicular components of the\nelectric field E⊥to be expressed in terms of changes in the parallel component of the\nmagnetic field δB∥, as shown by (A 7). In a system where the gyroaverage has not been\nperformed, the work done by TTD is actually mediated (at the position of the particle)\nby the perpendicular component of the electric field E⊥(Howes et al. 2024). Therefore,\nthe perpendicular electric field correlation, given by summing the two perpendicular\ncontributions to the electric field correlation, CE⊥(r0,v, t;τ) (Klein et al. 2020; Afshari\net al. 2023), can be used to seek the velocity-space signature of TTD at the parallel\nresonant phase velocity, as seen recently in hybrid particle-in-cell simulations of plasma\nturbulence (Cerri et al. 2021).\n2.2.Prediction of the Velocity-Space Signature of Transit-Time Damping\nTo predict the velocity-space signature of TTD, we begin with a simple model of a\nmagnetic field with an amplitude variation that varies along the mean field direction z,\ngiven in cylindrical coordinates ( r, ϕ, z ) by\nB(r, ϕ, z ) =−δBz\n4krsin(kz′)ˆr+\u001a\nB0+δBz\n2[1−cos(kz′)]\u001b\nˆz (2.8)\nwhere the wavenumber kof the spatial variation of the magnetic field magnitude is\nalong the mean field direction z, and z′=z−Ut, such that that pattern moves in the\n+zdirection with a phase speed U⩾0. The corresponding electric field variation can\nbe determined by the Lorentz transform from the primed (wave) frame K′in which\nthe magnetic field pattern is stationary (and therefore E′= 0) to the unprimed (lab)\nframe K. This Lorentz transformation in the non-relativistic limit U/c≪1 is given by\nE=E′−U×BandB=B′(Howes et al. 2014), where the transformation velocity is\njustU=Uˆz. The resulting induced electric field in the lab frame Kis given by\nE(r, ϕ, z ) =UδBz\n4krsin(kz′)ˆϕ. (2.9)\nWith this simple model, we can illustrate how a single particle is accelerated into different\nregions of velocity space by the electromagnetic fields. Extending this approach to con-\nsider a distribution of particles will enable us to predict the qualitative and quantitative\nfeatures of the velocity-space signature of TTD.\nConsider first the acceleration of a single particle in a stationary mirror field with\nU= 0, as shown in Figure 2(a), for a “wave” amplitude of δBz/B0= 0.2, giving a mirror\nratio of Bmax/Bmin= 1.2. The particle begins at the minimum in the magnetic field at\nz= 0 with an initial perpendicular velocity v⊥and an initial parallel velocity v∥<0,\ngiven by the red + in the figure at the tip of the initial velocity vector vi(blue). As the\nparticle moves into the increasing magnetic field at z <0, the mirror force increases v⊥\nand decreases v∥such that the particle moves through velocity space (green arrow) on a\ncircle of constant total velocity v=q\nv2\n∥+v2\n⊥(black dashed circle). The particle will be\nreflected by the mirror field if the particle has an initial pitch angle α= tan−1(v⊥/v∥)\nlarger than the loss cone angle αloss= sin−1\u0010p\nBmin/Bmax\u0011\n. For Bmax/Bmin= 1.2, the\nloss cone angle is αloss= 66◦; the particle depicted in Figure 2(a) has an initial pitchTransit-Time Damping 7\nangle α > αlossand is therefore reflected by the mirror field. The particle follows this\ncircular trajectory in ( v∥, v⊥) velocity space until it returns to its initial axial position\nz= 0, ending up with a final velocity vf(blue) with the same perpendicular component\nbut an equal and opposite parallel component. Thus, the kinetic energy of the particle\ndoes not change, consistent with the fact that magnetic fields do no work on particles.\nThe particle has simply been reflected by the mirror, reversing the sign of its parallel\nvelocity.\nNext, we consider the case for a magnetic mirror field moving with velocity U=Uˆz,\nwhere U > 0. In the wave frame, moving at velocity U=Uˆzin which the magnetic field is\nstationary, the acceleration of the particle by the magnetic field must be the same as the\nstationary case in Figure 2(a). But, in the lab frame, depicted in Figure 2(b), the particle\nnow moves on a circular trajectory in velocity space centered about the mirror velocity U,\ngiven by a constant magnitude of velocity in the wave frame v0=q\n(v∥−U)2+v2\n⊥. Here\nthe particle is initially moving in the same direction as the mirror field but with a slower\ninitial parallel velocity 0 ⩽v∥⩽U, given by the red + in the figure at the tip of the initial\nvelocity vector vi(blue). If the pitch angle in the wave frame αw= tan−1[v⊥/(v∥−U)] is\ngreater than the loss-cone angle, αw> αloss, the particle will be reflected by the moving\nmirror field, leading to a net acceleration in the axial direction, ultimately ending up with\na parallel velocity greater than the mirror velocity v∥> U, with a final velocity vector vf\n(blue). In this case, the induced electric field given by (2.9) has done work on the particle\n(Howes et al. 2024), ultimately leading to a net acceleration in the axial direction. This\nprocess is the fundamental energy transfer underlying the physics of TTD.\nFinally, we consider how this understanding of the single particle motion and accel-\neration can be combined with a distribution of initial particle velocities to predict the\nvelocity-space signature of TTD. Note that for the sinusoidally oscillating magnetic field\nmagnitude given by (2.8), the long time evolution of the particle in velocity space would\noscillate back and forth between viandvfalong the green trajectory shown in Fig-\nure 2(a) and (b); for example, if the particle started with initial velocity vf, the other\nside of the mirror field would lead to a reflection in the opposite direction, ultimately\nresulting in the particle ending up with a final velocity vi. In the case of a moving mirror\nfield, for a realistic velocity distribution there will be more particles with parallel veloc-\nitiesv∥< U than with v∥> U, so the net effect is that more particles will gain energy\nthan lose energy, leading to a net energization of the particles and consequent damping\nof the electromagnetic wave. Only particles with pitch angles in the wave frame larger\nthan the loss cone angle, αw> αloss, will undergo the mirror reflection, so that net effect\non the distribution is an acceleration of particles from v∥< U tov∥> U. The resulting\nchange in the phase-space energy density leads to the prediction of the velocity-space sig-\nnature of TTD depicted in Figure 2(c): a loss of phase-space energy density (blue) in the\nregion v∥< U, and a gain of phase-space energy density (red) in the region v∥> U. The\nextent of this velocity-space signature in ( v∥, v⊥) velocity space is confined by two effects:\n(i) only particles outside of the loss cone will experience a net acceleration; and (ii) the\nsignature is weighted by the v2\n⊥weighting in (2.7) for the rate of change of phase-space\nenergy density by TTD (which arises from the magnetic moment µ=mv2\n⊥/(2B) depen-\ndence of the mirror force) combined with the reduced perpendicular velocity distribution\nf(v⊥), where this net weighting of v2\n⊥f(v⊥) is shown in Figure 2(d). Thus, the velocity-\nspace signature of TTD in Figure 2(c) is restricted to “Landau resonant” particles with\nparallel velocities near the velocity of the magnetic field pattern v∥∼Uand to a region\naway from the v⊥= 0 axis, unlike the velocity-space signature of LD (Klein & Howes\n2016; Howes et al. 2017; Klein et al. 2017) which extends down to v⊥= 0.8\n−0.5 0.0 0.5 1.0\nv/bardbl0.00.20.40.60.81.01.2v⊥\nαloss>>\nvi vfU= 0\n−0.5 0.0 0.5 1.0\nv/bardbl0.00.20.40.60.81.01.2v⊥\nαloss>>\nvi vfU > 0 U(a) (b)\n−0.5 0.0 0.5 1.0\nv/bardbl0.00.20.40.60.81.01.2v⊥\nαlossU\n0.0 0.1\nv2\n⊥f(v⊥)0.000.250.500.751.001.251.501.75v⊥\n(c) (d)\nFigure 2. Diagram of the magnetic mirror force and prediction for the velocity-space signature\nof TTD: (a) v⊥versus v∥for the single particle motion in a static magnetic mirror field; (b) v⊥\nversus v∥for the single particle motion in a moving magnetic mirror field, where the vertical\nblack dashed line denotes the wave phase velocity U; (c) the predicted velocity-space signature\nfor a Maxwellian velocity distribution function, where the phase-space energy density decreases\natv∥< U(blue) and increases at v∥> U(red); (d) effective v⊥weighting of correlation v2\n⊥f(v⊥),\nwhich constrains the velocity-space signature in the v⊥direction.\n3. Single Kinetic Alfv´ en Wave (KAW) Simulations\nHere we perform numerical simulations of single kinetic Alfv´ en waves to determine the\nvelocity-space signature of TTD using the Astrophysical Gyrokinetics Code, AstroGK\n(Numata et al. 2010). AstroGK evolves the gyroaveraged scalar potential ϕ(r), parallel\nvector potential A∥(r), and the parallel magnetic field fluctuation δB∥(r), as well as\nthe gyrokinetic distribution function hs(r, v⊥, v∥), in a triply-periodic slab geometry of\nsizeL2\n⊥×L∥elongated along the straight, uniform mean magnetic field B0=B0ˆz.\nThe domain-scale wavenumbers are defined by k∥0= 2π/L∥andk⊥0= 2π/L⊥. The\ngyrokinetic expansion parameter is defined by ϵ∼k∥0/k⊥0≪1 (Howes et al. 2006),\nand all quantities are scaled to accommodate an arbitrary value of ϵ. The gyrokinetic\ndistribution function is related to the total distribution function fsvia\nfs(r,v, t) =F0s(v)\u0012\n1−qsϕ(r, t)\nTs\u0013\n+hs(Rs, v⊥, v∥, t) +δf2s+... (3.1)Transit-Time Damping 9\nwhere F0sis the equilibrium distribution, ris the spatial position, Rsis the associated\nspecies gyrocenter related to rbyr=Rs−v׈z/Ωs, and δf2sare corrections second-\norder in the gyrokinetic expansion parameter ϵwhich are not retained (Howes et al.\n2006). The code employs a pseudospectral method in the ( x, y) (perpendicular) plane\nand finite differencing in the z-direction. The velocity distribution is resolved on a grid\nin energy E=v2\n∥+v2\n⊥and pitch angle λ=v2\n⊥/v2space, with the points selected on a\nLegendre polynomial basis. A fully conservative, linearized, gyroaveraged collision oper-\nator is employed (Abel et al. 2008; Barnes et al. 2009) to ensure velocity-space structure\nishsremains resolved throughout the simulation evolution. We normalize time using\nthe domain-scale Alfv´ en wave frequency ωA≡k∥0vA, and particle velocity is normalized\nto the ion thermal velocity vti=p\n2Ti/mi, where the Boltzmann constant has been\nabsorbed to yield temperature in units of energy.\n3.1.Single Wave Simulation Set Up\nWe first determine the velocity-space signature of TTD by performing simulations of\nsingle kinetic Alfv´ en waves (KAWs) for a fully ionized proton-electron plasma with\nMaxwellian equilibrium velocity distributions with a temperature ratio Ti/Te= 1 and a\nrealistic ion-to-electron mass ratio of mi/me= 1836. We perform three simulations with\nion plasma beta βi= 0.3,1,3 and sample the time evolution of the electromagnetic fields\nand gyrokinetic distribution function at discrete single points throughout the simulation\ndomain with dimensions L⊥= 2πρiandL∥= 2πa0, yielding an arbitrary expansion pa-\nrameter ϵ=ρi/a0≪1. Here ρi≡vti/Ωiis the ion Larmor radius, where the angular ion\n(proton) cyclotron frequency is Ω i=qiB0/mi. The dimensions of these single-wave simu-\nlations are ( nx, ny, nz, nλ, nE, ns) = (10 ,10,32,128,64,2), where nsdenotes the number\nof species. Using the solutions to the linear gyrokinetic dispersion relation (Howes et al.\n2006), a single plane-wave KAW with wavevector ( kxρi, kyρi, k∥a0) = (1 ,0,1) is ini-\ntialized throughout the domain and allowed to evolve linearly for 5 wave periods with\nenhanced collisionality to eliminate any transients associated with the initialization, yield-\ning a clean, single KAW with k⊥ρi= 1. The simulation is then restarted with lowered\ncollisionalities νs/(k∥0vti) = 2 ×10−3and evolved to allow collisionless wave-particle\ninteractions to damp the wave. We have verified that the collisionless damping rates\nof the initialized KAWs agree with the analytical predictions from Vlasov-Maxwell and\ngyrokinetic linear dispersion relations.\n3.2.The Velocity-Space Signature of Transit-Time Damping\nWe select the βi= 1 single KAW simulation as a fiducial case to determine the velocity-\nspace signature of TTD and compare it to the known velocity-space signature of LD\n(Klein & Howes 2016; Howes et al. 2017, 2018; Klein et al. 2017; Chen et al. 2019;\nKlein et al. 2020; Horvath et al. 2020; Afshari et al. 2021). The linear Vlasov-Maxwell\ndispersion relation yields a parallel phase velocity normalized to the Alfv´ en velocity of\nω≡ω/(k∥vA) = 1 .137 for a KAW with k⊥ρi= 1, βi= 1, and Ti/Te= 1, corresponding\nto a normalized wave period of TωA= 5.526.\nA key step in the field-particle correlation analysis is to choose an appropriate correla-\ntion interval τover which to time-average the rate of energization to eliminate a possibly\nlarger amplitude signal of oscillatory energy transfer in order to reveal the often smaller\nsecular rate of energy transfer that corresponds to the collisionless damping of the wave\n(Klein & Howes 2016; Howes et al. 2017; Klein et al. 2017). In Figure 3, we present a\ntest of different correlation intervals over the range 0 ⩽τωA⩽10 for the βi= 1 single\nKAW simulation. In panel (a), we plot the velocity-space integrated rate of ion ener-\ngization due to TTD, ( ∂Wi/∂t)TTD=R\nCδB∥,i(v∥, v⊥)d3v, vs. time. The unaveraged10\n60 70 80\ntωA−0.50.00.51.0∂Wi/∂t×10−5\n0.02.55.07.510.0\nτωA\n60 70 80\ntωA−1.5−1.0−0.50.00.51.0CδB/bardbl,i(v/bardbl= 1.1vti)×10−5\n0.02.55.07.510.0\nτωA\n(a) (b)\n−2 0 2\nv/bardbl/vti6065707580tωACδB/bardbl,i(v/bardbl),τωA= 0.0\n−1.0−0.50.00.51.0×10−5\n−2 0 2\nv/bardbl/vti6065707580tωACδB/bardbl,i(v/bardbl),τωA= 5.5\n−6−4−20246×10−6\n(c) (d)\nFigure 3. Analysis of correlation interval selection for βi= 1AstroGK single KAW simulation.\nTop row: time evolution of (a) the rate of change of ion kinetic energy density due to TTD,\ndenoted as ∂Wi/∂t, and (b) the reduced correlation CδB∥,i(v∥, t) atv∥= 1.1vti. Both quantities\nare presented over a range of τωAvalues from 0 to 10. The selected τωAvalue of 5.5 is marked\nwith a black line. Bottom row: timestack plots of the reduced correlation CδB∥,i(v∥, t) for (c)\nτωA= 0 and (d) τωA= 5.5, with the vertical dashed line at v∥/vti= 1.137 labelling the\nnormalized parallel phase velocity ω/(k∥vti).\ncorrelation ( τ= 0, dark blue) exhibits pronounced oscillations of the net energy transfer\nvs. time. Setting the correlation interval to one linear wave period τωA=TωA≃5.5\n(black) minimizes the oscillations, providing an optimal choice for τfor a single wave\nwith a well-defined period, as might be expected on theoretical grounds. We show in\npanel (b) the evolution of the reduced parallel correlation CδB∥,i(v∥, t) at a parallel ve-\nlocity v∥= 1.1vtislightly below the resonant velocity. Here again, setting τωA= 5.5\n(black) effectively minimizes the oscillations, revealing clearly the rate of secular energy\ntransfer at that parallel velocity. We illustrate the impact of choosing an appropriate\ncorrelation interval τon the timestack plot of the correlation CδB∥,i(v∥, t) by compar-\ning (c) the instantaneous ( τ= 0) field-particle correlation to (d) the correlation using\nτωA= 5.5, showing a clear bipolar signature about the normalized parallel phase velocity\nω/(k∥vti) = 1 .137 that persists over the course of the simulation.\nSpecifying the correlation interval to be approximately one wave period τωA= 5.5,†\nwe now present in Figure 4 the velocity-space signatures of (a) TTD and (b) LD from\ntheβi= 1 single KAW simulation. Each panel presents three sub-plots, explained here\nfor the TTD case in panel (a): (i) the main plot presents the gyrotropic velocity-space\nsignature CδB∥,i(v∥, v⊥) at time tωA= 63.57, with the parallel phase velocity indicated\n†Note that the velocity-space signature for a single KAW is independent of the probe position\nif the correlation interval is taken an integral multiple of the wave period.Transit-Time Damping 11\n(a)\n012345v⊥/vtiFrame: 150/501 tωA= 63.57CδB/bardbl,i(v/bardbl,v⊥)\n−6−4−20246×10−6\n−3−2−1 0 1 2 3\nv/bardbl/vti05\nCδB/bardbl,i(v/bardbl)×10−60 2\n∂Wi/∂t×10−66065707580tωA\n(b)\n012345v⊥/vtiFrame: 150/501 tωA= 63.57CE/bardbl,i(v/bardbl,v⊥)\n−1.5−1.0−0.50.00.51.01.5×10−5\n−3−2−1 0 1 2 3\nv/bardbl/vti01\nCE/bardbl,i(v/bardbl)×10−60 2\n∂Wi/∂t×10−76065707580tωA\nFigure 4. Velocity-space signatures of (a) transit-time damping (TTD) and (b) Landau damp-\ning (LD), from the AstroGK simulation of a single kinetic Alfv´ en wave with k⊥ρi= 1, βi= 1,\nandTi/Te= 1, each showing the gyrotropic signatures in the main panel, the time-integrated\nreduced parallel signatures in the lower panel, and the net rate of ion energization vs. time for\neach mechanism in the left panel. The correlation interval is chosen as τωA= 5.5. The nor-\nmalized parallel phase velocity is labeled by the two vertical dashed lines at v∥/vti=±1.137.12\n(vertical dotted line); (ii) the lower plot shows the time-integrated parallel velocity-\nspace signature CδB∥,i(v∥) to highlight the variation of the net energy transferred as a\nfunction of v∥, showing a clear bipolar signature at the parallel phase velocity; and (iii)\nthe left plot shows the velocity-space integrated net energy density transfer rate due\nto TTD ( ∂Wi(t)/∂t)TTDvs. time, with the centered time of the correlation interval\nshown in the gyrotropic signature indicated (horizontal solid line). Panel (b) presents\nthe corresponding plots for the LD case.\nA key result of this paper is the velocity-space signature of transit time damping plotted\non gyrotropic velocity space CδB∥,i(v∥, v⊥) in Figure 4(a). From the same simulation, the\ngyrotropic velocity-space signature of LD CE∥,i(v∥, v⊥), given by the parallel contribution\nto the dot-product in (2.4), is presented in (b) for comparison. The TTD signature agrees\nqualitatively with our prediction presented in Figure 2(c), showing the key features: (i)\nthe bipolar signature of the rate of loss (blue) and gain (red) of phase-space energy density\nis centered about the parallel wave phase velocity v∥∼ω/k∥(vertical dotted black line);\n(ii) the gyrokinetic velocity-space signature does not extend down to v⊥= 0, due to a\ncombination of the v2\n⊥weighting arising from the magnetic moment µ=mv2\n⊥/(2B) in the\nmirror force and from the loss cone angle of the mirror force, as explained in §2.2. The LD\nsignature in (b) likewise yields a bipolar signature near the parallel wave phase velocity.\nBesides the fact that E∥governs energization through LD and E⊥governs energization\nthrough TTD (Howes et al. 2024), a key way to distinguish these two mechanisms is that\nin gyrotropic velocity space the TTD signature does not extend down to v⊥= 0, whereas\nthe LD signature extends right down to v⊥= 0.\n3.3.Variation of Signature with Ion Plasma Beta βi\nResonant damping of electromagnetic fluctuations through TTD and LD depends strongly\non the plasma beta, typically with LD dominant at βi≪1 and TTD dominant at βi≫1\n(Quataert 1998), so we vary the value of βihere to determine its impact on the character-\nistics of the velocity-space signature of the Landau-resonant damping mechanisms. Using\nthe same parameters Ti/Te= 1, mi/me= 1836, and k⊥ρi= 1, we perform additional\nsingle KAW simulations with βi= 0.3 and βi= 3. For βi= 0.3, the normalized parallel\nphase velocity is given by ω≡ω/(k∥vA) = 1 .267 yielding a normalized wave period of\nTωA= 4.959; for βi= 3, we obtain ω= 1.009 and TωA= 6.227.\nIn Figure 5, we plot the velocity-space signatures for (a) TTD and (b) LD for the\nβi= 0.3 case and for (c) TTD and (d) LD for the βi= 3 case, where each panel has\nthree subplots in the same format as in Figure 4. These bipolar velocity-space signatures\nlook qualitatively similar to the βi= 1 case in Figure 4, with two important quantitative\ndifferences.\nFirst, the position of the bipolar signature in v∥/vtichanges as βiis varied, consistent\nwith the variation of the parallel phase velocity normalized to the ion thermal velocity\nasβivaries, given by ω/(k∥vti) =ω/β1/2\ni. For βi= 0.3, we obtain ω/(k∥vti) = 2 .313,\nindicated by the vertical black dashed line in Figure 5(a) and (b); for βi= 3, we obtain\nω/(k∥vti) = 0 .583, as shown in (c) and (d). In both cases, the bipolar signature remains\nclosely associated in v∥with the parallel phase velocity ω/k∥, as expected for a Landau-\nresonant energy transfer between the fields and the ions.\nSecond, looking at the vertical subplots on the left for each panel, which shows the\nnet rate of change of ion energy density Wimediated by each mechanism averaged over\na correlation interval equal to one wave period τ=T, we find the surprising result for\nβi= 0.3 that, although LD leads to a net gain of energy by the ions (as expected for\ncollisionless damping of a wave), TTD leads to a net lossof energy from the ions. This\nfinding suggests that the ions are losing energy through the magnetic mirror force whileTransit-Time Damping 13\n012345v⊥/vtiFrame: 150/501 tωA= 57.03CδB/bardbl,i(v/bardbl,v⊥)\n−6−4−20246×10−6\n−3−2−1 0 1 2 3\nv/bardbl/vti−2.50.0\nCδB/bardbl,i(v/bardbl)×10−6−5 0\n∂Wi/∂t×10−755606570tωA\n012345v⊥/vtiFrame: 150/501 tωA= 57.03CE/bardbl,i(v/bardbl,v⊥)\n−6−4−20246×10−5\n−3−2−1 0 1 2 3\nv/bardbl/vti−101\nCE/bardbl,i(v/bardbl)×10−50 5\n∂Wi/∂t×10−655606570tωA(a) (b)\n012345v⊥/vtiFrame: 150/501 tωA= 71.63CδB/bardbl,i(v/bardbl,v⊥)\n−1.8−0.90.00.91.8×10−10\n−3−2−1 0 1 2 3\nv/bardbl/vti02\nCδB/bardbl,i(v/bardbl)×10−100 2\n∂Wi/∂t×10−10657075808590tωA\n012345v⊥/vtiFrame: 150/501 tωA= 71.63CE/bardbl,i(v/bardbl,v⊥)\n−4−2024×10−11\n−3−2−1 0 1 2 3\nv/bardbl/vti−2.50.0\nCE/bardbl,i(v/bardbl)×10−11−5 0\n∂Wi/∂t×10−11657075808590tωA\n(c) (d)\nFigure 5. Velocity-space signatures of transit-time damping (TTD, left column) and Landau\ndamping (LD, right column) in AstroGK single KAW simulations with k⊥ρi= 1,Ti/Te= 1, and\nβi= 0.3 (top row) and βi= 3 (bottom row). The correlation intervals are set to the correspond-\ning linear wave periods, with τωA= 5.0 for βi= 0.3 case and τωA= 6.2 for βi= 3 case. The\nnormalized parallel phase velocity is labeled by the two vertical dashed lines at v∥/vti=±2.313\nforβi= 0.3 and v∥/vti=±0.583 for βi= 3. Each panel follows the layout format of Figure 4.\ngaining energy through acceleration by the parallel electric field. The rate of ion energy\ngain by LD is about ten times larger than the rate of loss by TTD, so the summed effect\nof these two mechanisms is energization of the ions by collisionless damping of the wave,\nas expected. For the βi= 3 case, we find the equally surprising result that although TTD\nleads to a net gain of energy by the ions, LD is leading to a net loss of energy from the\nions; the rate of ion energy gain by TTD is larger than the rate of ion energy loss by LD,\nso the summed contributions yield a net gain of ion energy as expected for collisionless\ndamping.\nHow do we reconcile these surprising results with the general expectation of Landau-\nresonant collisionless damping of waves for a Maxwellian equilibrium velocity distribu-\ntion? As it turns out, this behavior is exactly what is predicted by the linear Vlasov-\nMaxwell dispersion relation. To be specific, we take the complex eigenfrequency from the\nlinear dispersion relation to be given by ω+iγ, so that time evolution of a plane-wave\nmode is given by exp( −iωt) exp( γt): positive imaginary components γ >0 correspond to\ngrowth of the wave, and negative imaginary components γ <0 correspond to damping of\nthe wave. In Figure 6, we plot the normalized absolute value of the imaginary component\nof the wave frequency |γ|/ωvs.k⊥ρifor KAWs using the PLUME solver (Klein & Howes\n2015) for a fully ionized proton-electron plasma with isotropic Maxwellian velocity dis-\ntributions with Ti/Te= 1, mi/me= 1836, vti/c= 10−4,k∥ρi= 10−3over the range\n10−3⩽k⊥ρi⩽102for the ion plasma beta values (a) βi= 0.3, (b) βi= 1, and (c) βi= 3.14\n10−310−210−1100101102\nk⊥ρi10−910−710−510−310−1|γ|/ωβi= 0.3,mi/me= 1836\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD<0\nγiTTD>0\n10−310−210−1100101102\nk⊥ρi10−910−710−510−310−1|γ|/ωβi= 1,mi/me= 1836\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD<0\nγiTTD>0\n10−310−210−1100101102\nk⊥ρi10−910−710−510−310−1|γ|/ωβi= 3,mi/me= 1836\nγ=γi+γe\nγi<0\nγiLD<0\nγiLD>0\nγiTTD>0\nγiTTD<0\n(a)\n(b)\n(c)\nFigure 6. Linear dispersion relations for KAWs from PLUME calculations with the realistic\nmass ratio mi/me= 1836, showing the absolute value of the normalized wave growth rate |γ|/ω\nas a function of the dimensionless perpendicular wave vector k⊥ρifor (a) βi= 0.3, (b) βi= 1,\nand (c) βi= 3. The vertical black dashed line at k⊥ρi= 1 indicates the values used in the\nsingle KAW AstroGK simulations. We plot γ(total damping rate, black), γi(total ion damping\nrate, green), γiTTD (ion growth or damping rate via the magnetic mirror force, red), and γiLD\n(ion growth or damping rate via the electrostatic force, blue). Line styles—solid, dashed, and\ndotted—represent the total damping rates, damping rates separated by mechanism, and growth\nrates separated by mechanism, respectively.Transit-Time Damping 15\nWe plot separately the total collisionless damping rate (black solid) due to both ions and\nelectrons, the total ion damping rate (green solid), and the separate contributions to\nthe ion damping rate from TTD (red) and LD (blue). For the separated TTD and LD\ncontributions, we plot negative imaginary components (which correspond to collisionless\ndamping of the wave) using dashed lines, and positive imaginary components (which\ncorrespond to collisionless growth of the wave) using dotted lines.\nAlthough all of the KAW dispersion relations plotted in Figure 6 yield a net effect of\ncollisionless damping by the ions, over some ranges in k⊥ρi(dotted lines) either TTD\nor LD individually may lead to a net transfer of energy from the ions tothe waves over\nthe course of a single wave period. For example, for the βi= 3 case in Figure 6(c), the\nimaginary component due to LD is positive over 1 .4×10−3≲k⊥ρi≲2.1, corresponding\nto a transfer of energy from the ions to the wave, and is negative outside of that range,\ncorresponding to a transfer of energy from the wave to the ions. A transfer from ions to\nthe wave would lead to a growth of the wave, but the sum of both of the LD and TTD\ncontributions is always negative for these cases with a Maxwellian velocity distribution,\nleading to a net collisionless damping of the wave.\nThe perpendicular wave number k⊥ρi= 1 of our simulated single waves is indicated\nin Figure 6 by the vertical black dashed line, and with these plots we can understand\nthe results presented in Figure 6. For the βi= 0.3 case, at k⊥ρi= 1 we have γiLD<0\n(blue dashed) and γiTTD >0 (red dotted), suggesting that ions gain energy due to E∥\nbut lose energy due to the mirror force when averaged over the full wave period. This\nfinding agrees with the net gain of ion energy by LD in Figure 5(b) and the net loss of\nion energy by TTD in (a). Similarly, for the βi= 3 case, at k⊥ρi= 1 we have γiTTD <0\n(red dashed) and γiLD>0 (blue dotted), suggesting that ions gain energy due to the\nmirror force but lose energy due to E∥when averaged over the full wave period. Again,\nthis finding from the linear Vlasov-Maxwell dispersion relation agrees with the net gain\nof ion energy by TTD in Figure 5(c) and the net loss of ion energy by LD in (d).\nTo further understand the physical meaning of γiTTD >0 in the βi= 0.3 case, we\nfirst point out that the only mechanism that can change the particle energy is work done\nby the electric field, and the rate of change of the ion energy density Wiis given by\nji·E. For TTD, this energization arises from the component of the electric field that is\nperpendicular to the magnetic field (Howes et al. 2024) (whereas LD energizes particles\nthrough the parallel component of the electric field), so here we consider the perpendicular\ncontribution to the energization j⊥iE⊥. For a single plane wave, the net transfer of energy\nto or from the ions by j⊥iE⊥over a single wave period depends on the phase of the\nperpendicular electric field fluctuation E⊥relative to the phase of the self-consistent\nperpendicular component of the ion current density associated with the wave, j⊥i. If the\nphase difference δϕis such that there is an in-phase component 0 < δϕ < π/ 2, there will\nbe a net energization of the ions; if there is an out-of-phase component π/2< δϕ < π ,\nthe ions will lose energy. The eigenfunctions arising from solutions of the linear Vlasov-\nMaxwell dispersion relation dictate the phases of the components of the electric field and\ncurrent density. For the βi= 0.3 case, this eigenfunction dictates that j∥iandE∥are\nin-phase, leading to ion energization and wave damping by E∥(yielding LD), but j⊥i\nandE⊥are out-of-phase, so the magnetic mirror force partly counteracts the damping\nof the wave.16\n4. Turbulence Simulations\nNow that we have determined the gyrotropic velocity-space signature of TTD for single\nKAWs, with the fiducial example for βi= 1 shown in Figure 4(a), we will seek similar\nsignatures of TTD in simulations of strong plasma turbulence.\n4.1.Turbulence Simulation Set Up\nWe perform kinetic simulations of strong plasma turbulence using the Astrophysical\nGyrokinetics Code AstroGK (Numata et al. 2010) for three values of the ion plasma\nbeta βi= 0.3,1,3. Each simulation has numerical resolution ( nx, ny, nz, nλ, nE, ns) =\n(96,96,32,64,32,2) within a simulation domain L2\n⊥×L∥= (8πρi)2×(2πa0), where the\nelongation of the domain along the equilibrium magnetic field B0=B0ˆzis characterized\nby the arbitrary gyrokinetic expansion parameter ϵ∼ρi/a0≪1. The proton-to-electron\ntemperature ratio of the Maxwellian equilibrium is Ti/Te= 1, and we choose a reduced\nmass ratio mi/me= 36 to ensure that we fully resolve the kinetic damping mechanisms\nneeded to achieve a steady-state turbulent cascade in a driven simulation, as discussed\nin Howes et al. (2018). These parameters lead to a fully resolved range of perpendicular\nwavenumbers 0 .25⩽k⊥ρi⩽7.75, or 0 .042⩽k⊥ρe⩽1.29. For the βi= 0.3 and βi= 1\nsimulations, the proton and electron collisionalities are set to νs/(k∥0vti) = 0 .1, and for\ntheβi= 3 simulation, νs/(k∥0vti) = 0 .05. These collisionalities ensure weakly collisional\nplasma conditions, yet prevent the small-scale variations that develop in velocity space\nfrom becoming unresolved on the velocity grid.\nTurbulence in the simulations is driven from zero initial conditions using an oscillating\nLangevin antenna (TenBarge et al. 2014) to drive four Alfv´ en wave modes at the domain\nscale with wave vectors ( kxρi, kyρi, kza0) = (0 .25,0,±1) and (0 ,0.25,±1), generating\nfour perpendicularly polarized Alfv´ en waves propagating in both directions along the\nequilibrium magnetic field. This driving has been shown to generate effectively a strong\nplasma turbulent cascade to small scales in previous kinetic simulations through nonlinear\ninteractions between the counterpropagating Alfv´ en waves (Howes et al. 2008; Howes\net al. 2011; Howes & Nielson 2013; TenBarge & Howes 2013; TenBarge et al. 2013; Howes\net al. 2018; Verniero et al. 2018; Verniero & Howes 2018; Horvath et al. 2020; Conley\net al. 2023). To provide the data needed to apply the field-particle correlation analysis,\nthe electromagnetic fluctuations and proton velocity distributions are sampled at a high\ncadence at twenty-four probe points that are distributed throughout the domain, sixteen\nin the xy-plane at z= 0, and the remaining eight along the z-axis, as illustrated in Fig. 2\nof Horvath et al. (2020).\nThe timescale associated with outer scale of the turbulent cascade in each simulation\nis the wave period of the domain-scale Alfv´ en wave T, and the simulations are run for\n6.78Tfor the βi= 0.3 simulation, 5 .51Tfor the βi= 1 simulation, and 3 .61Tfor the\nβi= 3 simulation. The perpendicular magnetic energy spectrum EB⊥(k⊥) at the end of\neach of the simulations is shown in Figure 7, showing the spectrum for βi= 0.3 (red),\nβi= 1 (black), and βi= 3 (blue). These spectra demonstrate that each simulation yields\na broadband turbulent spectrum, with the spectral slope for each simulation consistent\nwith the expectation of −5/3 for strong plasma turbulence (Goldreich & Sridhar 1995)\nin the inertial range at k⊥ρi<1, and steepening of the spectrum at the transition to\nthe dissipation range at k⊥ρi∼1. In the dissipation range at k⊥ρi>1, the spectral\nslopes begin around −3.2, steepening as k⊥ρe→1 due to the resolved kinetic dissipation\nmechanisms that remove energy from the turbulent cascade. Note that these dissipation\nrange slopes are slightly steeper than the values ranging from −2.7 to−3.1 typically\nobserved in the solar wind (Sahraoui et al. 2013), but this is to be expected due to the\nunphysical mass ratio of mi/me= 36, which effectively enhances the damping rate dueTransit-Time Damping 17\nFigure 7. Perpendicular magnetic energy spectra at the end of each of the turbulence simula-\ntions, showing βi= 0.3 (red), βi= 1 (black), and βi= 3 (blue). Vertical dotted lines indicate\nthe limit of fully resolved perpendicular wavenumbers in the simulation, 0 .25⩽k⊥ρi⩽7.75, or\n0.042⩽k⊥ρe⩽1.29.\nto electrons relative to the realistic mass ratio case (TenBarge et al. 2013), leading to\nslightly steeper dissipation range spectra for stronger damping (Howes et al. 2011).\nIn Figure 8, we plot the normalized damping rates |γ|/ωfrom the linear dispersion re-\nlation for the simulation parameters with the reduced mass ratio mi/me= 36, presenting\nthe results for (a) βi= 0.3, (b) βi= 1, and (c) βi= 3, in the same format as presented\nin Figure 6. Here we plot vertical black dashed lines at the perpendicular wavenumber\nlimits of the simulation at k⊥ρi= 0.25 and k⊥ρi= 7.75. Note the salient features that\nTTD yields a loss of ion energy for βi= 0.3 in (a), and LD yields a loss of ion energy for\nβi= 3 at k⊥ρi≲1.5. These calculations of the linear wave properties and the effective\nion energization rates by TTD and LD provide an important theoretical framework for\nthe interpretation of our field-particle analysis results.\n4.2.Choosing the Correlation Interval τ\nUnlike in the single-KAW simulations, where the single wave period Tis the obvious\nchoice for the correlation interval τto eliminate the oscillatory contribution to the transfer\nof energy from fields to particles, choosing τfor a plasma supporting broadband turbulent\nfluctuations is less straightforward. The longest wave period for an Alfv´ en wave at the\ndomain scale in our βi= 1 simulation is τωA≃6.28. In Figure 9, for a single probe\nposition in the βi= 1 simulation, we present (a) the total energy transfer rate to ions\ndue to TTD ( ∂Wi/∂t)TTDand (b) the energy transfer rate at v∥/vti=−1.3, for a\ncorrelation interval spanning 0 ⩽τωA⩽15. The instantaneous values of the local energy\ntransfer rates (with τ= 0, blue) exhibit large fluctuations with both positive and negative\nsigns, but for τωA= 6.4 (black) and longer correlation intervals, those large fluctuations\nare averaged out, leading to a time-averaged energy transfer rate that is about an order-\nof-magnitude smaller in amplitude than the peaks of the instantaneous value. In Figure 9,\nwe also show timestack plots CδB∥,i(v∥, t;τ) for (c) τ= 0 and (d) τωA= 6.4, showing\nthat a relatively persistent bipolar signature of TTD is revealed at v∥/vti=−1.1 in\ntheτωA= 6.4 case with peak amplitudes about an order-of-magnitude smaller than18\n10−1100101\nk⊥ρi10−610−410−2100|γ|/ωβi= 0.3,mi/me= 36\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD>0\n10−1100101\nk⊥ρi10−610−410−2100|γ|/ωβi= 1,mi/me= 36\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD<0\nγiTTD>0\n10−1100101\nk⊥ρi10−610−410−2100|γ|/ωβi= 3,mi/me= 36\nγ=γi+γe\nγi<0\nγiLD>0\nγiLD<0\nγiTTD<0\nγiTTD>0\n(a)\n(b)\n(c)\nFigure 8. Linear dispersion relations for KAWs from PLUME calculations with the reduced\nmass ratio mi/me= 36, showing the absolute value of the normalized wave damping or growth\nrate|γ|/ωas a function of the dimensionless perpendicular wavenumber k⊥ρifor (a) βi= 0.3,\n(b)βi= 1, and (c) βi= 3. The two vertical black dashed lines at k⊥ρi= 0.25 and 7 .75 label the\nrange consistent with the AstroGK turbulence simulations, and the two vertical green dashed\nlines mark the range of 1 /eof the peak value of γi. Each panel follows the layout format of\nFigure 6.Transit-Time Damping 19\n10 20 30\ntωA−4−2024∂Wi/∂t\n0.003.757.5011.2515.00\nτωA\n10 20 30\ntωA−1.5−1.0−0.50.00.51.01.5CδB/bardbl,i(v/bardbl=−1.3vti)\n0.003.757.5011.2515.00\nτωA\n(a) (b)\n−2 0 2\nv/bardbl/vti102030tωACδB/bardbl,i(v/bardbl),τωA= 0.0\n−12−9−6−3036912\n−2 0 2\nv/bardbl/vti15202530tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.6−0.4−0.20.00.20.40.6\n(c) (d)\nFigure 9. Analysis of correlation interval selection for the βi= 1 AstroGK turbulence simu-\nlation. Top row: time evolution of (a) the rate of change of ion kinetic energy density due to\nTTD, denoted as ∂Wi/∂t, and (b) the reduced correlation CδB∥,i(v∥, t) atv∥=−1.3vti. Both\nquantities are presented over a range of τωAvalues from 0 to 15. The selected τωAvalue of 6.4\nis marked with a black line. Bottom row: timestack plots of the reduced correlation CδB∥,i(v∥, t)\nfor (c) τωA= 0 and (d) τωA= 6.4, where the range of parallel phase velocities of KAWs that\nexperience significant damping by ions is indicated by vertical dashed lines at v∥/vti=±1.020\nandv∥/vti=±1.704.\nthe instantaneous case with τ= 0. Thus, we choose a correlation interval τωA= 6.4 to\nperform the field-particle correlation analysis of our βi= 1 turbulence simulation.\n4.3.Results for βi= 1Simulation\nPerforming the field-particle correlation analysis with τωA= 6.4 for TTD and LD at all\n24 probe positions in our βi= 1 turbulence simulation, we seek the gyrotropic velocity-\nspace signatures of TTD and LD shown in Figure 4. In Figure 10(a), we plot the gy-\nrotropic velocity-space signature CδB∥,i(v∥, v⊥;τ) at one of the 24 probes, centered in\ntime at tωA= 19.71, showing a clear bipolar signature comparable to that shown in Fig-\nure 4(a). The range of resonant parallel phase velocities for the kinetic Alfv´ en wave mode\nover which significant ion collisionless damping is expected is indicated by the vertical\ndashed lines; specifically, these lines mark the resonant velocities of the kinetic Alfv´ en\nwave mode at k⊥ρi≃0.5 and k⊥ρi≃2.3, the points at which the total ion collisionless\ndamping rate drops to a factor 1 /eof its peak value at k⊥ρi≃1.3, as illustrated by\nthe vertical green dashed lines on the linear dispersion relation plot for mi/me= 36 in\nFigure 8(b). In the lower panel is shown the reduced parallel velocity-space signature\nCδB∥,i(v∥;τ) integrated over v⊥, yielding a clear bipolar signature with a zero crossing\natv∥/vti≃ −1.1. Note that the zero crossing of this velocity-space signature falls within\nthe expected range of resonant velocities (vertical dashed lines), as expected theoretically20\n0 1\n∂Wi/∂t1015202530tωA\n−2 0 2123v⊥/vtiFrame: 140/339 tωA= 19.707CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.3−0.2−0.10.00.10.20.3\n−2 0 2\nv/bardbl/vti−0.250.000.25CδB/bardbl,i(v/bardbl)\n−2 0 215202530tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.6−0.4−0.20.00.20.40.6\n−2 0 2\nv/bardbl/vti05/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(a) (b)\n0 20\n∂Wi/∂t1015202530tωA\n−2 0 2123v⊥/vtiFrame: 200/339 tωA= 24.267CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−9−6−30369\n−2 0 2\nv/bardbl/vti05CE/bardbl,i(v/bardbl)\n−2 0 215202530tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−6−4−20246\n−2 0 2\nv/bardbl/vti−25025/integraltext\nCE/bardbl,i(v/bardbl)dt\n(c) (d)\nFigure 10. Velocity-space signatures of transit-time damping (TTD, top row) and Landau\ndamping (LD, bottom row) in AstroGK turbulence simulation with 0 .25⩽k⊥ρi⩽7.75,\nTi/Te= 1, and βi= 1. The correlation interval is set as τωA= 6.4. The left column presents\ngyrotropic plane ( v∥, v⊥) signatures, following the layout format of panels in Figure 4. The right\ncolumn features the timestack plots of the v⊥−integrated reduced correlation; the main panel\nhere shows the reduced correlation on ( v∥, t) grids, and the lower panel shows the time-inte-\ngrated reduced correlation. Four vertical dashed lines at v∥/vti=±1.020 and v∥/vti=±1.704\nindicate the resonant parallel phase velocity ranges where significant ion damping occurs.\nfor a resonant collisionless damping mechanism. The left panel shows the net rate of ion\nenergization due to TTD ( ∂Wi/∂t)TTDas a function of time, showing net positive ion\nenergization due to TTD at this position over almost the full duration of the simulation.\nThus, Figure 10(a) demonstrates that TTD indeed plays a role in the damping of tur-\nbulent fluctuations and consequent energization of the ions, a second key result of this\npaper.\nIn Figure 10(b), we show a timestack plot CδB∥,i(v∥, t;τ) of the ion energization by\nTTD at the same probe position as in (a), showing the persistence in time of the reduced\nparallel velocity-space signature with the zero crossing at v∥/vti≃ − 1.1. Note that\nthe zero crossing at v∥/vti≃ −1.1 in this timestack plot shifts to slightly lower phase\nvelocities near the end of the simulation at tωA>25, likely due to damping associated\nwith KAWs that have slightly lower perpendicular wavenumbers k⊥ρi, as can occur in\nbroadband turbulence. There is also a relatively short-lived bipolar signature observed atTransit-Time Damping 21\nv∥/vti≃1 at times tωA<15, indicating that TTD is also acting on KAWs propagating\nthe other direction along the magnetic field (Afshari et al. 2021).\nWe perform the analogous field-particle correlation for LD, showing CE∥,i(v∥, v⊥;τ)\nin Figure 10(c), yielding a bipolar gyrotropic velocity-space signature at v∥/vti≃1.2 at\ntime tωA= 24.3, also within the expected range of resonant parallel phase velocities. This\nfinding of the velocity-space signature of LD confirms previous field-particle correlation\nanalyses showing that ion LD plays a role in the dissipation of plasma turbulence (Klein\net al. 2017; Howes et al. 2018; Klein et al. 2020; Cerri et al. 2021). The timestack plot\nCE∥,i(v∥, t;τ) in Figure 10(d) shows this strong bipolar signature of LD at v∥/vti≃1.2\npersists over 22 ≲tωA≲30. In closing, it is worthwhile noting that, to distinguish\nthe velocity-space signature of TTD from that of LD, it is necessary to examine the\nsignatures in gyrotropic velocity space ( v∥, v⊥), showing that ion energization is limited\nto ions with v⊥/vti≳1 for TTD, but that ion energization extends down to v⊥→0 for\nLD, as expected by the physical arguments outlined in §3.2.\n4.4.Variations with Ion Plasma Beta βi\nNext, we explore how the velocity-space signatures of TTD and LD in plasma turbulence\nvary with changing the ion plasma beta βi=v2\nti/v2\nA. Because βiis a function of the\nratio of the ion thermal velocity to the Alfv´ en velocity, it directly characterizes where\nthe parallel wave phase velocity falls with the ion velocity distribution, making it the\nmost important parameter controlling resonant wave-particle interactions in a weakly\ncollisional plasma. Specifically, ω/(k∥vti) =ωβ−1/2\ni, where the parallel phase velocity\nnormalized to the Alfv´ en velocity ω≡ω/(k∥vA) typically has a value ω∼1 for the\nperpendicular wavelengths k⊥ρi∼1 at which the ions strongly interact with the waves.\nNote that, at the perpendicular scale of the domain k⊥0ρi= 0.25, the Alfv´ en wave has\ntheω≃1 for all values of βi, so we simply choose a correlation interval τωA= 6.4 for\nall of the turbulence simulation analysis below.\nWe plot some typical velocity-space signatures for TTD and LD for the βi= 0.3\nsimulation in Figure 11. Note that, for this value of βi= 0.3, the contribution of TTD\nis always to remove energy from the ions, as shown in Figure 8(a), so we expect only\nnegative TTD signatures, similar to that shown in Figure 5(a). Consequently, we expect\nLD to dominate the removal of energy from the turbulence. Performing the TTD analysis\nto determine CδB∥,i(v∥, v⊥;τ), we display a typical gyrotropic velocity-space signature\nin Figure 11(a) with the associated timestack plot at the same probe in (b), showing\ntwo bipolar signatures with a net negative energy transfer rate and zero crossings at\nv∥/vti≃ ±0.7. These look like typical reversed TTD signatures, but the phase velocity\nisnotwithin the expected range 1 .8≲|v∥/vti|≲2.4 for a KAW with βi= 0.3. Only\n3 of the 24 probes showed reversed TTD signatures with phase velocities closer to the\nexpected range, as shown in Figure 11(c) and (d) with a bipolar zero crossing around\nv∥/vti≃ −1.5.\nPerforming the LD analysis to determine CE∥,i(v∥, v⊥;τ) on the same βi= 0.3 tur-\nbulence simulation, we find a similar intriguing result that we commonly find bipolar\nsignatures associated with positive energy transfer to ions, but with zero crossings well\nbelow the expected range of 1 .8≲|v∥/vti|≲2.4. In Figure 11(e), we see two bipo-\nlar signatures at v∥/vti≃ −0.7 and v∥/vti≃1.2. Only 2 of the 24 probes recover LD\nvelocity-space signatures in the expected range, such as that shown in Figure 11(g) and\n(h).\nTurning next to the field-particle correlation analysis of the βi= 3 turbulence simula-\ntion, the linear dispersion relation plot for KAWs in Figure 8(c) shows that LD removes\nenergy from ions for waves with k⊥ρi≲1.7, but TTD positively energizes ions over per-22\n−5 0\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 340/396 tωA= 35.337CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.3−0.2−0.10.00.10.20.3\n−2 0 2\nv/bardbl/vti−0.250.00CδB/bardbl,i(v/bardbl)\n−2 0 21520253035tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.4−0.20.00.20.4\n−2 0 2\nv/bardbl/vti−2.50.0/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(a) (b)\n−2.5 0.0\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 200/396 tωA= 25.113CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.06−0.04−0.020.000.020.040.06\n−2 0 2\nv/bardbl/vti−0.10.0CδB/bardbl,i(v/bardbl)\n−2 0 21520253035tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.2−0.10.00.10.2\n−2 0 2\nv/bardbl/vti−20/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(c) (d)\n050\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 220/396 tωA= 26.574CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−15−10−5051015\n−2 0 2\nv/bardbl/vti05CE/bardbl,i(v/bardbl)\n−2 0 21520253035tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−10−50510\n−2 0 2\nv/bardbl/vti050/integraltext\nCE/bardbl,i(v/bardbl)dt\n(e) (f)\n0 25\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 300/396 tωA= 32.416CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−6−4−20246\n−2 0 2\nv/bardbl/vti0.02.5CE/bardbl,i(v/bardbl)\n−2 0 21520253035tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−6−4−20246\n−2 0 2\nv/bardbl/vti025/integraltext\nCE/bardbl,i(v/bardbl)dt\n(g) (h)\nFigure 11. Velocity-space signatures of transit-time damping (TTD, top two rows) and\nLandau damping (LD, bottom two rows) sampled from AstroGK turbulence simulation with\n0.25⩽k⊥ρi⩽7.75,Ti/Te= 1, and βi= 0.3. The correlation interval is set as τωA= 6.4.\nThe left column presents gyrotropic plane ( v∥, v⊥) signatures, and the right column features the\ntimestack plots of the v⊥−integrated reduced correlation; both following the layout format of\nFigure 10. The resonant parallel phase velocity ranges are marked by the four vertical dashed\nlines at v∥/vti=±1.832 and v∥/vti=±2.373.Transit-Time Damping 23\n0.0 0.5\n∂Wi/∂t7.510.012.515.017.520.0tωA\n−2 0 2123v⊥/vtiFrame: 110/286 tωA= 12.119CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.04−0.020.000.020.04\n−2 0 2\nv/bardbl/vti0.000.05CδB/bardbl,i(v/bardbl)\n−2 0 210.012.515.017.5tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.09−0.06−0.030.000.030.060.09\n−2 0 2\nv/bardbl/vti0.00.5/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(a) (b)\n−2 0\n∂Wi/∂t7.510.012.515.017.520.0tωA\n−2 0 2123v⊥/vtiFrame: 97/286 tωA= 11.518CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.2−0.10.00.10.2\n−2 0 2\nv/bardbl/vti−0.20.0CE/bardbl,i(v/bardbl)\n−2 0 210.012.515.017.5tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−0.3−0.2−0.10.00.10.20.3\n−2 0 2\nv/bardbl/vti−101/integraltext\nCE/bardbl,i(v/bardbl)dt\n(c) (d)\nFigure 12. Velocity-space signatures of transit-time damping (TTD, top row) and\nLandau damping (LD, bottom row) sampled from AstroGK turbulence simulation with\n0.25⩽k⊥ρi⩽7.75,Ti/Te= 1, and βi= 3. The correlation interval is set as τωA= 6.4.\nThe left column presents gyrotropic plane ( v∥, v⊥) signatures, and the right column features the\ntimestack plots of the v⊥−integrated reduced correlation; both following the layout format of\nFigure 10. The resonant parallel phase velocity ranges are marked by the four vertical dashed\nlines at v∥/vti=±0.583 and v∥/vti=±0.936.\npendicular wavenumbers with k⊥ρi≲4. At all probes in the simulation, we find bipolar\nvelocity-space signatures of positive energy transfer by TTD to ions at both positive\nand negative parallel velocities near the expected range 0 .58≲|v∥/vti|≲0.94, with a\ntypical case illustrated in Figure 12, showing (a) the gyrotropic velocity-space signature\nCδB∥,i(v∥, v⊥;τ) and (b) the timestack plot CδB∥,i(v∥, t;τ) indicating that these signa-\ntures are persistent in time. On the other hand, analyzing the signatures of LD using\n(c) the gyrotropic velocity-space signature CE∥,i(v∥, v⊥;τ) and (d) the timestack plot\nCE∥,i(v∥, t;τ), the pattern of energy transfer to ions is widely variable, with ions losing\nenergy more than gaining energy, in agreement with the expectations from the linear\ndispersion relation. During the majority of the time when clear reversed bipolar patterns\nare visible for LD, they appear to have zero crossings that are close to the expected range\nof parallel resonant velocities.\nSince the field-particle correlation analysis of our single-wave simulations in §3.3 shows\nthat the velocity-space signatures of TTD and LD appear near the resonant parallel ve-24\n1 2 3\nβi−5.0−2.50.02.55.0Ion Energization\nWi,TTD/Wi\nWi,LD/Wi\nFigure 13. Ratio of the change of the ion kinetic energy density due to TTD and LD to the\ntotal change of the ion kinetic energy density during the analysis time, both averaged over all\n24 probes, plotted against βi. The error bars represent the standard deviations calculated across\nall probes.\nlocities of the waves, as illustrated in Figure 5, it raises the question of why the velocity-\nspace signatures in the βi= 0.3 simulation in Figure 11 often do not appear in the\nexpected range of resonant velocities. There exist several possible explanations for this\nfinding. First, it is important to emphasize that the velocity-space signature created by\nthe field-particle correlation analysis shows the net rate of energy transfer to or from ions\nas a function of velocity space. At velocities v/vti>1 there are fewer ions in the un-\nderlying Maxwellian equilibrium velocity distribution, so we may expect the net energy\ntransfer to the ions at those suprathermal velocities to be smaller than the transfer to\nthe large number of ions in the core of the distribution at v/vti≲1. Due to the predom-\ninantly Alfv´ enic nature of fluctuations in space plasma turbulence (Tu & Marsch 1995;\nSchekochihin et al. 2009; Bruno & Carbone 2013), we focus on the collisional damping\nrates of Alfv´ enic fluctuations by TTD and LD in this study. However, the plasma tur-\nbulence in our gyrokinetic simulations is broadband and may contain slow magnetosonic\nfluctuations that arise from nonlinear couplings among the turbulent fluctuations. Al-\nthough slow magnetosonic waves are not generated by nonlinear couplings among the\ndominantly Alfv´ enic fluctuations in the MHD limit at k⊥ρi≪1 (Schekochihin et al.\n2009), at the ion kinetic scales k⊥ρi∼1 it is possible that energy can be nonlinearly\ntransferred into slow magnetosonic fluctuations. These kinetic slow wave fluctuations\nmay have a different parallel phase velocity and thereby mediate damping of the turbu-\nlent energy through either TTD or LD. Another possibility is that nonlinear transit-time\ndamping may occur, whereby the beat mode fluctuations (which are not natural wave\nmodes of the system)—arising from nonlinear interactions between Alfv´ enic fluctuations\nwith different frequencies and wave vectors—can have an effective phase velocity that\nfalls in the core of the velocity distribution, leading to efficient net energy transfer to\nthe ions via collisionless wave-particle interactions. In closing, a final point to empha-\nsize is that the clear bipolar signatures of negative energy transfer from ions by TTD in\nFigure 11(a) and of positive energy transfer to ions by LD in Figure 11(e) for βi= 0.3\nsuggest that a resonant energy transfer mechanism is governing the energization in these\nsimulations; the details of this turbulent damping mechanism, which may include non-\nlinear couplings to other linear wave modes or nonlinear beat modes, are clearly a ripe\navenue for exploration in future workTransit-Time Damping 25\nTo determine the net energy density transfer to or from ions mediated by TTD or LD,\nwe integrate the energy density transfer rate due to each mechanism over the duration of\neach turbulence simulation (after the energy spectra have reached a statistically steady\nstate, and average over all 24 probes for each simulation, yielding the change in the ion\nenergy density due to each mechanism, denoted as Wi,TTD andWi,LD. We compute\nthe ratio of these ion energy density changes by each mechanism to the total ion energy\ndensity change Wi=Wi,TTD +Wi,LD, and plot Wi,TTD /Wi(green) and Wi,LD/Wi(blue)\nversus the ion plasma beta βiin Figure 13. Variability in the energy density changes due\nto TTD and LD is indicated by error bars on each point, computed using the standard\ndeviation of the time-integrated energy density changes at all 24 probes. The red solid\nline at 1 presents the sum of Wi,TTD /WiandWi,LD/Wi, and the black dashed line\nat zero highlights whether a mechanism yields a net positive transfer to ions or net\nnegative energy transfer from ions. For βi= 0.3, the ion energization is dominated by\nLD, with TTD yielding a small and slightly negative energy transfer, consistent with the\nexpectations from the linear dispersion relation in Figure 8(a). For βi= 1, both LD and\nTTD yield net positive energization of the ions, as expected from the linear dispersion\nrelation in Figure 8(b), but LD dominates over TTD once more. At βi= 3, on the other\nhand, TTD mediates the positive energy transfer to the ions, while LD counteracts with\nenergy transfer from the ions back to the waves; in this higher βicase, the time-integrated\nenergy transfer varies much more widely from probe to probe, as indicated by the larger\nstandard deviation, compared to the unity or low βicases.\n5. Conclusion\nTransit-time damping is a well-known mechanism for the resonant collisionless damp-\ning of electromagnetic waves exhibiting variations of the magnetic field magnitude along\nthe mean magnetic field direction, mediated by the magnetic mirror force. This mecha-\nnism has been proposed as a possible means for removing energy from the fluctuations\nin weakly collisional plasma turbulence, but to date there exists little direct evidence\nclearly showing the damping of turbulence via transit-time damping. Here we employ\nthe recently developed field-particle correlation technique to use measurements of the\ngradient of the magnetic field magnitude and the ion velocity distribution at a single\npoint to determine a velocity-space signature that can be used to identify definitively\nthat transit-time damping plays a role in the damping of plasma turbulence.\nWe first derive the particular mathematical form of the field-particle correlation for\nthe rate of energy transfer due to transit-time damping in §2.1, and then we predict the\nvelocity-space signature of the rate of change of phase-space energy density due to transit-\ntime damping using a simple model in §2.2. Next, we perform gyrokinetic simulations of\nsingle kinetic Alfv´ en waves to determine the resulting velocity-space signature of transit-\ntime damping numerically, confirming the qualitative features of our prediction, and\npresenting the first key result of this study: the gyrotropic velocity-space signature of\ntransit-time damping in Figure 4(a).\nWe contrast the velocity-space signature of transit-time damping with the known bipo-\nlar velocity-space signature of Landau damping, showing the same bipolar pattern of\nphase-space energy density loss below and gain above the resonant parallel phase veloc-\nity, but the transit-time damping signature does not extend down to v⊥→0 because\nit is mediated via the magnetic moment of the charged particle µ=mv2\n⊥/(2B); thus,\nsignatures of transit-time damping and Landau damping can be distinguished in gy-\nrotropic velocity space by examining the behavior at the resonant parallel phase velocity\nasv⊥→0. Furthermore, we find the unexpected result that transit-time damping can26\nlead to a net loss of ion energy over the period of the wave for βi<1 and Landau\ndamping can lead to a net lossof ion energy over the period of the wave for βi>1.\nThis surprising result is explained, however, by examining the separate contributions\nof transit-time damping and Landau damping to ion damping from the linear Vlasov-\nMaxwell dispersion relation: the net effect of transit-time damping and Landau damping\ncombined for a plasma with a Maxwellian equilibrium ion velocity distribution always\nleads to a net damping of the wave and net gain of energy by the ions.\nNext, we perform three gyrokinetic simulations of weakly collisional plasma turbulence\nwith three values of βi= 0.3,1,3 to seek the velocity-space signature of transit-time\ndamping in the damping of the strong turbulent fluctuations. In the βi= 1 turbulence\nsimulation, we indeed find a velocity-space signature of transit-time damping as shown\nin Figure 10(a), indicating that this mechanism does indeed play a role in the dissipation\nof kinetic plasma turbulence, along with confirming previously demonstrated signatures\nof Landau damping with ions in Figure 10(c). This second key result of this paper shows\nclearly the transit-time damping does serve to damp the fluctuations in weakly collisional\nplasma turbulence.\nThe relative strength of transit-time damping and Landau damping is predicted to be a\nstrong function of βi(Quataert 1998), so we analyze our βi= 0.3 and βi= 3 simulations\nto confirm this prediction. For βi= 3, we indeed find signatures of transit-time damping\nin the predicted range of resonant parallel phase velocities in Figure 12(a), but Landau\ndamping signatures vary widely, with both positive and negative energy transfer rates\nto the ions, and a negative overall average consistent with expectations from the linear\ndispersion relation in Figure 8(c). For βi= 0.3, however, we discover puzzling bipolar\nvelocity-space signatures of negative energy transfer but with a zero-crossing well below\nthe parallel phase velocity of kinetic Alfv´ en waves. This may indicate that energy transfer\nvia transit-time damping is occurring through alternative wave modes, such as kinetic\nslow magnetosonic fluctuations, or through nonlinear transit-time damping via beat wave\nmodes that are generated by nonlinear interactions among the turbulent fluctuations.\nThese possibilities will be explored in future work.\nDetermining the integrated change of ion kinetic energy density due to transit-time\ndamping and Landau damping as a function of βifrom the three simulations, we find\nresults in Figure 13 that are generally consistent with the expectations from the linear\ndispersion relation: (i) at βi= 0.3, transit-time damping is small and slightly negative,\nwhile Landau damping is about an order-of-magnitude larger and positive; (ii) at βi= 1,\nboth transit-time damping and Landau damping are positive, but again Landau damp-\ning is about an order-of-magnitude larger than transit-time damping; and (iii) at βi= 3,\ntransit-time damping is large and positive while Landau damping is somewhat smaller\nand negative. Note that despite one of the mechanisms possibly leading to a net negative\ntransfer of energy from ions to waves, it is always the subdominant mechanism that is\nnegative, so the net effect of the sum of both of these n= 0 Landau resonant collision-\nless wave-particle interactions (transit-time damping and Landau damping) is always a\ndamping of the turbulence for equilibrium Maxwellian velocity distributions.\nAcknowledgements\nNumerical simulations were performed using the Extreme Science and Engineering\nDiscovery Environment (XSEDE), which is supported by National Science Foundation\ngrant number ACI-1548562, through allocation TG-PHY090084.Transit-Time Damping 27\nFunding\nSupported by NASA grants 80NSSC18K0643, 80NSSC18K1217, and 80NSSC18K1371\nand NSF grant AGS-1842561.\nDeclaration of interests\nThe authors report no conflict of interests.\nAppendix A. Explicit Form of Landau Damping and Transit-Time\nDamping Terms in Nonlinear Gyrokinetics\nThe nonlinear, collisionless gyrokinetic equation (Howes et al. 2006) can be manipu-\nlated into a form in which the terms governing Landau damping (LD) and transit-time\ndamping (TTD) are readily apparent. We begin with the nonlinear, collisionless gyroki-\nnetic equation in cgs units, Eq. (25) in Howes et al. (2006),\n∂hs\n∂t+v∥∂hs\n∂z+c\nB0[⟨χ⟩Rs, hs] =qsF0s\nT0s∂⟨χ⟩Rs\n∂t(A 1)\nwhere the nonlinear term is expressed in the Poisson bracket, defined by\n[U, V] =ˆz·\u0014∂U\n∂Rs×∂V\n∂Rs\u0015\n=∂U\n∂X∂V\n∂Y−∂U\n∂Y∂V\n∂X(A 2)\nwhere the guiding center coordinates are given by Rs= (X, Y, z ). The gyroaverage of\na given quantity at the guiding center position for a particle of species sis denoted\nby⟨. . .⟩Rs. The gyrokinetic potential is defined by χ(r, t) =ϕ−v·A/c, where ϕ(r, t)\nis the scalar electrostatic potential and A(r, t) is the vector potential. In this formula-\ntion, the total velocity distribution function for species sis separated into fs(r,v, t) =\nF0s(v) + (−qsϕ(r, t)/Ts)F0s(v) +hs(Rs, v∥, v⊥, t) +O(ϵ2), where F0s(v) is the spatially\nhomogeneous and temporally constant equilibrium Maxwellian velocity distribution and\nhs(Rs, v∥, v⊥, t) is the perturbed gyrokinetic distribution function at the particle guiding\ncenter position Rs(independent of gyrophase θin cylindrical velocity space).\nWe transform from the perturbed gyrokinetic distribution function hsto the comple-\nmentary perturbed gyrokinetic distribution function gs(Schekochihin et al. 2009), given\nby\ngs(Rs, v∥, v⊥, t) =hs(Rs, v∥, v⊥, t)−qsF0s\nT0s\u001c\nϕ−v⊥·A⊥\nc\u001d\nRs, (A 3)\nsubstituting for hseverywhere in the nonlinear gyrokinetic equation (A 1). After some\nsimplification, the equation can be rearranged to obtain\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] =−qsF0s\nT0s∂\n∂t\u001cv∥A∥\nc\u001d\nRs(A 4)\n−qsF0s\nT0sv∥∂\n∂z\u001c\nϕ−v⊥·A⊥\nc\u001d\nRs+qsF0s\nT0sc\nB0\"\u001cv∥A∥\nc\u001d\nRs,\u001c\nϕ−v⊥·A⊥\nc\u001d\nRs#\nWe can rearrange the terms on the right-hand side to obtain a more physically illumi-28\nnating form,\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] (A 5)\n=qsF0s\nT0sv∥\"\n−∂\n∂z⟨ϕ⟩Rs−1\nc∂\nA∥\u000b\nRs\n∂t#\n+qsF0s\nT0sv∥1\nB0\" \n∂\nA∥\u000b\nRs\n∂Rs׈z!\n·−∂⟨ϕ⟩Rs\n∂Rs#\n+qsF0s\nT0sv∥∂\n∂z\u001cv⊥·A⊥\nc\u001d\nRs+qsF0s\nT0sv∥1\nB0\" \n∂\nA∥\u000b\nRs\n∂Rs׈z!\n·1\nc∂⟨v⊥·A⊥⟩Rs\n∂Rs#\nUsing the following relations,\n∂\nA∥\u000b\nRs\n∂Rs׈z=⟨δB⊥⟩Rs, (A 6)\nqs\nc⟨v⊥·A⊥⟩Rs=−1\n2qs\ncv2\n⊥\nΩs⟨δB∥⟩Rs=−µs⟨δB∥⟩Rs, (A 7)\n−∂\n∂z⟨ϕ⟩Rs−1\nc∂\nA∥\u000b\nRs\n∂t=\nE∥\u000b\nRs, (A 8)\n−∂⟨ϕ⟩Rs\n∂Rs=⟨E⊥⟩Rs, (A 9)\nwe can simplify the result to obtain\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] (A 10)\n=qsF0s\nT0sv∥\u0012\nˆz+⟨δB⊥⟩Rs\nB0\u0013\n· ⟨E⟩Rs+F0s\nT0sv∥\"\n−µs\u0012\nˆz+⟨δB⊥⟩Rs\nB0\u0013\n·∂\nδB∥\u000b\nRs\n∂Rs#\nFinally, to put this into a more concise form, we recognize that the direction of the total\nmagnetic field (including the perturbation) to the first order is B=B0ˆz+⟨δB⊥⟩Rs, so\nwe can define the unit vector of the total magnetic field direction as ˆb\nˆb=B0ˆz+⟨δB⊥⟩Rs\nB0. (A 11)\nWith this final simplification, we obtain the final result for the nonlinear gyrokinetic\nequation,\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] =qsF0s\nT0sv∥ˆb· ⟨E⟩Rs−F0s\nT0sv∥µsˆb· ∇Rs\nδB∥\u000b\nRs(A 12)\nThis equation has a simple physical interpretation with respect to work done on the\ndistribution functions by the fields: the first term on the right-hand side is the effect of\nLandau damping by electric field parallel to the total magnetic field; and the second term\non the right-hand side is the effect of transit-time damping by the magnetic mirror force\ndue to the gradient of the magnetic field magnitude along the total magnetic field, which\nto lowest order is just due to the parallel magnetic field perturbations, as shown in (2.6).\nNote also that the nonlinear term involves interactions between the electromagnetic fields\nand the plasma particles, but when integrated over all guiding-center space Rs, it leads\nto zero net energy change.Transit-Time Damping 29\nAppendix B. Gyrokinetic Form of the Field-Particle Correlation\nIn gyrokinetics, a form of conserved energy inspired from the definition of entropy is\ncalculated by multiplying the complementary perturbed gyrokinetic distribution function\ngsbyT0sgs/F0sand integrating over all velocity and physical space (Howes et al. 2006;\nBrizard & Hahm 2007; Schekochihin et al. 2009; Li et al. 2016; Howes et al. 2018). Using\na similar approach, we can obtain an energy equation for the gyrokinetic phase-space\nenergy density ws(Rs, v∥, v⊥, t) =Tsg2\ns/(2F0s) by multiplying (A 12) by T0sgs/F0sto\nobtain\n∂ws\n∂t+v∥∂ws\n∂z+T0sc\nB0F0s\u0014\n⟨χ⟩Rs,g2\ns\n2\u0015\n=v∥ˆb· ⟨qsE⟩Rsgs−v∥µsˆb· ∇Rs\nδB∥\u000b\nRsgs(B 1)\nIn this formulation, the gyrokinetic form of the field-particle correlation for Landau\ndamping would be given by\nCE∥,s(R0,s, v∥, v⊥, t) =1\nτZt+τ/2\nt−τ/2v∥ˆb· ⟨qsE⟩Rsgsdt′, (B 2)\nand for transit-time damping would be given by\nCδB∥,s(R0,s, v∥, v⊥, t) =−1\nτZt+τ/2\nt−τ/2v∥µsˆb· ∇Rs\nδB∥\u000b\nRsgsdt′. (B 3)\nNote that the energy transfer in the nonlinear case is simply “linear” collisionless\ndamping occurring along the local total magnetic field direction (which is a nonlinear\ncorrection from the equilibrium magnetic field direction B0=B0ˆz). 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Lett. 114(6), 065002."    },    {        "title": "2104.10918v1.Impact_of_Fe___80__B___20___insertion_on_the_properties_of_dual_MgO_perpendicular_magnetic_tunnel_junctions.pdf",        "content": "1\nImpact of Fe 80B20 insertion on the properties of dual-MgO perpendicular \nmagnetic tunnel junctions \nEnlong Liu1, Taeyoung Lee2 and Hyunsoo Yang1 \n1 Department of Electrical and computer Engineering, National University of Singapore, \n117576, Singapore \n2 GLOBALFOUNDRIES Singapore Pte. Ltd., Singapore 738 406, Singapore \nE-mail: eleyang@nus.edu.sg \n\nAbstract \nWe explore the impact of Fe 80B20 inserted at both Co 20Fe60B20/MgO interfaces of dual-MgO \nfree layers (FLs) in bottom-pinned magnetic tunne l junctions (MTJs). MTJ stacks are annealed \nfor 30 min at 350 °C and 400 °C in a vacuum after film deposition. Current-in-plane tunneling \nmeasurements are carried out to characteri ze magnetotransport properties of the MTJs. \nConventional magnetometry measurements and ferromagnetic resonance are conducted to \nestimate the saturation magnetization, the effective perpendicular anisotropy field and the Gilbert damping of dual-MgO FLs as a function of the Fe\n80B20 thickness and annealing \ntemperatures. With ultrathin Fe 80B20 (0.2  0.4 nm)  inserted, perpendicular magnetic \nanisotropy (PMA) of FLs increases with si milar tunnel magneto-resistance (TMR) and low \ndamping values. As Fe 80B20 layer thickness further increases (0.6  1.2 nm), both TMR and \nPMA degrade, and damping increases dramatically. This study demonstrates a novel approach \nto tune properties of MTJ stacks with dual-MgO FLs up to 400 °C annealing, which enables \nMTJ stacks for various applications. \n\n1. Introduction \nMagnetic tunnel junctions (MTJs) with perpendi cular magnetic anisotropy (PMA) have been \nstudied in the recent decades as the crucial el ement for next generation memory applications, \nsuch as spin-transfer-torque and spin-orb it-torque magnetic random access memory \n(STT/SOT-MRAM), due to their non-volatility, en ergy effectiveness, high endurance, and \nscalability [1–3]. Many efforts have been made in the engineering of the data-storage layer in MTJs, i.e. the free layer (FL), whose magnetic moment can be switched by the writing current. \nEspecially, dual-MgO FLs with a structure as MgO/CoFeB/spacer/CoFeB/MgO have been under intense development [4–6]. On the one hand, dual-MgO FLs can provide high PMA and \nlow damping to guarantee the high thermal stability and low switching current, after scaling-\ndown of MTJ devices [7]. On the other hand, the aformentioned properties of dual-MgO FLs \ncan be maintained after post-annealing up to 400 °C, which is required for the CMOS back-end-of-line (BEOL) process [8]. \nTo further optimize dual-MgO FLs performanc e, previous studies focused on different \ntopics. Among them, non-magnetic spacer engineering and element composition effects  have \ndrawn lots of interest. Researches on non-ma gnetic spacer sandwiched between two CoFeB \nlayers in dual-MgO FLs have been widely conducte d. Materials such as Mo [9–11], Ta [8,12], \nand W [13–15] were explored as the spacer to identify its impact on PMA and damping before 2\nand after annealing. Other works examined the effect of element (Fe or B) composition in \nCoFeB layers in MTJ stacks on several parameters, including tunnel magneto-resistance \n(TMR) [16], PMA [17–19], and annealing stability  [20]. It has been demonstrated that a \nthickness gradient in the B content can modify the properties of CoFeB/MgO bilayer system \nsuch as damping and anisotropy [21]. PMA of th e FL was also reported to be improved with \nincreasing the Fe composition in CoFeB, on which its TMR is almost independent [22].\nHowever, it is still an open question how a gradient of Fe in dual-MgO FLs impacts the overall \nproperties of MTJ stacks. In such a case, th e PMA of dual-MgO FLs would benefit from an \nincreased Fe concentration, while the TMR of the MTJs is expected to be improved due to the \nformation of Fe/MgO/Fe interface after annealing [23].  \nHere we propose an insertion of ultrathin Fe 80B20 (hereafter FeB) at the interface of \nCo20Fe60B20/MgO in dual-MgO FLs to achieve tunable magnetic properties and annealing \nstability. By optimizing the thickness of the FeB insertion layers at both CoFeB/MgO \ninterfaces, a large PMA can be obtained after 350 °C annealing and further improved at 400 \n°C, which is accompanied with a low damping constant. The result of a low saturation \nmagnetization, large anisotropy field and low damping at the same time in the FeB-inserted dual-MgO FLs makes it promising for low switching currents in MTJ devices without reducing \nthe thermal stability [24]. In addition, the tunability of FL performance by FeB insertion enlarges its potential for various spintronic appl ications where CoFeB-based MTJs are present, \nsuch as SOT-MRAM, spin logic devices and STT nano-oscillators. \n2. Experimental \nBottom-pinned perpendicular MTJs with [Co/Pt ] multilayers as a perpendicular synthetic \nantiferromagnet (p-SAF) were in-situ  deposited at room temperature by magnetron sputtering \non W/Ru/W/Ru/W bottom electrodes (BE) and ca pped by the Ta/Ru top electrode (TE) in a \nULVAC Magest S200 multi-chamber machine. All samples were first annealed with a 0.5 T \nmagnetic field perpendicular to film plane in a magnetic vacuum annealing oven at 350 °C for 30 min. The TMR and resistance-area product (RA) of the MTJ stack was measured via current-\nin-plane tunnelling method (CIPT) [25]. The hyst eresis loops of blanket stacks were measured \nby a vibrating sample magnetometer (VSM) with the magnetic field perpendicular to the \nsample plane. Field-modulated ferromagneti c resonance (FMR) measurements with the \nfrequency range of 10-25 GHz were conducted to ex tract the resonance field and linewidth of \nthe FL versus the frequency, from which the effective perpendicular anisotropy field and Gilbert damping of the FL can be estimated.  All FMR measurements were conducted with \nsamples placed film-side down on a coplanar waveguide in an electromagnet with a field range \nup to 0.5 T and perpendicular to the sample pl ane.  The same batch of samples were then \nannealed at 400 °C for 30 min to study the annealing impact. \n3. Results and discussion \n3.1 Stack characterization without FeB insertion \nThe detailed stack structure of MTJs used in this  study is provided schematically in figure 1(a). \nIt consists of (thickness in nm): \n Hard layer (HL): Pt (5)/[Co (0.25)/Pt (0.2)]/Co (0.6) \n Reference layer (RL): Co (0.6)/Pt (0.2)/ Co (0.3)/Pt (0.2)/Co (0.5)/W (0.3)/Co\n20Fe60B20 \n(0.8) 3\n Free layer (FL): Co 20Fe60B20 (1.2)/W (0.4)/ Co 20Fe60B20 (0.8). \n \nThe FL is sandwiched between the MgO tunnel barrier and the 2nd MgO layer to form a dual-\nMgO structure. The magnetic hysteresis  loop of  the full stack in figure 1(b) indicates good \nPMA in each functional layer after 350 °C annealing. From the minor hysteresis loop shown \nin figure 1(c), PMA of the FL is still maintained  after 400 °C annealing, and the coercive field \nalso increases slightly. The reduction of the saturation magnetization per area ( 𝑀௦∙𝑡) of the \nFL after 400 °C annealing can be attributed to magnetic dead layer formation, which will be \ndiscussed in the following sections.   \nAfter 400 °C annealing, a sloped plateau is observed around 500 mT in the red curve in \nfigure 1(b), indicating a decreased PMA in the RL  [26]. In addition, the TMR of the stack is \nreduced from 110% to 50%. Since the FL PMA is maintained, the TMR reduction is attributed \nto a PMA loss in the RL. Thus, the impact of Fe B insertion in dual-MgO FLs on TMR is studied \nin stacks after 350 °C annealing. \n3.2 Impact of FeB insertion on TMR and RA \nFigure 2(a) and (b) show schematically the FeB insertion position in the dual-MgO FL. Top \nFeB is inserted between the 2nd MgO layer and CoFeB above W spacer, while bottom FeB is \nbetween the MgO tunnel barrier and CoFeB below W spacer. To eliminate the difference in \nthe thickness of FL after insertion, the total thickness of FeB insertion plus remaining CoFeB \nis kept at 0.8 nm and 1.2 nm for layers above  and below W spacer, respectively. As such, the \nthickness of the top FeB insertion layer is chosen  as 0.2, 0.4, 0.6 and 0.8 nm. For the bottom \nFeB insertion layer, its thickness options are 0.2, 0.4, 0.6, 0.8, 1.0 and 1.2 nm. \nThe TMR and RA values as a function of the FeB insertion layer at different CoFeB/MgO \ninterfaces are summarized in figure 2(c) and (d), respectively. The trend for both TMR and RA \nis similar regardless of FeB insertion position. The TMR value is similar for the thickness of FeB < 0.4 nm, but the RA value reduces for thin FeB insertion < 0.6 nm and increases with \nthicker FeB. In addition, it is found that top FeB insertion leads to a more pronounced RA \nreduction. This phenomenon indicates that with top FeB insertion, the contribution to RA from \nthe 2\nnd MgO layer can be reduced. Since the MgO tunnel barrier (0.85 nm) is thicker than the \n2nd MgO layer (0.7 nm), its RA change due to bottom FeB insertion is not as significant as in \nthe top FeB case. Overall, a similar TMR value at low RA is realized when ultrathin FeB (0.2 \n 0.4 nm) is inserted, suggesting that th e formation of Fe/MgO interface benefits \nmagnetotransport properties of MTJ stacks. \nThe TMR starts to drop when FeB thicker than 0.6 nm was inserted at the bottom \nCoFeB/MgO interface, but the RA value does not show any significant change. This TMR drop \nis attributed mainly to a lower spin polarization when thicker FeB replaces CoFeB in the FL \n[27]. \n3.3 Impact of FeB insertion and annealing on FL magnetic properties \nIn figure 1(c), an example of hysteresis loop of  the FL is shown. The saturation magnetization \nper area of the FL can be estimated by using 𝑀௦∙𝑡ൌ𝑚 /𝐴, where 𝑚 is the magnetic moment, \n𝑡 is the thickness of FL, and 𝐴 is the sample area. The effective anisotropy field ( 𝜇𝐻) is \nderived from FMR measurements, as shown by exemplary data in figure 3(a) from dual-MgO \nFLs without FeB insertion, with 0.2 nm top FeB insertion and 0.2 nm bottom FeB insertion, all \nafter 350 C annealing. For each sample, the power absorption by FL versus the applied field 4\nscan is measured at various frequencies, fro m which the ferromagne tic resonance field ( 𝜇𝐻௦) \nand linewidth ( 𝜇Δ𝐻) is estimated. The relation between 𝜇𝐻௦ and frequency 𝑓 is described \nby the Kittel equation for the out-of-plane applied field [28]: \n𝑓ൌఊ\nଶగ൫𝜇𝐻௦𝜇𝐻൯                             (1) \nwhere 𝛾 is the gyromagnetic ratio. From figure 3(a), the x-intercept is the 𝜇𝐻. From 𝑀௦∙𝑡 \nand 𝜇𝐻, the effective perpendicular anisotropy energy can be calculated as \n𝐾∙𝑡ൌଵ\nଶ𝜇𝐻ሺ𝑀௦∙𝑡ሻ.                        (2)  \nFigure 4 summarized the results calculated by the above method. First, 𝑀௦∙𝑡 of FLs can be \ndescribed simply by 𝑀௦∙𝑡ൌ𝑀 ௦ி∙2.0െ𝑡ி∙ሺ𝑀௦ிെ𝑀௦ிሻ, where 𝑡ி is the \ninserted thickness of FeB. It is found in figure 4(a) and (b) that 𝑀௦∙𝑡 decreases with thicker \ninserted FeB, i.e. the slope is negative. Thus, in our stack 𝑀௦ி is smaller than 𝑀௦ி. Next, \nfor the top FeB insertion case in figure 4(a), the amount of change in 𝑀௦∙𝑡 at the same FeB \ninsertion thickness after 350 °C annealing is larger  than that of the bottom insertion case in \nfigure 4(b). This indicates that the top FeB inserted layer is more damaged than the bottom \ninserted FeB. Another major difference between top and bottom insertion cases is a larger 𝑀௦∙\n𝑡 reduction after 400 ºC annealing compared to that after 350 °C annealing, with thick bottom \nFeB inserted (> 0.4 nm). As the bottom-inserted FeB is less damaged, it still reduces the 𝑀௦∙𝑡 \nvalue on further annealing at 400 ºC. However, for the top inserted case, there is not much \nchange after 400 ºC annealing, supporting that the top FeB insertion layer is damanged, mainly \ndue to the 2nd MgO deposition.  \nThe FeB thickness dependence of  𝜇𝐻 and 𝐾∙𝑡 can be discussed together. In the top \nFeB insertion case after 350 ºC annealing, 𝜇𝐻 in figure 4(c) increases monotonically with \nFeB insertion. Due to the reduction in 𝑀௦∙𝑡, however, this increase cannot lead to a higher \nPMA. In figure 4(e), the PMA of FL after 350 ºC  annealing changes little, and even slightly \ndecreases with thicker FeB. However, after 400 ºC annealing, 𝜇𝐻 overall increases without \nany clear dependence on the FeB thickness, and the PMA of FL reaches the maximum value \nwhen 0.2 nm FeB is inserted.  \nOn the other hand, the behavior of 𝜇𝐻 and 𝐾∙𝑡 in bottom-inserted FeB cases is \ndifferent. In general, the PMA can be improved more significantly with bottom-inserted FeB \nthan top-inserted FeB cases. In 350 ºC  annealing cases, both 𝜇𝐻 and 𝐾∙𝑡 increase till \n1.0 nm FeB insertion as shown in figure 4(d) and figure 4(f), respectively. While in 400 ºC \nannealing cases, 𝐾∙𝑡 is significantly improved in thin FeB (0.2 – 0.6 nm) cases. To \nsummarize, the impact of FeB on the PMA of FL differs according to the insertion position. \nRegardless of annealing temperature, FeB insertion at the bottom CoFeB/MgO interface \ninduces a larger PMA, probably due to less damage and the formation of Fe/MgO interface. By changing the FeB insertion position and thickness, magnetic properties of dual-MgO FL \ncan be tuned in a wide range.  \nIt can be noticed that the vertical error bar of  anisotropy field is huge when the FeB thickness \nis 0.8 nm in the top insertion case, and 1.0 or 1.2 nm in bottom insertion cases. It reflects that the resonance field is difficult to be determined precisely in those cases, and thus the fits of \nEq.(1) contains large uncertainties. It is probabl y due to a large damping constant in thick FeB \ninsertion cases, which will be discussed in the following section.  \n3.4 Impact of FeB insertion layer on FL damping 5\n \nIn order to evaluate the effect of FeB insertion on the FL damping, FMR measurements are \nconducted. In figure 3(b), 𝜇Δ𝐻 versus the external excitation frequency for the three samples \nwas plotted. The linewidth of the resonance is linear in frequency: \n𝜇Δ𝐻 ൌ 𝜇 Δ𝐻ସగఈ\nఊ𝑓                          (3) \nwhere 𝜇Δ𝐻 is the inhomogeneous linewidth broadening and 𝛼 is the Gilbert damping \ncoefficient.  \nFigure 5 summarizes 𝛼 as a function of top or bottom FeB insertion thickness under different \nannealing conditions. For top FeB insertion (figure 5(a)), the influence of FeB insertion on 𝛼 \nshows a moderate dependence on its thickness after 350 C annealing. For bottom FeB insertion \n(figure 5(b)), however, 𝛼 reaches the minimum at 0.4 nm FeB insertion and increases \ndramatically beyond the measurement range w ith the FeB thickness in both annealing \nconditions. For those cases, the resonance is to o broadened to be resolved, reflecting a very \nlarge damping [21]. It also leads to a huge uncertainty in the resonance field and hence 𝜇𝐻 \ndetermination, as mentioned in the preivous section.  \nAn increase of 𝛼 is observed at 0.6 nm FeB in the 400 C annealing condition. However, a \nreduction in 𝛼 after 400 C annealing is obtained when ultrathin FeB (0.2  0.4 nm) is inserted \nat either interface. This suggests that the annealing treatment has different effects on 𝛼 of the \nsamples with various FeB insertion thicknesses. Perhaps different amount of FeB insertion leads to changes in the Fe concentration, micr ostructures and crystallization of dual-MgO FLs \nafter boron depletion upon annealing and thereby to different damping behaviors [29,30]. \nFinally, Table 1 summarizes and compares MTJ stacks after 350 C annealing with FeB \ninsertion at top, bottom, or both CoFeB/MgO interfaces. As the FeB insertion thickness and \nposition differ, the PMA of the dual-MgO FL can be tuned in a wide range, while the damping \nis almost independent. From the systematic studies on insertion thickness in the previous \nsections, top and bottom FeB are optimized to be 0.2 nm and 0.4 nm, respectively. As a result, \nthe MTJ stack with such FeB insertion at both interfaces in dual-MgO FLs can be engineered \nwith a high TMR, low RA, large 𝐾\n∙𝑡, low 𝑀௦∙𝑡, high 𝜇𝐻, and low damping constant.  \n4. Conclusion \nIn this paper, we explore the impact of Fe 80B20 layer inserted at two interfaces of \nCo40Fe60B20/MgO in dual-MgO FLs in MTJ stacks and its annealing stability. With ultrthin \nFeB (0.2  0.4 nm) inserted at the top or bottom CoFeB/MgO interface, the TMR can be \nmaintained with lower RA values, while the top-FeB insertion results in a more RA drop with \na similar TMR. In both cases, the FL saturation magnetization reduces with increasing the Table1. Comparison of magnetotransport and magnetic proper ties of dual-MgO FLs with different FeB insertion \nafter 350 C 30 min annealing. \n \nFL types TMR RA 𝑀௦∙𝑡 𝜇𝐻 𝐾∙𝑡  \n% m2 10-5 A mT mJ∙m2 10-3 \nMgO/CoFeB(1.2)/W/CoFeB(0.8)/MgO 109.3 8.5 184.0  2.7 320  1 0.294  0.004 12.5  2.6 \nMgO/CoFeB(1.2)/W/CoFeB(0.6)/ FeB(0.2) /MgO 114.9 6.9 177.7  3.1 340  5 0.302  0.007 15.0  4.9 \nMgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.8)/MgO 108.6 8.3 181.2  1.7 349  2 0.316  0.006 11.3  5.0 \nMgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.6)/ FeB(0.2) /MgO 108.2 7.5 175.7  2.6 381  2 0.335  0.005 12.1  0.7 \nMgO/ FeB(0.4) /CoFeB(0.8)/W/CoFeB(0.6)/ FeB(0.2) /MgO 110.5 8.6 168.3  2.9 434  2 0.365  0.006  4.4  0.7 \n 6\ninserted FeB thickness, while the FL effective anisotropy field increases. However, the PMA \nof dual-MgO FLs with FeB inserted at the bo ttom interface shows a larger improvement than \nits top FeB insertion counterpart, even after 400 C annealing. At the same time, the FeB (0.2 \n 0.4 nm) insertion at either interface reduces the damping constant in the FL. By optimizing \nthe FeB insertion layer thickness, the dual-MgO FL with a low saturation magnetization, high \neffective anisotropy field and low damping can be achieved after 400 C annealing. However, \nthe performance degrades if a thicker FeB is used to replace CoFeB in dual-MgO FLs. \nThis study demonstrates a novel approach to tune dual-MgO FL properties other than typical \nboron composition or non-magnetic spacer engineering. By using the FeB insertion layer at \nCoFeB/MgO interfaces, magnetic properties of the FL and the magnetotransportation of MTJs \ncan be engineered in a wide range, which enables MTJs to meet different performance requirements for various spintronic applications. \nAcknowledgements \nThis work was supported by NRF In vestigatorship (NRFI06-2020-0015). \nReferences \n[1] Ikeda S, Miura K, Yamamoto H, Mizunuma K, Gan H D, Endo M, Kanai S, Hayakawa \nJ, Matsukura F and Ohno H 2010 A perpendicular-anisotropy CoFeB-MgO magnetic \ntunnel junction. Nat. 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Express  6 073002 \n\n9\n \nFigure1. (a) Stack layout of blanket MTJs without FeB insertion. Free layer (FL), reference \nlayer (RL) and hard layer (HL) are indicated. Thicknesses of sublayers are shown in nm with \nparentheses. Blanket films were annealed first at 350 C and then at 400 C, both for 30 min. \n(b) Major loop and (c) minor loop of the stack in  (a) measured by VSM after different annealing \nconditions with the magnetic field perpendicular to the sample plane. \n\n \nFigure2. Schematic of FeB insertion at (a) top and (b) bottom CoFeB/MgO interface in dual-\nMgO FL in the stack shown in figure 1(a). The total thickness of inserted FeB plus remaining \nCoFeB is kept at 0.8 nm and 1.2 nm for layers above and below W spacer, respectively. The \nimpact of FeB insertion on TMR (c) a nd RA (d) of the MTJ stacks after 350 C annealing are \nshown. \n\n10\n \nFigure3. (a) The external excitation frequency as a function of ferromagnetic resonance field \nand (b) the linewidth versus frequency of FL in MTJ stacks without FeB insertion (black open \ncircles), with 0.2 nm top FeB insertion (red open triangles), and with 0.2 nm bottom FeB \ninsertion (blue open squares). Solid lines are fits. \n \n \nFigure4. Effect of FeB insertion and annealing conditions on 𝑀௦∙𝑡 ((a) and (b)), 𝜇𝐻 ((c) \nand (d)), and 𝐾∙𝑡 ((e) and (f)). (a), (c) and (e) show th e impact from FeB insertion at the \ntop CoFeB/MgO interface, while (b), (d) and (f) show the impact from bottom interface. \n11\n\n \nFigure5. Gilbert damping as a function of FeB insertion thickness at (a) top interface and (b) \nbottom interface under two annealing conditions. \n\n"    },    {        "title": "2001.11929v2.Dynamo_in_weakly_collisional_nonmagnetized_plasmas_impeded_by_Landau_damping_of_magnetic_fields.pdf",        "content": "Dynamo in Weakly Collisional Nonmagnetized Plasmas Impeded by Landau Damping\nof Magnetic Fields\nIstv\u0013 an Pusztai,1,\u0003James Juno,2Axel Brandenburg,3Jason M. TenBarge,4, 5\nAmmar Hakim,6Manaure Francisquez,7and Andr\u0013 eas Sundstr om1\n1Department of Physics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden\n2Institute for Research in Electronics and Applied Physics,\nUniversity of Maryland, College Park, Maryland 20742, USA\n3Nordita , KTH Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden\n4Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08543, USA\n5Princeton Plasma Physics Laboratory, Princeton, New Jersey 08544, USA\n6Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA\n7Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA\n(Dated: June 29, 2020)\nWe perform fully kinetic simulations of \rows known to produce dynamo in magnetohydrodynam-\nics (MHD), considering scenarios with low Reynolds number and high magnetic Prandtl number,\nrelevant for galaxy cluster scale \ructuation dynamos. We \fnd that Landau damping on the elec-\ntrons leads to a rapid decay of magnetic perturbations, impeding the dynamo. This collisionless\ndamping process operates on spatial scales where electrons are nonmagnetized, reducing the range\nof scales where the magnetic \feld grows in high magnetic Prandtl number \ructuation dynamos.\nWhen electrons are not magnetized down to the resistive scale, the magnetic energy spectrum is\nexpected to be limited by the scale corresponding to magnetic Landau damping or, if smaller, the\nelectron gyroradius scale, instead of the resistive scale. In simulations we thus observe decaying\nmagnetic \felds where resistive MHD would predict a dynamo.\nThe energy density corresponding to the microgauss\n(10\u000010T) magnetic \feld permeating the Universe at\ngalaxy [1] and galaxy cluster [2] scales is comparable\nto that of the turbulent \rows [3] on these scales. This\napproximate equipartition of magnetic and directed ki-\nnetic energies is consistent with the \feld being gener-\nated and maintained by a turbulent dynamo (see Ref. [4]\nand references therein). Small seed \felds are ampli\fed\nby the dynamo until they become dynamically signi\f-\ncant, after which the \feld strength nonlinearly saturates\nin a self-consistent turbulent state. Because of the multi-\nscale and inherently three-dimensional [5, 6] nature of dy-\nnamos, they have almost exclusively been studied within\nthe framework of magnetohydrodynamics (MHD). Al-\nthough MHD is well justi\fed for the modeling of dynamos\nin dense and collisional stellar interiors, it breaks down\nwhen the mean free path of the plasma particles becomes\ncomparable with the scales of interest, such as in galaxy\nclusters.\nRecent e\u000borts have started to shed light on turbulent\ndynamos in the collisionless regime. Using kinetic tools\nfor the ion dynamics and isothermal \ruid models for the\nelectrons, dynamo ampli\fcation of magnetic \felds has\nbeen demonstrated [7]. The role of pressure anisotropy\ninstabilities, such as \frehose and mirror instabilities, has\nbeen shown to be critical for dynamo ampli\fcation [8],\nleading to the development of sharp magnetic \feld line\nfeatures, thereby breaking magnetic moment conserva-\ntion and alleviating the issue of related stringent con-\nstraints [9] on \feld growth. While the role of kinetic ions\nin the context of the dynamo is only just beginning to beexplored, what e\u000bects, if any, kinetic electrons have on\nthe dynamo have yet to be studied.\nIn this Letter, we consider a kinetic electron e\u000bect on\ndynamos: the Landau damping of magnetic \ructuations.\nThis enhances the decay of magnetic perturbations com-\npared to resistive di\u000busion, thereby reducing the range of\nscales where \feld ampli\fcation occurs. We also show that\nthis e\u000bect impedes dynamo \feld ampli\fcation in fully ki-\nnetic simulations of weakly collisional nonmagnetized hy-\ndrogen plasmas. The possibility of Landau damping of\nmagnetic \felds has not received wide attention in the lit-\nerature, except for a few sporadic applications, a\u000becting,\ne.g., the persistence of magnetic \ructuations downstream\nof ultrarelativistic pair-plasma shock waves with conse-\nquences on synchrotron emission in gamma-ray bursts\n[10].\nThe turbulent dynamo is a multiscale problem: Ki-\nnetic energy injected into \rows at the outer scale l0, non-\nlinearly cascades down to viscous scales l\u0017\u0018Re\u00003=4l0,\nwhere the energy is dissipated. The scale separation,\nl0=l\u0017, is characterized by the \ruid Reynolds number,\nRe =u0l0=\u0017, where\u0017is the kinematic viscosity and u0\nis the characteristic \row velocity at scale l0. The dis-\nsipation scale of magnetic \ructuations, below which re-\nsistive di\u000busion of the \felds dominates, is the resistive\nscalel\u0011. A key dimensionless quantity in dynamo the-\nory is the magnetic Reynolds number, Rm = u0l0=\u0011, as\ndynamo \feld ampli\fcation requires a minimum Rm that\ndepends on the properties of the \row. Here \u0011= (\u001b\u00160)\u00001\nis the magnetic di\u000busivity, with the Spitzer conductivity\n\u001b, and the magnetic permeability \u00160. When the mag-arXiv:2001.11929v2  [physics.plasm-ph]  26 Jun 20202\nnetic Prandtl number, Pm = Rm =Re =\u0017=\u0011is large,\nas in galaxies, galaxy clusters, the intracluster medium,\nand in some hot accretion disks [11], then l\u0011\u001cl\u0017, and\nmagnetic \feld growth mostly takes place in the range be-\ntween thel\u0017andl\u0011scales [4]. In astrophysical systems\nof interest, Pm can be extremely large.\nThe physics is kinetic for scales comparable to or\nsmaller than the Coulomb mean free path \u0015. Using\nl\u0017\u0018Re\u00003=4l0and\u0015\u0018Re\u00001l0M0, whereM0is the Mach\nnumber corresponding to u0, for a moderate Re and a\nM0\u00181, we see that \u0015andl\u0017are comparable. Therefore,\nthe scales of interest for Pm \u001d1 are kinetic.\nSeveral processes have been proposed to generate the\nseed \feld for dynamos (see Ref. [12] and references\ntherein). One of the leading candidates is the Biermann\nbattery [13] at ionization fronts in the early Universe,\nthought to produce a typical seed \feld of B\u001810\u000024T\n[14{16]. While galaxy clusters are magnetized down to\nthe resistive scale at current magnetic \feld levels, at the\ntime when the \feld was comparable to that of seed \felds,\nthe electron Larmor radius was comparable to the mean\nfree path. That is, electrons were not magnetized on\nkinetic scales for the Biermann seed case, allowing for\nmagnetic perturbations to be Landau damped.\nWe consider here fully kinetic simulations of spatially\nperiodic \rows, which are known to produce a dynamo in\nMHD simulations. As has been done in several dynamo\nstudies [4, 8, 17], we sacri\fce the \ruid cascade for numer-\nical feasibility and focus on subviscous scales. Accord-\ningly,l0andl\u0017are comparable to our simulation box size\nL0. The simulations employ the kinetic-Maxwell solver\n[18] of the Gkeyll [19] plasma physics simulation frame-\nwork, which applies a discontinuous Galerkin method to\nsolve the kinetic equation\n@tfa+v\u0001rfa+aa\u0001rvfa=C[fa]; (1)\nfor all species a, with mass ma, chargeea, and dis-\ntribution function fa. In the acceleration term, aa=\nfa=ma+ (ea=ma)(E+v\u0002B), the electric and magnetic\n\felds, EandB, are computed from Maxwell's inductive\nequations, and fa(x;t) is an externally prescribed forc-\ning. Inter- and intraspecies Coulomb collisions are mod-\neled by a conservative Dougherty (or Lenard-Bernstein)\noperator [20, 21], C[fa]. The simulations are initialized\nwith Maxwellian electrons ( e) and protons ( i), with tem-\nperatureTa= 1 keV, density na= 2:3\u00021028m\u00003, and\na \row with a characteristic speed u0=M0p\nTe=miand\nM0= 0:35. Our baseline plasma parameters are not rep-\nresentative of astrophysical plasmas, rather they are cho-\nsen to give estimated values of Rm \u001913 (with Spitzer re-\nsistivity) and Re \u00190:64 (with nonmagnetized collisional\nviscosity), thus Pm \u001920 for a box size of L0= 9:73\u0016m\nand an assumed Coulomb logarithm of 10. The collisional\nmean free path is \u0015= 1:25\u0016m.\nFirst, we consider the time-dependent Galloway-\nProctor (GP) \row [22] that produces a fast dynamo (Rm-\n0.0 0.5 1.0 1.5 2.05.×10-81.×10-75.×10-71.×10-65.×10-6\nt/ttEB[J]kineticMHD\nFIG. 1. Volume integrated magnetic energy. Solid lines: ki-\nnetic simulation; dashed lines: resistive MHD induction equa-\ntion. Red, blue, and green correspond to the contributions\nfromx,y, andz\feld components to the total (black). For\nreference, (3 =2)niTiL3\n0= 5:1\u000210\u00003J.\nindependent growth rate, for Rm \u001d1) and requires a low\ncritical Rm,\nuGP(x;t) =u0fsin(k0z+ sin!t) + cos(k0y+ cos!t);\ncos(k0z+ sin!t);sin(k0y+ cos!t)g;(2)\nwherek0= 2\u0019=L 0,!= 2\u0019=tt, andtt=L0=u0\u0019\n9\u000210\u000011s is the turnover time. The \row is sustained\nby exerting a force of fi=Cfmiu(x;t)=tion the ions,\nwith the thermal ion passing time ti=L0=p\n2Ti=mi; we\nsetCf= 1. The magnetic \feld is initialized as Bi=\nB0P\nj6=i;nbij;ncos[nk(xi+'ij;n)], wherebij;nand'ij;n\nare uniform random numbers on [0 ;1],n= 1;2;:::;N\nwithN= 4, andB0= 40 (the thermal electron Larmor\nradius at this \feld strength is 2 :7\u0016m). In addition to\nuGP, the initial electron \row velocity also has a compo-\nnent producing a current consistent with the magnetic\nseed \feld.\nThe value Rm\u001913 is su\u000eciently large for the GP\n\row to produce magnetic \feld growth in resistive MHD.\nIndeed, solving the MHD induction equation @tB=\nr\u0002(u\u0002B) +\u0011r2Bwithu=uGP, using the high order\n\fnite-di\u000berence MHD solver Pencil Code [23] at spatial\nresolutions between 123and 323, we \fnd that, after a\nslight decay, the magnetic \feld starts to grow exponen-\ntially, as shown in Fig. 1 (dashed). Additional MHD sim-\nulations (not shown here) also evolving the \row produce\nsimilar results. However, in the kinetic simulation, the\n\feld energy is observed to monotonically decay (solid).\nThe magnetic energy in the kinetic simulation rapidly de-\nvelops a strongly decaying wave number spectrum (solid\nlines in Fig. 2). In contrast, the spectrum correspond-\ning to the MHD induction equation quickly assumes its\nweakly decaying shape (dashed), which is then preserved\nin the phase of exponential growth (dotted line). The\nkinetic simulations used 12 grid cells in each direction\nof the con\fguration space, 10 in velocity space extend-\ning between\u00003 and 3 times the thermal speed of each3\n1 2 3 4 510-610-510-40.0010.0100.1001\nk/k0EB(k)/EB(0)(k=k 0)\nFIG. 2. Wave number spectra of magnetic energy, EB(k)\n(normalized to its value at k0,t= 0), fort=f0;1;2;:::;6g\u0002\n10\u000011s (lines lightening). Solid lines: kinetic simulation;\ndashed lines: MHD induction equation; dotted line: MHD\ninduction equation in the growing phase t= 2\u000210\u000010s.\nspecies, and employed a set of basis functions of polyno-\nmial order 1, i.e., a resolution equivalent to 24 and 20\ngrid points, respectively, in a \fnite-di\u000berence scheme.\nThe decay of the magnetic \feld energy in the kinetic\nsimulation is caused by Landau damping of the mag-\nnetic \ructuations. To elaborate on this e\u000bect, we per-\nformed decaying magnetic \feld simulations in 1 spatial\nand 2 velocity coordinates, initialized with Bz(x;t= 0) =\nB0cos(kx), and the corresponding current deposited as\na \row of Maxwellian electrons in the ydirection. The\nplasma parameters are similar to the GP \row simula-\ntion, and the simulations use up to 40 spatial and 20\nvelocity cells, with a polynomial order of 2. For an\nelementary magnetic perturbation of this form, resis-\ntive magnetic di\u000busion @tB=\u0011r2Bleads to a decay\nBz/exp(\u0000\rt) = exp(\u0000k2\u0011t). In a weakly collisional\nplasma, i.e., \u0017ei!+0, where\u0017eiis the electron-ion col-\nlision frequency, such a \ructuation decays due to Lan-\ndau damping with a decay rate \r=jkj3c2ve=(p\u0019!2\npe) =\njkj3veme=(p\u0019\u00160nee2) [24], where !pe=p\nnee2=(\u000f0me),\nve=p\n2Te=meis the electron thermal speed, \u0000eand\nneare the electron charge and density, and \u000f0denotes\nthe vacuum permittivity. We would get this decay rate\nfrom resistive di\u000busion, if we replaced \u001b\u00001with a scale-\ndependent e\u000bective resistivity \u001b\u00001\ne\u000b=jkjveme=(p\u0019nee2),\nwhich corresponds to an e\u000bective magnetic di\u000busivity\n\u0011e\u000b\u0018\u0011\u0015=l, where\u0015=ve=\u0017ei, and 2\u0019=l=jkj.\nWe introduce an overall collisionality scaling factor,\nC\u0017, that multiplies all inter- and intraspecies collision\nfrequencies calculated for the given plasma parameters.\nFigigure. 3 shows the C\u0017dependence of \u001b\u00001\ne\u000bthat is cal-\nculated as an instantaneous value of jy=Ey, and is con-\nsistent with the exponential decay rate of current per-\nturbations. At the longest wavelength considered ( L0=\n9:73\u0016m, dark solid curve) the e\u000bective resistivity starts\ndeviating from the Spitzer resistivity below C\u0017= 0:5,\nand forC\u0017!+0 it asymptotes to a collisionality in-\ndependent value determined by Landau damping. As\n0.0 0.2 0.4 0.6 0.8 1.01.×10-95.×10-91.×10-85.×10-8\nCν1/σeffSpitzerL0=8\nL0=4\nL0=2\nL0\nFIG. 3. E\u000bective resistivity, 1 =\u001be\u000b[\nm], as a function of col-\nlision scaling factor C\u0017, for four values of the wavelength of the\ncurrent perturbation, increasing from L0=8 toL0= 9:73\u0016m\n(solid lines darkening). The Spitzer resistivity (dotted line),\nand the collisionless, unmagnetized theoretical limits (dashed\nlines) are also indicated.\nthe wavelength of the perturbations is decreased (lighter\ncurves) the e\u000bective resistivity increases; in particular\nwhenk= 2k0,\u001b\u00001\ne\u000bremains already above the Spitzer\nlevel over the collisionality range plotted. We note that\nperfectly collisionless simulations exhibit an echolike re-\ncurrence of the magnetic \feld energy, unlike the weakly\ncollisional simulations shown here, where a simple expo-\nnential decay is observed.\nThe simple physical picture behind the magnetic \feld\ndecay in the collisionless regime is the following. A cur-\nrent perturbation of wave number kwould, without the\nself-consistent electromagnetic \felds, decay on a time\nscale\u0018(vek)\u00001due to free streaming; however, the cor-\nresponding @tBinduces an electric \feld that inhibits this\ncurrent decay. The induced electric \feld being propor-\ntional to the current can be thought of as an e\u000bective re-\nsistivity, which leads to a di\u000busion, and thus a decay, of\nthe magnetic \feld perturbation. In a collisional plasma,\nthe electric \feld is balanced by collisional friction, result-\ning in a Spitzer response. In the weakly collisional case,\nhowever, the electric \feld is balanced by a viscous stress\ncorresponding to an o\u000b-diagonal element of the electron\npressure tensor, analogously to collisionless reconnection\n[25{27]. This viscous balance is illustrated in Fig. 4(a),\nwhere the ratio of the relevant viscous stress component\nto the electric force is shown as a function of C\u0017for var-\nious wavelengths. In all cases, the small C\u0017limit is close\nto unity, within a small di\u000berence due to electron inertia.\nThe contribution from the viscous stress monotonically\ndecreases with C\u0017as the friction on ions becomes more\nimportant in balancing the electric \feld; at the longest\nwavelength (darkest curve) the viscous stress contribu-\ntion is negligibly small for C\u0017= 1, consistently with the\nSpitzer response observed in Fig. 3.\nFree streaming of electrons across the current pertur-\nbation is inhibited when the electrons are magnetized and\nare thus con\fned to magnetic \feld lines. Therefore, the4\nL0=8\nL0=4\nL0=2\nL0\n(a) (b)\nFIG. 4. Solid lines: The ratio of the relevant component\nof the electron viscous stress and the electric \feld force. In\n(a) the ratio is shown as a function of collision scaling factor\nC\u0017, for four values of the wavelength of the current pertur-\nbation, increasing from L0=8 toL0= 9:73\u0016m (lines darken-\ning). In (b) the ratio is shown as a function of the electron\nmagnetization L0=\u001ae0, where\u001ae0is the electron Larmor ra-\ndius at a \feld strength of B0. Here, the e\u000bective resistivity\nis also shown (dashed curve, normalized to its highest value,\n1:65\u000210\u00008\nm); the wavelength is L0, andC\u0017= 0:05.\nLandau damping of magnetic \feld \ructuations becomes\nunimportant with increasing magnetic \feld strength, as\nillustrated in Fig. 4(b), showing the reduction of the ef-\nfective resistivity with increasing B0(\u001ae0is the electron\nthermal Larmor radius at B0). For lowL0=\u001ae0, the\u001b\u00001\ne\u000b\nis comparable to the theoretical collisionless value from\nLandau damping, and it drops rapidly with increasing\nL0=\u001ae0. As for its relevance in dynamos, when the mag-\nnetic \feld energy grows, the range of scales where Lan-\ndau damping of magnetic \ructuations are important de-\ncreases with the electron Larmor radius.\nNote that accurate interpretation of fully kinetic dy-\nnamo simulations is made di\u000ecult by currents unavoid-\nably driven by the forcing. Even exerting a force on\nions and electrons appropriately scaled by their masses\nleads to a current, as the momentum transport proper-\nties of the two species are di\u000berent (and magnetization-\ndependent); in weakly collisional plasmas, the corre-\nsponding driven current is comparable to that when forc-\ning only acts on the ions. Therefore, a magnetic \feld is\nbeing generated that may be larger than the initial seed\n\felds. This e\u000bect is illustrated in Fig. 5, which shows the\nmagnetic \feld energy in a simulation with a driven, time\nindependent Roberts \row [28]\nuR(x;t) =u0fcos(k0y)\u0000cos(k0z);sin(k0z);sin(k0y)g:\n(3)\nIn these simulations, L0= 1:22\u0016m,B0= 10 T, the\ncollisionality is scaled as C\u0017= 0 (solid) and C\u0017= 0:3\n(dashed), and the \row is more strongly forced Cf= 3,\notherwise the parameters are similar to those of the\nGalloway-Proctor \row simulation. The magnetic \feld\nenergies level o\u000b after an initial growth phase in both\ncases. We \fnd that the \fnal \feld strength is of the size\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.71.×10-115.×10-111.×10-105.×10-101.×10-95.×10-91.×10-8\nt/ttEB[J]FIG. 5. Volume integrated magnetic energy in a forced\nRoberts \row simulation, for C\u0017= 0 (solid) and 0 :3 (dashed).\nRed, blue, and green correspond to the contributions from x,\ny, andz\feld components to the total (black). For reference,\n(3=2)niTiL3\n0= 9:94\u000210\u00006J.\n\u0018eu0ni\u00160L0, which is expected to arise from the forcing\nof the ion \row. Indeed, at the end of the simulations, the\ncurrent density has a form close to uR(as does B, since\nthe \feld is essentially force free). When simulations are\nstarted from a higher initial B0, the magnetic energy de-\ncays down to the same level, where the continuous drive\nis balanced by the e\u000bect of Landau damping and colli-\nsions. For these parameters no dynamo ampli\fcation is\nobserved in the simulation.\nFinally, we consider the implication of Landau damp-\ning on \ructuation dynamos with asymptotically large\nPm. In the MHD framework, l\u0011is estimated by balancing\nthe rate of stretching of magnetic \ructuations at the vis-\ncous scaleu\u0017=l\u0017with the dissipation rate at the resistive\nscale\u0011=l2\n\u0011, yieldingl\u0011\u0018l0Re\u00003=4Pm\u00001=2\u0018l\u0017Pm\u00001=2\n[4]. In a weakly collisional plasma, we may introduce\nthe analogous Landau dissipation scale lL, where mag-\nnetic \feld growth due to stretching at the viscous scale\nbalances decay due to Landau damping. Thus, recall-\ning\u0011e\u000b(l) =\u0011\u0015=l, we balance u\u0017=l\u0017\u0018u0=(l0Re1=2) and\n\u0011e\u000b(lL)=l2\nL\u0018\u0011\u0015=l3\nL\u0018\u0017\u0015=(l3\nLPm). This result, combined\nwith\u0015\u0018l0M0=Re, yields the estimate\nlL\u0018l0M1=3\n0\nRe5=6Pm1=3\u0018l\u0017M1=3\n0\nRe1=12Pm1=3: (4)\nWhen Re1=2=M2\n0\u001cPm, as for instance in galaxy clus-\nters,l\u0011\u001clL, implying that the range of scales over which\nmagnetic \feld growth can occur is reduced compared to\nthe prediction of resistive di\u000busion.\nIn conclusion, considering weakly collisional, nonmag-\nnetized initial conditions, we have performed fully kinetic\ncontinuum simulations of model \rows known to produce\ndynamo ampli\fcation of the magnetic \feld in resistive\nMHD. The magnetic \feld energy|apart from that cor-\nresponding to a current caused by the forcing of the ion\n\row|in these cases is observed to decay due to the Lan-\ndau damping of the magnetic perturbations. Demon-\nstrating dynamo growth in this setting will demand an5\nincreased scale separation between the \rows and the ef-\nfective magnetic dissipation. The computational feasi-\nbility of greater scale separation would require employ-\ning reduced physics parameters, which we avoided here.\nThe e\u000bect of the Landau damping is similar to that of\na magnetic di\u000busivity that scales with the wave number\nof the perturbation jkj. In high magnetic Prandtl num-\nber plasmas (such as on galactic scales and above), the\ndamping is expected to lead to a peak of the magnetic\nspectrum at lL, a scale larger than that given by resistive\ndi\u000busion,l\u0011, potentially reducing the total energy in mag-\nnetic \ructuations. As the magnetic \feld grows during the\ndynamo process, the scale at which electrons demagne-\ntize decreases, shrinking the region where this process is\noperational. While the maximum of the saturated mag-\nnetic energy spectrum in kinetic ion hybrid simulations\nappears at the ion gyroradius scale [8], our results sug-\ngest that a resolved and saturated fully kinetic dynamo\nsimulation would produce a magnetic spectrum peaked\naround the electron gyroradius scale, or lL, whichever\nis smaller. On scales where electrons are magnetized,\nthe issue of magnetic moment conservation potentially\nimpeding dynamo growth [9] becomes relevant. It is pos-\nsible that, similarly to ions [29], electrons develop their\nown instabilities and corresponding sharp phase space\nstructures, leading to breaking magnetic moment conser-\nvation, and alleviating this problem. This remains to be\ndemonstrated.\nThe simulation data presented in this article is avail-\nable at Zenodo [30].\nThe authors are grateful for fruitful discussions with\nS. L. Newton, L. Gremillet, S. Tobias, F. I. Parra,\nM. Jenab, and T. F ul op. This project has received fund-\ning from the European Research Council (ERC) under\nthe European Union's Horizon 2020 research and inno-\nvation programme under Grant Agreement No 647121.\nJ.M.T. was supported by the NSF (SHINE award AGS-\n1622306), J.J. by NASA (Earth and Space Science Fel-\nlowship, No. 80NSSC17K0428), and M.F. by the U.S. De-\npartment of Energy (Grants No. DOE-SC-0010508 and\nNo. DE-FC02-08ER54966). Simulations were performed\non resources provided by Swedish National Infrastruc-\nture for Computing (SNIC) at HPC2N, and the Extreme\nScience and Engineering Discovery Environment, which\nis supported by National Science Foundation (No. ACI-\n1548562).\n\u0003pusztai@chalmers.se\n[1] R. 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Zweibel, The As-\ntrophysical Journal 539, 505 (2000).\n[17] R. M. Kinney, B. Chandran, S. Cowley, and J. C.\nMcWilliams, Astrophys. J. 545, 907 (2000).\n[18] J. Juno, A. Hakim, J. TenBarge, E. Shi, and W. Dorland,\nJournal of Computational Physics 353, 110 (2018).\n[19] \\ Gkeyll home page,\" https://gkyl.readthedocs.io/\nen/latest/ , accessed: 2020.\n[20] A. Hakim, M. Francisquez, J. Juno, and G. W. Ham-\nmett, \\Conservative discontinuous Galerkin schemes for\nnonlinear Fokker-Planck collision operators,\" (2019),\narXiv:1903.08062 [physics.comp-ph].\n[21] J. P. Dougherty, The Physics of Fluids 7, 1788 (1964),\nhttps://aip.scitation.org/doi/pdf/10.1063/1.2746779.\n[22] D. J. Galloway and M. R. E. Proctor, Nature 356, 691\n(1992).\n[23] \\ Pencil Code home page,\" http://pencil-code.\nnordita.org , accessed: 2020.\n[24] A. B. Mikhailovskii, Plasma Physics 22, 133 (1980).\n[25] V. M. Vasyliunas, Reviews of Geophysics 13, 303 (1975),\nhttps://doi.org/10.1029/RG013i001p00303.\n[26] J. Birn and E. R. Priest, eds., Reconnection of Magnetic\nFields (Cambridge University Press, 2006).\n[27] M. Hesse, T. Neukirch, K. Schindler, M. Kuznetsova,\nand S. Zenitani, Space Sci. Rev. 160, 3 (2011).\n[28] G. O. Roberts, Philosophical Transactions of the Royal\nSociety of London. Series A, Mathematical and Physical\nSciences 271, 411 (1972).\n[29] D. A. St-Onge, Fluctuation Dynamo in Collisionless and\nWeakly Collisional Magnetized Plasmas , Ph.D. thesis,6\nPrinceton University (2019), arXiv:1912.11072.\n[30] \\Simulation data for this article,\" https://doi.org/10.5281/zenodo.3886562 , accessed: 2020."    },    {        "title": "2107.13305v1.Magnetic_field_induced_asymmetric_splitting_of_the_output_signal.pdf",        "content": "arXiv:2107.13305v1  [physics.class-ph]  28 Jul 2021Magnetic field induced asymmetric splitting of the output si gnal\nL. R. Rahul Biswas, Joydip Das and Bidhan Chandra Bag∗\nDepartment of Chemistry, Visva-Bharati, Santiniketan 731 235, India\nAbstract\nIn this paper we have investigated the dynamics of a damped ha rmonic oscillator in the presence\nof an electromagnetic field. Thetransients for the two dimen sional harmonic oscillator imply about\nthe modulation of the frequency of the oscillator by the velo city dependent non conservative force\nfrom an applied magnetic field. Except a special condition, t he motion is in general quasi periodic\nnature even in the absence of damping. Another interesting fi nding is that the magnetic field may\ninduce an asymmetric splitting of the spectrum of the output signal with two peaks in the case\nof a driven damped two dimensional harmonic oscillator. One more additional peak may appear\nfor the three dimensional case. In some cases the spectrum ma y have similarity with the Normal\nZeeman Effect. At the same time one may observe to appear the ant i resonance phenomenon even\nfor the driven damped cyclotron motion where the system with the purely non conservative force\nfields is driven by an electric field. Finally, our calculatio n exhibits how the magnetic field can\nmodulate the phasedifference (between inputand outputsigna ls) and the efficiency like quantity of\nthe energy storing process. Thus the present study might be a pplicable in the areas related to the\nrefractive index, the barrier crossing dynamics and autono mous stochastic resonance, respectively.\nKeywords: Isotropic,anisotropic,chargedparticle,harmonicos cillator,magneticfield, periodicforce,electric\nfield, resonance frequency\n∗Author for correspondence, e-mail:bidhanchandra.bag@visva-bh arati.ac.in\n1I. INTRODUCTION\nAlthough the study on the dynamics in the presence of an electroma gnetic field is an\nold issue but due to the following facts it may be still an important issue in the recent\ntechnology. The investigation on the ion conducting electrolytic mat erials is a key area in\nphysics and chemistry [1–7]. The materials have potential application s in a diverse range of\nall-solid-statedevices, such asrechargeablelithium batteries, flex ible electrochromic displays\nand smart windows [1]. The properties of the electrolytes are tuned by varying chemical\ncomposition to a large extent and hence are adapted to specific nee ds [6, 7]. High ionic\nconductivity is needed for optimizing the glassy electrolytes in variou s applications. Then it\nwould be very interesting if one can tune the ionic conductivity accor ding to specific need by\na physical method. In this context, very recent studies [8–16] s how that the conductivity of\nan electrolytic material can be tuned by an applied magnetic field. To t une the conductivity\nof ions in the solid electrolytes the combination of both magnetic field a nd time-dependent\nelectric field may be an important choice. Then one may be interested to know the basic\ndynamics aswell asenergetics ofananalyticallysolvablemodel like driv en dampedharmonic\noscillator in the presence of a magnetic field. The driven damped harm onic oscillator is a\nwell studied text book material [17, 18]. At the same time, the dyna mics of a particle in the\npresence of both constant magnetic and electric fields is also a well s tudied issue in the text\nbook [18]. This study has been extended in different contexts[19, 20 ]. But to the best of our\nknowledge, the study on the dynamics of periodically driven damped h armonic oscillator\nin the presence of a magnetic field was not addressed. It does not m ean that this issue is\nnot an important one. In other words, the model study may be a ve ry relevant one in the\ncontext of barrier crossing dynamics as mentioned above. At thes ame time it mayalso bean\nimportant one to explain the refractive of a dielectric material in the presence of a magnetic\nfield[21]. Thus our objective is to explore distinguishable feature (if a ny) of this dynamics\nincluding the energetics and the resonance at the steady state. T hen we start with the\ntransients of the two dimensional damped harmonic oscillator (havin g same frequency along\nbothxandy-directions) in the presence of a magnetic field. x(t) andy(t) are superposition\nof two periodic terms in the absence of damping. Thus the motion may be quasi periodic in\nnature. Then we determine the condition for simple periodic motion. A t the same time we\ndetermine how the frequencies of the periodic terms in the damped o scillation may depend\non the damping strength. These calculations corroborate to the r esonance conditions which\nare determined based on the steady state dynamics for the driven system. Here we find\nthat the magnetic field induces an asymmetric splitting of the spectr um of the output signal\nwith the finite values of the amplitude at the resonance conditions. B ut the values of the\namplitudes at resonance condition become infinite in the absence of d amping. It is to be\nnoted here that for this case, the phase shift between the input a nd output signals at the\nresonance condition is similar as that of the driven damped harmonic o scillator. It proofs\nindirectly that the finite value of the amplitude at the resonance con dition in the presence\n2of damping is solely due to the dissipation of energy. In other words, the phase shift has no\nsignificant role in this context. Major points like these have been inclu ded in the conclusion\nsection.\nBefore leaving this section we would mention that in recent past the m agnetic field has\nbeenconsideredindifferentcontextssuchasbarriercrossingdyn amics[8–16],nonMarkovian\ndynamics of a Brwonian particle in the presence of a magnetic field [13 , 15, 16, 22–26],\nstochastic thermodynamics [27], nonlinear dynamics [28] and others [29–32]. The present\nstudy may be relevant in some of these areas. In the conclusion sec tion we have mentioned\nabout the possible applications of the present investigation.\nThe outlay of the paper is as follows. In Sec.II, we have presented t he dynamics of a\ntwo dimensional harmonic oscillator in the presence of an electromag netic field. The steady\nstate dynamics of a driven damped three dimensional harmonic oscilla tor in the presence of\na magnetic field has been addressed in the next section. The paper is concluded in Sec.IV.\nII. DYNAMICS OF A TWO DIMENSIONAL HARMONIC OSCILLATORIN THE\nPRESENCE OF AN ELECTROMAGNETIC FIELD\nA. Transients: Implication of the resonance condition\nThe dynamics of a oscillating particle (with angular frequency ωand mass m) in the\npresence of a magnetic field B= (0,0,Bz) and the frictional force can be described by\nm˙ux=−mω2x+mΩuy−γux, (1)\nand\nm˙uy=−mω2y−mΩux−γuy. (2)\nHereuxanduycorrespond to the components of the velocity for the motion in the x-y\nplane. The magnetic force from the given field is confined in this plane a s implied in the\nabove equations of motion. Here the parameter, Ω =qBz\nmis the cyclotron frequency for the\nrotational motion of the particle (with charge, q) which is driven by only the magnetic force.\nThe remaining parameter, γin Eqs. (1-2) measures the damping strength. However, the\nabove coupled equations of motion can be solved using the transfor mation,ξ=x+iy[33].\nThen we have\n¨ξ=−ω2ξ−β˙ξ , (3)\nwhereβ=γ+iΩ. This leads to have the solution of the above equation as\nξ(t) =ξ(0)e−β/2tcos/parenleftBig/radicalbig\nω2−β2/4t/parenrightBig\n. (4)\n3Hereξ(0) can be identified as, x(0) +iy(0). However, the argument of cos(/radicalbig\nω2−β2/4t)\n(which may be represented by θ) restricts us to proceed further analytically. It can be\nexpressed as\nθ=√\nA+iBt . (5)\nHere we have used A=ω2−γ2/4 + Ω2/4 andB=−γΩ\n2. Ifγas well as Bis zero then\ndecomposing the above solution one may read time dependence of x(t) andy(t) as\nx(t) =x(0)\n2/bracketleftBig\ncos/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4+Ω/2/parenrightBig\nt/bracerightBig\n+cos/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4−Ω/2/parenrightBig\nt/bracerightBig/bracketrightBig\n+y(0)\n2/bracketleftBig\nsin/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4+Ω/2/parenrightBig\nt/bracerightBig\n−sin/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4−Ω/2/parenrightBig\nt/bracerightBig/bracketrightBig\n,(6)\nand\ny(t) =y(0)\n2/bracketleftBig\ncos/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4+Ω/2/parenrightBig\nt/bracerightBig\n+cos/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4−Ω/2/parenrightBig\nt/bracerightBig/bracketrightBig\n−x(0)\n2/bracketleftBig\nsin/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4+Ω/2/parenrightBig\nt/bracerightBig\n−sin/braceleftBig/parenleftBig/radicalbig\nω2+Ω2/4−Ω/2/parenrightBig\nt/bracerightBig/bracketrightBig\n.(7)\nThe above relations (6-7) satisfy the initial condition. For further check, one can show easily\nthat Eq. (6-7) reduce to the expected results at the limit Ω = 0 .0. However, it is apparent in\nthe above solutions that how the coupling of the two dimensional mot ion through the cross\neffect of the velocity dependent magnetic force modifies the respe ctive amplitudes. Another\nimportant point is to be noted here that x(t) as well as y(t) are superposition of periodic\nterms with periods\nT1=2π/radicalbig\nω2+Ω2/4+Ω/2, (8)\nand\nT2=2π/radicalbig\nω2+Ω2/4−Ω/2. (9)\nTheir ratio can be read asT2\nT1=ω2+Ω2/2+Ω√\nω2+Ω2/4\nω2 . If the ratio is an integer then motion\nwould be a simple periodic one otherwise the dynamics may be quasi per iodic in nature.\nOne may obtain the condition for the integer ratio in the following way. Letn≥1 is a ratio\nbetween the two periods. Then we have\nΩ2+2Ω/radicalbigg\nω2+Ω2\n4−2(n−1)ω2= 0. (10)\nThe solution of the above equation can be read as\n4Ω =(n−1)ω√n. (11)\nThe above relation suggests that for n= 1, Ω = 0. It is a check of the above calculation. For\nΩ =ω,nis3+√\n5\n2≃2.618033989 ..... Thus the motion seems to be quasi periodic in nature\nfor Ω =ω. In Fig.1, we have demonstrated the feature of the dynamics base d on Eqs. (6-7)\nfor different Ω. It is fully consistent with Eq. (11).\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s45/s48/s46/s48/s53 /s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s45/s48/s46/s48/s53 /s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s117\n/s120\n/s120/s40/s97/s41 /s32/s110/s32/s61/s32 /s40/s98/s41\n/s117\n/s120\n/s120/s32/s110/s32/s61/s32\n/s40/s99/s41\n/s117\n/s120\n/s120/s32/s110/s32/s61/s32 /s40/s100/s41\n/s117\n/s120\n/s120/s32/s32 /s110/s32/s61/s32\nFigure 1: Plot ofuxvs.xfor different values of n along with the relevant parameter set, ω2= 2.0,x(0) =\n0.1,y(0) = 0.0,ux(0) = 0.0 anduy(0) = 0.0 (Units are arbitrary)\nWe now consider another condition, γ= 0 and ω= 0. Then from Eq. (4) we have\nx(t) =x(0)\n2/bracketleftbig\n1+cos(Ω t)/bracketrightbig\n−y(0)\n2sin(Ωt), (12)\nand\ny(t) =y(0)\n2/bracketleftbig\n1+cos(Ω t)/bracketrightbig\n−x(0)\n2sin(Ωt). (13)\n5Ify(0) = 0 then above equations (12-13) imply expected cyclotron mot ion with radius\n=x(0)/2, frequency = Ω and speed = {x(0)Ω}/2 =radius×cyclotronfrequency . Similarly\nforx(0) = 0 Eqs.(12-13) also imply expected cyclotron motion. These are the checks of the\npresent calculation. However, if both x(0) andy(0) are not zero then center of the cycle will\noscillate in time along a line in the x−yplane. Thus we may have a complicated cycloid\ntype motion.\nFor further check, we consider the condition, Ω = 0. Then Eq. (4) r educes to\nx(t) =x(0)e−(γt)/2cos/parenleftBig/radicalbig\nω2−γ2/4t/parenrightBig\n, (14)\nand\ny(t) =y(0)e−(γt)/2sin/parenleftBig/radicalbig\nω2−γ2/4t/parenrightBig\n. (15)\nThese relations imply the expected damped oscillation at each directio n. Forγ= 0, the\nabove relations reduce to the well known results. For direct check , one may arrive to these\nresults putting Ω = 0 and γ= 0 in Eq.(4).\nWe now consider the approximate solutions at the limit,B\nA→0, when both Ω and γmay\nnot be zero. Then from Eq. (5) we have\nθ≃√\nA/parenleftbigg\n1+iB\n2A/parenrightbigg\nt . (16)\nUsing this relation in Eq.(4) one may have\nx(t)≃x0\n4p0e−γt\n2/parenleftBig\neBt\n2√\nA+e−Bt\n2√\nA/parenrightBig\n−x0\n4q0e−γt\n2/parenleftBig\neBt\n2√\nA−e−Bt\n2√\nA/parenrightBig\n+y0\n4s0e−γt\n2/parenleftBig\neBt\n2√\nA−e−Bt\n2√\nA/parenrightBig\n−y0\n4r0e−γt\n2/parenleftBig\neBt\n2√\nA−e−Bt\n2√\nA/parenrightBig\n(17)\nand\ny(t)≃y0\n2p0e−γt\n2/parenleftBig\neBt\n2√\nA+e−Bt\n2√\nA/parenrightBig\n−y0\n4q0e−γt\n2/parenleftBig\neBt\n2√\nA−e−Bt\n2√\nA/parenrightBig\n+x0\n4s0e−γt\n2/parenleftBig\neBt\n2√\nA−e−Bt\n2√\nA/parenrightBig\n−x0\n4r0e−γt\n2/parenleftBig\neBt\n2√\nA−e−Bt\n2√\nA/parenrightBig\n(18)\nwherep0= cos/braceleftBig/parenleftBig√\nA+Ω\n2/parenrightBig\nt/bracerightBig\n+ cos/braceleftBig/parenleftBig√\nA−Ω\n2/parenrightBig\nt/bracerightBig\n,q0= cos/braceleftBig/parenleftBig√\nA+Ω\n2/parenrightBig\nt/bracerightBig\n−\ncos/braceleftBig/parenleftBig√\nA−Ω\n2/parenrightBig\nt/bracerightBig\n,r0= sin/braceleftBig/parenleftBig√\nA+Ω\n2/parenrightBig\nt/bracerightBig\n+ sin/braceleftBig/parenleftBig√\nA−Ω\n2/parenrightBig\nt/bracerightBig\nands0=\n6sin/braceleftBig/parenleftBig√\nA+Ω\n2/parenrightBig\nt/bracerightBig\n−sin/braceleftBig/parenleftBig√\nA−Ω\n2/parenrightBig\nt/bracerightBig\n. The above equations imply the damped os-\ncillation which is composed of two frequencies,√\nA+Ω\n2and√\nA−Ω\n2, respectively. Now\none can check easily that they reduce to all the exact results at th e appropriate limits\nas mentioned above. Furthermore, from these approximate solut ions one may infer the\ndamped cyclotron motion for A= Ω2/4−γ2/4.\nBefore leaving this part we would mention that the argument in the sin usoidal function\nimplies the condition at which the resonance may appear if the dynamic al system (1-2)\nis driven periodically. In the next subsection we will check it based on t he steady state\ndynamics.\nB. Steady state dynamics: Resonance condition and energetics\nWe are now in a position to include an electric field (which is periodic in time w ith the\nangular frequency, ωE) in Eqs. (1-2) to explore the resonance condition and the related\naspect. Let the electric field applied be as follows\nE=ˆiE0xcos(ωEt)+ˆjE0ycos(ωEt). (19)\nHereEx(t) =E0xcos(ωEt) andEy(t) =E0ycos(ωEt) are the components of the applied\nelectric field. Then the Eqs. (1-2) of motion become\nm¨x=−mω2x−mγ˙x+mΩ˙y+qE0xcos(ωEt), (20)\nand\nm¨y=−mω2x−mγ˙y−mΩ˙x+qE0ycos(ωEt). (21)\nThus the components of driving force are Fx=qE0xcos(ωEt) andFy=qE0ycos(ωEt),\nrespectively. One may now choose the following particular solutions f or the above equations\nx(t) =acos(ωEt−φ1), (22)\nand\ny(t) =bcos(ωEt−φ2). (23)\nHereφ1andφ2are two relevant phase constants. Using relations (22-23) into Eq s. (20-21)\nwe have\n/braceleftbig\na/parenleftbig\nω2−ω2\nE/parenrightbig\ncosφ1/bracerightbig\ncos(ωEt)+/braceleftbig\na/parenleftbig\nω2−ω2\nE/parenrightbig\nsinφ1/bracerightbig\nsin(ωEt)\n= (aγωEcosφ1)sin(ωEt)−(aγωEsinφ1)cos(ωEt)−(bΩωEcosφ2)sin(ωEt)\n+ (bΩωEsinφ2)cos(ωEt)+q\nmE0xcos(ωEt). (24)\n7and\n/braceleftbig\nb/parenleftbig\nω2−ω2\nE/parenrightbig\ncosφ2/bracerightbig\ncos(ωEt)+/braceleftbig\nb/parenleftbig\nω2−ω2\nE/parenrightbig\nsinφ2/bracerightbig\nsin(ωEt)\n= (bγωEcosφ2)sin(ωEt)−(bγωEsinφ2)cos(ωEt)+(aΩωEcosφ1)sin(ωEt)\n−(aΩωEsinφ1)cos(ωEt)+q\nmE0ycos(ωEt). (25)\nComparing the coefficients of cos( ωEt) and sin( ωEt) in the both sides of Eq. (24) we get the\nfollowing relations\na/parenleftbig\nω2−ω2\nE/parenrightbig\ncosφ1=q\nmE0x−aγωEsinφ1+bΩωEsinφ2, (26)\nand\na/parenleftbig\nω2−ω2\nE/parenrightbig\nsinφ1=aγωEcosφ1−bΩωEcosφ2. (27)\nSimilarly from Eq. (25) we get the following relations\nb/parenleftbig\nω2−ω2\nE/parenrightbig\ncosφ2=q\nmE0y−bγωEsinφ2−aΩωEsinφ1, (28)\nand\nb/parenleftbig\nω2−ω2\nE/parenrightbig\nsinφ2=bγωEcosφ2+aΩωEcosφ1. (29)\nNow from coupled Eqs. (26-29) we have\na=/radicalbig\nH2\n1+H2\n2\nH0, (30)\nb=/radicalbig\nH2\n3+H2\n4\nH0, (31)\ntanφ1=H2\nH1. (32)\nand\ntanφ2=H4\nH3. (33)\nHere we have used\nH0=/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightBig2\n+4γ2Ω2ω4\nE, (34)\nH1=q\nm/parenleftbig\nω2−ω2\nE/parenrightbig/bracketleftBig/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightBig\nE0x+2γΩω2\nEE0y/bracketrightBig\n,(35)\n8H2=q\nmωE/bracketleftBig/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightBig\n(γE0x−ΩE0y)+2γΩω2\nE(γE0y+ΩE0x)/bracketrightBig\n,\n(36)\nH3=q\nm/parenleftbig\nω2−ω2\nE/parenrightbig/bracketleftBig/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightBig\nE0y−2γΩω2\nEE0x/bracketrightBig\n,(37)\nand\nH4=q\nmωE/bracketleftBig/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightBig\n(γE0y+ΩE0x)−2γΩω2\nE(γE0x−ΩE0y)/bracketrightBig\n.\n(38)\nThus the amplitudes of the particular solutions carry the interestin g signature of the mag-\nnetic field induced coupling of the two dimensional motion through the similarity in their\nstructures in terms of the relevant parameters, frequencies of the harmonic oscillator and the\ndriving field, amplitudes of the driving fields and strength of the magn etic field, respectively.\nabecomes bon replacement of Ω by −Ω and interchange of position between E0xandE0y.\nThus even for E0x=E0y,a/ne}ationslash=b. It is a signature of the magnetic force induced breakdown\nof the equivalence between the motions along the two directions. Eq s. (30-33) imply the role\nof another velocity dependent dissipative force in this context. Sim ilarly one may expect\nthe signature of the breakdown of the equivalence in terms of the p hase shift between input\nand output signals. Shortly we will discuss this in detail.\nWe are now in a position to dertermine the resonance conditions. It s eems to be difficult\nto find the resonance condition since the derivative of aorbwith respect to the driving\nfrequency may correspond to an algebraic equation which is not solv able analytically. Before\ngoing to predict the approximate resonance condition, we show tha t Eqs. (30-33) reduce to\nthe well known results for Ω = 0. At this limit Eqs. (34-38) become\nH0=/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2+γ2ω2\nE/bracerightBig2\n, (39)\nH1=q\nmE0x/parenleftbig\nω2−ω2\nE/parenrightbig/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2+γ2ω2\nE/bracerightBig\n, (40)\nH2=q\nmE0xγωE/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2+γ2ω2\nE/bracerightBig\n, (41)\nH3=q\nmE0y/parenleftbig\nω2−ω2\nE/parenrightbig/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2+γ2ω2\nE/bracerightBig\n, (42)\nand\nH4=q\nmE0yγωE/braceleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2+γ2ω2\nE/bracerightBig\n. (43)\nThen Eqs.(30-33) become\n9a=qE0x\nm/bracketleftBig\n(ω2−ω2\nE)2+γ2ω2\nE/bracketrightBig1/2, (44)\nb=qE0y\nm/bracketleftBig\n(ω2−ω2\nE)2+γ2ω2\nE/bracketrightBig1/2, (45)\ntanφ1= tanφ2=γωE\nω2−ω2\nE. (46)\nEqs. (44-46) imply a very good check of the present calculation. Fo r further check, one may\ndetermine easily the following resonance condition from Eqs. (44-45 ),\nωE=ω/parenleftbigg\n1−γ2\n2ω2/parenrightbigg1/2\n. (47)\nThis condition is implied by the transient motion (14-15) in the presenc e of the driving\nforce. It reduces to another well known result, ωE=ωforγ= 0. Then a=qE0x\nm(ω2−ω2\nE),\nb=qE0y\nm(ω2−ω2\nE)andφ1=φ2= 0. For the direct check, one may arrive to these results using\nγ= Ω = 0 in Eqs. (30-33). However, comparing Eqs. (45-46) we find a c ontrast result\nin the presence of a magnetic field. The effect of dissipative force ma y also appear in the\nnumerator of the amplitude function as a signature of the modulatio n of the frequency of the\ndynamics by the field. In other words, the interference between t he two driving components\nthrough velocity dependent coupling results to appear γ, Ω,E0xandE0yin the numerator\nof the amplitude functions, aandb, respectively.\nNow we have to determine the resonance condition in the presence o f a magnetic field.\nOne may determine it applying a trick by inspection of both numerator and denominator\nin Eqs. (30-31). At the weak damping limit when the resonance pheno menon may appear\nthenH0may be minimum around the following condition,\n/parenleftbig\nω2−ω2\nE/parenrightbig2−Ω2ω2\nE= 0, (48)\nThus the approximate resonance conditions may be read as\nωL≃/radicalbigg\nω2+Ω2\n4−Ω\n2, (49)\nand\nωR≃/radicalbigg\nω2+Ω2\n4+Ω\n2. (50)\nIt is to be noted here that the above conditions are very closed to t hose which are implied\nin Eqs. (17-18). To check the validity of our calculation we have demo nstrated the exact\nresults (30-31) in Fig.2.\n10/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51/s52/s53/s54/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51\n/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51/s52/s53/s54/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51\n/s32/s87 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53\n/s32/s87 /s32/s61/s32/s49/s46/s48\n/s32/s87 /s32/s61/s32/s49/s46/s53\n/s32/s87 /s32/s61/s32/s50/s46/s48\n/s119\n/s69/s40/s99/s41\n/s97/s32/s87 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53\n/s32/s87 /s32/s61/s32/s49/s46/s48\n/s32/s87 /s32/s61/s32/s49/s46/s53\n/s32/s87 /s32/s61/s32/s50/s46/s48/s40/s97/s41\n/s119\n/s69/s97\n/s98\n/s119\n/s69/s32/s87 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53\n/s32/s87 /s32/s61/s32/s49/s46/s48\n/s32/s87 /s32/s61/s32/s49/s46/s53\n/s32/s87 /s32/s61/s32/s50/s46/s48/s40/s100/s41/s40/s98/s41\n/s98\n/s119\n/s69/s32 /s87 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53\n/s32/s87 /s32/s61/s32/s49/s46/s48\n/s32/s87 /s32/s61/s32/s49/s46/s53\n/s32/s87 /s32/s61/s32/s50/s46/s48\nFigure 2: Plot ofaandbvs.ωEfor different values of Ω. (a) ω2= 2.0,γ= 0.1 andE0x=E0y= 0.25 (b)\nω2= 2.0,γ= 0.1 andE0x=E0y= 0.25 (c)ω2= 2.0,γ= 0.0 andE0x=E0y= 0.25 (d)ω2= 2.0,γ= 0.0\nandE0x=E0y= 0.25 (Units are arbitrary)\nTable I: Comparison between theoretically calculated reso nating frequencies and the exact result\nfor the driven damped two dimensional harmonic oscillator\nValue of Resonance at ωLResonance at ωR\nΩTheoretical ExactTheoretical Exact\n0.5 1.190 1.189 1.680 1.679\n1.0 1.002 1.000 1.997 1.996\n1.5 0.852 0.849 2.348 2.347\n2.0 0.732 0.729 2.730 2.730\nThe resonance conditions according to this figure are compared wit h the analytically calcu-\nlated results in Table I. It shows that there is a very good agreemen t between the theoretical\nand the exact results. Another important point is to be noted here that the Fig.2 exhibits\nan asymmetric splitting of the spectrum with the output signal. The s plitting is implied by\nthe solutions (17-18) which are composed of two frequencies. Follo wing the above approxi-\n11mation, the amplitudes at the resonance condition may be read as\na≃q\nm/radicalBig\n(ω2−ω2\nE)2(γE0x+2ΩE0y)2+ω2\nE[γ(γE0x−ΩE0y)+2Ω(γE0y+ΩE0x)]2\nγ3ω2\nE+4γΩ2ω2\nE,\n(51)\nand\nb≃q\nm/radicalBig\n(ω2−ω2\nE)2(γE0y−2γΩE0x)2+ω2\nE[γ(γE0y+ΩE0x)−2γΩ(γE0x−ΩE0y)]2\nγ3ω2\nE+4γΩ2ω2\nE,\n(52)\nThe above equations imply that at the resonance condition, the amp litude may decreases\nwith increase in the resonating frequency. In panels (a) and (b) of Fig.2 we have demon-\nstrated nature of the asymmetric splitting for different strength of the magnetic field.\nWe now check the fate of the asymmetric splitting in the absence of d amping. In the\nabsence of dissipative force, Eqs. (30-33) become\na=q/radicalBig\nE2\n0x(ω2−ω2\nE)2+E2\n0yΩ2ω2\nE\nm/braceleftBig\n(ω2−ω2\nE)2−Ω2ω2\nE/bracerightBig, (53)\nb=q/radicalBig\nE2\n0y(ω2−ω2\nE)2+E2\n0xΩ2ω2\nE\nm/braceleftBig\n(ω2−ω2\nE)2−Ω2ω2\nE/bracerightBig, (54)\ntanφ1=−E0yΩωE\nE0x(ω2−ω2\nE)(55)\nand\ntanφ2=E0xΩωE\nE0y(ω2−ω2\nE). (56)\nThe denominator of the amplitude functions imply that the resonatin g frequencies may be\nequal to those which are given by Eqs. (49-50). These are consist ent with Eqs. (6-7).\nThen amplitude becomes infinity as suggested by Eqs. (51-52). To c heck this we have\ndemonstrated Eqs. (30-31) in panels (c) and (d) of Fig.2. It exact ly corresponds to Eqs.\n(51-52). Thus in the absence of damping, the asymmetric nature o f the splitting is not clear\nas like as the previous case.\nWe now consider the phase shift between the input and output signa ls. Using Eqs. (32-\n33), thesignatureofthebreakdown oftheequivalence intermsof thephaseshift between the\ninputandtheoutputsignalfortherespective directionshasbeen demonstratedinFig.3both\nfor dissipative and non-dissipative systems. Panel (a) of this figur e shows that the magnetic\nfield induces the phase shift even in the absence of damping. Here th e magnetic force takes\n12/s48 /s49 /s50 /s51 /s52/s45/s112 /s45/s112 /s47/s50/s48/s112 /s47/s50/s112 \n/s48 /s49 /s50 /s51 /s52/s45/s112 /s47/s50/s48/s112 /s47/s50/s112 /s51/s112 /s47/s50/s50/s112 \n/s40/s97/s41/s102\n/s49/s44/s102\n/s50\n/s119 \n/s69/s32/s102 \n/s49 /s59 /s32 /s32/s102 \n/s50 /s59 /s32 /s87 /s32/s61 /s32/s48 /s46/s48 \n/s32/s102 \n/s49 /s59 /s32 /s32/s102 \n/s50 /s59 /s32 /s87 /s32/s61 /s32/s48 /s46/s53 \n/s32/s102 \n/s49 /s59 /s32 /s32/s102 \n/s50 /s59 /s32 /s87 /s32/s61 /s32/s49 /s46/s48 \n/s32/s102 \n/s49 /s59 /s32 /s32/s102 \n/s50 /s59 /s32 /s87 /s32/s61 /s32/s49 /s46/s53 \n/s32/s102 \n/s49 /s59 /s32 /s32/s102 \n/s50 /s59 /s32 /s87 /s32/s61 /s32/s50 /s46/s48 /s32 /s102 \n/s49 /s59 /s32 /s32 /s102 \n/s50 /s59 /s32 /s87 /s32 /s61/s32 /s48/s46 /s48\n/s32 /s102 \n/s49 /s59 /s32 /s32 /s102 \n/s50 /s59 /s32 /s87 /s32 /s61/s32 /s48/s46 /s53\n/s32 /s102 \n/s49 /s59 /s32 /s32 /s102 \n/s50 /s59 /s32 /s87 /s32 /s61/s32 /s49/s46 /s48\n/s32 /s102 \n/s49 /s59 /s32 /s32 /s102 \n/s50 /s59 /s32 /s87 /s32 /s61/s32 /s49/s46 /s53\n/s32 /s102 \n/s49 /s59 /s32 /s32 /s102 \n/s50 /s59 /s32 /s87 /s32 /s61/s32 /s50/s46 /s48\n/s119 \n/s69/s102\n/s49/s44/s102\n/s50/s40/s98/s41\nFigure 3: Plot ofφ1andφ2vs.ωEfor different values of Ω along with the relevant parameter set, ω2= 2.0\nandE0x=E0y= 0.25. (a)γ= 0.0 (b)γ= 0.1 (Units are arbitrary).\nthe similar role as like as the dissipative one [17, 18]. This is explicit in Eqs.( 55-56). Thus\nthe phase shift at ω=ωE, may not depend on the strength of the applied magnetic field\nas a signature of similarity between damping strength and MF. Now on e may address the\ndifference between these as we expect from the equations of motio n. The relevant points\nare to be noted here. First, the phase shift may depend on the amp litude of the driving\nforce in the presence of the magnetic field. It is a sharp contrast b ehavior compared to the\ncase where the damping induces the phase shift which is independent on the amplitude of\nthe driving force (46). Eq. (55) implies that the phase shift for the x-component motion\nis enhanced by the motion of the other direction in the presence of o pposition from the\ndriving force along the x-direction. Similarly, one may interpret Eq. (56). Thus here the\norigin of the phase shift is the effect of the cross coupling which may m imic the role of\ndissipative action in the same context. Then it is expected that in the presence of damping\nforce, the phase shift may depend on the amplitude of the periodic e lectric field in a more\ncomplicatedwayasimpliedinEqs. (32-33)aswell aspanel(b)ofFig.3 . Onemaynoticehere\nthe nature of change of the phase shits around the resonance co nditions. It may be useful\nto corroborate the appearance of the magnetic field induced anti resonance phenomenon.\nShortly we will consider this issue. Second, although there is a phase difference (at the\nresonance condition in the absence of damping force) between the input and outputs as like\nas the driven damped harmonic oscillator but the amplitudes of the ou tput signal is still\ninfinite as shown in panels (c) and (d) of Fig.2. The nature of divergen ce is quite similar to\nthe driven harmonic oscillator in the absence of damping when the pha se shift is zero. Thus\nthe finite amplitude at the resonance condition in the presence of da mping for the driven\nharmonic oscillator may be due to the dissipation of energy and here t he phase shift may\nnot have any role. It is an indirect conclusion with the help of driven ha rmonic oscillator\nin the presence of a magnetic field. Here the non zero phase shift giv es the indication that\n13the resonating frequency may be different from the periodically driv en harmonic oscillator.\nHowever, as the amplitudes becomes infinite at the resonance cond ition for this case, the\nasymmetric nature (as expected from the numerator of the amplit ude of the output signal)\nof the splitting is not prominent in the panels (c) and (d) as like as the o ther panels in\nthe same figure. Thus the terms, γ2ω2\nEand 4γ2Ω2ω4\nE(which appears in the denominator\nof the amplitude functions (30-31) due to dissipative action) are th e leading quantities to\nmodulated the nature of the asymmetric splitting of the spectrum o f the output signal. In\nother words, one may observe a single peak instead of two at relativ ely high strength of the\napplied magnetic field as implied in panel (a) of Fig.2.\nC. Phase difference between the input signals: Anti resonance\nIn the earlier discussion we have noted that the interference betw een the two driving\ncomponents through the velocity dependent coupling may have impo rtant consequences.\nThen we consider the phase difference between the two input signals with the following\ndriving electric field,\nE=ˆiE0xcos(ωEt−φ)+ˆjE0ycos(ωEt). (57)\nThen equations (20-21) of motion become\nm¨x=−mω2x−mγ˙x+mΩ˙y+qE0xcos(ωEt−φ), (58)\nand\nm¨y=−mω2x−mγ˙y−mΩ˙x+qE0ycos(ωEt). (59)\nChoosing the particular solutions as like as given by Eqs. (22-23) and following the above\nprocedure we have\na=/radicalbig\nH2\n1+H2\n2\nH0, (60)\nb=/radicalbig\nH2\n3+H2\n4\nH0, (61)\ntanφ1=H2\nH1(62)\nand\ntanφ2=H4\nH3. (63)\n14/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s40/s97/s41\n/s97\n/s119\n/s69/s32 /s102 /s32/s61/s32/s48/s46/s48\n/s32 /s102 /s32/s61/s32/s49/s46/s48\n/s32 /s102 /s32/s61/s32/s49/s46/s51\n/s32 /s102 /s32/s61/s32 /s112 /s47 /s50\n/s32 /s102 /s32/s61/s32/s51 /s112 /s47 /s50/s40/s98/s41\n/s98\n/s119\n/s69/s32 /s102 /s32/s61/s32/s48/s46/s48\n/s32 /s102 /s32/s61/s32/s49/s46/s48\n/s32 /s102 /s32/s61/s32/s49/s46/s51\n/s32 /s102 /s32/s61/s32 /s112 /s47 /s50\n/s32 /s102 /s32/s61/s32/s51 /s112 /s47 /s50\n/s40/s99/s41/s97/s44/s98\n/s119\n/s69/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s52\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s52/s53\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s53\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s53/s53\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s52\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s52/s53\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s53\n/s32/s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s53/s53\n/s97/s44/s98\n/s119\n/s69/s40/s100/s41\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s54\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s54/s53\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s55\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s49/s46/s55/s53\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s54\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s54/s53\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s55\n/s32/s98/s59 /s32/s102 /s32/s61/s32/s49/s46/s55/s53\n/s40/s101/s41 /s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s55/s53\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s56\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s56/s53\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s57\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s55/s53\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s56\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s56/s53\n/s32/s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s57\n/s119\n/s69/s97/s44/s98/s40/s102/s41 /s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s53/s53\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s54\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s54/s53\n/s32 /s97/s59 /s32/s102 /s32/s61/s32/s52/s46/s55\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s53/s53\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s54\n/s32 /s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s54/s53\n/s32/s98/s59 /s32/s102 /s32/s61/s32/s52/s46/s55/s97/s44/s98\n/s119\n/s69\nFigure 4: Plot ofaandbvs.ωEfor different values of φalong with the parameter set, ω2= 2.0,γ= 0.1\nandE0x=E0y= 0.25 (Units are arbitrary).\nIt is to be noted here that H0remains same as given by Eq. (34) but Eqs. (35-38) are\nmodified as\n15H1=q\nmE0x/bracketleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracketrightBig/braceleftbig/parenleftbig\nω2−ω2\nE/parenrightbig\ncosφ−γωEsinφ/bracerightbig\n+ 2q\nmγΩω2\nE/braceleftbig/parenleftbig\nω2−ω2\nE/parenrightbig\nE0y−E0xΩωEsinφ/bracerightbig\n,(64)\nH2=1\n(ω2−ω2\nE)/parenleftBigq\nmE0xH0sinφ+γωEH1−ΩωEH3/parenrightBig\n, (65)\nH3=q\nm/bracketleftBig/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracketrightBig/braceleftbig/parenleftbig\nω2−ω2\nE/parenrightbig\nE0y−E0xΩωEsinφ/bracerightbig\n−2q\nmE0xγΩω2\nE/braceleftbig/parenleftbig\nω2−ω2\nE/parenrightbig\ncosφ−γωEsinφ/bracerightbig\n(66)\nand\nH4=ωE\n(ω2−ω2\nE)(γH3+ΩH1). (67)\nPuttingφ= 0 in Eqs. (64)-(67) one can show easily that Eqs. (60-63) reduce to Eqs.\n(30-33). It constitutes an important check of the present calcu lation.\nIt is really difficult to find out the resonant condition from Eqs. (60-6 1). But one may\nanticipate it by inspection of both numerator and denominator in Eqs . (60-61) as before.\nThe condition would be remain same as that of φ= 0 case since transients are same for\nboth the situations. As a matter of fact same denominator ( H0) appears in Eqs. (30-31)\nand Eqs. (60-61), respectively. Thus the phase difference betwe en the input signals may\nmodulate amplitude of the output signals (60-61) without affecting t he resonance condition\nas a signature of their interference through the velocity depende nt coupling. This has been\ndemonstrated in panels (a) and (b) of Fig.4. Another point is to be n oted from this figure.\nComparing it with Fig.2 we find that in the asymmetric splitting process, the peak height\nat higher resonating frequency may be greater than that of the o ther. An extreme case such\nas one of the peaks may disappear as shown in panels (a) and (b). Fin ally, the remaining\npanels (c) to (f) exhibits that the magnetic field may induce anti res onance phenomenon.\nThus for a certain phase difference between the input signals the re duced amplitude at\nanti resonance can be regarded as due to destructive interfere nce or cancellation of forces\nacting on the oscillator. In this context we have demonstrated var iation of the relevant\nphase constants φ1andφ2with the driving frequency in Fig.5. Comparing it with Fig.3 we\nfind that the nature of change of phase constants around the re sonating driving frequency\ncorroborates the appearance of the anti resonance. It is to be noted here that both the\nresonance and the anti resonance phenomena may appear at the same driving frequency\ndepending on the phase difference between the input signals. This is c onsistent with the\ntransient motion as well as Fig.2. As the anti resonance phenomeno n is driven by the\nphase difference between the input signals then one may expect to a ppear it at the relevant\n16/s48 /s49 /s50 /s51 /s52/s45/s112 /s47/s50/s48/s112 /s47/s50/s112 \n/s48 /s49 /s50 /s51 /s52/s48/s112 /s47/s50/s112 /s51 /s112 /s47/s50\n/s48 /s49 /s50 /s51 /s52/s48/s112 /s47/s50/s112 /s51 /s112 /s47/s50\n/s48 /s49 /s50 /s51 /s52/s45/s112 /s47/s50/s48/s112 /s47/s50/s112 /s32/s102 /s32/s61 /s32/s49/s46/s53\n/s32/s102 \n/s49 \n/s32/s102 \n/s50 \n/s119\n/s69/s102\n/s49/s44/s102\n/s50\n/s102\n/s49/s44/s102\n/s50\n/s119\n/s69/s32/s102 /s32/s61 /s32/s49/s46/s55\n/s32/s102 \n/s49 \n/s32/s102 \n/s50 /s102\n/s49/s44/s102\n/s50\n/s119\n/s69/s32/s102 /s32/s61 /s32/s52/s46/s54\n/s32/s102 \n/s49 \n/s32/s102 \n/s50 \n/s119\n/s69/s102\n/s49/s44/s102\n/s50/s32/s102 /s32/s61 /s32/s52/s46/s56\n/s32/s102 \n/s49 \n/s32/s102 \n/s50 /s40/s97/s41 /s40/s98/s41\n/s40/s99/s41/s40/s100/s41\nFigure 5: Plot ofφ1andφ2vs.ωEfor different values of φalong with the relevant parameter set, ω2= 2.0,\nγ= 0.1, Ω = 1.0 andE0x=E0y= 0.25. (Units are arbitrary).\nfrequency which may correspond to a normal mode. This is sharp co ntrast to the case of\ncoupled oscillators having coupling throughtheconservative force field[34]. As a signature of\nthis kind of coupling the system may constitute one or more frequen cies which are different\nfrom the relevant normal mode(s) and anti resonance may appea r at these frequencies. For\nthis case resonance and anti resonance may appear alternatively in the output signal. Thus\nthemagnetic field induced anti resonance is a distinct one. It is to be notedhere that there is\na similarity between the two case. In the case of velocity dependent coupling, if a resonating\nfrequency become anti resonating one then another frequency behaves as a resonating one.\n1. The energetics\nAlthough the magnetic force does not work but it may modulate the d ynamics of a\ndynamical system through the modification of the characteristic f requency of the system\nas implied in the transient dynamics. Then the work done on the syste m by the driving\nforce may depend on it. An indication regarding this already we have s hown through the\ninvestigation on amplitude of the oscillation at the steady state. Thu s we are in a position\n17to calculate the power ( P) due to the work done by the driving force [17, 18]. It is the\nproduct between the force and the velocity. For the two dimension al motion the power is\ngiven by\nP=Fxdx\ndt+Fydy\ndt. (68)\nIt is an oscillating quantity. Then one may be interested to know the a verage power, ( /an}bracketle{tP/an}bracketri}ht).\nUsing Eqs. (22-23) into the above equation we obtain\n/an}bracketle{tP/an}bracketri}ht=1\n2mγω2\nE/parenleftbig\na2+b2/parenrightbig\n. (69)\nSimilarly one can determine the average energy as\n/an}bracketle{tU/an}bracketri}ht=1\n2mω2\nE1\n2/parenleftbig\na2+b2/parenrightbig\n+1\n2mω21\n2/parenleftbig\na2+b2/parenrightbig\n. (70)\nEq. (22) implies that the first term corresponds to the stored mea n potential energy and\nthe other one is due to the average kinetic energy.\nIn Fig.6, we have shown that how the average power and the mean st ored energy depend\non the driving frequency. Panel (a) shows that the asymmetric na ture of the splitting of\nthe spectrum of the average power is not so prominent as for the c ase of amplitude pro-\nvided the phase difference between the two driving component is zer o. It is a consequence\nof the following fact. The amplitude at the resonance condition decr eases with increase in\nresonating frequency as the frictional loss of energy becomes hig her. The average power\n(69) depends on both these quantities as implied by Eqs. (22-23, 68 ). Then the strongly\nasymmetric splitting of the amplitudes results the work done per unit time at lower resonat-\ning frequency with higher amplitude may be comparable to the same at higher frequency\nhaving lower amplitude. But the panel (b) implies that the spectrum o f the stored mean\nenergy may mimic the splitting pattern as that of the amplitudes. It m ay happen because\nthe stored mean potential energy is not an explicit function of drivin g frequency (70) as like\nas the other part of the average energy.\nNow one may be interested to compare the stored energy ( /an}bracketle{tU/an}bracketri}ht) with the amount of work\nthat is done in one cycle. Then the following relevant quantity [17] of interest is given by\nQ= 2π/an}bracketle{tU/an}bracketri}ht\n/an}bracketle{tP/an}bracketri}ht2π/ωe=ω2+ω2\nE\n2γωE. (71)\nThusQis defined as 2 πtimes the mean stored energy, divided by the work done per cycle.\nIt may be an very useful quantity at resonance condition. In the p resence of a magnetic\nfield, for the resonance at lower frequency we represent QbyQL. Other one is denoted by\nQR. Similarly one may represent /an}bracketle{tP/an}bracketri}htand/an}bracketle{tU/an}bracketri}htas/an}bracketle{tP/an}bracketri}htL/an}bracketle{tP/an}bracketri}htR,/an}bracketle{tU/an}bracketri}htLand/an}bracketle{tU/an}bracketri}htR, respectively.\nPanels (c), (d), (e) and (f) exhibit how these quantities depend on the different parameters.\nIt is apparent here that at the resonance condition, the process of storing of the energy may\nbe more efficient as the magnetic field becomes more stronger. It is a lso apparent here that\n18/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48 /s49 /s50 /s51 /s52/s48/s50/s52/s54/s56\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s48 /s49 /s50 /s51 /s52 /s53/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s119\n/s69 /s60/s80/s62/s32/s87/s32 /s61 /s32 /s48/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s48/s46 /s48\n/s32/s87/s32 /s61 /s32 /s49/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s48/s46 /s48\n/s32/s87/s32 /s61 /s32 /s49/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s112 /s47/s50\n/s32/s87/s32 /s61 /s32 /s49/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s51/s112 /s47/s50\n/s32/s87/s32 /s61 /s32 /s50/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s48/s46 /s48\n/s40/s102 /s41 /s40/s101/s41/s40/s100/s41 /s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s119\n/s69 /s60/s85/s62/s32/s87/s32 /s61 /s32 /s48/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s48/s46 /s48\n/s32/s87/s32 /s61 /s32 /s49/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s48/s46 /s48\n/s32/s87/s32 /s61 /s32 /s49/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s112 /s47/s50\n/s32/s87/s32 /s61 /s32 /s49/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s51/s112 /s47/s50\n/s32/s87/s32 /s61 /s32 /s50/s46 /s48/s59 /s32 /s102 /s32 /s61 /s32 /s48/s46 /s48\n/s87/s60/s80/s62/s32 /s119\n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s48/s46/s48/s32/s32/s32/s32 /s32 /s119\n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s48/s46/s48\n/s32 /s119\n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47/s52 /s32/s32/s32 /s32 /s119\n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47/s52 \n/s32 /s119\n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47/s50/s32/s32/s32/s32 /s32 /s119\n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47/s50 \n/s32 /s119\n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s32/s32/s32/s32/s32/s32/s32 /s32 /s119\n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s112 \n/s32 /s119\n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s51 /s112 /s47/s50 /s32/s32 /s32 /s119\n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s51 /s112 /s47/s50 \n/s87/s60/s85/s62/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s48 /s46 /s48 /s32 /s32 /s32 /s32 /s32 /s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s48 /s46 /s48 \n/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47 /s52 /s32/s32/s32 /s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47 /s52 \n/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47/s50 /s32 /s32 /s32 /s32 /s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47 /s50 \n/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s112 \n/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s32/s32 /s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s60/s85/s62/s32 /s119 \n/s76 /s59/s32/s69 \n/s48 /s120 /s61 /s32/s48/s46/s50/s53/s44/s32 /s69 \n/s48 /s121 /s61 /s32/s48/s46/s50/s53\n/s32 /s119 \n/s76 /s59/s32 /s69 \n/s48 /s120 /s61 /s32/s48/s46/s51/s53/s44/s32 /s69 \n/s48 /s121 /s61 /s32/s48/s46/s50/s53\n/s32 /s119 \n/s76 /s59/s32 /s69 \n/s48 /s120 /s61 /s32/s48/s46/s51/s53/s44/s32 /s69 \n/s48 /s121 /s61 /s32/s48/s46/s51/s53\n/s32 /s119 \n/s82 /s59/s32 /s69 \n/s48 /s120 /s61 /s32/s48/s46/s50/s53/s44/s32 /s69 \n/s48 /s121 /s61 /s32/s48/s46/s50/s53\n/s32 /s119 \n/s82 /s59/s32 /s69 \n/s48 /s120 /s61 /s32/s48/s46/s51/s53/s44/s32 /s69 \n/s48 /s121 /s61 /s32/s48/s46/s50/s53\n/s32 /s119 \n/s82 /s59/s32 /s69 \n/s48 /s120 /s61 /s32/s48/s46/s51/s53/s44/s32 /s69 \n/s48 /s121 /s61 /s32/s48/s46/s51/s53\n/s87/s81\n/s87/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \n/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \n/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47 /s50 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \n/s32 /s119 \n/s76 /s59 /s32 /s102 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \n/s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \n/s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \n/s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s112 /s47 /s50 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \n/s32 /s119 \n/s82 /s59 /s32 /s102 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s59 /s32 /s69 \n/s48 /s120 /s61 /s32/s48 /s46/s50 /s53 /s44 /s32 /s69 \n/s48 /s121 /s61 /s32/s48 /s46/s50 /s53 \nFigure 6: Demonstration about average power, average stored energy an d their ratio. Common parameter\nset for panels (a), (b), (c) and (d) : ω2= 2.0,γ= 0.1 andE0x=E0y= 0.25. Common parameter set for\npanels (e) and (f): ω2= 2.0 andγ= 0.1. (Units are arbitrary)\nalthough the stored energy depends on the amplitudes of the drivin g components and the\nphase difference between therm but the efficiency like quantity ( Q) does not depend on these\nparameters. But the efficiency depends on the parameters, freq uency of the oscillator and\ndamping strength, respectively. This aspect has been demonstra ted in Fig.6.\n192. Driven damped cyclotron motion\nWe now consider nature of the resonance phenomenon for the driv en damped cyclotron\nmotion. For ω= 0, Eqs. (34)-(38) become\nH0=/braceleftbig\nω4\nE−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightbig2+4γ2Ω2ω4\nE, (72)\nH1=−qω2\nE\nm/bracketleftbig/braceleftbig\nω4\nE−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightbig\nE0x+2γΩω2\nEE0y/bracketrightbig\n, (73)\nH2=q\nmωE/bracketleftbig/braceleftbig\nω4\nE−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightbig\n(γE0x−ΩE0y)+2γΩω2\nE(γE0y+ΩE0x)/bracketrightbig\n,(74)\nH3=−qω2\nE\nm/bracketleftbig/braceleftbig\nω4\nE−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightbig\nE0y−2γΩω2\nEE0x/bracketrightbig\n, (75)\nand\nH4=q\nmωE/bracketleftbig/braceleftbig\nω4\nE−/parenleftbig\nΩ2−γ2/parenrightbig\nω2\nE/bracerightbig\n(γE0y+ΩE0x)−2γΩω2\nE(γE0x−ΩE0y)/bracketrightbig\n.(76)\nMakinguseoftheaboverelationsinEqs. (30-31)wefindagainthatit isdifficulttodetermine\nthe resonance condition maximizing aandbwith respect to the driving frequency. Then one\nmay adopt the previous trick. At the weak damping limit when the reso nance phenomenon\nmay be appear then H0in Eq. (72) may be minimum around the following condition,\nω4\nE−Ω2ω2\nE= 0, (77)\nThus the approximate resonance condition may be read as\nωE≃Ω. (78)\nAnother root as a solution of Eq. (78), ωE= 0. It corresponds to the diverging motion of\nthe charged particle inthe presence of a constant electromagnet ic field with E=ˆiE0x+ˆjE0y.\nOnemay verify it easily considering therelevant equations of motion. Thus we may have two\npeaks at ωE= 0 andωE= Ω, respectively. This is implied by Eqs.(6-7). To check accuracy\nof the approximation we have demonstrated the exact results in Fig .7. The resonance\nconditions according to this figure compared with the analytically calc ulated results in Table\nII. It shows that there is a very good agreement between the app roximate and the exact\nresults.\nThe remaining panels of Fig.7 imply that the anti resonance phenomen on may appear\neven for a system with only non conservative force fields. For this s ystem both resonance\nand anti resonance phenomena may appear at the same driving fre quency depending upon\n20Table II: Comparison between theoretically calculated res onating frequency and the exact result\nfor the driven damped cyclotron motion\nValue of Ω Theoretical Exact\n0.5 0.50 0.48\n1.0 1.00 0.99\n1.5 1.50 1.49\n2.0 2.00 1.99\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s52/s54/s56/s49/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s52/s54/s56/s49/s48\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s40/s97/s41\n/s119 \n/s69 /s97/s32 /s87 /s32/s61/s32/s48/s46/s53/s59 /s32 /s102 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53 /s59 /s32 /s102 /s32 /s61/s32 /s112 /s47 /s50\n/s32/s87 /s32/s61/s32/s48/s46/s53 /s59 /s32 /s102 /s32/s61/s32 /s112 \n/s32/s87 /s32/s61/s32/s48/s46/s53 /s59 /s32 /s102 /s32/s61/s32/s51 /s112 /s47 /s50\n/s32/s87 /s32/s61/s32/s49/s46/s48 /s59 /s32 /s102 /s32/s61/s32/s48/s46/s48/s40/s98/s41\n/s119 \n/s69 /s98/s32 /s87 /s32/s61/s32/s48/s46/s53/s59 /s32 /s102 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53 /s59 /s32 /s102 /s32 /s61/s32 /s112 /s47 /s50\n/s32/s87 /s32/s61/s32/s48/s46/s53 /s59 /s32 /s102 /s32/s61/s32 /s112 \n/s32/s87 /s32/s61/s32/s48/s46/s53 /s59 /s32 /s102 /s32/s61/s32/s51 /s112 /s47 /s50\n/s32/s87 /s32/s61/s32/s49/s46/s48 /s59 /s32 /s102 /s32/s61/s32/s48/s46/s48\n/s40/s99/s41\n/s97\n/s119 \n/s69 /s32 /s102 /s32/s61/s32/s52/s46/s52/s53\n/s32 /s102 /s32/s61/s32/s53/s46/s48/s40/s100/s41\n/s119 \n/s69 /s98/s32 /s102 /s32/s61/s32/s52/s46/s52/s53\n/s32 /s102 /s32/s61/s32/s53/s46/s48\nFigure 7: Demonstration about amplitudes and anti resonance for the drive n damped oscillator. (a) and\n(b) Plot of aandbvs.ωEfor different values of Ω and φalong with the paramete set: ω2= 0.0,γ= 0.1\nandE0x=E0y= 0.25; (c) and (d) Plot of aandbvs.ωEfor different values of φalong with the paramete\nset:ω2= 0.0,γ= 0.1 Ω = 0.5 andE0x=E0y= 0.25 (Units are arbitrary).\nthe phase difference between the input signals. Thus the underlying reason to appear the\nanti resonance for the driven damped cyclotron motion may be the same as that of driven\ndamped harmonic oscillator in the presence of a magnetic field.\n21Beforeleaving thispartwewouldmentionherethatmakinguseofrela tions(30-31)withEqs.\n(72-76) into the previous subsection one may study the energetic s for the driven damped\ncyclotron motion.\nIII. STEADY STATE DYNAMICS OF A THREE DIMENSIONAL HARMONIC\nOSCILLATOR IN THE PRESENCE OF AN ELECTROMAGNETIC FIELD: SIMI-\nLARITY OF THE OUTPUT SPECTRUM TO THE NORMAL ZEEMAN EFFECT\nWe now consider the steady state dynamics for the general case, driven damped three\ndimensional harmonic oscillator in the presence of a magnetic field ( B= (Bx,By,Bz)) which\nmay be at arbitrary direction. For this case we choose a periodic elec tric field as\nE=ˆiE0xcos(ωEt−φx)+ˆjE0ycos(ωEt−φy)+ˆkE0zcos(ωEt−φz) (79)\nwhereφx,φyandφzarethephaseconstantsassociatedwitheachcomponentalongre spective\ndirections. Then one may write the equations of motion as\nm¨x=−mω2\nxx−mγ˙x+mΩz˙y−mΩy˙z+qE0xcos(ωEt−φx), (80)\nm¨y=−mω2\nyy−mγ˙y+mΩx˙z−mΩz˙x+qE0ycos(ωEt−φy), (81)\nm¨z=−mω2\nzz−mγ˙z+mΩy˙x−mΩx˙y+qE0zcos(ωEt−φz). (82)\nHere we have used Ω x=qBx/m,Ωy=qBy/mand Ω x=qBz/m. Increase in dimension of\nthe system makes difficult to study the dynamics at short time in the a bsence of electric\nfield. However, to characterize the steady state dynamics we cho ose the following particular\nsolutions to solve the above equations\nx(t) =acos(ωEt−φ1), (83)\ny(t) =bcos(ωEt−φ2), (84)\nand\nz(t) =ccos(ωEt−φ3). (85)\nFollowing the earlier case now one may determine the amplitudes and th e phase constants\nas\na=/radicalbig\nA2\n1+A2\n2\nD1, (86)\n22b=/radicalbig\nB2\n1+B2\n2\nD1, (87)\nc=/radicalbig\nC2\n1+C2\n2\nD2. (88)\ntanφ1=A2\nA1, (89)\ntanφ2=B2\nB1, (90)\nand\ntanφ3=C2\nC1. (91)\nIn these relations we have used\nA1= (HcHx−HzH2)(HbHc+H3H6)+(HcHy+HzH3)(HcH1+H2H6),(92)\nA2= (HcH′\nx−H′\nzH2)(HbHc+H3H6)+/parenleftbig\nHcH′\ny+H′\nzH3/parenrightbig\n(HcH1+H2H6),(93)\nB1= (HcHy+HzH3)(HaHc+H2H5)−(HcHx−HzH2)(HcH4−H3H5),(94)\nB2=/parenleftbig\nHcH′\ny+H′\nzH3/parenrightbig\n(HaHc+H2H5)−(HcH′\nx−H′\nzH2)(HcH4−H3H5),(95)\nC1= (HbHz−HyH6)(HaHb+H1H4)+(HbHx+HyH1)(HbH5+H4H6),(96)\nC2=/parenleftbig\nHbH′\nz−H′\nyH6/parenrightbig\n(HaHb+H1H4)+(HbH′\nx+HyH1)(HbH5+H4H6),(97)\nD1= (HaHc+H2H5)(HbHc+H3H6)+(HcH1+H2H6)(HcH4−H3H5).(98)\nD2= (HaHb+H1H4)(HbHc+H3H6)+(HbH5+H4H6)(HbH2−H1H3),(99)\n23with\nHa=/parenleftbig\nω2\nx−ω2\nE/parenrightbig2/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n+/braceleftBig\nγ2/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n−Ω2\ny/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n−Ω2\nz/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig/bracerightBig\nω2\nE, (100)\nHb=/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig2/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n+/braceleftBig\nγ2/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n−Ω2\nz/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n−Ω2\nx/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/bracerightBig\nω2\nE, (101)\nHc=/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig2+/braceleftBig\nγ2/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n−Ω2\nx/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n−Ω2\ny/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig/bracerightBig\nω2\nE, (102)\nHx=q\nm/braceleftBig\nE0x/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\ncosφx\n−E0xγωE/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nsinφx\n+E0yΩzωE/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\nsinφy\n−E0zΩyωE/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\nsinφz/bracerightBig\n, (103)\nHy=q\nm/braceleftBig\nE0y/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\ncosφy\n−E0xΩzωE/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nsinφx\n−E0yγωE/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\nsinφy\n+E0zΩxωE/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\nsinφz/bracerightBig\n, (104)\n24Hz=q\nm/braceleftBig\nE0z/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\ncosφz\n+E0xΩyωE/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nsinφx\n−E0yΩxωE/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\nsinφy\n−E0zγωE/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\nsinφz/bracerightBig\n, (105)\nH′\nx=q\nm/braceleftBig\nE0x/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nsinφx\n+E0xγωE/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\ncosφx\n−E0yΩzωE/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\ncosφy\n+E0zΩyωE/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\ncosφz/bracerightBig\n, (106)\nH′\ny=q\nm/braceleftBig\nE0y/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nsinφy\n+E0xΩzωE/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\ncosφx\n+E0yγωE/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\ncosφy\n−E0zΩxωE/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\ncosφz/bracerightBig\n, (107)\nH′\nz=q\nm/braceleftBig\nE0z/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nsinφz\n−E0xΩyωE/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\ncosφx\n+E0yΩxωE/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\ncosφy\n+E0zγωE/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\ncosφz/bracerightBig\n, (108)\n25H1=/braceleftBig\nγΩz/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n+γΩz/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n−ΩxΩy/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/bracerightBig\nω2\nE,\n(109)\nH2=/braceleftBig\nγΩy/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n+γΩy/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n+ΩzΩx/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig/bracerightBig\nω2\nE,\n(110)\nH3=/braceleftBig\nγΩx/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n+γΩx/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n−ΩyΩz/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig/bracerightBig\nω2\nE,\n(111)\nH4=/braceleftBig\nγΩz/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n+γΩz/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n+ΩxΩy/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/bracerightBig\nω2\nE,\n(112)\nH5=/braceleftBig\nγΩy/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n+γΩy/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\n−ΩzΩx/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig/bracerightBig\nω2\nE,\n(113)\nand\nH6=/braceleftBig\nγΩx/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n+γΩx/parenleftbig\nω2\nz−ω2\nE/parenrightbig/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n+ΩyΩz/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig/bracerightBig\nω2\nE.\n(114)\nOne may now show easily that Eqs. (86)-(91) reduce to the results of the previous section.\nIt constitutes an important check of our calculation.\nWe are now in a position to determine the resonance condition. The na ture of the amplitude\nfunctions implies that it is very difficult to have the condition even apply ing the previous\ntechnique. According to the earlier procedure, we have to solve th e following equation,\n/parenleftbig\nω2\nx−ω2\nE/parenrightbig2/parenleftbig\nω2\ny−ω2\nE/parenrightbig2/parenleftbig\nω2\nz−ω2\nE/parenrightbig2/bracketleftbigg/braceleftBig/parenleftbig\nω2\nx−ω2\nE/parenrightbig2/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig2−Ω2\nx/parenleftbig\nω2\nx−ω2\nE/parenrightbig2/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nω2\nE\n−2Ω2\ny/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nω2\nE−Ω2\nz/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig2ω2\nE+Ω2\nxΩ2\ny/parenleftbig\nω2\nx−ω2\nE/parenrightbig\nω4\nE\n+ Ω4\ny/parenleftbig\nω2\ny−ω2\nE/parenrightbig\nω4\nE+Ω2\nyΩ2\nz/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nω4\nE/bracerightBig\n×/braceleftBig/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig2/parenleftbig\nω2\nz−ω2\nE/parenrightbig2\n−2Ω2\nx/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nω2\nE−Ω2\ny/parenleftbig\nω2\ny−ω2\nE/parenrightbig2/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nω2\nE−Ω2\nz/parenleftbig\nω2\ny−ω2\nE/parenrightbig/parenleftbig\nω2\nz−ω2\nE/parenrightbig2ω2\nE\n+ Ω4\nx/parenleftbig\nω2\nx−ω2\nE/parenrightbig\nω4\nE+Ω2Ω2\ny/parenleftbig\nω2\ny−ω2\nE/parenrightbig\nω4\nE+Ω2\nzΩ2\nx/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nω4\nE/bracerightBig\n−Ω2\nxΩ2\ny/braceleftBig\nΩ2\nx/parenleftbig\nω2\nx−ω2\nE/parenrightbig\nω2\nE\n+ Ω2\ny/parenleftbig\nω2\ny−ω2\nE/parenrightbig\nω2\nE+Ω2\nz/parenleftbig\nω2\nz−ω2\nE/parenrightbig\nω2\nE−/parenleftbig\nω2\nx−ω2\nE/parenrightbig2/parenleftbig\nω2\ny−ω2\nE/parenrightbig2/parenleftbig\nω2\nz−ω2\nE/parenrightbig2/bracerightBig2\nω4\nE/bracketrightbigg\n= 0, (115)\n26/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s40/s97/s41\n/s119\n/s69/s97/s32 /s87\n/s120 /s32/s61/s32/s48/s46/s53/s44/s32 /s87\n/s121 /s32/s61/s32/s49/s46/s48 /s44/s32 /s87\n/s122 /s32/s61/s32/s49/s46/s53\n/s32 /s87\n/s120 /s32/s61/s32/s48/s46/s56/s44/s32 /s87\n/s121 /s32 /s61/s32/s49/s46/s51 /s44/s32 /s87\n/s122 /s32/s61/s32/s49/s46/s56\n/s32 /s87\n/s120 /s32/s61/s32/s49/s46/s48/s44/s32 /s87\n/s121 /s32/s61/s32/s49/s46/s53 /s44/s32 /s87\n/s122 /s32/s61/s32/s50/s46/s48\n/s119\n/s69/s98/s40/s98/s41\n/s32 /s87\n/s120 /s32/s61/s32/s48/s46/s53/s44/s32 /s87\n/s121 /s32/s61/s32/s49/s46/s48 /s44/s32 /s87\n/s122 /s32/s61/s32/s49/s46/s53\n/s32 /s87\n/s120 /s32/s61/s32/s48/s46/s56/s44/s32 /s87\n/s121 /s32 /s61/s32/s49/s46/s51 /s44/s32 /s87\n/s122 /s32/s61/s32/s49/s46/s56\n/s32 /s87\n/s120 /s32/s61/s32/s49/s46/s48/s44/s32 /s87\n/s121 /s32/s61/s32/s49/s46/s53 /s44/s32 /s87\n/s122 /s32/s61/s32/s50/s46/s48\n/s119 \n/s69/s99/s40/s99/s41\n/s32/s87\n/s120/s32/s61/s32/s48/s46/s53/s44/s32 /s87\n/s121 /s32/s61/s32/s49/s46/s48 /s44/s32 /s87\n/s122/s32/s61/s32/s49/s46/s53\n/s32/s87\n/s120/s32/s61/s32/s48/s46/s56/s44/s32 /s87\n/s121 /s32/s61/s32/s49/s46/s51 /s44/s32 /s87\n/s122/s32/s61/s32/s49/s46/s56\n/s32/s87\n/s120/s32/s61/s32/s49/s46/s48/s44/s32 /s87\n/s121 /s32/s61/s32/s49/s46/s53 /s44/s32 /s87\n/s122/s32/s61/s32/s50/s46/s48\nFigure 8: Plot ofa,bandcvs.ωEfor different values of Ω x, Ωyand Ω zalon with the parameter set:\nω2\nx= 1.5,ω2\ny= 2.0,ω2\nz= 3.0,γ= 0.1,E0x=E0y=E0z= 0.25 andφx=φy=φz= 0.0 (Units are\narbitrary).\ntofindthevaluesof ωEforwhich aandbmaybemaximum. Similarly, onemayfacedifficulty\nto fine the conditions for which cmay be maximum. Then we may be interested to find the\nresonance condition for relatively simple case like ωx=ωy=ωz=ω, Ωx/ne}ationslash= Ωy/ne}ationslash= Ωz/ne}ationslash= Ω.\nFor this case to determine the conditions for which aandbmay be maximum we have to\nsolve the following equation,\n27ω16\nE−/parenleftbig\na1+a2+8ω2/parenrightbig\nω14\nE+/braceleftbig\n28ω4+6(a1+a2)ω2+(a1a2−a2a1+b1+b2)/bracerightbig\nω12\nE\n−/braceleftbig\n56ω6+15(a1+a2)ω4+4(a1a2−a2a1+b1+b2)ω2−(a1b2+a2b1+2a3b3)/bracerightbig\nω10\nE\n+/braceleftbig\n70ω8+20(a1+a2)ω6+6(a1a2−a2a1+b1+b2)ω4+2(a1b2+a2b1+2a3b3)ω2+/parenleftbig\nb1b2−b2\n3/parenrightbig/bracerightbig\nω8\nE\n−/braceleftbig\n56ω6+15(a1+a2)ω4+4(a1a2−a2a1+b1+b2)ω2−(a1b2+a2b1+2a3b3)/bracerightbig\nω4ω6\nE\n−/braceleftbig\n28ω4+6(a1+a2)ω2+(a1a2−a2a1+b1+b2)/bracerightbig\nω8ω4\nE−/parenleftbig\na1+a2+8ω2/parenrightbig\nω12ω2\nE+ω16= 0 (116)\nwherea1= Ω2\nx+ 2Ω2\ny+ Ω2\nz,a2= 2Ω2\nx+ Ω2\ny+ Ω2\nz,a3= ΩxΩy,b1= Ω2\ny/parenleftbig\nΩ2\nx+Ω2\ny+Ω2\nz/parenrightbig\n,\nb2= Ω2\nx/parenleftbig\nΩ2\nx+Ω2\ny+Ω2\nz/parenrightbig\nandb3= ΩxΩy/parenleftbig\nΩ2\nx+Ω2\ny+Ω2\nz/parenrightbig\n. It seems to be difficult to solve the\nabove equation. Similarly, one may face difficulty to find the conditions for which cmay\nbe maximum. Then we may determine the resonance condition from th e plot as shown in\nthe panels (a), (b) and (c) in Fig.8. It exhibits an interesting featur e that in the process\nof asymmetric splitting one more additional peak may appear around the frequency of the\nrelevant vibrational mode. To understand the appearance of the three peaks we consider\nthe simplest case, ωx=ωy=ωz=ω, Ωx= Ωy= Ωz= Ω. For this condition, Eqs. (92)-(99)\nbecome\nA1= (HxH0−HzH2)/parenleftbig\nH2\n0+H1H2/parenrightbig\n+(HyH0+HzH1)/parenleftbig\nH2\n2+H0H1/parenrightbig\n,(117)\nA2= (H′\nxH0−H′\nzH2)/parenleftbig\nH2\n0+H1H2/parenrightbig\n+/parenleftbig\nH′\nyH0+H′\nzH1/parenrightbig/parenleftbig\nH2\n2+H0H1/parenrightbig\n,(118)\nB1= (HyH0+HzH1)/parenleftbig\nH2\n0+H1H2/parenrightbig\n+(HxH0−HzH2)/parenleftbig\nH2\n1−H0H2/parenrightbig\n,(119)\nB2=/parenleftbig\nH′\nyH0+H′\nzH1/parenrightbig/parenleftbig\nH2\n0+H1H2/parenrightbig\n+(H′\nxH0−H′\nzH2)/parenleftbig\nH2\n1−H0H2/parenrightbig\n,(120)\nC1= (HzH0+HxH1)/parenleftbig\nH2\n0+H1H2/parenrightbig\n+(HyH0−HxH2)/parenleftbig\nH2\n1−H0H2/parenrightbig\n,(121)\nC2= (H′\nzH0+H′\nxH1)/parenleftbig\nH2\n0+H1H2/parenrightbig\n+/parenleftbig\nH′\nyH0−H′\nxH2/parenrightbig/parenleftbig\nH2\n1−H0H2/parenrightbig\n,(122)\nD1=D2=D=/parenleftbig\nH2\n0+H1H2/parenrightbig2−/parenleftbig\nH2\n1−H0H2/parenrightbig/parenleftbig\nH2\n2+H0H1/parenrightbig\n,(123)\n28with\nH0=/parenleftbig\nω2−ω2\nE/parenrightbig2−/parenleftbig\n2Ω2−γ2/parenrightbig\nω2\nE, (124)\nH1= Ωω2\nE(2γ−Ω), (125)\nH2= Ωω2\nE(2γ+Ω), (126)\nHx=q\nmE0x/parenleftbig\nω2−ω2\nE/parenrightbig\n, (127)\nHy=q\nmE0y/parenleftbig\nω2−ω2\nE/parenrightbig\n, (128)\nHz=q\nmE0z/parenleftbig\nω2−ω2\nE/parenrightbig\n, (129)\nH′\nx=q\nmωE(γE0x−ΩE0y+ΩE0z), (130)\nH′\ny=q\nmωE(γE0y−ΩE0z+ΩE0x), (131)\nand\nH′\nz=q\nmωE(γE0z−ΩE0x+ΩE0y). (132)\nTo determine the resonance condition we now follow the previous sec tion. At the weak\ndamping limit when the resonance phenomenon may appear then Dmay be minimum\naround the following condition,\n/parenleftbig\nH2\n0+H1H2/parenrightbig2−/parenleftbig\nH2\n1−H0H2/parenrightbig/parenleftbig\nH2\n2+H0H1/parenrightbig\n= 0. (133)\nwhere\nH0=/parenleftbig\nω2−ω2\nE/parenrightbig2−2Ω2ω2\nE, (134)\nH1=−Ω2ω2\nE, (135)\nand\nH2= Ω2ω2\nE, (136)\nRearranging Eq. (133) we have\n29H0(H0−H1+H2)/braceleftbig\nH0(H0−H1+H2)+3Ω4ω4\nE/bracerightbig\n= 0.\nIt implies the following resonance conditions\nωm=ω , (137)\nωL=/radicalbigg\nω2+Ω2\n2−/radicalbigg\nΩ2\n2(138)\nand\nωR=/radicalbigg\nω2+Ω2\n2+/radicalbigg\nΩ2\n2(139)\nThese conditions are consistent with panels (a), (b) and (c) in Fig.9 . We compare the above\nconditions with the exact results in Table III. It shows a fair agreem ent between them.\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s40/s97/s41\n/s97\n/s119\n/s69/s32/s87 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53\n/s32/s87 /s32/s61/s32/s49/s46/s48\n/s32/s87 /s32/s61/s32/s49/s46/s53\n/s32/s87 /s32/s61/s32/s50/s46/s48\n/s40/s101/s41/s98\n/s119\n/s69/s32/s87 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53\n/s32/s87 /s32/s61/s32/s49/s46/s48\n/s32/s87 /s32/s61/s32/s49/s46/s53\n/s32/s87 /s32/s61/s32/s50/s46/s48/s40/s99/s41\n/s99\n/s119\n/s69/s32/s87 /s32/s61/s32/s48/s46/s48\n/s32/s87 /s32/s61/s32/s48/s46/s53\n/s32/s87 /s32/s61/s32/s49/s46/s48\n/s32/s87 /s32/s61/s32/s49/s46/s53\n/s32/s87 /s32/s61/s32/s50/s46/s48\n/s40/s100/s41\n/s119\n/s69/s97/s32/s87\n/s120/s32 /s61 /s32 /s48/s46 /s53/s44 /s32 /s87\n/s121 /s32 /s61 /s32 /s49/s46 /s48 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s49/s46 /s53\n/s32/s87\n/s120/s32 /s61 /s32 /s48/s46 /s56/s44 /s32 /s87\n/s121 /s32/s61 /s32 /s49/s46 /s51 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s49/s46 /s56\n/s32/s87\n/s120/s32 /s61 /s32 /s49/s46 /s48/s44 /s32 /s87\n/s121 /s32 /s61 /s32 /s49/s46 /s53 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s50/s46 /s48/s40/s98/s41\n/s32/s87\n/s120/s32 /s61 /s32 /s48/s46 /s53/s44 /s32 /s87\n/s121 /s32 /s61 /s32 /s49/s46 /s48 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s49/s46 /s53\n/s32/s87\n/s120/s32 /s61 /s32 /s48/s46 /s56/s44 /s32 /s87\n/s121 /s32/s61 /s32 /s49/s46 /s51 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s49/s46 /s56\n/s32/s87\n/s120/s32 /s61 /s32 /s49/s46 /s48/s44 /s32 /s87\n/s121 /s32 /s61 /s32 /s49/s46 /s53 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s50/s46 /s48\n/s119\n/s69/s98/s40/s102/s41\n/s99\n/s119\n/s69/s32/s87\n/s120/s32 /s61 /s32 /s48/s46 /s53/s44 /s32 /s87\n/s121 /s32 /s61 /s32 /s49/s46 /s48 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s49/s46 /s53\n/s32/s87\n/s120/s32 /s61 /s32 /s48/s46 /s56/s44 /s32 /s87\n/s121 /s32/s61 /s32 /s49/s46 /s51 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s49/s46 /s56\n/s32/s87\n/s120/s32 /s61 /s32 /s49/s46 /s48/s44 /s32 /s87\n/s121 /s32 /s61 /s32 /s49/s46 /s53 /s44 /s32 /s87\n/s122 /s32 /s61 /s32 /s50/s46 /s48\nFigure 9: Plot ofa,bandcvs.ωEfor the parameter set: ω2= 2.0,γ= 0.1,E0x=E0y=E0z= 0.25 and\nφx=φy=φz= 0.0 (Units are arbitrary).\nAnother point is to be noted here. It is apparent in the resonance c onditions that in the\npresence of a magnetic field, a vibrational motion may be composed o f three frequencies and\none(ωm)ofthemmaynotdependontheappliedfieldasasignatureofthevelo citydependent\ncoupling in the equation of motion. This is also true even for the case, ωx=ωy=ωz=ω,\nΩx/ne}ationslash= Ωy/ne}ationslash= Ωz/ne}ationslash= Ω as shown in panels (d), (e) and (f) of Fig.9. Thus the appearance of\nthree peaks for the isotropic harmonic oscillator has a similarity with t he Normal Zeeman\nEffect in the sense that the middle peak does not depend on the applie d magnetic field. One\n30Table III: Comparison between theoretically calculated re sonating frequency and the exact result\nfor the driven damped isotropic harmonic oscillator\nValue of Resonance at ωLResonance at ωmResonance at ωR\nΩTheoretical ExactTheoretical ExactTheoretical Exact\n0.5 1.104 1.039 1.412 1.409 1.811 1.919\n1.0 0.874 0.789 1.412 1.409 2.288 2.529\n1.5 0.707 0.620 1.412 1.409 2.828 3.230\n2.0 0.586 0.500 1.412 1.409 3.414 3.970\nmay find all the peaks making use of oscillating electric field which may be unpolarized or\npolarized at any of the three directions. However, for the driven d amped isotropic harmonic\noscillator in the presence of a magnetic field along the z-direction, Eqs. (138-139) reduce\nto Eqs. (49-50) and one may find three peaks at the frequencies g iven by Eqs. (137, (49-\n50). This is also similar to the Normal Zeeman Effect. To avoid any conf usion we would\nmentionherethatifthedrivendampedthreedimensional anisotrop icoscillatorexperiences a\nmagnetic field along the z-direction then the characteristics of the spectrum also may mimic\nthe Normal Zeeman Effect. Shortly we will show that one may determ ine approximately\nthe field dependent resonating frequencies. For these cases the motion along the z-direction\nis decoupled from the motion in x-yplane. Then three peaks may appear for the polarized\nelectric field which may lying either in x-zory-zplane. Of course one may find all the\nthree peaks making use of unpolarized electric field. Finally, it is to be n oted here that the\nposition of the three peaks may depend on the direction of the applie d field (if it is not along\nthez-direction) for the three dimensional anisotropic oscillator as implied in Fig.8.\nWe are now in a position to demonstrate the effect of phase differenc e among the com-\nponents of the input signal on the output signal. Fig.10 has been inc luded in this context.\nAgain this figure implies that the phase difference between the input s ignals may modulate\namplitude of the output signals as a signature of the interference b etween the input signals\nthrough the velocity dependent coupling. As a result of that the sp ecial features for the\ntwo dimensional case is also continued for three dimensional motion. One more additional\nfeature, anti resonance may appear at nearby the resonating f requency for the latter one.\nBefore leaving this section we show that one may determine the reso nance condition for\nthe driven damped anisotropic oscillator at relatively simple situation s uch asωz= 0,\nΩx= Ωy= 0 and Ω z= Ω. Then the relevant amplitudes and phases of the output signal\ncan be read from Eqs. (86, 87, 89, 90) as\na=/radicalbig\nH2\n1+H2\n2\nH0, (140)\n31/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51/s52\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s119\n/s69/s97/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32 /s112 /s47/s52 \n/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32 /s112 /s47/s50 \n/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32 /s112 \n/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32/s51 /s112 /s47/s50 \n/s97\n/s119\n/s69/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 /s47 /s52\n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 /s47 /s50\n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 \n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32/s51 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32/s51 /s112 /s47 /s50/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32 /s112 /s47/s52 \n/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32 /s112 /s47/s50 \n/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32 /s112 \n/s32 /s102 \n/s120 /s32/s61 /s32/s48/s46/s48/s59 /s32/s102 \n/s121 /s32/s61 /s32 /s112 /s47/s50 /s59 /s32/s102 \n/s122 /s32/s61 /s32/s51 /s112 /s47/s50 \n/s119\n/s69/s98\n/s32 /s102\n/s120 /s32/s61 /s32 /s48/s46 /s48/s59 /s32 /s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32 /s102\n/s122 /s32 /s61 /s32 /s112 /s47 /s52 \n/s32 /s102\n/s120 /s32/s61 /s32 /s48/s46 /s48/s59 /s32 /s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32 /s102\n/s122 /s32 /s61 /s32 /s112 /s47 /s50 \n/s32 /s102\n/s120 /s32/s61 /s32 /s48/s46 /s48/s59 /s32 /s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32 /s102\n/s122 /s32 /s61 /s32 /s112 \n/s32 /s102\n/s120 /s32/s61 /s32 /s48/s46 /s48/s59 /s32 /s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32 /s102\n/s122 /s32 /s61 /s32 /s51 /s112 /s47 /s50 \n/s119\n/s69/s98/s119\n/s69/s99/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 /s47 /s52\n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 /s47 /s50\n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 \n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32/s51 /s112 /s47 /s50\n/s119\n/s69/s99/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 /s47 /s52\n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 /s47 /s50\n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32 /s112 \n/s32 /s102\n/s120 /s32 /s61 /s32/s48 /s46/s48 /s59 /s32/s102\n/s121 /s32 /s61 /s32 /s51/s112 /s47 /s50 /s59 /s32/s102\n/s122 /s32/s61 /s32/s51 /s112 /s47 /s50\nFigure 10: Plot ofa,bandcvs.ωEfor different values of φx,φyandφzalong with the parameter set:\nω2\nx= 1.5,ω2\ny= 2.0,ω2\nz= 3.0,γ= 0.1 andE0x=E0y=E0z= 0.25 (Units are arbitrary).\nb=/radicalbig\nH2\n3+H2\n4\nH0, (141)\ntanφ1=H2\nH1, (142)\nand\n32tanφ2=H4\nH3. (143)\nHere we have used\nH0=H0xH0y+γ2Ω2ω4\nE/parenleftbig\nω2\nx+ω2\ny−2ω2\nE/parenrightbig2, (144)\nH1=q\nm/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/braceleftbig\nE0xH0y+E0yγΩω2\nE/parenleftbig\nω2\nx+ω2\ny−2ω2\nE/parenrightbig/bracerightbig\n,(145)\nH2=q\nmωE/parenleftbig\nω2\ny−ω2\nE/parenrightbig/braceleftbig\nE0xH0yγ−E0yH0xΩ+γΩω2\nE/parenleftbig\nω2\nx+ω2\ny−2ω2\nE/parenrightbig\n(E0yγ+E0xΩ)/bracerightbig\n,\n(146)\nH3=q\nm/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig/braceleftbig\nE0yH0x−E0xγΩω2\nE/parenleftbig\nω2\nx+ω2\ny−2ω2\nE/parenrightbig/bracerightbig\n,(147)\nH4=q\nmωE/parenleftbig\nω2\nx−ω2\nE/parenrightbig/braceleftbig\nE0yH0xγ+E0xH0yΩ+γΩω2\nE/parenleftbig\nω2\nx+ω2\ny−2ω2\nE/parenrightbig\n(E0yΩ−E0xγ)/bracerightbig\n.\n(148)\nwith\nH0x=/parenleftbig\nω2\nx−ω2\nE/parenrightbig2/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n+γ2ω2\nE/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n−Ω2ω2\nE/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n,(149)\nand\nH0y=/parenleftbig\nω2\ny−ω2\nE/parenrightbig2/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n+γ2ω2\nE/parenleftbig\nω2\nx−ω2\nE/parenrightbig\n−Ω2ω2\nE/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n,(150)\nIt is to be noted here that for the purpose of the determination of the resonance condition\nwe have chosen no phase difference between the driving component s. However, one can show\nthat forωx=ωy, Eqs. (140-143) reduce to Eqs. (30-33). It is an important chec k for this\nsection.\nFollowingtheearlier calculationwenowfindtheresonance condition. S incetheresonance\nphenomenonmayappearat γ→0thentheamplitudeoftheoutputsignalmaybemaximum\naround the following condition\n/parenleftbig\nω2\nx−ω2\nE/parenrightbig/parenleftbig\nω2\ny−ω2\nE/parenrightbig\n−Ω2ω2\nE= 0. (151)\nIfωy= 0 then the solution of the above can be read as\nωE=/radicalbig\nω2\nx+Ω2. (152)\n33/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51/s52/s53/s54\n/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51/s52/s53/s54/s55\n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s51/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 /s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s51/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 /s40/s97/s41\n/s97\n/s119\n/s69 /s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s48 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s48/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s32 /s119\n/s120 /s50 \n/s32 /s61 /s32 /s49/s46 /s53/s44 /s32 /s119\n/s121 /s50 \n/s32 /s61 /s32 /s50/s46 /s48/s44 /s32 /s119\n/s122 /s50 \n/s32 /s61 /s32 /s51/s46 /s48/s44 /s32 /s87 \n/s120 /s32 /s61/s32 /s49 /s46 /s48 /s44 /s32 /s87 \n/s121 /s32 /s61/s32 /s49 /s46 /s53 /s44 /s32 /s87 \n/s122 /s32 /s61/s32 /s50 /s46 /s48 \n/s119\n/s69 /s98/s40/s98/s41\n/s99/s40/s99/s41\n/s119 \n/s69 \nFigure 11: Plot ofa,bandcvs.ωEfor the parameter set: γ= 0.1,E0x=E0y=E0z= 0.25 and\nφx=φy=φz= 0.0 (Units are arbitrary).\nand\nωE= 0. (153)\nSimilarly for ωx= 0 we have\nωE=/radicalBig\nω2y+Ω2. (154)\nand\nωE= 0. (155)\nThe above Eqs. (152-155) imply that for the one dimensional harmo nic oscillator in the\npresence of a magnetic field (which is perpendicular to the direction o f the oscillator) only\none resonance peak may appear as shown red curve in panel (a) of Fig.10. For the given\n34parameter set for this curve, the location of the peak at ωE= 2.345. This is very much\nconsistent with the theoretical value. To avoid any confusion we wo uld mention here that\nthe peak at ωE= 0 as implied by Eq.(153) is corresponding to the diverging motion in\nthe presence of the constant electric field. One may verify it easily c onsidering the relevant\nequations of motion. However, Fig.10 is a typical demonstration of s teady state dynamics\nof driven damped anisotropic oscillators. It shows that if the field is t ilted from the perpen-\ndicular direction then one more additional peak may appear. Panels ( b) and (c) imply that\nthe additional peak appears as a signature of a complex motion whos e projection in y−z\nplane is like a damped driven cyclotron motion. Then continuation of th e discussion for a\nhigher dimensional oscillator is straight forward.\nThen for ωx=ωy= 0 one may show easily from Eq. (151) that\nωE= Ω2. (156)\nThus it constitutes an important check for the present approxima tion calculation. We now\nconsider the condition when both ωxandωyare not zero. If ωy≪ωEthen one may show\neasily from Eq. (151) that\nωR≃/radicalbig\nω2\nx+Ω2. (157)\nSimilarly for ωx≪ωEwe get\nωR≃/radicalBig\nω2y+Ω2. (158)\nTable IV: Comparison between theoretically calculated res onating frequency and the exact result\nfor the driven dampedan-isotropic harmonicoscillator. Fo r therelation, ωR≃/radicalbig\nω2x+Ω2,ω2\nx= 2.0\nandω2\ny= 0.1. Similarly for ωR≃/radicalBig\nω2y+Ω2,ω2\nx= 0.1 andω2\ny= 3.0.\nValueResonance at ωR≃/radicalbig\nω2x+Ω2Resonance at ωR≃/radicalBig\nω2y+Ω2\nof ΩTheoretical Exact Theoretical Exact\n0.51.500 1.500 1.803 1.799\n1.01.732 1.740 2.000 2.000\n1.52.061 2.069 2.291 2.299\n2.02.449 2.460 2.646 2.649\nAgain for ωx≫ωE, we have from Eq. (151)\nωL≃ωxωy/radicalbig\nω2\nx+Ω2. (159)\nSimilarly for ωy≫ωE, we get\n35/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s97/s40/s98/s41/s32 /s32/s61/s32/s48/s46/s53\n/s32 /s32/s61/s32/s49/s46/s48\n/s32 /s32/s61/s32/s49/s46/s53\n/s32 /s32/s61/s32/s50/s46/s48\n/s98/s40/s97/s41 /s32 /s32/s61/s32/s48/s46/s53\n/s32 /s32/s61/s32/s49/s46/s48\n/s32 /s32/s61/s32/s49/s46/s53\n/s32 /s32/s61/s32/s50/s46/s48\n/s97/s40/s100/s41/s32 /s32/s61/s32/s48/s46/s53\n/s32 /s32/s61/s32/s49/s46/s48\n/s32 /s32/s61/s32/s49/s46/s53\n/s32 /s32/s61/s32/s50/s46/s48\n/s98/s40/s99/s41/s32 /s32/s61/s32/s48/s46/s53\n/s32 /s32/s61/s32/s49/s46/s48\n/s32 /s32/s61/s32/s49/s46/s53\n/s32 /s32/s61/s32/s50/s46/s48\nFigure 12: Plot ofaandbvs.ωEfor different values of Ω at various limits of ω2\nxandω2\ny: (a)ω2\nx= 2.0,ω2\ny=\n0.1, (b)ω2\nx= 0.1,ω2\ny= 3.0, (c)ω2\nx= 50.0,ω2\ny= 3.0, and (d) ω2\nx= 2.0,ω2\ny= 50.0. Other common parameters\nare:E0x=E0y= 0.25 (Units are arbitrary).\nωL≃ωxωy/radicalbigω2y+Ω2. (160)\nTo check the accuracy of our calculation we have demonstrated th e exact results (140-143) in\nFig.12. The resonance conditions according to this figure are compa red with the analytical\nresults in Table IV and V. These show that there is a very good agree ment between the ap-\nproximate and the exact results. Further more, one may explain an y distinguishable feature\nsuch as the peak height of asymmetric spectrum of amplitudes cons idering the interference\nbetween the input signal.\nA. The energetics:\nExtension of Eqs. (69-71) for three dimensional harmonic oscillato r can be read as\n36Table V: Comparison between theoretically calculated reso nating frequency and the exact result\nfor the driven damped an-isotropic harmonic oscillator. Fo r the relation, ωL≃ωxωy√\nω2x+Ω2,ω2\nx= 50.0\nandω2\ny= 3.0. Similarly for ωL≃ωxωy√\nω2y+Ω2,ω2\nx= 2.0 andω2\ny= 50.0.\nValue Corresponding to the left peak\nofResonance at ωL≃ωxωy√\nω2x+Ω2Resonance at ωL≃ωxωy√\nω2y+Ω2\nΩTheoretical Exact Theoretical Exact\n0.51.723 1.730 1.407 1.409\n1.01.715 1.709 1.400 1.399\n1.51.707 1.689 1.393 1.379\n2.01.698 1.659 1.387 1.360\n/an}bracketle{tP/an}bracketri}ht=1\n2mγω2\nE/parenleftbig\na2+b2+c2/parenrightbig\n,\n/an}bracketle{tE/an}bracketri}ht=1\n2m/parenleftbig\nω2\nx+ω2\nE/parenrightbig1\n2a2+1\n2m/parenleftbig\nω2\ny+ω2\nE/parenrightbig1\n2b2+1\n2m/parenleftbig\nω2\nz+ω2\nE/parenrightbig1\n2c2\nand\nQ=(ω2\nx+ω2\nE)a2+/parenleftbig\nω2\ny+ω2\nE/parenrightbig\nb2+(ω2\nz+ω2\nE)c2\n2γωE(a2+b2+c2)\nMaking use of Eqs. (86-88) into the above relations one may find the role of magnetic field\nin the process of energy storing.\nIV. CONCLUSION\nIn the present study we have considered various aspects of drive n damped harmonic\noscillator in the presence of a magnetic field. Our investigation include s the following points.\n(A) Dynamics of a two dimensional harmonic oscillator (having same fr equency along\nboth the directions) in the presence of magnetic and dissipative for ces:\n(i) The exact solution of the equations of motion in the absence of da mping exhibits a\ntime dependent position which is composed of two frequencies as a sig nature of the cross\neffect of the non conservative magnetic force. Then we have dete rmined the condition for a\nsimple periodic motion.\n(ii)Wehave determined thesolutions ofequationsofmotionatweak d amping limit which\nexhibitsthatthedampedoscillationisacomposedoftwofrequencies . Thesesolutionsreduce\nto all the standard results for the cases like damped harmonics osc illator and cyclotron\nmotion, respectively.\n37Thus the above cases may imply the resonance conditions for the pe riodically driven\ndamped harmonic oscillator in the presence of a magnetic field. Regar ding this the following\nmajor points are given below.\n(B) Dynamics at steady state for the periodically driven damped two dimensional har-\nmonic oscillator in the presence of a magnetic field:\n(i) We have determined the relevant amplitudes and phase constant s for the oscillation\nat steady state. The values of the amplitudes at resonance condit ion become infinite in\nthe absence of damping. It is to be noted here that for this case, t he phase shift between\nthe input and output signals at the resonance condition is similar as th at of the driven\ndamped harmonic oscillator. It proofs indirectly that the finite value of the amplitude at\nthe resonance condition in the presence of damping is solely due to th e dissipation of energy.\nIn other words, the phase shift has no significant role in this contex t.\n(ii) Magnetic field induces an asymmetric splitting of the spectrum of o utput signal with\ntwo peaks.\n(iii) The relevant resonance conditions has been determined with a ve ry good approxi-\nmation which describes how the resonating frequency depends on t he magnetic field and the\nforce constant of the harmonic oscillator. This leads to have the re sonance condition for the\ndriven damped cyclotron motion.\n(iv) There is a magnetic field induced phase shift between the input an d output signals\nbears a clear signature of the magnetic field induced breakdown of t he equivalence of the\ntwo dimensional motion.\n(v) Using the phase difference between the components of the driv ing field one may\nmodulate the amplitude of the oscillation at the steady state. As a co nsequence of the\ninterference between the two driving components through the ve locity dependent coupling\nthe following special features may appear. (a) The peak height at h igher frequency may be\nhigher compared to the other (b) One of the two pics may disappear for a certain phase\ndifference. (c) The anti resonance phenomenon may occur. Even it may appear for a driven\ndamped cyclotron motion where the system with purely non conserv ative force fields is\ndriven by an electric field. It is to be noted here that for these case s both resonance and anti\nresonance phenomena may appear at the same driving frequency d epending upon the phase\ndifference between the input signals. But for a three dimensional an isotropic oscillator the\nanti resonance may appear at near by the resonating frequency .\n(vi) Our calculation shows that the stored average energy closely m imics the variation\nof amplitude as a function of driving frequency but the average pow er does not follow this\npattern. At the resonance condition, the latter decreases mono tonically with increase in the\nstrength of the applied field but the former pass through a minimum. For a given damped\nharmonic oscillator the efficiency like quantity in the energy storing pr ocess does not depend\non the parameters related to driving electric field. But it depends on the damping strength\nand the frequency of the oscillator, respectively.\n(C) Dynamics at steady state for the periodically driven damped thr ee dimensional har-\n38monic oscillator in the presence of a magnetic field at arbitrary direct ion:\n(i) For this very general case, we have calculated the relevant amp litudes and the phase\nconstants for the steady state dynamics. Here we find that magn etic field may induce one\nmore additional peak compared to the previous case. It is interest ing to be noted here that\nthe position of the additional peak for the isotropic harmonic oscillat or does not depend\non the strength and the direction of the applied magnetic field . Thus this observation has\na similarity with the Normal Zeeman Effect. Again it is to be noted here t hat if a driven\ndamped isotropic harmonic oscillator experiences a magnetic field alon gz-direction then one\nmay find three peaks and the phenomenon is also similar to the Normal Zeeman Effect. To\navoid any confusion we would mention here that if the driven damped t hree dimensional an-\nisotropic oscillator experiences a magnetic field along the z-direction then the characteristics\nof the spectrum also may mimic the Normal Zeman Effect. Finally, it is to be noted here\nthat the position of the three peaks may depend on the direction of the applied field (if it is\nnot along the z-direction) for the three dimensional an-isotropic oscillator\n(ii) The generalization of the problem restricts us to determine the r elevant resonance\nconditions only for the special cases like (1) the periodically driven da mped isotropic har-\nmonic oscillator in the presence of a magnetic field whose all the compo nents are same, (2)\nthe periodically driven damped two dimensional harmonic oscillator (ha ving different force\nconstant along each direction) whose motion is confined in x−yplane in the presence of\na magnetic field along z-direction and (3) the periodically driven damped one dimensional\nharmonicoscillator inthepresence ofa magneticfield which isperpend icular tothedirection\nof the harmonic motion.\n(iii) Finally, we have determine how the quantities like the phase shift, t he average power\nand the average stored energy depend on the direction of the mag netic field, the damping\nstrength and the frequencies of the three dimensional oscillator.\nBefore leaving this section we would mention the possible applications o f the present\nstudy. In this context one may address the following issues. First, effect of magnetic field\non the vibrational resonance. Second, the magnetic field may induc e a phase shift between\nthe input and output signals even in the absence of damping. Thus inv estigation on the\nmodulation of the refractive index of a material by virtue of the Lor entz force may be an\nworthyissue. Third, modulationofthefrequencyaswellasenergy ofaharmonicoscillatorby\nan applied magnetic field clearly requires a detail study on the therma lly activated barrier\ncrossing dynamics in the presence of an electromagnetic field. One m ay be interested to\ncheck whether the asymmetric splitting of the spectrum of the bar rier crossing rate constant\noccurs or not in the presence of the field. Finally, in a recent study [3 5] we have shown that\na colored noise can recognize the character of a dynamical system in terms of autonomous\nstochastic resonance which caries the sense of the dynamical res onance. 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E98, 012120 (2018).\n41"    },    {        "title": "1604.02998v1.All_Optical_Study_of_Tunable_Ultrafast_Spin_Dynamics_in__Co_Pd__NiFe_Systems__The_Role_of_Spin_Twist_Structure_on_Gilbert_Damping.pdf",        "content": "All-Optical Study of Tunable Ultrafast Spin Dynamics in [Co/Pd]-NiFe Systems: The\nRole of Spin-Twist Structure on Gilbert Damping\nChandrima Banerjee,1Semanti Pal,1Martina Ahlberg,2T. N.\nAnh Nguyen,3, 4Johan \u0017Akerman,2, 4and Anjan Barman1,\u0003\n1Department of Condensed Matter Physics and Material Sciences,\nS. N. Bose National Centre for Basic Sciences, Block JD, Sec. III, Salt Lake, Kolkata 700 098, India\n2Department of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden\n3Laboratory of Magnetism and Superconductivity,\nInstitute of Materials Science, Vietnam Academy of Science and Technology,\n18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam.\n4Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden\n(Dated: April 12, 2016)\nWe investigate optically induced ultrafast magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) exchange-spring samples with tilted perpendicular magnetic anisotropy using a time-\nresolved magneto-optical Kerr e\u000bect magnetometer. The competition between the out-of-plane\nanisotropy of the hard layer, the in-plane anisotropy of the soft layer and the applied bias \feld reor-\nganizes the spins in the soft layer, which are modi\fed further with the variation in t. The spin-wave\nspectrum, the ultrafast demagnetization time, and the extracted damping coe\u000ecient all depend on\nthe spin distribution in the soft layer, while the latter two also depend on the spin-orbit coupling\nbetween the Co and Pd layers. The spin-wave spectra change from multimode to single-mode as t\nincreases. At the maximum \feld reached in this study, H=2.5 kOe, the damping shows a nonmono-\ntonic dependence on twith a minimum at t= 7.5 nm. For t<7.5 nm, intrinsic e\u000bects dominate,\nwhereas for t>7.5 nm, extrinsic e\u000bects govern the damping mechanisms.\nI. INTRODUCTION\nNonuniform magnetic structures, including exchange\nbias (ferromagnet/antiferromagnet)3,24and exchange-\nspring (ferromagnet/ferromagnet)5{8systems, have\nrecently been explored extensively on account of their\nintrinsic advantages for applications in both permanent\nmagnets and recording media. Exchange-spring (ES)\nmagnets are systems of exchanged-coupled hard and soft\nmagnetic layers that behave as a single magnet. Here,\nthe high saturation magnetization ( Ms) of the soft phase\nand the high anisotropy ( Hk) of the hard phase result in\na large increase in the maximum energy product. This\nmakes them useful as permanent magnets in energy ap-\nplications such as engines or generators in miniaturized\ndevices. On the other hand, for spintronic applications,\nthe soft phase is used to improve the writability of\nthe magnetic media, which in turn is stabilized by the\nmagnetic con\fguration of the hard layer. Consequently,\na wealth of research has been devoted to investigating\nthe static and dynamic magnetic properties, including\nthe switching behavior and exchange coupling strength,\nin ES systems.\nIn case of ES systems with tilted anisotropy, the hard\nand soft phases consist of materials with out-of-plane\n(OOP) and in-plane (IP) anisotropies, respectively. This\ncombination results in a canting of the magnetization\nof the soft layer with a wide and tunable range of tilt\nangles. The advantage of such a hybrid anisotropy sys-\ntem is that it is neither plagued by the poor writability\nand thermal instability of systems with IP anisotropy,\nnor does it lead to very high switching \felds, as in OOPsystems. As a result, these materials provide additional\ndegrees of freedom to control the magnetization dynam-\nics in magnetic nanostructures, and hint at potential\napplications in novel spintronic devices utilizing the\nspin-transfer torque (STT) e\u000bect|such as spin-torque\noscillators (STOs)25,26and STT-MRAMs.\nSo far, numerous studies have been performed on\nsuch systems where the exchange coupling between\nthe hard and soft layers has been tailored by varying\nthe layer thickness,12,13layer composition,19number\nof repeats,15and interfacial anisotropy.13The litera-\nture describes investigations of domain structure and\nother static magnetic properties for [Co/Pd]/Co,14\n[Co/Pd]/NiFe,12,14,19,21[Co/Pd]/CoFeB,14,15,20\n[Co/Pd]-Co-Pd-NiFe,13[Co/Ni]/NiFe,4and CoCrPt-\nNi11|these systems being studied with static mag-\nnetometry, magnetic force microscopy (MFM), and\nmicromagnetic simulations. The magnetization dy-\nnamics in such systems have also been measured using\nBrillouin light scattering (BLS)19,20and ferromagnetic\nresonance (FMR)21experiments, where the spin-wave\n(SW) modes have been investigated by varying the thick-\nness of the soft layer and changing the con\fguration of\nthe hard layer. In any process involving magnetization\ndynamics, the Gilbert damping constant ( \u000b) plays a key\nrole in optimizing writing speeds and controlling power\nconsumption. For example, in case of STT-MRAM\nand magnonic devices, low \u000bfacilitates a lower writing\ncurrent and the longer propagation of SWs, whereas a\nhigher\u000bis desirable for increasing the reversal rates and\nthe coherent reversal of magnetic elements, which are\nrequired for data storage devices.arXiv:1604.02998v1  [cond-mat.mtrl-sci]  11 Apr 20162\n46810121416350400450500\n  )sf( emit noitazitengameDt (nm)(d)(a) \n-202 600 1200 1800-2-10\n  Kerr rotation (a rb. unit)\nTime (ps)(b)0 10 20 30\n  Power (arb. unit)\nFrequency (GHz)(c)\n(b)\n-202 60012001800-2-10Kerrrotation(arb.unit)\nTime(ps)\nFigure 1. (color online) (a) Schematic of the two-color pump-\nprobe measurement of the time-resolved magnetization dy-\nnamics of exchange-spring systems. The bias \feld is applied\nwith a small angle to the normal of the sample plane. (b)\nTypical time-resolved Kerr rotation data revealing ultrafast\ndemagnetization, fast and slow relaxations, and precession\nof magnetization for the exchange-spring system with t=\n7.5 nm at H= 2.5 kOe. (c) FFT spectrum of the background-\nsubtracted time-resolved Kerr rotation. (d) Variation of de-\nmagnetization time with t.\nIn this paper, we present all-optical excitation and de-\ntection of magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) tilted anisotropy ES systems, with varying\nsoft layer thickness ( t), using a time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer. The dy-\nnamical magnetic behavior of similar systems has previ-\nously been studied using BLS19and FMR21measure-\nments. However, a detailed study of the precessional\nmagnetization dynamics and relaxation processes in such\ncomposite hard/soft systems is yet to be carried out.\nThe advantage of implementing TR-MOKE is that here\nthe magnetization dynamics can be measured on di\u000ber-\nent time scales and the damping is measured directly\nin the time domain, and is therefore more reliable. We\ninvestigate the ultrafast magnetization dynamics over pi-\ncosecond and picosecond time scales. The ultrafast de-\nmagnetization is examined and found to change due to\nthe modi\fed spin structure in the soft layer for di\u000berent\ntvalues. The extracted SW spectra are strongly depen-\ndent on t. An extensive study of the damping coe\u000ecient\nreveals that the extrinsic contribution to the damping\nis more dominant in the higher thickness regime, while\nintrinsic mechanisms govern the behavior at lower thick-\nnesses.II. EXPERIMENTAL DETAILS\nA. Sample fabrication\nThe samples were fabricated using dc mag-\nnetron sputtering and have the following structure:\nTa(5nm)/Pd(3nm)/[Co(0.5nm)/Pd(1nm)] \u00025=Ni80Fe20(t)\n/Ta(5nm), where t= 4{20 nm. The chamber base pres-\nsure was below 3 \u000210\u00008Torr, while the Ar work\npressure was 2 and 5 mTorr for the Ta, NiFe and Co,\nPd layers, respectively. The samples were deposited\nat room temperature on naturally oxidized Si(100)\nsubstrates. The 5 nm Ta seed layer was used to induce\nfcc-(111) orientation in the Pd layer, which improves\nthe perpendicular magnetic anisotropy of the Co/Pd\nmultilayers; a Ta cap layer was used to avoid oxidation,\nwhich has been reported in previous studies.12{14The\nlayer thicknesses are determined from the deposition\ntime and calibrated deposition rates.\nB. Measurement technique\nTo investigate the precessional frequency and damp-\ning of these samples, the magnetization dynamics were\nmeasured by using an all-optical time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer2based on\na two-color optical pump-probe experiment. The mea-\nsurement geometry is shown in Fig. 1(a). The magne-\ntization dynamics were excited by laser pulses of wave-\nlength (\u0015) 400 nm (pulse width = 100 fs, repetition rate\n= 80 MHz) of about 16 mJ/cm2\ruence and probed by\nlaser pulses with \u0015= 800 nm (pulse width = 88 fs, rep-\netition rate = 80 MHz) of about 2 mJ/cm2\ruence. The\npump and probe beams are focused using the same micro-\nscope objective with N.A. of 0.65 in a collinear geometry.\nThe probe beam is tightly focused to a spot of about\n800 nm on the sample surface and, as a result, the pump\nbecomes slightly defocused in the same plane to a spot\nof about 1 \u0016m. The probe beam is carefully aligned at\nthe centre of the pump beam with slightly larger spot\nsize. Hence, the dynamic response is probed from a ho-\nmogeneously excited volume. The bias \feld was tilted\nat around 15\u000eto the sample normal (and its projection\nalong the sample normal is referred to as Hin this ar-\nticle) in order to have a \fnite demagnetizing \feld along\nthe direction of the pump beam. This \feld is eventually\nmodi\fed by the pump pulse which induces precessional\nmagnetization dynamics in the samples. The Kerr rota-\ntion of the probe beam, back-re\rected from the sample\nsurface, is measured by an optical bridge detector us-\ning phase sensitive detection techniques, as a function of\nthe time-delay between the pump and probe beams. Fig-\nure 1(b) presents typical time-resolved Kerr rotation data\nfrom the ES sample with t= 7.5 nm at a bias \feld H=\n2.5 kOe. The data shows a fast demagnetization within\n500 fs and a fast remagnetization within 8 ps, followed by\na slow remagnetization within 1800 ps. The precessional3\n(b) \n010 20 30 0 1 2\nPower (arb. unit)  Kerr Rotation(arb. unit)\n  \n  \n    \n  \n    \n  \nFrequency (GHz)4.5 nm\n5.5 nm\n7.5 nm\n8 nm\n15 nm\n  \nTime (ns)20 nm \n  \n  B\nA \n   \nNiFe  (t = 20 nm)  \nCo/Pd \n1 -1 Normalized Mz Co/Pd NiFe  (t = 6 nm)  \nCo/Pd NiFe ( t = 10 nm) (a) \nFigure 2. (color online) (a) Background-subtracted time-\nresolved Kerr rotation and the corresponding FFT spectra\nfor samples with di\u000berent tvalues at H= 2.5 kOe. The\nblack lines show the \ft according to Eq. 1. (b) Simulated\nstatic magnetic con\fgurations for samples with t= 20, 10,\nand 6 nm with a bias \feld H= 2.5 kOe in the experimental\ncon\fguration. The simulated samples are not to scale. The\ncolor map is shown at the bottom of the \fgure.\ndynamics appear as an oscillatory signal above the slowly\ndecaying part of the time-resolved Kerr rotation data.\nThis part was further analyzed and a fast Fourier trans-\nform (FFT) was performed to extract the corresponding\nSW modes, as presented in Fig. 1(c).III. RESULTS AND DISCUSSIONS\nIn order to closely observe the ultrafast demagnetiza-\ntion and fast remagnetization, we recorded the transient\nMOKE signals for delay times up to 30 ps at a resolution\nof 50 fs. In Fig. 1(d), the demagnetization times are plot-\nted as a function of t. We observe that the demagnetiza-\ntion is fastest in the thinnest NiFe layer ( t= 4 nm) and\nincreases sharply with the increase in t, becoming con-\nstant at 500 fs at t= 5 nm. At t= 10 nm, it decreases\ndrastically to 400 fs and remains constant for further in-\ncreases in t. For t<5 nm, the laser beam penetrates\nto the Co/Pd layer. In this regime, the large spin-orbit\ncoupling of Pd enhances the spin-\rip rate, resulting in a\nfaster demagnetization process. As tincreases, the top\nNiFe layer is primarily probed. Here, the spin con\fgura-\ntion across the NiFe layer, which is further a\u000bected by the\ncompetition between the in-plane and the out-of-plane\nanisotropies of the NiFe and [Co/Pd] layers, governs the\ndemagnetization process. Qualitatively, ultrafast demag-\nnetization can be understood by direct transfer of spin\nangular momentum between neighboring domains10,23.\nwhich may be explained as follows: For t>8 nm, the\nmagnetization orientation in the NiFe layer varies over a\nwide range of angles across the \flm thickness, where the\nmagnetization gradually rotates from nearly perpendicu-\nlar at the Co/Pd and NiFe interface to nearly parallel to\nthe surface plane in the topmost NiFe layer. Such a spin\nstructure across the NiFe layer thickness can be seen as a\nnetwork of several magnetic sublayers, where the spin ori-\nentation in each sublayer deviates from that of the neigh-\nboring sublayer. This canted spin structure accelerates\nthe spin-\rip scattering between the neighboring sublay-\ners and thus results in a shorter demagnetization time,\nsimilar to the work reported by Vodungbo et al.23On the\nother hand, for 5 nm <t<8 nm, the strong out-of-plane\nanisotropy of the Co/Pd layer forces the magnetization\nin the NiFe \flm towards the direction perpendicular to\nthe surface plane, giving rise to a uniform spin structure.\nThe strong coupling reduces the transfer rate of spin an-\ngular momentum and causes the demagnetization time\nto increase.\nTo investigate the variation of the precessional dynamics\nwith t, we further recorded the time-resolved data for a\nmaximum duration of 2 ns at a resolution of 10 ps. Fig-\nure 2(a) shows the background-subtracted time-resolved\nKerr rotation data for di\u000berent values of tatH=\n2.5 kOe and the corresponding fast Fourier transform\n(FFT) power spectra. Four distinct peaks are observed\nin the power spectrum of t= 20 nm, which reduces to\ntwo for t= 15 nm. This is probably due to the rela-\ntive decrease in the nonuniformity of the magnetization\nacross the NiFe thickness, which agrees with the varia-\ntion in the demagnetization time, as described earlier. To\ncon\frm this, we simulated the static magnetic con\fgura-\ntions of these samples in a \feld of H= 2:5 kOe using the\nLLG micromagnetic simulator.18Simulations were per-\nformed by discretizing the samples in arrays of cuboidal4\ncells with two-dimensional periodic boundary conditions\napplied within the sample plane. The simulations assume\nthe Co/Pd multilayer as an e\u000bective medium16with sat-\nuration magnetization Ms= 690 emu/cc, exchange sti\u000b-\nness constant A= 1.3\u0016erg/cm, and anisotropy constant\nKu1= 5.8 Merg/cc along the (001) direction, while the\nmaterial parameters used for the NiFe layer were Ms=\n800 emu/cc, A= 1.3\u0016erg/cm, and Ku1= 0.17The in-\nterlayer exchange between Co/Pd and NiFe is set to 1.3\n\u0016erg/cm and the gyromagnetic ratio \r= 18.1 MHz Oe\u00001\nis used for both layers. The Co/Pd layer was discretized\ninto cells of dimension 5 \u00025\u00022 nm3and the NiFe layer\nwas discretized into cells of dimension 5 \u00025\u00021 nm3.\nThe results are presented in Fig. 2(b) for t= 20, 10, and\n6 nm samples. The nonuniform spin structure is promi-\nnent in the NiFe layer of the t= 20 nm sample, which\nmodi\fes the SW spectrum of this sample, giving rise to\nthe new modes.17With the reduction of t, the spin struc-\nture in the NiFe layer gradually becomes more uniform,\nwhile at t= 7.5 nm it is completely uniform over the\nwhole thickness pro\fle. Hence, for low values of t, the\npower spectra shows a single peak due to the collective\nprecession of the whole stack. The variation in precession\nfrequency with tis plotted in Fig. 3(a). The frequency\nof the most intense mode shows a slow decrease down to\nt= 7.5 nm, below which it increases sharply down to the\nlowest thickness t= 4.5 nm, exhibiting precessional dy-\nnamics. This mode is basically the uniform mode of the\nsystem and follows Kittel's equation.9The variation in\nfrequency depicts the evolution of the e\u000bective anisotropy\nfrom OOP to IP with increasing t, which is in agreement\nwith previously reported results.21For lower t, the sys-\ntem displays an OOP easy axis, owing to the strong OOP\nanisotropy of the Co/Pd multilayer. This is manifested\nas a sharp increase in the frequency with decreasing t\nbelow 7.5 nm. For greater thicknesses, the e\u000bect of the\nperpendicular anisotropy of the Co/Pd multilayer gradu-\nally decreases and the e\u000bect of the in-plane NiFe becomes\nmore prominent. These two anisotropies cancel near t=\n7.5 nm, resulting in a minimum frequency, as shown in\nFig. 3(a).\nTo extract the damping coe\u000ecient, the time domain\ndata was \ftted with an exponentially damped harmonic\nfunction given by Eq. 1.\nM(t) =M(0)e\u0000t\n\u001csin(2\u0019ft\u0000\u001e) (1)\nwhere the relaxation time \u001cis related to the Gilbert\ndamping coe\u000ecient \u000bby the relation \u001c= 1/(2\u0019f\u000b).\nHere, fis the experimentally obtained precession fre-\nquency and \u001eis the initial phase of the oscillation. The\n\ftted data for various values of tis shown by the solid\nblack lines in Fig. 2. We did not extract a value of \u000bfort\n= 15 and 20 nm due to the occurrence of multimode os-\ncillations, which may lead to an erroneous estimate of the\ndamping. The extracted \u000bvalues are plotted against tin\nFig. 3(b) for two di\u000berent \feld values of 2.5 and 1.3 kOe.\nThe evolution of \u000bas a function of tdepends signi\fcantly\n5 6 7 8 9 100.0160.0200.0240.0280.032\n5 10 15 2041216\nt(nm) t(nm)(a) (b)Frequency(GHz)Figure 3. (color online) Evolution of (a) spin-wave frequency\nand (b) Gilbert damping constant as a function of tat 1.3 kOe\n(green circles) and 2.5 kOe (violet circles).\nonH, as can be seen from the \fgure. This is because of\nthe di\u000berent mechanisms responsible for determining the\ndamping in di\u000berent samples, as will be discussed later.\nAn interesting trend in the \u000bvs.tplot is observed\nforH= 2.5 kOe. For 10 nm \u0014t\u00147.5 nm,\u000bdecreases\nwith decreasing tand reaches a minimum value of about\n0.014 for t= 7.5 nm. Below this thickness, \u000bincreases\nmonotonically and reaches a value of about 0.022 for the\nlowest thickness. This variation of \u000bis somewhat cor-\nrelated with the variation of precession frequency with\nthickness. In the thinner regime, we probe both the\nNiFe layer and a fraction of the Co/Pd multilayer and\nthe relative contribution from the latter increases as t\ndecreases. The occurrence of a single mode oscillation\npoints towards a collective precession of the stack, which\nmay be considered a medium with e\u000bective magnetic pa-\nrameters consisting of both NiFe and Co/Pd layers. The\nvariation in damping may be related to the variation in\nthe anisotropy of the material. The competing IP and\nOOP anisotropies of the NiFe and Co/Pd layers lead\nto the appearance of a minimum in the damping. The\ndamping in this system may have multiple contributions,\nnamely (a) dephasing of the uniform mode in the spin-\ntwist structure1(b) interfacial d-dhybridization at the\nCo/Pd interface16, and (c) spin pumping into the Pd\nlayer.22The \frst is an extrinsic mechanism and is dom-\ninant in samples with higher NiFe thicknesses, while the\nother two mechanisms are intrinsic damping mechanisms.\nFort>7.5 nm, due to the nonuniformity of the spin\ndistribution, the dominant mode undergoes dynamic de-\nphasing and the damping thus increasescompared to the\nmagnetically uniform samples. With the increase in NiFe\nthickness, the nonuniformity of spin distribution and the\nconsequent mode dephasing across its thickness increases,\nleading to an increase in the damping value. Hence, in\nsamples with higher tvalues, dephasing is the dominant\nmechanism, while at lower tvalues|i.e., when the con-\ntribution from the Co/Pd multilayer is dominant|the\nspin-orbit coupling and spin pumping e\u000bects dominate.\nAt intermediate tvalues, the extrinsic and intrinsic ef-\nfects compete with each other, leading to a minimum\nin the damping. However, the damping increases mono-\ntonically with tin a lower \feld of H=1.3 kOe. For a\ndeeper understanding of this e\u000bect, we have measured \u000b5\n24681012140.0120.0160.0200.0240.0280.0324\n56789100.0140.0210.0280.0350.042(b) \n  \n5nm \n5.5nm \n6.5nm \n7nm/s61537F\nrequency (GHz)(a) \n  \n10nm \n8.5nm \n8nm \n7.5nm \n7nm/s61537F\nrequency (GHz)\nFigure 4. (color online) Dependence of Gilbert damping co-\ne\u000ecient on soft layer thickness ( t) for (a) 7{10 nm and (b)\n5{7 nm, respectively.\nas a function of precession frequency f. Figures 4(a){(b)\nshow the variation of \u000bwith f. Two di\u000berent regimes in\nthe thickness are presented in (a) and (b) to show the\nrate of variation more clearly. For 10 nm \u0014t\u00147 nm,\u000b\ndecreases strongly with the decrease in fand the rate of\nvariation remains nearly constant with t. This is the sig-\nnature of extrinsic damping generated by the nonuniform\nspin distribution. However, for t= 6.5 nm, the rate falls\ndrastically and for t\u00145.5 nm,\u000bbecomes nearly indepen-\ndent of t, which indicates that purely intrinsic damping is\noperating in this regime. This con\frms the competition\nbetween two di\u000berent types of damping mechanisms in\nthese samples.\nThe study demonstrates that various aspects of ul-\ntrafast magnetization dynamics|namely demagnetiza-\ntion time, precession frequency, number of modes, and\ndamping|are in\ruenced by the spin distribution in the\nsoft magnetic layer, as well as by the properties of the\nhard layer. By changing the thickness of the soft layer,\nthe relative contributions of these factors can be tuned\ne\u000bectively. This enables e\u000ecient control of the damp-\ning and other magnetic properties over a broad range,\nand will hence be very useful for potential applications\nin spintronic and magnonic devices.IV. CONCLUSION\nIn summary, we have employed the time-resolved\nMOKE technique to measure the evolution of ul-\ntrafast magnetization dynamics in exchange-coupled\n[Co/Pd] 5/NiFe( t) multilayers, with varying NiFe layer\nthicknesses, by applying an out-of-plane bias magnetic\n\feld. The coupling of a high-anisotropy multilayer with\na soft layer allows broad control over the spin struc-\nture, and consequently other dynamic magnetic prop-\nerties which are strongly dependent on t. The ultra-\nfast demagnetization displayed a strong variation with\nt. The reason for this was ascribed to the chiral-spin-\nstructure-dependent spin-\rip scattering in the top NiFe\nlayer, as well as to interfacial 3 d-4dhybridization of\nCo/Pd layer. The precessional dynamics showed mul-\ntiple spin-wave modes for t= 20 nm and 15 nm, whereas\na single spin-wave mode is observed for thinner NiFe lay-\ners following the change in the magnetization pro\fle with\ndecreasing t. The precession frequency and the damp-\ning show strong variation with the thickness of the NiFe\nlayer. The changes in frequency are understood in terms\nof the modi\fcation of the anisotropy of the system, while\nthe variation in damping originates from the competition\nbetween intrinsic and extrinsic mechanisms, which are\nsomewhat related to the anisotropy. The observed dy-\nnamics will be important for understanding the utiliza-\ntion of tilted anisotropy materials in devices such as spin-\ntransfer torque MRAM and spin-torque nano-oscillators.\nV. ACKNOWLEDGEMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Coun-\ncil (VR), the Knut and Alice Wallenberg Foundation\n(KAW), and the Swedish Foundation for Strategic Re-\nsearch (SSF). This work was also supported by the Euro-\npean Research Council (ERC) under the European Com-\nmunity's Seventh Framework Programme (FP/2007{\n2013)/ERC Grant 307144 \"MUSTANG\". AB acknowl-\nedges the \fnancial support from the Department of Sci-\nence and Technology, Government of India (Grant no.\nSR/NM/NS-09/2011(G)) and S. N. Bose National Centre\nfor Basic Sciences, India (Grant no. SNB/AB/12-13/96).\nC.B. thanks CSIR for the senior research fellowship.\n\u0003abarman@bose.res.in\n1A. Barman and S. Barman. Dynamic dephasing of mag-\nnetization precession in arrays of thin magnetic elements.\nPhys. Rev. B , 79:144415, 2009.\n2A. Barman and A. Haldar. 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Phys. , 105:07D116, 2009."    },    {        "title": "1508.01478v1.Phenomenological_description_of_the_nonlocal_magnetization_relaxation_in_magnonics__spintronics__and_domain_wall_dynamics.pdf",        "content": "Phenomenological description of the nonlocal magnetization relaxation in magnonics,\nspintronics, and domain-wall dynamics\nWeiwei Wang,1Mykola Dvornik,2, 3Marc-Antonio Bisotti,1Dmitri Chernyshenko,1\nMarijan Beg,1Maximilian Albert,1Arne Vansteenkiste,2Bartel V. Waeyenberge,2\nAndriy N. Kuchko,4, 5Volodymyr V. Kruglyak,6and Hans Fangohr1\n1Engineering and the Environment, University of Southampton, Southampton, UK\n2DyNaMat Lab, Ghent University, Gent, Belgium\n3Physics Department, University of Gothenburg, 412 96, Gothenburg, Sweden\n4Physical and Technical Department, Donetsk National University, Donetsk, Ukraine\n5Institute of Magnetism of NAS of Ukraine, 36b Vernadskogo Avenue, Kiev, 03142, Ukraine\n6School of Physics, University of Exeter, Exeter, UK\nA phenomenological equation called Landau-Lifshitz-Baryakhtar (LLBar) equation, which could\nbe viewed as the combination of Landau-Lifshitz (LL) equation and an extra \\exchange damping\"\nterm, was derived by Baryakhtar using Onsager's relations. We interpret the origin of this \\exchange\ndamping\" as nonlocal damping by linking it to the spin current pumping. The LLBar equation is\ninvestigated numerically and analytically for the spin wave decay and domain wall motion. Our\nresults show that the lifetime and propagation length of short-wavelength magnons in the presence\nof nonlocal damping could be much smaller than those given by LL equation. Furthermore, we \fnd\nthat both the domain wall mobility and the Walker breakdown \feld are strongly in\ruenced by the\nnonlocal damping.\nPACS numbers: 75.78.Cd, 76.50.+g, 75.60.Ch\nI. INTRODUCTION\nThe genuine complexity of magnetic and spintronic\nphenomena occurring in magnetic samples and devices\nimposes both fundamental and technical limits on the\napplicability of quantum-mechanical and atomistic the-\nories to their modeling. To a certain degree, this chal-\nlenge can be circumvented by exploiting phenomenologi-\ncal theories based on the continuous medium approxima-\ntion. The theories operate with the magnetization (i.e.\nthe magnetic moment density) and the e\u000bective magnetic\n\feld as generalized coordinates and forces respectively\n[1, 2]. The e\u000bective magnetic \feld is de\fned in terms\nof various magnetic material parameters, which are de-\ntermined by \ftting theoretical results to experimental\ndata, and at least in principle, can be calculated using\nthe quantum-mechanical or atomistic methods. However,\nsolving the phenomenological models analytically is still\na formidable task in the majority of practically impor-\ntant cases. The di\u000eculty is primarily due to the pres-\nence of the long range magneto-dipole interaction and\nassociated non-uniformity of the ground state con\fgura-\ntions of both the magnetization and e\u000bective magnetic\n\feld. Hence, the phenomenological models are solved in-\nstead numerically, using either \fnite-di\u000berence or \fnite-\nelement methods realized in a number of micromagnetic\nsolvers [3{7].\nTraditionally, the software for such numerical micro-\nmagnetic simulations of magnetization dynamics is based\non solving the Landau-Lifshitz equation [1] with a trans-\nverse magnetic relaxation term, either in the original\n(Landau) [1] or \"Gilbert\" [8] form. Over time, dictated\nby the experimental and technological needs, the solvershave been modi\fed to include \fnite temperature e\u000bects\n[9] and additional contributions to the magnetic energy\n(and therefore to e\u000bective magnetic \feld) [10]. The re-\ncent advances in spintronics and magnonics have led to\nthe implementation of various spin transfer torque terms\n[11, 12] and periodic boundary conditions [13{15]. Fur-\nthermore, the progress in experimental investigations of\nultrafast magnetization dynamics [16] has exposed the\nneed to account for the variation of the length of the mag-\nnetization vector in response to excitation by femtosec-\nond optical pulses, leading to inclusion of the longitudi-\nnal relaxation of the magnetization within the formalism\nof numerical micromagnetics [17]. Provided that a good\nagreement between the simulated and measured results\nis achieved, a microscopic (i.e. quantum-mechanical or\natomistic) interpretation of the experiments can then be\ndeveloped.\nThe described strategy relies on the functional com-\npleteness of the phenomenological model. For instance, a\nforceful use of incomplete equations to describe phenom-\nena originating from terms missing from the model may\nresult in false predictions and erroneous values of \ftted\nparameters, and eventually in incorrect conclusions. The\nnature of the magnetic relaxation term and associated\ndamping constants in the Landau-Lifshitz equation is of\nparamount importance both fundamentally and techni-\ncally. It is this term that is responsible for establish-\nment of equilibrium both within the magnetic sub-system\nand with its environment (e.g. electron and phonon sub-\nsystems), following perturbation by magnetic \felds, spin\ncurrents, and/or optical pulses [16]. Moreover, it is the\nsame term that will eventually determine the energy e\u000e-\nciency of any emerging nano-magnetic devices, including\nboth those for data storage [18] and manipulation [19].arXiv:1508.01478v1  [cond-mat.mtrl-sci]  6 Aug 20152\nIn this report, we demonstrate how the phenomeno-\nlogical magnetic relaxation term derived by Baryakhtar\nto explain the discrepancy between magnetic damping\nconstants obtained from ferromagnetic resonance (FMR)\nand magnetic domain wall velocity measurements in di-\nelectrics [20{22] can be applied to magnetic metallic sam-\nples. We show that the Landau-Lifshitz equation with\nBaryakhtar relaxation term (Landau-Lifshitz-Baryakhtar\nor simply LLBar equation) contains the Landau-Lifshitz-\nGilbert (LLG) equation as a special case, while also natu-\nrally including the contribution from the nonlocal damp-\ning in the tensor form of Zhang and Zhang [23] and De\nAngeli [24]. The e\u000bects of the longitudinal relaxation\nand the anisotropic transverse relaxation on the magne-\ntization dynamics excited by optical and magnetic \feld\npulses, respectively, in continuous \flms and magnetic ele-\nments were discussed e.g. in Ref. [17, 25, 26]. So, here we\nfocus primarily on the manifestations of the Baryakhtar\nrelaxation in problems speci\fc for magnonics [19] and\ndomain wall dynamics [27, 28]. This is achieved by in-\ncorporating the LLBar equation within the code of the\nObject Oriented Micromagnetic Framework (OOMMF)\n[3], probably the most popular micromagnetic solver cur-\nrently available, and by comparing the results of simula-\ntions with those from simple analytical models. Specif-\nically, we demonstrate that the Baryakhtar relaxation\nleads to increased damping of short wavelength spin\nwaves and to modi\fcation of the domain wall mobility,\nthe latter being also a\u000bected by the longitudinal relax-\nation strength.\nThe paper is organized as follows. In Sec. II, we re-\nview and interpret the Baryakhtar relaxation term. In\nSec. III, we calculate and analyze the spin wave decay in\na thin magnetic nanowire. In Sec. IV, we simulate the\nthe suppression of standing spin waves in thin \flm. In\nSec. V, we analyze the domain wall motion driven by the\nexternal \feld and compare the relative strength of con-\ntributions from the longitudinal and nonlocal damping.\nWe conclude the discussion in Sec. VI.\nII. BASIC EQUATIONS\nIn the most general case, the LLBar equation can be\nwritten as [20, 25]\n@M\n@t=\u0000\rM\u0002He\u000b+R (1)\nwhere\r(>0) is the gyromagnetic ratio and the relaxation\ntermRis\nR=^\u0003r\u0001He\u000b\u0000^\u0003e;sp@2He\u000b\n@xs@xp: (2)\nHere and in the rest of the paper, the summation is\nautomatically assumed for repeated indices. The two\nrelaxation tensors ^\u0003rand ^\u0003edescribe relativistic and\nexchange contributions, respectively, as originally intro-\nduced in Ref. [21].To facilitate comparison with the LLB equation as\nwritten in Ref. [29], the magnetic interaction energy of\nthe sample is de\fned as\nw=w\u0016+\u00160\n8\u001f(M2\u0000M2\ne)2\nM2e; (3)\nwhereMeis the equilibrium magnitude of the magne-\ntization vector at a given temperature and zero micro-\nmagnetic e\u000bective \feld, i.e. the e\u000bective \feld derived\nfrom the micromagnetic energy density w\u0016, as used in\nstandard simulations at constant temperature under con-\nditionjMj=Me= const (i.e. with only the trans-\nverse relaxation included). The second term in right-\nhand side of Eq. (3) describes the energy density induced\nby the small deviations of the magnetization length from\nits equilibrium value Meat the given temperature, i.e.,\njM2\u0000M2\nej\u001cM2\ne, and\u001fis the longitudinal magnetic sus-\nceptibility. Therefore, the associated e\u000bective magnetic\n\feld is\nHe\u000b=\u00001\n\u00160\u000ew\n\u000eM=H\u0016+1\n2\u001f(1\u0000n2)M (4)\nwhere n=M=Me,H\u0016is the e\u000bective magnetic \feld\nassociated to w\u0016. Hereafter we assume that our system\nis in contact with the heat bath, so that the equilibrium\ntemperature and associated value of Meand\u001fremain\nconstant irrespective of the magnetization dynamics.\nIn accordance with the standard practice of both mi-\ncromagnetic simulations and analytical calculations, to\nsolve LLBar equations (1-4), one \frst needs to the cor-\nresponding static equations obtained by setting the time\nderivatives to zero and thereby to derive the spatial dis-\ntribution of the magnetization in terms of both its length\nand direction. We note that, in general (e.g. as in the\ncase of a domain wall), the resulting distribution of the\nlongitudinal e\u000bective \feld and therefore also of the equi-\nlibrium magnetization length is nonuniform, so that the\nlength is not generally equal to Me. With the static\nsolution at hands, the dynamical problem is solved so\nas to \fnd the temporal evolution of the magnetization\nlength and direction following some sort of a perturba-\ntion. Crudely speaking, the e\u000bect of the relaxation terms\nis that, at each moment of time, the magnetization di-\nrection relaxes towards the instantaneous direction of the\ne\u000bective magnetic \feld, while the magnetization length\nrelaxes towards the value prescribed by the instantaneous\nlongitudinal e\u000bective magnetic \feld. The e\u000bective \feld\nitself varies with time, which makes the problem rather\ncomplex. However, this is the same kind of complexity as\nthe one that has always been inherent to micromagnet-\nics. The account of the longitudinal susceptibility within\nthe LLBar equation only brings another degree of free-\ndom (the length of the magnetization) into the discus-\nsion. One should note however that the longitudinal sus-\nceptibility has a rather small value at low temperature\nand so its account is only required at temperatures of the\norder of the Curie temperature.3\nWe neglect throughout the paper any e\u000bects due to\nthe anisotropy of relaxation, which could be associated\ne.g. with the crystalline structure of the magnetic ma-\nterial [20, 25]. This approximation is justi\fed for poly-\ncrystalline and amorphous soft magnetic metals, as has\nbeen con\frmed by simulations presented in Ref. [25].\nHence, we represent the relaxation tensors as ^\u0003r=\u0015r^I\nand^\u0003e=\u0015e^Iwhere parameters \u0015rand\u0015eare the rela-\ntivistic and exchange relaxation damping constants and\n^Iis the unit tensor. Then, Eq. (1) is reduced to\n@tM=\u0000\rM\u0002He\u000b+\u0015rHe\u000b\u0000\u0015er2He\u000b:(5)\nWe separate the equations describing the dynamics and\nrelaxation of the length and direction of the magnetiza-\ntion vector. Representing the latter as a product of its\nmagnitude and directional unit vector M=Mm, we can\nwrite\nM@m\n@t+m@M\n@t=\u0000\rM\u0002He\u000b+R: (6)\nWe multiply this equation by mto obtain,\n@M\n@t=m\u0001R: (7)\nThen, subtracting the product of equation (7) and m\nfrom equation (6), we obtain\n@m\n@t=\u0000\rm\u0002He\u000b+1\nMR? (8)\nwhere R?=\u0000m\u0002(m\u0002R). In the rest of the paper, we\nwill use A?\u0011(A)?\u0011A\u0000(A\u0001m)mto represent the\ncomponent of the vector Athat is perpendicular (trans-\nverse) to vector m. Note that only the perpendicular\ncomponent of the torque contributes to @tm\u0011@m=@t.\nFor given temperature, Meis constant and we can de\fne\n\u000b=\u0015r=(\rMe). In the limiting case of \u001f!0,M!Me\nand thus\u000bis recognized as the Gilbert damping con-\nstant from the LLG equation. Let us now consider the\ncase of ^\u0003e6= 0. The corresponding contribution to the\nrelaxation term, which we denote here as BBar, can be\nwritten as\nBBar=\u0000\u0015er2He\u000b\u0011\u0000@iji; (9)\nwhere@i\u0011@=@xiand the quantity ji=\u0000\u0015e@iHe\u000bhas\nthe form of some magnetization current density (magne-\ntization \rux).\nFor the following, it is useful to split the e\u000bective \feld\ninto its perpendicular (relative to m) part ( H?\ne\u000b, \\perpen-\ndicular \feld\") and parallel part ( Hk\ne\u000b, \\parallel \feld\"),\ni.e.,He\u000b=H?\ne\u000b+Hk\ne\u000b, and then to consider the associ-\nated magnetic \ruxes and torques separately. The mag-\nnetic \rux of jk;i=\u0000\u0015e@iHk\ne\u000band then the contribution\nof the associated torque \u001ck=\u0000@ijk;iontomis\n(\u001ck)?=\u00002\u0015e@iHk\ne\u000b@im\u0000\u0015eHk\ne\u000b(r2m)?: (10)The perpendicular \feld can be represented as\nH?\ne\u000b=1\n\rM2\u0014\nM\u0002@M\n@t\u0015\n+O(R)\u00191\n\r[m\u0002@tm]:(11)\nSo, we can write for the magnetization \rux associated\nwith the perpendicular \feld\nj?;i=\u0000(\u0015e=\r)@i(m\u0002@tm): (12)\nThe right-hand side of Eq. (12) could be regarded as the\ntorque generated by spin current pumping since m\u0002@tm\ncan be considered as the exchange spin current [30], and\nthen for the associated perpendicular torque \u001c?, we ob-\ntain,\n\u001c?=\u0000@ij?;i=\u0000\u001bMe@i@i(m\u0002@tm); (13)\nwhere we have introduced variable \u001b=\u0015e=(\rMe). We\nshow that the torque ( \u001c?)?could be written as (see Ap-\npendix A for details)\n(\u001c?)?=Me\u0002\nm\u0002(D\u0001@tm)\u0000\u001bm\u0002r2@tm\u0003\n(14)\nwhereDis a 3\u00023 tensor [23, 31],\nD\u000b\f= 2\u001b(m\u0002@im)\u000b(m\u0002@im)\f\u0000\u001b(@im\u0001@im)\u000e\u000b\f:\n(15)\nIn the limit of \u001f!0, we assume Hk\ne\u000b= 0 and therefore\nobtain\n@tm=\u0000\rm\u0002He\u000b\u0000\r\u000bm\u0002(m\u0002He\u000b)\n+m\u0002(D\u0001@tm)\u0000\u001bm\u0002r2@tm:(16)\nAt the same time, Eq. (8) can then be written as\n@m\n@t=\u0000\rm\u0002He\u000b\u0000\rm\u0002(m\u0002HB\ne\u000b); (17)\nwhere\nHB\ne\u000b=\u000bHe\u000b\u0000\u001br2H?\ne\u000b; (18)\nandH?\ne\u000bis the transverse component of the e\u000bective\n\feld. The \frst term in Eq. (18) is kept as He\u000bsince\nm\u0002He\u000b=m\u0002H?\ne\u000b. In practice, we use Eq. (17) rather\nthan Eq. (16) for numerical implementation. As shown\nin Eq. (16) the damping terms contain both the form\n\u0000m\u0002r2@tm[30, 32] and tensor form m\u0002(D\u0001@tm) [23].\nHence, we conclude that the exchange damping can\nbe explained as the nonlocal damping, and Eq. (17) is\nthe phenomenological equation to describe the nonlocal\ndamping.\nThe intrinsic Gilbert damping is generally considered\nto have the relativistic origin [1, 33]. Phenomenologically,\nthe Gilbert damping is local and the damping due to the\nnonuniform magnetization dynamics being ignored [8].\nThe exchange relaxation term in the LLBar equation de-\nscribes the nonlocal damping due to the nonuniform ef-\nfective \feld. Despite the complexity of various damping\nmechanisms, the spin current jin conducting ferromag-\nnets can be calculated, e.g. using the time-dependent4\nPauli equation within the s-d model. The spin current\nis then given by ji= (g\u0016B~G0=4e2)(@tm\u0002@im), where\nG0is the conductivity [23], and thus the nonlocal damp-\ning of the tensor form can be obtained [23, 31]. As\nwe can see from Appendix A, this spin current densi-\ntiesjiandja\nihave the same form, and therefore, we\ncan establish that \u001b\u0018g\u0016B~G0=4e2Me. The spin cur-\nrent component jb\ni(see Appendix A) gives the term\n\u0000m\u0002r2@tm[30], and the value of \u001bcan be therefore\ninterpreted as \u001b\u0018(\r=\u00160Me)(~=2)2ne\u001csc=m\u0003, wherene\nis the conduction electron density, m\u0003the e\u000bective mass\nand\u001cscis the transverse spin scattering time [34].\nIt is of interest to compare Eq. (5) with Landau-\nLifshitz-Bloch (LLB) equation [29], which could be writ-\nten as\n@n\n@t=\u0000\rn\u0002He\u000b+\r\u000bk\nn2[n\u0001He\u000b]n\u0000\r\u000b?\nn2n\u0002(n\u0002He\u000b)\n(19)\nwhere n=M=Me(T) is the reduced magnetization and\nMe(T) is the equilibrium magnetization value at tem-\nperatureT. The e\u000bective \feld He\u000bcontains the usual\nmicromagnetic contributions Hintas well as the contri-\nbution from the temperature,\nHe\u000b=Hint+me\n2~\u001fk(1\u0000n2)n (20)\nwhereme=Me(T)=Me(0) and ~\u001fk=@m=@H withm=\nM=Me(0) [29]. By substituting Eq. (20) into Eq. (19),\none arrives at\n@n\n@t=\u0000\rn\u0002Hint+\r\u000bk(Hint)k+\r\u000b?(Hint)?\n+\u000bk\rme\n2~\u001fk(1\u0000n2)n:(21)\nMeanwhile, if we neglect the \u0015eterm in Eq. (5) and insert\nthe e\u000bective \feld Eq. (3) into Eq. (5), we obtain\n@n\n@t=\u0000\rn\u0002Hint+\r\u0015rHint+\u0015r\r\n2\u001f(1\u0000n2)n:(22)\nAs we can see, Eq. (22) is a special case of LLB equation\nwith the assumption that \u000b?=\u000bk=\u0015r=(\rMe) and\u001f=\nMe(0)~\u001fk. However, the LLB equation does not contain\nthe\u0015e-term (nonlocal damping term) which is the main\nfocus in this work.\nIII. SPIN WAVE DECAY\nTo perform the micromagnetic simulation for the spin\nwave decay, we have implemented Eq. (17) as an ex-\ntension for the \fnite di\u000berence micromagnetics package\nOOMMF. A new variable \ffor the exchange damping is\nintroduced with \u001b=\fG, whereGis a coe\u000ecient to scale\n\fto the same order as \u000b. In practice, Gwas chosen to\nbeG=A=(\u00160M2\ne).\nThe simulation geometry has dimensions Lx=\n2002 nm,Ly= 2 nm and Lz= 2 nm, and the cellsize is 1\u00022\u00022 nm3. The magnetization aligns along\ntheexdirection for the equilibrium state and the pa-\nrameters are typical of Permalloy: the exchange con-\nstantA= 1:3\u000210\u000011J=m, the saturation magnetization\nMe= 8:6\u0002105A=m and the Gilbert damping damping\ncoe\u000ecient\u000b= 0:01. The spin waves are excited locally\nin the region 0\u0014x\u00142 nm, and to prevent the spin wave\nre\rection the damping coe\u000ecient is increased linearly [35]\nfrom 0.01 at x= 1802 nm to 0.5 at x= 2002 nm.\n−0.04−0.02 0 0.02 0.04\n 0  500  1000  1500  2000my\nx (nm)Simulation\nExponential fitting\nFIG. 1. The spin wave amplitude decay along the rod, for a\nspin wave was excited locally by applying a microwave H=\nH0sin(2\u0019ft)eyof frequency f= 30 GHz and amplitude H0=\n1000 Oe in the region 0 \u0014x\u00142 nm. The data were \ftted\nusing Eq. (23) with \f= 0:02 and\u000b= 0:01.\nFigure 1 illustrates the spin wave amplitude decay\nalong the rod. The ycomponent of magnetization unit\nvectormydata for 30\u0014x\u00141800 nm were \ftted using\n(23) to extract the wave vector kand the decay constant\n\u0015, and good agreement is observed due to the e\u000bective\nabsence of spin wave re\rection. We use data after having\ncomputed the time development of the magnetization for\n4 ns to reach a steady state. The injected spin wave en-\nergy is absorbed e\u000eciently enough within the right 200\nnm of the rod due to the increased damping.\nTo analyze the simulation data, we exploit the uniform\nplane wave assumption with its exponential amplitude\ndecay due to energy dissipation, i.e. magnetization with\nthe formei(kx\u0000!t)e\u0000\u0015x, where\u0015is the characteristic pa-\nrameter of the spin wave damping. For a small amplitude\nspin wave propagation we have [36]\nm=ex+m0ei(kx\u0000!t)e\u0000\u0015x(23)\nwherejm0j\u001c1, and the e\u000bective \feld of the long rod\ncan be expressed as\nHe\u000b=Hsmxex+Dr2m; (24)\nwhere the `easy axis' anisotropy \feld Hsmxexoriginates\nfrom the demagnetizing \feld, and the constant Dmea-\nsures the strength of the exchange \feld,\nHs=2K\n\u00160Me=1\n2Me; D =2A\n\u00160Me: (25)\nTo test the spin wave decay for this system, a sinusoidal\n\feldH=H0sin(2\u0019ft)eywas applied to the rod in the\nregion 0\u0014x\u00142 nm to generate spin waves.5\n20 40 60 80 100\nFrequency f (GHz)05001000150020002500λk  (µm−2)\nβ=0\nβ=0.01\nβ=0.02\nEq.(29)\nβ=0\nβ=0.01\nβ=0.02\nEq.(29)\nFIG. 2. The spin wave decay constant{wave vector product\n\u0015kas a function of the frequency for di\u000berent \fvalues. The\nslateblue line was drawn using Eq. (29) for the case \f= 0:01.\nFigure 2 shows the product of spin wave decay con-\nstant\u0015and wave vector kas a function of the frequency.\nThe dependence is linear for the \f= 0 case, which is\nagreement with the zero adiabatic spin torque case [36].\nThe addition of a nonzero \fterm leads to a nonlinear\nrelation, and the amplitude of the spin wave decay con-\nstant that is signi\fcantly larger than that given by the\nlinear dependence. We also performed the simulation for\n\u001f>0 case by using Eq. (5) which shows that \fterm is\nthe leading factor for this nonlinearity (the relative error\nis less than 1% for \u001f= 1\u000210\u00003). To analyze the nonlin-\near dependence, we introduce the complex wave vector\nek,\nek=k+\u0015i: (26)\nBy linearizing Eq. (17) and setting the determinant of the\nmatrix to zero we obtained (see Appendix B for details):\n(!+ ~!0+i~!1)(!\u0000~!0+i~!1) = 0; (27)\nwhere ~!0=\r(Hs+D~k2) and ~!1=\u000b~!0+\fG~k2~!0. The\nsecond term of the Eq. (27) is expected to be equal to\nzero, i.e., ~!1\u0000i!+i~!0= 0. There are two scenarios to\nconsider: First is the \f= 0 case,k\u0015could be extracted\nby taking the imaginary part of ~k2at Eq. (26):\nk\u0015=1\n2Imn\n~k2o\n=\u000b!\n2(1 +\u000b2)\rD: (28)\nThe linear dependence of k\u0015as a function of frequency\nmatches the data plotted in Fig. 2. For the \f >0 case,\nsolving Eq. (27) yields in the linear with respect to the\ndamping constants approximation,\nk\u0015\u0019!\n2\rD(\u000b+\fGk2) (29)\nwhere the dispersion relation for the rod is !=\r(Hs+\nDk2). Eq. (29) shows there is an extra k2term associated\nwith the exchange damping term besides the linear de-\npendence between k\u0015and!. The slateblue line in Fig. 2is plotted using Eq. (29) with \f= 0:01 and\u000b= 0:01,\nwhich shows a good approximation for the simulation\ndata. Besides, this exchange damping could be impor-\ntant in determining the nonadiabatic spin torque. We\ncould establish the value of \fusing the existing exper-\nimental data, such as the transverse spin current data\n[34] gives\f\u00180:1 which hints the lifetime and propaga-\ntion length of short-wavelength magnons could be much\nshorter than those given by the LLG equation [37].\nIV. SUPPRESSION OF STANDING SPIN\nWAVES\nIn the presence of nonlocal damping, the high fre-\nquency standing spin waves in the thin \flms are sup-\npressed [37]. If the magnetization at the surfaces are\npinned, the spin wave resonance can be excited by a uni-\nform alternating magnetic \feld [38]. With given out-of-\nplane external \feld Hzin thez-direction, the frequencies\nof the excited spin waves of the \flm are given by [39],\n!n=!0+!M\u00152\nex\u0010n\u0019\nd\u00112\n(30)\nwheredis the \flm thickness, !0=\r(Hz\u0000Me),!M=\n\rMeand\u0015ex=p\n2A=(\u00160M2e). The excited spin wave\nmodes are labeled by the integer n, and the odd nhas\na nonvanishing interaction with the given uniform alter-\nnating magnetic \feld [38].\n2 4 6 8 10 12\nFrequency f (GHz)0100200300Im χyyn=1\nn=3n=5 n=7(a)\nβ=0\nβ=0.1\n0 5 10 15 20 25 30\nFrequency f (GHz)050100150200Im χyyn=1\nn=3(b)\nβ=0\nβ=0.1\n25 3005\nFIG. 3. Imaginary parts of the dynamical susceptibility \u001fyy\nof the \flm for (a) thickness d= 300 nm, and (b) thickness\nd= 60 nm.\nTo reduce the simulation time, we consider a system\nwith cross-sectional area 4 \u00024 nm2inxy-plane and apply\nthe two-dimensional periodic boundary conditions [14] to\nthe system. We use the Permalloy as the simulation ma-\nterial with external \feld Hz= 1\u0002106A/m and the cell\nsize is 4\u00024\u00022 nm3. Instead of applying microwaves to\nthe system, we calculate the magnetic absorption spec-\ntrum of the \flm by applying a sinc-function \feld pulse6\nh=h0sinc(!0t) to the system [40]. With the collected\naverage magnetization data, the dynamic susceptibility\n\u001fis computed using Fourier transformation [41]. For ex-\nample, the component \u001fyyis computed using mywhen\nthe pulse is parallel to the y-axis.\nFigure 3(a) shows the imaginary part of the dynamic\nsusceptibility \u001fyyfor a \flm with d= 300 nm. As we\ncan see, the spin wave of modes n= 1;3;5;::: are ex-\ncited, and the in\ruence of the \\exchange damping\" is\nsmall. However, the presence of the \\exchange damp-\ning\" suppresses the spin wave excitation ( n > 1 mode)\nsigni\fcantly for the \flm with thickness d= 60 nm, as\nshown in Fig. 3(b). The reason is because the damping\nof the standing spin waves is proportional to k4in the\npresence of exchange damping [37].\nV. DOMAIN WALL MOTION\nWe implemented Eq. (5) in a \fnite element based mi-\ncromagnetic framework to study the e\u000bect of parallel re-\nlaxation process on domain wall motion. The simulated\nsystem for the domain wall motion is a one-dimensional\n(1D) mesh with length 20000 nm and discretization size\n4 nm, a head-to-head domain wall is initialized with its\ncenter near x= 500 nm. In this section, the demagnetiz-\ning \felds are simpli\fed as Hd=\u0000NMand the demag-\nnetizing factors are chosen to be Nx= 0,Ny= 0:2 and\nNz= 0:8, respectively. The domain wall moves under\nthe applied \feld for 50 ns and the domain wall velocities\nat di\u000berent external \feld strengths are computed. Fig-\nure 4 shows the simulation results of domain wall motion\nunder external \felds for di\u000berent susceptibilities with-\nout consideration of exchange damping, i.e., \f= 0. For\nNickel and Permalloy, the longitudinal susceptibility is\naround 10\u00004at room temperature and increases with\nthe temperature up until the Curie point [29]. We \fnd\nthat the longitudinal susceptibilities have no in\ruence on\nthe maximum velocity but change the Walker breakdown\n\feldHwsigni\fcantly. The domain wall velocity in the\nlimit\u001f!0 is almost the same with the case of \u001f= 10\u00004,\nwhich could be explained by the relation that the di\u000ber-\nence proportional to the ratio of ( \u001f=\u000b)2in Eq. (53).\nTo investigate the e\u000bect of longitudinal magnetic sus-\nceptibility\u001fand exchange relaxation damping \u001bon the\ndomain wall motion, we use the remainder of this sec-\ntion for analytical studies. We start from the constant\nsaturation magnetization of one-dimensional domain wall\nmodel, such as the 1D head-to-head wall [42]. The static\n1D domain wall pro\fle can be expressed as\nmx=\u0000tanh\u0012x\u0000q\n\u0001\u0013\n; mt= sech\u0012x\u0000q\n\u0001\u0013\n(31)\nwheremtis the perpendicular component of the unit\nmagnetization vector, \u0001 is the wall width parameter and\nqis the position of the domain wall center.\nWe consider the case that the system is characterized\nby two anisotropies, easy uniaxial anisotropy Kand hard\n0 200 400 600 800 1000 1200\nApplied field (A/m)0100200300400500DW Velocity (m/s)\nχ=1×10−3\nχ=5×10−4\nχ=1×10−4\nχ=1×10−3\nχ=5×10−4\nχ=1×10−4FIG. 4. Simulations results of domain wall velocities for var-\nious susceptibilities. The parameters used are: \u000b= 0:001,\n\f= 0,Ny= 0:2 andNz= 0:8. The vertical dash lines are\nthe breakdown \felds computed using Eq. (53).\nplane anisotropy K?, which originate from demagnetiza-\ntion. The aim is to analyze the impact of the longitudinal\nmagnetic susceptibility under the 1D domain wall model,\nthe demagnetization energy density could be written as\nEan=\u0000K\nM2eM2\nx+K?\nM2eM2\nz (32)\nwhereK= (1=2)(Ny\u0000Nx)\u00160M2\neandK?= (1=2)(Nz\u0000\nNy)\u00160M2\ne. In the limit case \u001f!0 case, the e\u000bective\nanisotropy energy density Eancan be rewritten as\nE0\nan=Ksin2\u0012(1 +\u0014sin2'); (33)\nwhere m= (cos\u0012;sin\u0012cos';sin\u0012sin') is used and\n\u0014=K?=Kis the ratio between hard plane anisotropy\nstrength and easy uniaxial anisotropy strength.\nThe dynamics of the domain wall with 1D pro\fle can\nbe described using 3 parameters [43]: the domain width\n\u0001, the domain wall position qand the domain wall tilt\nangle\u001e. In this domain wall model, one can assume\nthat'(x;t) =\u001e(t) is only a function of time. Thus, the\nmagnetization pro\fle for the head-to-head domain wall\nis given by\n\u0012(x;t) = 2 tan\u00001exp\u0012x\u0000q(t)\n\u0001(t)\u0013\n; ' (x;t) =\u001e(t):\n(34)\nUsing the magnetization unit vector to calculate the\nexchange energy is a good approximation for the case\n\u001f\u001c1, thus, the total energy density can be rewritten as\nEtot=\u00160\n8\u001f(M2\u0000M2\ne)2\nM2e+M2w\u0016(m); (35)\nwhere\nw\u0016(m) =A\nM2e(rm)2\u0000K\nM2em2\nx+K?\nM2em2\nz: (36)7\nWithin the 1D domain wall pro\fle, Hm, the longitudinal\ncomponent of the e\u000bective \feld is obtained:\nHm=m\u0001He\u000b=M\n2\u001fM2e(M2\ne\u0000M2)\u00002MPsin2\u0012(37)\nwherePis de\fned as\nP=1\n\u00160M2e\u0014A\n\u00012+K(1 +\u0014sin2\u001e)\u0015\n: (38)\nAs we can see, Pis a function of the tilt angle \u001eand the\ndomain wall width \u0001. At the static state, Hmshould\nequal zero, i.e., dM=dt = 0, which gives\nM2= (1\u00004\u001fPsin2\u0012)M2\ne: (39)\nEq. (39) shows that the di\u000berence between magnetization\nlengthMandMereaches its maximum at the center of\nthe domain wall due to the e\u000bect of the exchange \feld,\nwhich also peaks in the centre of the domain wall. Ac-\ncording to Eq. (39), we can estimate that the magneti-\nzation length di\u000berence is \u000eM\u0019\u00002\u001fPsin2\u0012for\u001f\u001c1\ncase. Figure 5 shows the magnetization length di\u000ber-\nences of a 1D domain wall for various \u001f, it can be seen\nthat this approximation for \u000eMagrees very well with the\nsimulation results.\n460 480 500 520 540 560 580\nxs (nm)800\n600\n400\n200\n0δM (A/m)\nχ=1×10−3\nχ=5×10−4\nχ=1×10−4\nEq. (39)\nχ=1×10−3\nχ=5×10−4\nχ=1×10−4\nEq. (39)\nFIG. 5. Simulation results of the magnetization length di\u000ber-\nence\u000eMfor a 1D domain wall located at x= 500 nm with\nMe= 8:6\u0002105A=m andA= 1:3\u000210\u000011J/m. The demagne-\ntizing factors are selected to be Nx= 0 andNy=Nz= 0:5.\nIn the dynamic case, Hmis not equal to zero. If we\nwrote Eq. (37) as Hm=FM, we can \fnd that the non-\ntrivial term that contributes to Hmis\nF=1\n2\u001f(1\u0000M2=M2\ne)\u00002Psin2\u0012: (40)\nAs an approximation for Hm, we expect dF=dt = 0 [44],\nwhich gives\nHm=4P\n\u0001\u001f\n\u000b_q\n\rm2\ntmx: (41)\nIn this approximation, we have ignored the terms con-\ntainingdP=dt and thus the amplitude of Hmis in\ruencedby the domain wall velocity _ qonly. We employ the La-\ngrangian equation combined with dissipation function F\nto compute the domain wall dynamics [27]. The Lagrange\nequations are\n@L\n@X\u0000d\ndt\u0012@L\n@_X\u0013\n+@F\n@_X= 0; (42)\nwhereXrefers toq,\u001eand \u0001. The dissipation function\nis de\fned byF=R\nFdx where\nF=1\n2\u00160Me\r[\u000bH2\ne\u000b+\u001b(rHe\u000b)2] (43)\nis the dissipation density function.\nA. Parallel relaxation\nWe neglect the exchange damping term with assump-\ntion that\u001b\u001c\u000b\u00012and arrive at\nF=1\n2\u000b\u00160Me\rH2\ne\u000b=1\n2\u000b\u00160Me\r(H2\n?+H2\nm):(44)\nwhere H?andHmare the perpendicular and parallel\ncomponents of the e\u000bective \feld. If we also assume that\n\u000b\u0018\u001f\u001c1,H2\n?can be approximated by Eq. (11),\nH2\n?=1\n\r2_m2=1\n\r2(_\u00122+ sin2\u0012_\u001e2): (45)\nSubstituting Eq. (41) and Eq. (45) into Eq. (44) and\nintegrating over space, we obtain\nF=\u000b\u00160Me\n\r\u0014\n_\u001e2\u0001 +_q2\n\u0001(1 +Q)\u0015\n; (46)\nwhere we have ignored the _\u0001 term. This term leads to\nthe optimal domain wall width [27]:\n\u0001 =q\nA=(K+K?sin2\u001e) (47)\nand for\u0014= 0 the optimal domain wall width reduces\nto \u0001 0=p\nA=K . In what follows, the domain wall\nwidth parameter \u0001( t) is approximated by the optimal\nwall width. The parameter Pis then given by\nP=2K(1 +\u0014sin2\u001e)\n\u00160M2e=2\n\u00160M2eA\n\u00012; (48)\nand it is straightforward to \fnd its minimum P0=\n2K=(\u00160M2\ne), which corresponds to \u0001 = \u0001 0.\nThe introduced paramter Qin Eq. (46) is given by\nQ= (32=15)P2(\u001f=\u000b)2and its value is determined by the\nratio of\u001fand\u000b, which could be\u00181 although we assume\n\u001f\u0018\u000b\u001c1. Following the treatment of Ref. [27], the\nintegrated Lagrangian action Lis given by\nL=Z\n(Etot+\u00160Me\n\r_\u001ecos\u0012)dx\n=2A\n\u0001+ 2\u0001K(1 +\u0014sin2\u001e)(1\u0000V)\n\u00002\u00160MeHaq+2\u00160Me\n\r_\u001eq;(49)8\nwhere\u00160Me_\u001ecos\u0012=\ris the Berry phase term [45], V=\n8\u001fP=3 is a result of the varying magnetization that in-\ntroduced a pinning potential. However, the potential is\nfairly small and therefore is negligible since V\u001cQ. By\nsubstituting Eq. (49) and Eq. (44) into Eq. (42),\n_\u001e+\u000b_q\n\u0001(1 +Q) =\rHa;\n_q\n\u0001\u0000\u000b_\u001e=\rHk\n2sin 2\u001e:(50)\nwhereHk= 2K?=(\u00160Me). The domain wall dynamics\nis governed by Eq. (50), by eliminating _ qwe obtain an\nequation about \u001e,\n_\u001e=\r\n1 +\u000b2(1 +Q)[Ha\u0000Hw(1 +Q) sin 2\u001e] (51)\nwhereHw=\u000bHk=2 is the Walker breakdown \feld. From\nEq. (51) we can \fnd that the critical value of \u001eis ap-\nproximately equal to \u0019=4 ifQ\u001c1, which leads to the\nmaximum value of Pto beP1= 2K(1 +\u0014=2)=(\u00160M2\ne).\nThere exists an equilibrium state \u001e\u0003such that _\u001e= 0 if\nHa<Hw(1 +Q),\nsin(2\u001e\u0003) =h\u0011Ha\nHw(1 +Q); (52)\nwhich means the Walker breakdown \feld H0\nwfor\u001f > 0\ncase is increased to Hw(1 + maxfQg), i.e.,\nH0\nw=Hw\u0014\n1 +32\n15P2\n1\u0010\u001f\n\u000b\u00112\u0015\n; (53)\nwhereP1is the maximum vlaue of P. For this steady-\nstate wall motion, the domain wall velocity is\n_q=\rHa\n\u000b\u0001\u0003\n1 +Q(\u0001\u0003); (54)\nwhere\n\u0001\u0003= \u0001 0=r\n1 +\u0014\n2(1\u0000p\n1\u0000h2): (55)\nTherefore, \u0001\u0003!\u00010in the limit case Ha!0, and the\ndomain wall mobility \u0016is given by\n\u0016=\r\u0001\n\u000b\u0014\n1 +32\n15P2\n0(\u001f\n\u000b)2\u0015\u00001\n(56)\nwhereP0is the minimum value of P. In Fig. 4 the cor-\nresponding Walker breakdown \felds are plotted in ver-\ntical dash lines, which gives a good approximation for\n\u001f= 5\u000210\u00004and\u001f= 1\u000210\u00004cases. The simulation\nresults show that the Walker breakdown \feld Hwcould\nbe changed signi\fcantly if the longitudinal susceptibility\nis comparable to the damping constant.\n0200 400 600 800 1000 1200 1400\nApplied field (A/m)020406080100120140160DW Velocity (m/s)\nβ=0\nβ=0.01\nβ=0.02\nβ=0\nβ=0.01\nβ=0.02FIG. 6. Simulations results of domain wall velocities for the\nlimit case that \u001f!0 with various exchange dampings. The\nparameters used are: \u000b= 0:005,Ny= 0:4 andNz= 0:6. The\nvertical dash lines are the breakdown \felds computed with\nEq. (60).\nB. Nonlocal damping\nIn this part we consider the domain wall motion in-\n\ruenced by exchange damping for the case that \u001f!0.\nThe dissipation density function (43) thus becomes\nF=1\n2\u00160Me\r\u0002\n\u000bH2\n?+\u001b(rH\u0012)2+\u001b(rH\u001e)2\u0003\n(57)\nwhereH\u0012andH\u001eare the two components of the e\u000bec-\ntive \feld, and H?is computed using Eq. (45). After\ncalculation we obtain\nF=\u00160Me\n\r\u0014\n_\u001e2(\u000b\u0001 +1\n3\u001b\n\u0001) +_q2\n\u0001(\u000b+1\n3\u001b\n\u00012)\u0015\n:(58)\nWe take the same Lagrangian action (49) for \u001f= 0 and\narrive at\n_\u001e+ (\u000b+\u001b\n3\u00012)_q\n\u0001=\rHa;\n_q\n\u0001\u0000(\u000b+\u001b\n3\u00012)_\u001e=\rHk\n2sin 2\u001e:(59)\nSimilarly, the corresponding Walker breakdown \feld\nchanges to\nH0\nw=1\n2Hk\u0012\n\u000b+1\n3\u001b\n\u00012\n1\u0013\n; (60)\nwhere \u0001 1= \u0001 0p\n1=(1 +\u0014=2). The domain wall mobility\nis given by\n1\n\u0016=1\n\r\u00010\u0012\n\u000b+1\n3\u001b\n\u00012\n0\u0013\n: (61)\nAs we can see, the nonlocal damping term \u001bin\ruences\nthe domain wall motion as well, and we can establish\nthat\u001b=\u00012=\f(1 +\u0014=2)K=(\u00160M2\ne)/\f. Therefore, for9\nthe scenarios that K\u0018\u00160M2\ne, the contributions from\nthe Gilbert and nonlocal damping are of the order of\nmagnitude for both the domain wall mobility and Walker\nbreakdown \feld.\nFigure 6 shows the domain wall velocities for domain\nwall motion driven by external \felds in the limiting case\nof\u001f!0. The simulation results are based on a one-\ndimensional mesh with length 10000 nm with cell size of 2\nnm. The damping \u000bis set to 0:005 and the demagnetizing\nfactors are chosen to be Nx= 0,Ny= 0:4 andNz= 0:6.\nAs predicted by Eq. (60), the nonlocal damping \fleads to\nan increment of the Walker breakdown \feld, and Eq. (60)\n\fts the simulation results very well.\nVI. SUMMARY\nWe explain the \\exchange damping\" in the Landau-\nLifshitz-Baryakhtar (LLBar) equation as nonlocal damp-\ning by linking it to the spin current pumping, and there-\nfore the LLBar (17) can be considered as a phenomeno-\nlogical equation to describe the nonlocal damping. In the\npresence of nonlocal damping, the lifetime and propaga-\ntion length of short-wavelength magnons could be much\nshorter than those given by the LLG equation. Our simu-\nlation results show that the spin wave amplitude decays\nmuch faster in the presence of nonlocal damping when\nspin waves propagate along a single rod. The analyti-\ncal result shows that there is extra nonlinear dependence\nscaling with k2between\u0015k(the product of spin wave\ndecay constant \u0015and wave vector k) and frequency !\ndue to the nonlocal damping. Using the micromagnetic\nsimulation based on the LLBar equation, we show that\nthe di\u000berence between magnetization length MandMe\nreaches its maximum at the center of the domain wall.\nFor the cases that \u001f\u0018\u000bwhere\u001fis the longitudinal\nmagnetic susceptibility and \u000bis the Gilbert damping, the\nWalker breakdown \feld will increase signi\fcantly. By us-\ning a 1D domain wall model, we also show that both the\ndomain wall mobility and the Walker breakdown \feld are\nstrongly in\ruenced by the nonlocal damping as well.\nACKNOWLEDGMENTS\nWe acknowledge the \fnancial support from EPSRC's\nDTC grant EP/G03690X/1. W.W. thanks the China\nScholarship Council for \fnancial assistance. The re-\nsearch leading to these results has received funding\nfrom the European Community's Seventh Framework\nProgramme (FP7/2007-2013) under Grant Agreement\nn247556 (NoWaPhen) and from the European Union's\nHorizon 2020 research and innovation programme under\nthe Marie Sk lodowska-Curie grant agreement No 644348\n(MagIC).Appendix A: Derivation of equation (16)\nWe split the perpendicular spin current j?;iinto two\ncomponents,\nj?;i=ja\ni+jb\ni; (A1)\nwhere we write \u0015e=\ras ~\u001b,\nja\ni=\u0000~\u001b(@im\u0002@tm) (A2)\njb\ni=\u0000~\u001b(m\u0002@i@tm) (A3)\nThe torque \u001cagenerated by spin current ja\niis given by\n\u001ca= (@ija\ni)?, i.e.,\n\u001ca= ~\u001bm\u0002[@im\u0002(@tm\u0002@im)] (A4)\nwhere we have used the identities m\u0001@i@tm=\u0000@im\u0001@tm\nandm\u0001@i@im=\u0000@im\u0001@im. Meanwhile, the correspond-\ning torque \u001cbcan be computed by \u001cb= (@ijb\ni)?, which\ngives\n\u001cb=\u001ca\u0000~\u001b(@im\u0001@im)m\u0002@tm\u0000~\u001bm\u0002r2@tm(A5)\nNote that \u001ca= ~\u001b@im[(@tm\u0002@im)\u0001m] can be changed\ninto the tensor form,\n\u001ca=m\u0002(D0\u0001@tm); (A6)\nwhere\nD0\n\u000b\f= ~\u001b(m\u0002@im)\u000b(m\u0002@im)\f: (A7)\nTherefore, we obtain for \u001ca+\u001cb,\n\u001ca+\u001cb=m\u0002(D\u0001@tm)\u0000~\u001bm\u0002r2@tm (A8)\nwhereDis a 3\u00023 tensor,\nD\u000b\f= 2~\u001b(m\u0002@im)\u000b(m\u0002@im)\f\u0000~\u001b(@im\u0001@im)\u000e\u000b\f:\n(A9)\nAppendix B: Derivation of equation (27)\nWe introduce a new variable sto represent the second\nterm in the (23), i.e., s=m0ei(~kx\u0000!t), so we have\nm=ex+s; (B1)\ndm\ndt=\u0000i!s; (B2)\nHe\u000b=Hs(1 +s0\nx)ex\u0000D~k2s (B3)\nwheres0\nx\u0019(1=2)(s2\nx\u0000s2). Considering the fact jsj\u001c1\nand neglect the high order term s2, one obtains H?\ne\u000b=\n\u0000(Hs+D~k2)sand thus\nHb\ne\u000b=cex+ds (B4)\nwhere\nc=\u000bHs(1 +s0\nx); (B5)\nd=\u0000\fG~k2(D~k2+Hs)\u0000\u000bD~k2(B6)10\nSubstituting the above equations into (17), we have\ni!\n\r2\n4sx\nsy\nsz3\n5=f2\n40\nsz\n\u0000sy3\n5+ (c\u0000d)2\n4\u0000(s2\ny+s2\nz)\n(1 +sx)sy\n(1 +sx)sz3\n5;(B7)wheref=Hs(1+s0\nx)+D~k2. Neglecting high order terms\nsuch ass2\nxandsxsywe obtained,\n\u0014\n\r(\u000bHs\u0000d)\u0000i! ~w0\n\u0000~w0\r(\u000bHs\u0000d)\u0000i!\u0015\u0014\nsy\nsz\u0015\n=\u0014\n0\n0\u0015\n:\n(B8)\nTherefore, Eq. (27) can be obtained by setting the deter-\nminant of the matrix in (B8) to zero.\n[1] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 169, 14\n(1935).\n[2] A. I. Akhiezer, V. G. Baryakhtar, and S. V. Peletminskii,\nSpin waves (North-Holland, Amsterdam, 1968).\n[3] M. Donahue and D. 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Appl.\nPhys. 44, 384004 (2011)."    },    {        "title": "1111.3532v1.Spin_waves_in_nanosized_magnetic_films.pdf",        "content": "arXiv:1111.3532v1  [cond-mat.mes-hall]  15 Nov 2011Spin waves in nanosized magnetic films\nL.V. Lutsev\nA.F. Ioffe Physical-Technical Institute of the Russian Acad emy of Sciences, St Petersburg,\n194021, Russia\nE-mail: l lutsev@mail.ru\nAbstract\nWe have studied spin excitations in nanosized magnetic films in the Heisenberg model\nwith magnetic dipole and exchange interactions by the spin o perator diagram technique.\nDispersion relations of spin waves in thin magnetic films (in two-dimensional magnetic\nmonolayers and in two-layer magnetic films) and the spin-wav e resonance spectrum in N-\nlayer structures are found. For thick magnetic films general ized Landau-Lifshitz equations\nare derived from first principles. Landau-Lifshitz equatio ns have the integral (pseudodif-\nferential) form, but not differential one. Spin excitations a re determined by simultaneous\nsolutionoftheLandau-Lifshitzequationsandtheequation forthemagnetostatic potential.\nFor normal magnetized ferromagnetic films the spin wave damp ing has been calculated\nin the one-loop approximation for a diagram expansion of the Green functions at low\ntemperature. In thick magnetic films the magnetic dipole int eraction makes a major con-\ntribution to the relaxation of long-wavelength spin waves. Thin films have a region of low\nrelaxation of long-wavelength spin waves. In thin magnetic films four-spin-wave processes\ntake place and the exchange interaction makes a major contri bution to the damping. It is\nfound that the damping of spin waves propagating in magnetic monolayer is proportional\nto the quadratic dependence on the temperature and is very lo w for spin waves with small\nwavevectors. Spin-wave devices on the base of nanosized mag netic films are proposed –\ntunable narrow-band spin-wave filters with high quality at t he microwave frequency range\nand field-effect transistor (FET) structures contained nanos ized magnetic films under the\ngate electrode. Spin-wave resonances in nanosized magneti c films can beused to construct\nFET structures operating in Gigahertz and Terahertz freque ncy bands.\n75.10.Jm; 75.30.Ds\nHeisenberg model, diagram technique, spin waves, nanosized magne tic films, relaxation,\nspin waves devices in Gigahertz and Terahertz frequency bands\n1 Introduction\nNanosized magnetic films are of great interest due to their perspec tive applications in spin-wave\ndevices. At present, the most important spin waves – microwave filt ers, delay lines, signal-to-\nnoise enhancers, and optical signal processors have been realize d on the base of magnetic films\nof microwave thickness [1, 2, 3]. Nanosized films give us opportunity t o construct spin-wave\ndevices of small sizes and to design devices with new functional prop erties. Recently new\napplications of spin waves have been proposed – spin-wave computin g [4, 5], spin-wave filtering\nusing width-modulated nanostrip waveguides [6], and transmission of electrical signals by spin-\nwave interconversion in an insulator garnet Y 3Fe5O12(YIG) film based on the spin-Hall effect\n[7]. Spin-wave logicelements havebeendoneonthebaseofaMach-Ze hnder-typeinterferometer\n1[6, 8, 9] and can be realized on magnonic crystals [5]. Using nanosized m agnetic films, we have\nprobability to construct array of logic elements of small sizes.\nIn order to design new spin-wave devices based on nanosized magne tic films, it is necessary\nto determine the dispersion relations and damping of spin excitations in nanosized films. In the\nphenomenological model with the magnetic dipole interaction (MDI) a nd the exchange inter-\naction [10, 11, 12, 13] the magnetization dynamics in thick magnetic fi lms is described by the\nLandau-Lifshitz equations, which are differential with respect to s patial variables. The differ-\nential form of equations is postulated. In this connection, the follo wing question arises: is this\nform of Landau-Lifshitz equations correct for nanosized films? De termination of the dispersion\nrelations depends on the answer of this question. In phenomenolog ical models the spin-wave\ndampingisdescribedbyrelaxationtermsinGilbert, Landau-Lifshitz, orBlochforms[13]. Prop-\nerties of intrinsic relaxation processes are not taken into account in these terms and, therefore,\nthe calculated spin-wave damping may be incorrect. The above-men tioned leads us to the main\nquestion of thepaper: what are thedispersion relations and dampin g of spin waves innanosized\nfilms and can they be derived from first principles? In order to answe r this question, we develop\nthe Heisenberg model with the MDI and the exchange interaction. I n the framework of this\nmodel we consider spin excitations in nanosized films, relaxation of sp in waves, and generalize\nLandau-Lifshitz equations.\nThe above-mentioned problems have not yet been investigated com prehensively. One of the\ncause of these problems is the long-range action of the MDI. The sp in-wave relaxation and the\nspin-wave dynamics become dependent on the dimensions and shape s of ferromagnetic samples.\nInorder toanalyze theHeisenberg model withthe MDIandthe exch ange interaction we use the\nspin operator diagram technique [14, 15, 16, 17, 18]. Advantages o f the spin operator diagram\ntechnique are: the opportunity to calculate the spin wave damping a t high temperatures and\nmore exact relationships describing spin-wave scattering and excit ations in comparison with\nmethods based ondiagram techniques for creation and annihilation m agnon Bose operators [19,\n20, 21, 22, 23, 24, 25, 26, 27]. In [18, 28] the spin operator diagra m technique is generalized for\nmodels with arbitrary internal Lie-group dynamics.\nIn section 2 we consider spin operator diagram technique for the He isenberg model with\nthe MDI and the exchange interaction. Spin-wave excitations are d etermined by poles of the\nP-matrix – the matrix of the effective Green functions and interactio n lines. On the base of\nthis diagram technique dispersion relations of spin waves in a normal m agnetized monolayer\nand in a magnetized structure consisted of two monolayers and the spectrum of spin-wave\nresonances in a N-layer structure are found (section 3). For thick magnetic films it is more\nconvenient to present the P-matrix-pole equation describing spin-wave excitations in the form\nof the Landau-Lifshitz equations and the equation for the magnet ostatic potential (section 4).\nSpin excitations are determined by simultaneous solution of these eq uations. Landau-Lifshitz\nequations are integral (pseudodifferential) equations, but not diff erential ones with respect to\nspatial variables. The reduction of Landau-Lifshitz equations to d ifferential equations with\nexchange boundary conditions is incorrect and their solutions give d ispersion relations differed\nfrom dispersion relations calculated on the base of integral (pseud odifferential) Landau-Lifshitz\nequations. In section 5 we consider spin-wave relaxation in thick and thin magnetic films.\nIn thick films three-spin-wave processes take place and the MDI ma kes a major contribution\nto the relaxation of long-wavelength spin waves. Thin films have a reg ion of low relaxation\nof long-wavelength spin waves. In this case, three-spin-wave pro cesses are forbidden and the\nexchange interaction makes a major contribution tothe relaxation process. Nanosized magnetic\nfilms with low relaxation spin waves are applicable to microwave spin wave devices. Tunable\n2narrow-band spin-wave filters with high quality at the microwave fre quency range and field-\neffect transistor (FET) structures contained nanosized magnet ic films under the gate electrode\nare proposed in section 6. Spin-wave resonances of nanosized mag netic films can be used to\nconstruct FET structures operating in Gigahertz and Terahertz frequency bands.\n2 Heisenberg model with magnetic dipole and exchange\ninteractions\n2.1 Spin operator diagram technique\nLet us consider the Heisenberg model with the exchange interactio n and the MDI on a crystal\nlattice[17,18]. Theexchangeinteractionisshort-rangedandtheM DIislong-ranged. Operators\nS±=Sx±iSy,Szsatisfy the commutation relation\n[Sz(/vector1),S+(/vector1′)] =S+(/vector1)δ/vector1/vector1′\n[Sz(/vector1),S−(/vector1′)] =−S−(/vector1)δ/vector1/vector1′\n[S+(/vector1),S−(/vector1′)] = 2Sz(/vector1)δ/vector1/vector1′,\nwhere/vector1≡/vector r1,/vector1′≡/vector r1′is the abridged notation of crystal lattice sites.\nThe Hamiltonian of the Heisenberg model is\nH=−gµB/summationdisplay\n/vector1H(/vector1)Sz(/vector1)−gµB/summationdisplay\n/vector1hµ(/vector1)Sµ(/vector1)−1\n2/summationdisplay\n/vector1,/vector1′Jµν(/vector1−/vector1′)Sµ(/vector1)Sν(/vector1′),(1)\nwhereH(/vectorH/ba∇dblOz) is the external magnetic field, hµis the auxiliary infinitesimal magnetic\nfield,µ=−, +,z. It is supposed that the summation in (1) and in the all following relatio ns is\nperformed over all repeating indices µ,ν. The summation is carried out over the crystal lattice\nsites/vector1,/vector1′in the volume Vof the ferromagnetic sample. gandµBare the Land´ e factor and the\nBohr magneton, respectively. Jµν(/vector1−/vector1′) =Jνµ(/vector1′−/vector1) is the interaction between spins, which\nis the sum of the exchange interaction Iµνand the MDI\nJµν(/vector1−/vector1′) =Iµν(/vector1−/vector1′)−4π(gµB)2∇µΦ(/vector r−/vector r′)∇′\nν/vextendsingle/vextendsingle/vextendsingle\n/vector r=/vector1,/vector r′=/vector1′, (2)\nwhere Φ( /vector r−/vector r′) is determined by the equation\n∆Φ(/vector r−/vector r′) =δ(/vector r−/vector r′), (3)\n∇µ={∇−,∇+,∇z}=/braceleftigg1\n2/parenleftigg∂\n∂x+i∂\n∂y/parenrightigg\n,1\n2/parenleftigg∂\n∂x−i∂\n∂y/parenrightigg\n,∂\n∂z/bracerightigg\n.\nIn the 3-dimensional space Φ( /vector r−/vector r′) =−1/4π|/vector r−/vector r′|and the MDI term in the Hamiltonian\n(1) can be written as\nH(dip)=(gµB)2\n2/summationdisplay\n/vector1,/vector1′\n(/vectorS(/vector1),/vectorS(/vector1′))\n|/vector1−/vector1′|3−3(/vectorS(/vector1),/vector1−/vector1′)(/vectorS(/vector1′),/vector1−/vector1′)\n|/vector1−/vector1′|5\n.\n3For the following calculations of spin-wave dispersion relations in magn etic films we use more\nconvenient form of the MDI determined by relations (2), (3).\nSpin excitations, interaction of spin waves, spin wave relaxation and other parameters of\nexcitations in the canonical spin ensemble are determined by the gen erating functional [14, 29,\n28, 18]\nZ[h] = Spexp[ −βH(h)]\n=∞/summationdisplay\nn=0/summationdisplay\n/vector1,...,/vector n\nµ1,...,µnβ/integraldisplay\n0···β/integraldisplay\n0Qµ1,...,µn(/vector1,...,/vector n,τ 1,...,τ n)hµ1(/vector1,τ1)...hµn(/vector n,τn)dτ1...dτn,(4)\nwhereβ= 1/kT,kis the Boltzmann constant, Tis the temperature, h={hµi}. Coefficients\nQµ1,...,µnare proportional to the temperature Green function without vac uum loops\nGµ1...µn(/vector1,...,/vector n,τ 1,...,τ n)≡ /an}b∇acketle{t/an}b∇acketle{tTˆSµ1(/vector1,τ1)...ˆSµn(/vector n,τn)/an}b∇acket∇i}ht/an}b∇acket∇i}ht\n= (βgµB)−nZ−1 δnZ[h]\nδhµ1(/vector1,τ1)...δhµn(/vector n,τn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nh→0, (5)\nwhereˆSα(/vector n,τ) = exp(τH)Sα(/vector n)exp(−τH) are the spin operators in the Euclidean Heisenberg\nrepresentation, τ∈[0,β].Tistheτ-timeorderingoperator. Variable τisaddedintheauxiliary\nfieldhµin order to take into account T-ordering. /an}b∇acketle{t/an}b∇acketle{t.../an}b∇acket∇i}ht/an}b∇acket∇i}htdenotes averaging of spin operators\ncalculated with exp( −βH)/Spexp(−βH). The symbol Sp denotes the trace.\nThe frequency representation of the expansion (4) is more conve nient for calculations.\nThe Fourier transforms of Qµ1,...,µnare defined in terms of the Matsubara frequencies ω(1)\nm1=\n2πm1/¯hβ,...,ω(n)\nmn= 2πmn/¯hβ[30] (m1,...,mnare integers)\nQµ1,...,µn(/vector1,...,/vector n,ω(1)\nm1,...,ω(n)\nmn)\n=β/integraldisplay\n0···β/integraldisplay\n0Qµ1,...,µn(/vector1,...,/vector n,τ 1,...,τ n)exp[−i¯h(ω(1)\nm1τ1+...+ω(n)\nmnτn)]dτ1...dτn.(6)\nThe coefficients Qµ1,...,µncan be expanded with respect to the interaction Jµν(/vector1−/vector1′) (2) [17,\n18, 14, 15, 16, 28]. Each term of this expansion is represented by a diagram constructed of\npropagators, vertices, blocks and interaction lines.\n1. Propagators. Spin propagators\nD±(/vector1,/vector1′,ωm) =δ/vector1/vector1′\np0±iβ¯hωm, (7)\nwherep0=βgµBH, are determined for the spin ensemble without any interaction betw een\nspins. The propagators D±(/vector1,/vector1′,ωm) arerepresented by directed lines in diagrams(figure 1(a)).\nThe directions of arrows show the direction of growth of the frequ ency variable ωm.\n2. Vertices. There are five types of vertices (figure 1(b)). Vertices a,bare the start and\nend points of propagators, respectively. In analytical expressio ns of diagrams the vertex a\ncorresponds with the factor 2 and the vertex bwith the factor 1. The vertex cties three\n4a.\nb.\na b c\nd e\nc.\nd.D (1,1□,+ m’/c119 /c41/c32/c61\n1 1’\nD (1,1□,-’/c119 /c41/c32/c61m\n1 1’/c119m\n/c119m\nV (1-1’, )□= J/c109/c110 /c119 /c98m /c109/c110(1-1’)=(0)\n1 1’ /c119m\nFigure 1: (a) Propagators D±, (b) vertices, (c) block with isolated parts and (d) interaction\nlinesV(0)\nµν.\n5propagators and corresponds with the factor (-1) in analytical e xpressions. The vertex dwith\nthe factor 1 is defined as a single vertex. The vertex eties two propagators. The factor of the\ne-vertex is equal to (-1).\n3. Blocks. Blocks contain propagators and isolated vertices d(figure 1(c)). Propagators can\nbe connected through vertices c,e. In analytical expressions of the diagram expansion each\nblock corresponds with the block factor B[κ−1](p0), where κis the number of isolated parts in\nthe block. The factor B[κ−1](p0) is expressed by partial derivatives of the Brillouin function BS\nfor the spin Swith respect to p0\nB(p0) =/an}b∇acketle{t/an}b∇acketle{tSz/an}b∇acket∇i}ht/an}b∇acket∇i}ht0=SBS(Sp0)\nB[n](p0) =S∂nBS(Sp0)\n∂pn\n0, (8)\nwhere/an}b∇acketle{t/an}b∇acketle{t.../an}b∇acket∇i}ht/an}b∇acket∇i}ht0denotes the statistical averaging performed over the states de scribed by the\nHamiltonian H(1) without the interaction Jµνbetween spins. BS(x) = (1 + 1 /2S)coth[(1 +\n1/2S)x]−(1/2S)coth(x/2S).\n4. Interaction lines. The interaction line V(0)\nµν(/vector1−/vector1′,ωm) =βJµν(/vector1−/vector1′) connects two vertices\nin a diagram (figure 1(d)). The correspondence between the first indexµof the interaction line\nV(0)\nµνand the vertex type is the following. (1) If µ=−, then the left end point of V(0)\n−νis bound\nto the vertex a; (2) ifµ= +, then this end point is bound to the vertices borc; (3) ifµ=z,\nthen the end is bound to the vertices dore. The analogous correspondence is satisfied for the\nright end νofV(0)\nµν.\nCoefficients Qµ1,...,µnin the expansion (4) in the frequency representation (6) are the s um of\nNtopologically nontrivial diagrams/summationtext\nNQµ1,...,µn\nN. The general form of the analytical expression\nof the diagram in the frequency representation is written as [17, 1 8, 14, 15, 16]\nQµ1,...,µn\nN(/vector1,...,/vector n,ω(1)\nm1,...,ω(n)\nmn) = (−1)L2maPk\n2kk!/productdisplay\nlB[κl−1](p0)κl/productdisplay\n/vectori,/vectorj∈lδ/vectori/vectorj\n×/summationdisplay\n/vector1,.../vectork\n/vector1′.../vectork′/summationdisplay\nmiV(0)\nαγ(/vector1−/vector1′,ωm1)×...×V(0)\nρσ(/vectork−/vectork′,ωmk)ID/productdisplay\n/vector s,/vector s′D−(/vector s,/vector s′,ωms)Iv/productdisplay\nvδ/parenleftigg/summationdisplay\nr∈vβ¯hωmr/parenrightigg\n,(9)\nwhere/vector1,...,/vector n,ω(1)\nm1,...,ω(n)\nmnare the external lattice and frequency variables corresponded t o\nthe auxiliary fields hµiin the expansion (4). mais the number of a-vertices in a diagram. Lis\nthe number of cande-vertices. Pkis the number of topological equivalent diagrams. 2 kis the\nnumber of vertices connected with kinteraction lines V(0)\nαγ...V(0)\nρσ. The product/producttext\nlis performed\nover all blocks of a diagram. κlis the number of isolated parts in block l. The term/producttextκl\n/vectori,/vectorj∈lδ/vectori/vectorj\ndenotes that all isolated parts in block lare determined on a single crystal lattice site. IDis the\nnumber ofpropagatorsina diagram. Ivis the number of vertices ina diagram./summationtext\nmidenotes the\nsummation performed over all inner frequency variables. The term/producttextIv\nvδ(/summationtext\nr∈vβ¯hωmr) gives\nthe frequency conservation in each vertex v, i.e. the sum of frequencies of propagators and\ninteraction lines, which come in and go out from the vertex v, is equal to 0. The vertex dcan\nbe connected with the single interaction line. In the analytical expre ssion this corresponds to\nthe factor δ(β¯hωm). The lattice variables /vector s,/vector s′of propagators D−can be inner or external. In\nthe first case, end points of propagators are connected with the end points {/vector1,/vector1′,...,/vectork,/vectork′}of\n6interaction lines V(0)\nαγ...V(0)\nρσand the summation/summationtext\n/vector1,.../vectork\n/vector1′.../vectork′/summationtext\nmiis performed. In the second case,\nend points of propagators are not connected with interaction lines .\nThe first approximation of the diagram expansion (4) is the self-con sistent field approxi-\nmation, in which the effective field acting on spins is derived and the self -consistent field H(c)\nµ\ninduced by the neighboring spins is taken into account [14, 17, 18]. T his leads to the substi-\ntutionp0→p=βgµBH(c)\nzin the propagator D−in relation (7). The self-consistent field is the\nsum of exchange and magnetic dipole self-consistent fields, H(c)\nµ=H(exch)\nµ+H(m)\nµ, where\nH(exch)\nµ(/vector1) = (gµB)−1/summationdisplay\n/vector1′Iµν(/vector1−/vector1′)/an}b∇acketle{t/an}b∇acketle{tSν(/vector1′)/an}b∇acket∇i}ht/an}b∇acket∇i}ht\nH(m)\nµ(/vector1) =−4πgµB∇µ/summationdisplay\n/vector1′Φ(/vector r−/vector r′)∇′\nν/an}b∇acketle{t/an}b∇acketle{tSν(/vector r′)/an}b∇acket∇i}ht/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector r=/vector1\n/vector r′=/vector1′. (10)\nThe second approximation of the expansion (4) is the approximation of the effective Green\nfunctions and interactions. In this approximation, the poles of the matrix of the effective Green\nfunctions and interactions are determined and the dispersion curv es are obtained. The next\nterms in the diagram expansion determine the imaginary and real cor rections to the poles of\nthe matrix of the effective Green functions and interactions. The im aginary parts of the poles\ngive the relaxation parameters of spin excitations and the real par ts determine the corrections\nto the dispersion curves. In the next section we consider the appr oximation of the effective\nGreen functions and interactions.\n2.2 Effective Green functions and interaction lines\nIn the framework of this approximation the matrix of the effective G reen functions and effective\ninteractions P=/ba∇dblPAB(/vector1,/vector1′,ωm)/ba∇dblis introduced [17, 18]. We compose the P-matrix from\nanalytical expressions of connected diagrams with two external s ites. These sites are end points\nof propagators, single vertices d, or end points of interaction lines. Accordingly, multiindices\nA= (aµ),B= (bν) are the double indices, where µ,ν={−,+,z}and indices a,bpoint out\nthatA,Bbelong to a propagator or to a d-vertex (a,b= 1), or belong to an interaction line\n(a,b= 2). The zero-order approximation P(0)of theP-matrix is determined by the matrix of\nthe bare interaction V(0)=/ba∇dblV(0)\nµν(/vector1−/vector1′,ωm)/ba∇dbland by the two-site Green functions (5) in the\nself-consistent-field approximation G(0)=/ba∇dblG(0)\nµν/ba∇dbl, given on a crystal lattice site\nP(0)=\n/ba∇dblP(0)\n(1µ)(1ν)/ba∇dbl.../ba∇dblP(0)\n(1µ)(2ν)/ba∇dbl\n··· ··· ···\n/ba∇dblP(0)\n(2µ)(1ν)/ba∇dbl.../ba∇dblP(0)\n(2µ)(2ν)/ba∇dbl\n=\n/ba∇dblG(0)\nµν/ba∇dbl... 0\n··· ··· ···\n0.../ba∇dblV(0)\nµν/ba∇dbl\n,\nwhere\n/ba∇dblG(0)\nµν/ba∇dbl=\n0G(0)\n−+0\nG(0)\n+−0 0\n0 0 G(0)\nzz\n=\n0 2 B(p)D−(/vector1,/vector1′,ωm) 0\n2B(p)D+(/vector1,/vector1′,ωm) 0 0\n0 0 B[1](p)δ/vector1/vector1′δm0\n\n(11)\n7with the propagator (7), in which the substitution p0→p=βgµBH(c)\nzis performed.\nTheP-matrixis obtained by means of the summation of the P(0)-matrix – the summation of\nall diagram chains consisted of the bare Green functions G(0)\nµνand the bare interaction lines V(0)\nµν\n(figure 2). These chains of propagators and interaction lines do no t have any loop insertion.\nAnalyticalexpressionsoftheconsidereddiagramscanbewrittenin accordancewithrelation(9).\nThe summation gives equation of the Dyson type, which forms the re lationship between P(0)-\nandP-matrices\nP=P(0)+PσP(0), (12)\nwhere\nσ=\n0...E\n··· ··· ···\nE... 0\n,\nE=/ba∇dblδµν/ba∇dblis the diagonal matrix.\nTheP-matrix consists of the two-site effective Green functions G=/ba∇dblGµν/ba∇dbl=G(0)(E −\nV(0)G(0))−1, whereGµν=P(1µ)(1ν), effective interactions V=/ba∇dblVµν/ba∇dbl=V(0)(E − G(0)V(0))−1,\nwhereVµν=P(2µ)(2ν), and intersecting terms P(1µ)(2ν),P(2µ)(1ν)(figure 2). The effective Green\nfunctions, effective interactionsandintersecting termsaredeno tedindiagramsbydirectedthick\nlines, empty lines and compositions of the thick line - empty line, respec tively. The P-matrix\ndetermines the spectrum of quasi-particle excitations in the spin en semble. Spectrum relations\nfor spin excitations are given by the P-matrix poles – by zero eigenvalues of the operator\n1−σP(0)or, equivalently, by E −V(0)G(0)under the analytical continuation\niωm→ω+iεsignω\nδ(β¯hωm) =δm0→1\nβ¯h(ω+iεsignω)(ε→+0). (13)\nSince, zero eigenvalues of the operator E −V(0)G(0)can corresponds to different eigenfunc-\ntions and can determine different excitation modes, we introduce th e spectral parameter λfor\neigenfunctions h(λ)\nµ(/vector1,ωm) of the operator E −V(0)G(0). The spectral parameter λcan be dis-\ncrete or continuous. Taking into account the above-mentioned, w e get the equation describing\nspin-wave excitations\nh(λ)\nµ(/vector1,ωm)−/summationdisplay\n/vector1′,/vector1′′,ν,ρV(0)\nµν(/vector1−/vector1′,ωm)G(0)\nνρ(/vector1′,/vector1′′,ωm)h(λ)\nρ(/vector1′′,ωm)/vextendsingle/vextendsingle/vextendsingle\niωm→ω+iεsignω= 0.(14)\n3 Spin waves in nanosized magnetic films\n3.1 Spin-wave equations for magnetic films\nLet us consider spin waves with the wavevector /vector qin a normal magnetized film consisted of N\nmonolayers at low temperature. x-,y-axes are in the monolayer plane and the z-axis is normal\nto monolayers. The external magnetic field His normal to monolayers and is parallel to the\nz-axis. At low temperature derivatives of the Brillouin function in B[n](p) in relation (8) tend to\n0 exponentially with decreasing temperature. Thus, it follows that d iagrams containing blocks\nwith isolated parts can be dropped, the Green function G(0)\nzzin relation (11) is negligible and\n8a.\nG/c109/c110/c61(0)=/c109 /c110=□G+-(0)\nz z- +- +\n=□G-+(0)\n=□Gzz(0)\nP =□G =(1 )(1 )/c109 /c110 /c109/c110 /c109 /c110=/c109 /c110+/c83/c103/c32/c103/c49 /c50/c103/c32/c103/c49 /c50/c110 /c103/c50/c109 /c103/c49\nb.\nP =□V =(2 )(2 )/c109 /c110 /c109/c110/c109 /c110=/c109 /c110+/c83/c103/c32/c103/c49 /c50/c103/c32/c103/c49 /c50/c109 /c110 /c103/c49 /c103/c50\nc.\nP =(1 )(2 )/c109 /c110 = /c109 /c110/c83/c103/c110 /c109 /c103\nP =(2 )(1 )/c109 /c110/c109 =/c110/c103/c83 /c110 /c109 /c103\nFigure 2: (a) Definition of the effective Green functions P(1µ)(1ν)=Gµνvia the bare two-site\nGreen functions G(0)\nµν. (b) Definition of effective interaction lines P(2µ)(2ν)=Vµν. (c) Definition\nof intersecting terms P(1µ)(2ν),P(2µ)(1ν).\n9only the Green functions G(0)\n−+,G(0)\n+−are taken into account in equation (14). Indices µ,νof\ninteractions V(0)\nµνin equation (14) are {−,+}. We suppose that on monolayers spins are placed\nonquadraticcrystal latticesiteswiththelatticeconstant aandspinorientationisparalleltothe\nz-axis. The exchange interaction acts between neighboring spins an d is isotropic between spins\nin monolayers, 2 I−+= 2I+−=Izz=I0, and between neighboring layers, 2 I−+= 2I+−=Izz=\nId. Then, the Fourier transform of the exchange interaction with re spect to the longitudinal\nlattice variables /vector1xyis\n˜I(/vector q,1z−1′\nz) =/summationdisplay\n/vector1xy−/vector1′xyI(/vector1xy−/vector1′\nxy,1z−1′\nz)exp[−i/vector q(/vector1xy−/vector1′\nxy)]\n=I(0,1z−1′\nz)+2I0[cos(qxa)+cos(qya)]δ1z1′z,\nwhere/vector1xy,/vector1′\nxyarecrystal latticesites ina monolayer, 1 z, 1′\nzarez-positions of layers, /vector q= (qx,qy)\nis the longitudinal wavevector in monolayers, I(0,1z−1′\nz) is the exchange interaction at /vector q= 0,\nwhich is equal to Idbetween spins of neighboring layers. The corresponding exchange part of\nthe interaction line V(0)\nµν=V(exch)\nµν+V(dip)\nµν(figure 1(d)) is\nV(exch)\nµν(/vector q,1z−1′\nz) =β˜I(/vector q,1z−1′\nz). (15)\nThe MDI part V(dip)\nµνis determined by the Fourier transform of equation (3)\n/parenleftigg\n−q2+∂2\n∂z2/parenrightigg\nΦ(/vector q,z−z′) =S−1\naδ(z−z′)\nwith the solution\nΦ(/vector q,1z−1′\nz) = Φ(/vector q,z−z′)|z=1z,z′=1′z=−1\n2qSaexp(−q|1z−1′\nz|), (16)\nwhereSa=a2,q=|/vector q|. According to the solution (16), the corresponding MDI part of th e\ninteraction line is\nV(dip)\nµν(/vector q,1z−1′\nz) =−4πβ(gµB)2qµqν\nqSaexp(−q|1z−1′\nz|),(µ,ν={−,+}).(17)\nwhere\nq−=1\n2(qx+iqy)q+=1\n2(qx−iqy)\nTaking into account relations (15) and (17), from equation (14) we obtain equations for spin-\nwave modes with the wavevector /vector qinN-layer magnetic films\nh(λ)\nµ(/vector q,1z,ωm)−/summationdisplay\n/vector1′z/bracketleftig\nV(0)\nµ−(/vector q,1z−1′\nz,ωm)G(0)\n−+(1′\nz,1′\nz,ωm)h(λ)\n+(/vector q,1′\nz,ωm)\n+V(0)\nµ+(/vector q,1z−1′\nz,ωm)G(0)\n+−(1′\nz,1′\nz,ωm)h(λ)\n−(/vector q,1′\nz,ωm)/bracketrightig/vextendsingle/vextendsingle/vextendsingle\niωm→ω+iεsignω= 0, (18)\nwhere\nG(0)\n−+\n(+−)(1z,1′\nz,ωm) =2B(p)δ1z1′z\np±iβ¯hωm,\n10H\n0.0 1.0 2.0 3.0\naq0500100015002000Frequency /2 (GHz)\nFigure 3: Dispersion curve of spin waves propagating in the normal m agnetized monolayer film\nwith cubic lattice ( a= 0.4 nm) at the sum of magnetic fields H+H(m)= 3 kOe. Exchange\ninteraction I0is 0.085 eV.\nλ= 1,...,Nis the mode number, V(0)\nµν(/vector q,1z−1′\nz,ωm) =V(exch)\nµν(/vector q,1z−1′\nz)+V(dip)\nµν(/vector q,1z−1′\nz),\nµ,ν={−,+}. Eigenvalues of equations (18) give dispersion relations of spin wave s. In next\nsections we find spin-wave dispersion relations for the cases of mon olayer and two-layer films\nand spin-wave resonance relations for the case of N-layer structures.\n3.2 Spin waves in magnetic monolayer\nDispersion relations of spin waves in normal magnetized monolayer ar e determined by the\ndeterminant of equations (18) for variables h(1)\n−,h(1)\n+. Taking into account relations (15) and\n(17), we find\nω2(/vector q) = Ω(/vector q)[Ω(/vector q)+2πγσmq], (19)\nwhere\nΩ(/vector q) =γ(H+H(m))+2B(p)I0\n¯h[2−cos(qxa)−cos(qya)],\nγ=gµB/¯his the gyromagnetic ratio, H(m)is the depolarizing magnetic field (10), σm=\ngµBB(p)/Sais the surface magnetic moment density, q= (q2\nx+q2\ny)1/2. As one can see from\nrelation (19), in monolayer films spin waves have the one-mode chara cter. Figure 3 presents the\ndispersion curve (19) of spin waves propagating in the monolayer film with the lattice constant\na= 0.4 nm. The spin-wave wavevector /vector qis parallel to the x-axis (qx=q,qy= 0) and is in the\nrange [0,π/a]. Calculations have been done for the exchange interaction betwee n neighboring\nspinsI0= 0.085 eV, B(p) = 1/2 at the sum of magnetic fields H+H(m)= 3 kOe. The\nexchange interaction makes a major contribution to the dispersion . The relatively weak MDI\nis significant for the dispersion at small values of the wavevector /vector q.\n11IddH\n0.0 1.0 2.0 3.0\naq0100020003000Frequency /2 (GHz)12\nFigure 4: Dispersion curve of spin waves propagating in the normal m agnetized two-layer\nmagnetic film with quadratic lattice ( a= 0.4 nm) at the sum of magnetic fields H+H(m)=\n3 kOe. Exchange interactions I0=Id= 0.085 eV. Distance between monolayers dis equal to\nthe lattice constant a. 1, 2 are the first and the second modes of spin waves, respective ly.\n3.3 Spin waves in two-layer magnetic film\nLet us consider spin waves in a normal magnetized structure consis ted of two monolayers of\nthe quadratic lattice with the lattice constant a. The distance between layers is equal to dand\nthe exchange interaction between spins of layers is Id. Dispersion relations are determined by\neigenvalues of equations (18) for variables h(1)\n−,h(1)\n+,h(2)\n−,h(2)\n+and can be written as\nω(n)2(/vector q) = Ω(/vector q)[Ω(/vector q)+2πγσmq]+2B(p)Id\n¯h/bracketleftigg2B(p)Id\n¯h−2πγσmqexp(−qd)/bracketrightigg\n±2/bracketleftigg\n−2B(p)Id\n¯h(Ω(/vector q)+πγσmq)+πγσmqexp(−qd)Ω(/vector q)/bracketrightigg\n, (20)\nwhere\nΩ(/vector q) =γ(H+H(m))+2B(p)I0\n¯h[2−cos(qxa)−cos(qya)]+B(p)Id\n¯h,\nn= 1,2 is the mode number, q= (q2\nx+q2\ny)1/2. For the first mode spins in different layers\nchange their orientations in-phase. In this case, spin waves of the first mode correspond to\nspin waves in monolayer (19). For the second mode spins in different la yers change orientations\nin-anti-phase and the energy of the spin wave with the given longitud inal wavevector qis higher\nthan the energy of the spin wave of the first mode. Dispersion curv es of spin waves determined\nby relations (20) are shown in figure 4. Spin waves propagate along t hex-axis. Calculations\nhave been done for the exchange interactions I0=Id= 0.085 eV and for the distance between\nlayersd=a= 0.4 nm at the sum of magnetic fields H+H(m)= 3 kOe.\n123.4 Spin-wave resonance in N-layer structure\nInthissectionweconsiderspin-waveresonanceinastructurecon sistedofNuniformmonolayers\nwith the exchange interaction Idbetween spins of layers and with the distance dbetween layers.\nSpin-wave resonance is the limit case of a spin wave, when the longitud inal wavevector q→0.\nTherefore, the MDI terms V(dip)\nµν(/vector q,1z−1′\nz) in equations (18) can be dropped, equations with\nvariables h(λ)\n+andh(λ)\n−are separated and eigenvalues are determined by zero values of th e\ndeterminant (we write the determinant D(+)for equations with the h(λ)\n+)\nD(+)=G(0)(1)...G(0)(N)\n×det\n(G(0)−1(1)−V(0)(11)) −V(0)(12) 0...\n−V(0)(21) ( G(0)−1(2)−V(0)(22)) −V(0)(23)...\n0 −V(0)(32) ( G(0)−1(3)−V(0)(33))...\n··· ··· ··· ···\n,\nwhereV(0)(ij),G(0)(i) are the abridged notation of V(exch)\n+−(/vector q,iz−jz,ωm)|/vector q=0andG(0)\n−+(i,i,ωm)\natiωm→ω+iεsignω, respectively. ( i,j) are indices of layers. Taking into account that spins\nof outer layers ( i= 1,N) interact with spins of one inner layer and spins of inner layers intera ct\nwith spins of two layers and introducing the variable for inner layers in the determinant D(+)\nx=G(0)−1(i)−V(0)(ii)\n−V(0)(ji)=¯h\nB(p)Id[ω−γ(H+H(m))]−2 (i/ne}ationslash= 1,N, j=i±1),\nwe obtain that the spin-wave resonance spectrum is determined by roots of the polynomial\nRN(x) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x+1) 1 0 0... 0 0\n1x1 0... 0 0\n0 1 x1... 0 0\n0 0 1 x... 0 0\n··· ··· ··· ··· ··· ··· ···\n0 0 0 0...x1\n0 0 0 0... 1 ( x+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= (x+1)2PN−2(x)−2(x+1)PN−3(x)+PN−4(x) = 0 ( N≥2),\nwhereP−2(x) =−1,P−1(x) = 0,P0(x) = 1,PN(x) =xPN−1(x)−PN−2(x). Polynomial RN(x)\nhasNroots\nx(n)=−2cos/parenleftbiggπn\nN/parenrightbigg\n,\nwheren= 0,1,...,N−1. Taking into account the form of the roots x(n), we can introduce the\ntransverse wavevector q(n)\nz=πn/Nd. Then, the spin-wave resonance spectrum can be written\nas\nω(n)=γ(H+H(m))+2B(p)Id\n¯h[1−cos(q(n)\nzd)]. (21)\n13d\nIdD\n0.0 1.0 2.0 3.0\nq□□□d□=□□□□□n/N0.00.40.81.21.62.0\n1-□cos(q□□□d)h( - )/2B(p)I(n)\nd(0) -/c119 /c119\n(n)\nz(n)\nz\nFigure5: Spin-wave resonancespectrum ω(n)(n= 0,1,...,N−1)forthestructurewith N= 40\nlayers.q(n)\nzis the transverse wavevector, dis the distance between layers, Idis the exchange\ninteraction between spins of layers.\nFor the first mode ( n= 0) spins in different layers change their orientations in-phase. For the\nhighest mode ( n=N−1) spins in different layers change orientations in-anti-phase and th e\nenergy of spin-wave resonance is highest. Figure 5 presents the s pin-wave resonance spectrum\nforthestructurewith N= 40layers. Onecanseethatatlowvaluesofthetransversewavev ector\nthe resonance spectrum is proportional to the quadratic depend ence onq(n)\nz.\n4 Landau-Lifshitz equations and spin-wave excitations\nin thick magnetic films\n4.1 Linearized Landau-Lifshitz equations\nEquations (14), (18) describe spin-wave excitations. Solutions of these equations for magnetic\nsamples of great volumes and for thick N-layer magnetic films with N≫1 become difficult, be-\ncause determinants of equations (14), (18) have high orders. In order to overcome the difficulty\nand to find spin-wave spectrum for these samples, we derive Landa u-Lifshitz equations [17, 18].\nDispersion relations for spin excitations are determined by the P-matrix poles (12) which coin-\ncide with poles of the matrix Gof effective propagators. Accordingly, the dispersion relations\ncan be derived from the eigenvalues of equation\nG=G(0)+G(V(exch)+V(dip))G(0), (22)\nwhereG(0)=/ba∇dblG(0)\nµν/ba∇dblis the matrix of bare propagators (11). Since the considered inter action\nis the sum of exchange and magnetic dipole interactions, we can obta in the eigenvalues and\neigenfunctions of equation (22) by a two-step procedure. In the first stage, we perform the\nsummation of diagrams, taking into account the exchange interact ion, and find the propagator\n14matrixG(1)=/ba∇dblG(1)\nµν/ba∇dbl\nG(1)=G(0)+G(0)V(exch)G(1). (23)\nIn the second stage, the summation of diagrams with dipole interact ion lines is performed.\nThis gives the equation for the matrix Gof effective propagators expressed in terms of the\nmatrixG(1)\nG=G(1)+GV(dip)G(1). (24)\nThus, the solution of equation (22), which determines the matrix G, is equivalent to the\nsolution of equations (23), (24). After the performed two-step summation, equation (14) for\neigenfunctions h(λ)\nµis written in the more convenient form\nh(λ)\nµ(/vector1,ωm)−/summationdisplay\nρ,σ\n/vector1′/vector1′′V(dip)\nµρ(/vector1−/vector1′,ωm)G(1)\nρσ(/vector1′,/vector1′′,ωm)h(λ)\nσ(/vector1′′,ωm)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\niωm→ω+iεsignω= 0.(25)\nThe solution of simultaneous equations (23), (25) gives the dispers ion relations for spin\nexcitations. These equations can be reduced to linearized Landau- Lifshitz equations in the\ngeneralized form and the equation for the magnetostatic potentia l. In order to perform this\ntransformation one needs to make a transition to the retarded Gr een functions. We transform\nmatrix equation (23) to equations describing small variations of the magnetic moment density\n(or the variable magnetization), mν. The variable magnetization mνunder the action of the\nmagneticfield ¯hν,whichisgeneratedbytheMDI V(dip),isgivenbytheretardedGreenfunctions,\nwhich are determined by the analytical continued values of the prop agator matrix G(1)[31]\nmν(/vector1,ω) =β(gµB)2\nVa/summationdisplay\nρ,/vector1′G(1)\nνρ(/vector1,/vector1′,ωm)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\niωm→ω−iε¯hρ(/vector1′,ω), (26)\nwhereVais the atomic volume. The analytical continuation iωm→ω−iεdefines the retarded\nGreen functions. ¯hρ(/vector1,ω) is the field of the magnetic dipole-dipole interaction acting on spins.\nBymultiplying matrixequation(23)by G(0)−1fromtheleftandby ¯hρfromtheright, performing\nthe analytical continuation iωm→ω−iε,δ(β¯hωm)→[β¯h(ω−iε)]−1and taking into account\nrelation (26), we get matrix equation (23) in the form of simultaneou s equations\n/summationdisplay\nν,/vector1′[G(0)−1\nρν(/vector1,/vector1′,ω)−βIρν(/vector1−/vector1′)]mν(/vector1′,ω) =β(gµB)2\nVa¯hρ(/vector1,ω). (27)\nFor isotropic exchange interaction, 2 I−+= 2I+−=Izz=I, equations (27) have the form\nˆE±m±(/vector1,ω) = 2γM(/vector1)¯h∓(/vector1,ω) (28)\nˆEzmz(/vector1,ω) =B[1](p)\nB(p)γM(/vector1)¯hz(/vector1,ω), (29)\nwhereM(/vector1) =gµBB(p)/Vais the magnetic moment density at the low-temperature approxi-\nmation. We say that the operators ˆE±,ˆEz:\nˆE±m±(/vector1,ω) = [γ(H(/vector1)+H(m)(/vector1))±ω]m±(/vector1,ω)\n15+B(p)\n¯hVb/summationdisplay\n/vector1′/integraldisplay\nVb[¯I(0)−¯I(/vector q)]exp[i/vector q(/vector1−/vector1′)]m±(/vector1′,ω)d3q\nˆEzmz(/vector1,ω) =ωmz(/vector1,ω)−B[1](p)\n¯hVb/summationdisplay\n/vector1′/integraldisplay\nVb¯I(/vector q)exp[i/vector q(/vector1−/vector1′)]mz(/vector1′,ω)d3q\nare Landau-Lifshitz operators. ¯I(/vector q) =/summationtext\n/vector1I(/vector1)exp(−i/vector q/vector1) is the Fourier transform of the ex-\nchange interaction with respect to the lattice variables. The field H(m)(/vector1) is defined by relation\n(10) and depends on the magnetic moment density M(/vector1);Vb= (2π)3/Vais the volume of the\nfirst Brillouin zone. Equations (28), (29) have the generalized form of the Landau-Lifshitz\nequations [13]. Solutions m±of equations (28) depend on temperature, because β= 1/kT\nis contained in the variable pof the function B(p) (8), through which the magnetic moment\ndensityM(/vector1) is expressed. Equation (29) describes longitudinal variations of the variable mag-\nnetization under the influence of the field ¯hz. At low temperature the derivative of the function\nB[1](p) tends to 0 and the longitudinal variable magnetization mzis negligible.\nFrom the form of the magnetic dipole interaction in relations (2), (3) it follows that the\nfield¯hνin relation (26) is magnetostatic, i.e. it is expressed in terms of the ma gnetostatic\npotential ϕ:¯hν=−∇νϕ. We transform equation (25) to the equation for the magnetosta tic\npotential ϕ(/vector r,ω). Taking into account relation (26) and the explicit form of the magn etic\ndipoleinteractioninrelations(2), (3), performingthederivation ∇µ, theanalytical continuation\niωm→ω−iεand the summation of equation (25) over the index µ, we obtain the equation\nexpressed in terms of ϕ,mν\n−∆ϕ(/vector r,ω)+4π∇νmν(/vector1,ω)|/vector1→/vector r= 0. (30)\nThus, in consideration of the Landau-Lifshitz equations (28), (29 ), the dispersion relations of\nspin excitations are given by eigenvalues of equation (30).\n4.2 Exchange boundary conditions\nIf the scale of the spatial distribution of the variable magnetization mν(/vector1,ω) and the sample\nsize are much greater than the lattice constant a, then the sum over the lattice variables/summationtext\n/vector1\nin Landau-Lifshitz operators ˆE±,ˆEzcan be converted into an integral over the sample volume\nV−1\na/integraltextd3r. Let us consider the case when the temperature is low and the Four ier transform of\nthe exchange interaction is ¯I(/vector q) =¯I(0)−wq2. Then, we obtain that mz→0 and equation (29)\nis dropped. The operators ˆE±are pseudodifferential operators of order 2 [32]\nˆE±m±(/vector r,ω) = [γ(H(/vector r)+H(m)(/vector r))±ω]m±(/vector r,ω)\n+4πγαM(/vector r)\n(2π)3/integraldisplay\nV/integraldisplay\nVbq2exp[i/vector q(/vector r−/vectorr′)]m±(/vectorr′,ω)d3qd3r, (31)\nwhereα=wVa/4π(gµB)2is the exchange interaction constant, Vis the volume of the ferro-\nmagnetic sample. In [10, 11, 12, 13, 33, 34, 35, 36, 37] the pseudo differential Landau-Lifshitz\noperators are reduced to the differential operators with respec t to spatial variables\nˆE±(/vector r,ω) =γ[H(/vector r)+H(m)(/vector r)−4παM(/vector r)∆]±ω. (32)\n16For solvability of equations (28) with differential Landau-Lifshitz op erators (32) the ex-\nchange boundary conditions are imposed\n∂mν\n∂/vector n+ξmν|∂V= 0,\nwhere/vector nis the inward normal to the boundary ∂V, andξis the pinning parameter. This reduc-\ntion to differential Landau-Lifshitz operators is not correct. Figu re 6 presents exact dispersion\nrelations of spin excitations given by eigenvalues of equations (28), (30) with pseudodifferential\nand differential Landau-Lifshitz operators for the case of a norm al magnetized homogeneous\nfilm with the thickness D. The dispersion relations of spin waves have the form\nω(n)2(/vector q) = Ω(n)(Ω(n)+ΩMq2/q(n)2\n0), (33)\nwheren= 1,2,3,...is the mode number, /vector q= (qx,qy) is the two-dimensional longitudinal\nwavevector of spin waves, q=|/vector q|, Ω(n)=γ(H−4πM+ 4παMq(n)2\n0), ΩM= 4πγM,q(n)\n0=\n(q2+q(n)2\nz)1/2,q(n)\nzis the transverse vector. The magnetostatic potential over thic knessz∈\n[−D/2,D/2] of the magnetic film is\nϕ(n,/vector q)(x,y,z) = (2π)−1f(n)−1/2exp(iqxx+iqyy)cos[q(n)\nzz+π(n−1)/2], (34)\nwheref(n)=D/2+q/q(n)2\n0.\nFor the case of pseudodifferential Landau-Lifshitz operators (3 1), the transverse wavevector\nq(n)\nzis closely connected to the longitudinal wavevector qby the relation\n2cotq(n)\nzD=q(n)\nz\nq−q\nq(n)\nz. (35)\nFor the case of differential Landau-Lifshitz operators (32), the transverse wavevector is\ndetermined by the exchange boundary conditions and is given by the equation [10, 13]\n2cotq(n)\nzD=q(n)\nz\nξ−ξ\nq(n)\nz. (36)\nDispersion relations (33) of the first spin-wave mode propagating in the YIG film of the\nthickness D= 0.5µm with 4 πM= 1750 Oe, α= 3.2·10−12cm2at the applied magnetic\nfieldH= 3000 Oe are shown in figure 6 for the transverse wave vector q(1)\nz(35) and for the\ntransverse wave vector q(1)\nz(36) with different pinning parameters ξ. One can see that there\ndoes not exist any pinning parameter ξ, at which the curve Acalculated on the base of relation\n(35) coincides with the curves calculated on the base of the exchan ge boundary conditions.\nThus, we conclude that the reduction to differential Landau-Lifsh itz operators and the use of\nthe exchange boundary conditions are incorrect.\n5 Spin-wave relaxation\nIn this section we answer the question: what is the value of spin-wav e relaxation in the model\nwith magnetic dipole and exchange interactions derived from first pr inciples? The answer\ndepends on the ratio of the spin-wave energy to intervals between modes of the spin-wave\nspectrum and is different for thick and for thin magnetic films. In thic k films the spin-wave\n170.0 0.1 0.2 0.3 0.4 0.5\nqD35004000450050005500Frequency /2 (MHz)21\n3\n4A\nFigure 6: Dispersion curves of the first spin-wave mode propagatin g in the YIG film of the\nthickness D= 0.5µm with 4 πM= 1750 Oe, α= 3.2·10−12cm2at the applied magnetic\nfieldH= 3000 Oe. The curve Ais calculated on the base of relation (35) for the case of\npseudodifferential Landau-Lifshitz operators (31). Curves 1 - 4 are calculated for the case of\ndifferential Landau-Lifshitz operators (32) on the base of relatio n (36) with different pinning\nparameters ξ. (1)ξD= 0.01, (2) 0.1, (3) 1, (4) 10.\nenergy is greater than energy gaps between modes and a three-s pin-wave process takes place. If\nthe exchange interaction is isotropic, it cannot induce three-magn on processes and, therefore,\nthe MDI makes a major contribution to the relaxation. We consider t he spin-wave damping in\nthick films in the one-loop approximation. In thin magnetic films (for ex ample, in nanosized\nfilms) the energy of long-wavelength spin waves is less than energy g aps between modes and\nthree-spin-wave processes are forbidden. In this case, four-s pin-wave processes take place, the\nexchange interaction makes a major contribution to the relaxation , and the spin-wave damping\nhas lower values in comparison with the damping in thick films. We calculat e the spin-wave\nrelaxation for four-spin-wave processes in thin films for long-wave length spin waves in the two-\nloop approximation.\n5.1 Spin-wave relaxation in thick films\nThe spin-wave relaxation induced by a three-spin-wave process in n ormal magnetized homoge-\nneous ferromagnetic films is considered in [17, 18] in the one-loop app roximation for spin waves\nwith small longitudinal wavevectors at low temperature. The relaxa tion is determined by self-\nenergy diagram insertions Σ (1+)(1−)to theP-matrix given by relation (12) (figure 7). Damping\nof thej-mode excitation is defined by the imaginary part of the pole of the eff ective Green\nfunctions G−+=P(1−)(1+)with insertions Σ (1+)(1−)under the analytical continuation (13)\n∆(j)(/vector q) =δω(j)(/vector q)\nω(j)(/vector q)=2B(p)Va\nβ¯hω(j)(/vector q)ImΣ(1+)(1−)(j,j,/vector q,ω m)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\niωm→ω+iεsignω\n=Va\n2β¯hω(j)Im/summationdisplay\nn,i,k/integraldisplay\nF(i)F(k)[¯P(1−)(1+)(i,−/vector q1,−ωn)¯P(2z)(2z)(k,/vector q−/vector q1,ωm−ωn)\n18q,□j,/c119m q,j,/c119m/c83/c40/c49/c32/c32/c43/c41/c40/c49/c32/c32/c45/c41 /c40 /c119j,j,q, )m=\nq,j,/c119m q,□j,/c119m1\n2B +(2B)21\nFigure 7: Self-energy diagrams in the one-loop approximation at low t emperature. Bis deter-\nmined by relation (8).\n+1\n8B(p)¯P(1−)(2z)(i,/vector q1,ωn)¯P(2z)(1+)(k,/vector q−/vector q1,ωm−ωn)]N2(j,/vector q;i,/vector q1;k,/vector q−/vector q1)d2q1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\niωm→ω+iεsignω,\n(37)\nwhere\n¯P(1−)(1+)(j,/vector q,ωm) = 2ρV2\na(Ω(j)+2η(j)\n−++iωm)\n¯P(1−)(2z)(j,/vector q,ωm) =−2η(j)\n+z(Ω(j)+iωm)\n¯P(2z)(1+)(j,/vector q,ωm) =−2η(j)\nz−(Ω(j)+iωm)\n¯P(2z)(2z)(j,/vector q,ωm) =F(j)−1βVa˜I(q(j)\n0)−ρ−1η(j)\nzz(Ω(j)2+iω2\nm)\nF(j)= (ω(j)2+ω2\nm)−1, ρ=B(p)\nβ¯hVa,\nη(j)\nµν=ΩMqµqν\nq(j)2\n0(µ,ν=−,+,z)\nq±=1\n2(qx∓iqy),\n˜I(q(j)\n0) =˜I(0)−wq(j)2\n0.\nis the Fourier transform of the exchange interaction,\nN(j1, /vector q1;j2, /vector q2;j3, /vector q3)\n=1\n8πVa3/productdisplay\nk=11\nf(jk)1/2/summationdisplay\nσ1,σ2,σ3sin/bracketleftig/parenleftig/summationtext3\nk=1σkq(jk)\nz/parenrightig\nD/2/parenrightig\n]\n/summationtext3\nk=1σkq(jk)\nzexp/parenleftigg\ni3/summationdisplay\nk=1σkπ(jk−1)/2/parenrightigg\nis the block factor in the representation of the functions (34), f(j)=D/2+q/q(j)2\n0,σk=±1;/summationtext\nσ1,σ2,σ3denotes the summation over all sets {σ1,σ2,σ3}. The spin-wave frequency ω(j)and\nthe transverse wavevector q(j)\nzare determined by relations (33) and (35), respectively. The\ndamping ∆(j)increases directly proportionally to the temperature.\nRelation (37) describes relaxation of the spin-wave j-mode caused by inelastic scattering on\nthermal excited spin wave modes. Relaxation occurs through the c onfluence of the j-mode with\nthek-mode to form the i-mode. From the explicit form of the block factor Nin relation (37) it\nfollows that the confluence processes take place when the sum of m ode numbers j+i+kis equal\nto an odd number. The confluence processes are induced by the MD I and are accompanied by\ntransitions between thermal excited i- andk-modes. Transitions take place when equation\n190.0 0.1 0.2 0.3 0.4 0.5\nqD1.0E-51.0E-41.0E-31.0E-21.0E-1Spin wave damping12\n34/c68(1)\nFigure 8: Spin wave damping ∆(1)=δω(1)/ω(1)of the first mode in normal magnetized YIG\nfilm with the magnetization 4 πM= 1750 Oe and the exchange interaction constant α=\n3.2·10−12cm2atH= 3000 Oe, T= 300 K at different film thickness D. (1)D= 500 nm, (2)\n300 nm, (3) 200 nm, (4) 120 nm.\nω(j)(/vector q) =ω(i)(/vector q(s))−ω(k)(/vector q−/vector q(s)) (38)\nhas at least one solution /vector q(s)for the given /vector q,i,j,k. Existence of solutions /vector q(s)of equation (38)\ndepends on the thickness of the magnetic film. With decreasing film th icknessD, the density\nof dispersion curves of modes on the plane ( ω,q) decreases and the frequency of the spacings\nbetween curves increase. The least frequency spacing occurs be tween the first ( i= 1) and the\nthird (k= 3) modes. Figure 8 shows the damping ∆(1)of the first spin wave mode versus\nthe longitudinal wave vector qnormalized by the film thickness Dat different film thicknesses.\nCalculations have been done for a YIG film with the magnetization 4 πM= 1750 Oe and the\nexchange interaction constant α= 3.2·10−12cm2atH= 3000 Oe and T= 300 K. One can see\nthat for the YIG film with the thickness D= 120 nm in the region qD <0.14 the damping\n∆(1)is equal to 0 due to the absence of transitions between modes. Thu s, in thin magnetic\nfilms a low spin-wave relaxation region takes place. For the given j-mode this region appears,\nwhen the excitation frequency ω(j)(/vector q) is less than the difference ω(3)(/vector q(s))−ω(1)(/vector q−/vector q(s)) at\nany values of the wavevector /vector q(s). For the first mode ω(1)in the YIG film the low spin-wave\nrelaxation region is shown in figure 9 at q→0. If the film thickness D < D 0(ω(1)/2π), then\nω(1)(0)< ω(3)(/vector q(s))−ω(1)(/vector q(s)) and the first mode has low values of the spin wave damping\n∆(1).\n5.2 Relaxation in thin magnetic films\nWhat is the value of spin wave damping in the low relaxation region in thin m agnetic films?\nWe consider four-spin-wave processes in the normal magnetized m onolayer of the quadratic\nlattice with the lattice constant aat small longitudinal wavevector values /vector q= (qx,qy) at low\n200 4 8 12 16 20\nFrequency□□□□□□□/2□□□□□(GHz)050100150200Film thickness D (nm)0\nLow□relaxation\nregion, /c119 /c60/c32/c68/c119/c40/c49/c41\n(1)q3\n2\n1/c119\n/c119(1)/c68/c119\nFigure 9: Film thickness D0of YIG film versus the excitation frequency ω(1)(/vector q)/2πof the first\nmode at the wavevector /vector q→0. Low relaxation region of the first spin-wave mode exists for\nYIG films with the thickness D < D 0(ω(1)/2π).\ntemperature. As isotropy of the exchange interaction can not fo rbid four-spin-wave processes\nand the value of the exchange interaction much greater than the M DI, only the exchange\ninteraction will be taken into account in diagrams. We suppose that t he exchange interaction\nacts between neighboring spins and is equal to I0. In order to calculate self-energy diagram\ninsertions to the effective Green functions in the two-loop approxim ation, we use the ladder\nexpansion (figure 10). At small values of wavevectors the bare Γ 0-vertex (figure 10a) is\nΓ0(1,2;3,4)≡Γ0(/vectork,/vector s+/vector q−/vectork;/vector q,/vector s)\n=β[˜I(/vectork−/vector q)+˜I(/vectork−/vector s)−˜I(/vector s)−˜I(/vector q)] = 2βI0a2(/vector q,/vector s),\nwhere 1,2;3,4 is the abridged notation of 2-dimensional wavevectors, which are variables of\nΓ0-vertex;|/vectork|,|/vector q|,|/vector s| ≪a−1;\n˜I(/vector q) =/summationdisplay\n/vector1xy−/vector1′xyI(/vector1xy−/vector1′\nxy)exp[−i/vector q(/vector1xy−/vector1′\nxy)] = 2I0[cos(qxa)+cos(qya)].\nThe Γ-vertex in the ladder approximation (figure 10b) is determined by the relationship\nΓ(1,2;3,4)≡Γ(/vectork,ω1,/vector s+/vector q−/vectork,ω3+ω4−ω1;/vector q,ω3,/vector s,ω4) = Γ0(/vectork,/vector s+/vector q−/vectork;/vector q,/vector s)\n+1\n8B2(p)Sb/summationdisplay\nω(q)\nm/integraldisplay\nΓ0(/vectork,/vector s+/vector q−/vectork;/vectorq′,/vector s+/vector q−/vectorq′)G−+(/vectorq′,ω(q)\nm)G−+(/vector s+/vector q−/vectorq′,ω3+ω4−ω(q)\nm)\n21×Γ(/vectorq′,ω(q)\nm,/vector s+/vector q−/vectorq′,ω3+ω4−ω(q)\nm;/vector q,ω3,/vector s,ω4)d2q′,\nwhere\nG−+(/vector q,ωm) =2B(p)\nβ¯h(ω(/vector q)−iωm)\nis the effective Green function determined by the P-matrix (12), ω(/vector q) is the frequency of\nspin excitations in monolayer (19), Sbis the volume of the 2-dimensional first Brillouin zone.\nThe coefficient 1 /8B2(p) is due to the fact that the substitution of the bare Green functio n\nto effective ones in diagrams are performed inside blocks. The self-e nergy diagram insertion\n(figure 10c) is given by\nΠ(/vector q,ω(q)\nm) =1\n2Sb/summationdisplay\nω(k)\nn/integraldisplay\nΓ0(/vector q,/vectork;/vector q,/vectork)G−+(/vectork,ω(k)\nn)d2k\n+1\n16B2(p)S2\nb/summationdisplay\nω(k)\nn,ω(s)\nl/integraldisplay /integraldisplay\nΓ0(/vector q,/vector s+/vectork−/vector q;/vector s,/vectork)G−+(−/vector s−/vectork+/vector q,−ω(k)\nn−ω(s)\nl+ω(q)\nm)\n×G−+(/vectork,ω(k)\nn)G−+(/vector s,ω(s)\nl)Γ(/vector s,ω(s)\nl,/vectork,ω(k)\nn;/vector q,ω(q)\nm,/vector s+/vectork−/vector q,ω(k)\nn+ω(s)\nl−ω(q)\nm)d2kd2s.(39)\nThe damping of spin wave excitations is expressed by the imaginary pa rt of the self-energy\nΠ(/vector q,ω(q)\nm)\n∆(/vector q) =δω(/vector q)\nω(/vector q)=ImΠ(/vector q,ω(q)\nm)\nβω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\niωm→ω+iεsignω. (40)\nTaking into account the self-energy Π( /vector q,ω(q)\nm) in the Born approximation, namely, substituting\nΓ→Γ0in relation (39), integrating over /vectork,/vector sand summing over the frequency variables ω(k)\nn\nandω(s)\nlin equation (40), at ¯ hω(/vector q)< kTwe obtain\n∆(/vector q) =C(qa)2(kT)2\n16πB2(p)I0ε(0),\nwhereC= 1.12,kis the Boltzmann constant, ε(0)= ¯hγ(H+H(m)) is the Zeeman energy.\nIn order to evaluate the damping of spin waves, we calculate ∆( /vector q) for spin waves with the\nwavelength λ= 5µm propagating in the monolayer film with the lattice constant a= 0.4 nm\nand with the exchange interaction between neighboring spins I0= 0.085 eV, B(p) = 1/2 at\nT= 300 K. Then, taking into account that q= 2π/λ, forε(0)/h= 10 GHz we obtain ∆( /vector q) =\n4.28·10−6. Thus, one can see that the damping of spin waves of small wavevec tors is low.\n6 Spin-wave devices on the base of nanosized magnetic\nfilms\nAccording to the above-mentioned, spin excitations in nanosized film s have low damping. This\nproperty can be used in spin-wave devices. We consider spin-wave fi lters on the base of nano-\nsized magnetic films and field-effect transistors with magnetic films un der the gate contact.\n22a.\n/c71/c48(1,2;3,4)□= /c710=1\n23\n4+1\n23\n4+1\n23\n4+3\n41\n21 3\n2 4\nb.\n/c710 + /c710 /c71q,/c119/c40q)\n/c71 = /c71(1,2;3,4)□=3\n2 41 3 1 1 3\n2 4 2 4s,/c119/c40s)8B(p)21\nc.\n/c710 + /c710 /c7121/c80 /c119(q, )□=(q)\nq,/c119(q)q,/c119(q)k,/c119/c40/c107)\ns,/c119/c40s)k,/c119/c40/c107)\nq,/c119(q)q,/c119(q)k+s-q, /c119(k+s-q)\n16B(p)21\nFigure 10: (a) Bare Γ 0-vertex. (b) Ladder approximation. (c) Self-energy diagram inse rtion.\n236.1 Spin-wave filters\nUsing thin magnetic films, we can construct spin-wave devices of sma ll sizes and can integrate\nthem to semiconductor chips. Figure 11a presents tunable spin-wa ve filter on a piezoelectric\nsubstrate. Microwave frequency current flowing in the microstrip antenna of the width w\ngenerates spin waves in the magnetic film. Wavevector of spin waves is determined by the\nantenna width and is in the range [0, 2 π/w]. When spin waves come up to the second antenna,\nthe magnetic field of spin waves induces a current of the same frequ ency in this antenna. The\nwaveguide impedance of the filter depends on the antenna width, th e width of microstrips,\nthe dielectric constant of the film, and the thickness of the film betw een microstrips and the\nuppermetalcontact onthepiezoelectricsubstrate. Inorderto remove alatticemismatchandto\nreachdesirableimpedance, buffer layersbetween themagneticfilma ndthemetalcontactcanbe\nused. Tunability of the filter is provided by lattice variations of the pie zoelectric substrate. The\napplied voltage Uvaries the lattice constant of the substrate and, as a result of th is variation,\nvaries the lattice constant of the magnetic film. Compression and ex pansion of the lattice\nlead to the stress anisotropy H(a)in the magnetic film [38]. In this case, we must substitute\nH→H+H(a)in the spin propagators (7), in the Green functions (11), in the fre quency Ω( /vector q)\nin (19), (20), and in the Landau-Lifshitz equations (28), (29). Th e frequency ωof spin waves\nis varied.\nNarrow-bandfilterscanbeconstructed onthebaseofperiodican tennae(Figure11b). These\nantenna structure generate spin waves with the wavevector q= 2π/l, wherelis the period\nof the generating antenna. Filters with periodic antenna structur es have more selectivity in\ncomparison with filters with single antennae. The bandwidth of the filt er is given by\n∆ω=∂ω(q)\n∂q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq0·2π\nlN1N2\nwhereω(q) is dispersion relation (19), (20), or (33), q0= 2π/l,N1,N2are numbers of periods\nof generating and receiving antennae, respectively. Thin magnetic film used in the above-\nmentioned filters must be dielectric and can be garnet, spinel or hex aferrite films. These films\ncan be produced by the laser deposition method or by ion-beam sput tering with following\nannealing. In [39] band pass spin-wave filters at 5 −7 GHz have been fabricated on the base\nof submicron thick YIG films produced by the laser deposition on Gd 3Ga5O12substrates. In\nthe case, when spin-wave filters are integrated on semiconductor chips, the semiconductor\nmust endure the annealing procedure without any changes for the worse in the semiconductor\nstructure. For this purpose, Si and GaN can be used.\n6.2 Field-effect transistors with nanosized magnetic films\nAs we can see from relations (20), (21) and from figures 4, 5, spin- wave resonance peaks in thin\nmagnetic films have high frequencies and low numbers. For the excha nge interaction between\nlayersId=0.05-0.1eV(garnetfilms, lithiumferrospinel andhexaferritefilms[ 38,40])spin-wave\nresonance frequencies are in Gigahertz and Terahertz frequenc y bands. Distance between spin-\nwave resonance peaks is greater in thin films than in thick ones. Since in thin films spin-wave\nresonances (21) at the given high frequency have lower numbers nand have lower numbers of\nhalf-periods of waves on film thickness in comparison with resonance s at the same frequency in\nthick magnetic films, high-frequency resonances in thin films can be e xited more easily. This\ncan be used in field-effect transistor (FET) structures. In figure 12 thin dielectric magnetic\n24substratePiezoelectricinout\nwbuffer□layer magnetic□film\nmetal□layersUspin□wave\nSemiconductor\nsubstrateinout\nlbuffer□layermagnetic□film\nmetal□layerspin□wavea.\nb.\nFigure 11: (a). Spin-wave filter on the base of nanosized magnetic fi lm on a piezoelectric\nsubstrate. (b). Spin-wave filter with nanosized magnetic film on a se miconductor substrate.\n25Semiconductor\nsubstraten+n+\nChanneldrain sourcegate\nMagnetic\nfilm Multiferroic\nFigure 12: Field-effect transistor with nanosized magnetic film under the gate contact.\nfilm is placed under the gate electrode. Applied voltage of the microwa ve frequency on the\ngate generates electromagnetic field, which induces spin-wave res onances. In order to increase\nthe magnetic component of the electromagnetic field and to enhanc e amplitude of resonances,\nmultiferroic layer can be placed between the gate and the magnetic fi lm. The magnetic field of\nthe induced spin-wave resonances interacts with spins of electron s propagating in the channel\nand modulates the current of the transistor. Consequently, exis tence of spin-wave resonances\nin the gate circuit can be regarded as a filter. Choosing the certain s pin-wave resonance peak,\nwe can construct FET structure operating in a desirable range of G igahertz and Terahertz\nfrequency bands.\n7 Conclusions\nThe results of the investigations can be summarized as follows.\n(1) We have studied spin excitations in nanosized magnetic films in the H eisenberg model with\nmagnetic dipole and exchange interactions by the spin operator diag ram technique. Dispersion\nrelations of spin waves in two-dimensional magnetic monolayers and in two-layer magnetic films\nand the spin-wave resonance spectrum in N-layer structures are found.\n(2) Generalized Landau-Lifshitz equations for thick magnetic films w hich are derived from first\nprinciples, have the integral (pseudodifferential) form, but not diff erential one with respect to\nspatial variables. Spin excitations are determined by simultaneous s olution of the Landau-\nLifshitz equations and the equation for the magnetostatic potent ial. The use of exchange\nboundary conditions for solvability of the Landau-Lifshitz equation s is incorrect.\n(3) The magnetic dipole interaction makes a major contribution to th e relaxation of long-\nwavelength spin waves in thick magnetic films. The spin-wave damping is determined by\ndiagrams in the one-loop approximation, which correspond to three -spin-wave processes. The\nthree-spin-wave processes are accompanied by transitions betw een thermal excited spin-wave\n26modes. The damping increases directly proportionally to the temper ature.\n(4) Thin films have a region of low relaxation of long-wavelength spin wa ves. In thin magnetic\nfilms the energy of these waves is less than energy gaps between sp in-wave modes, therefore,\nthree-spin-wave processes are forbidden, four-spin-wave pro cesses take place and, as a result of\nthis, the exchange interaction makes a major contribution to the r elaxation. It is found that\nthe damping of spin waves propagating in a magnetic monolayer has th e form of the quadratic\ndependence on the temperature and is very low for spin waves with s mall wavevectors.\n(5)Nanosizedmagneticfilmscanbeusedinspin-wavedevices. Lowda mpingoflong-wavelength\nspin waves gives us opportunity to construct tunable narrow-ban d spin-wave filters with high\nquality at the microwave frequency range. Spin-wave resonances of nanosized magnetic films\nunder gate electrodes in field-effect transistor (FET) structure s can be used to construct FET\nstructures operating in Gigahertz and Terahertz frequency ban ds.\nAcknowledgment\nThis work was supported by the Russian Foundation for Basic Resea rch, grant 10-02-00516,\nand by the Ministry of Education and Science of the Russian Federat ion, project 2011-1.3-513-\n067-006.\nReferences\n[1] D.D. Stancil, Theory of Magnetostatic Waves (Springer, New York, 1993).\n[2] D.D. Stancil and A. Prabhakar, Spin Waves. Theory and Applications (Springer, New\nYork, 2009).\n[3] P. Kabos and V.S. Stalmachov Magnetostatic Waves and Their Applications (Chapman\n& Hall, New York, 1994).\n[4] A. Khitun, M. Bao, and K.L.Wang, J. Phys. D: Appl. Phys. 43(26), 264005 (2010).\n[5] B. Lenk, H. Ulrichs, F. Garbs, M. M¨ unzenberg, Physics Report s507, 107 (2011).\n[6] Sang-Koog Kim, J. Phys. D: Appl. Phys. 43, 264004 (2010).\n[7] Y. Kajiwara, K. Harii, S. 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Nguyen, and M.L. Plumer, Phys. Rev. B\n83(17), 174425 (2011).\n[28] L.V. Lutsev, J. Phys. A: Math. Theor. 40, 11791 (2007).\n[29] A.N. Vasil’ev, Functional Methods in Quantum Field Theory and Statistical Physics(Tay-\nlor & Francis Books, New York, 1997).\n[30] T. Matsubara, Prog. Theor. Phys. 14, 351 (1955).\n[31] D.N. Zubarev, Nonequilibrium Statistical Thermodynamics (Plenum, New York, 1974).\n[32] F. Treves, Introduction to Pseudodifferential and Fourier Integral Op erators, V.1 (Plenum\nPress, New York and London, 1982).\n[33] I.V. Rojdestvenski, M.G. Cottam, and A.N. Slavin, Phys. Rev. B 48(17), 12768 (1993).\n[34] S.O. Demokritov,B. Hillebrands, and A.N. Slavin, Physics Reports 348, 441 (2001).\n[35] K.Yu. Guslienko, S.O.Demokritov, B. Hillebrands, andA.N. Slavin, P hys. Rev. B 66(13),\n132402 (2002).\n28[36] K.Yu. Guslienko and A.N. Slavin, Phys. Rev. B 72(1), 014463 (2005).\n[37] N.Yu. Grigorieva and B.A. Kalinikos, Technical Physics 54(8), 1196 (2009).\n[38] S. Krupiˇ cka, Physik der Ferrite und der Verwandten Magnetischen Oxide (Academia\nVerlag der Tschechoslowakischen, Prag, 1973).\n[39] S. A. Manuilov, R. Fors, S. I. Khartsev, and A. M. Grishin, J. Ap pl. Phys. 105(3), 033917\n(2009).\n[40] W.A. Harrison, Electronic Structure and the Properties of Solids. The Phys ics of the\nChemical Bond (W.H. Freeman and Company, San Francisco, 1980).\n29"    },    {        "title": "1808.04385v2.Gilbert_damping_phenomenology_for_two_sublattice_magnets.pdf",        "content": "Gilbert damping phenomenology for two-sublattice magnets\nAkashdeep Kamra,1,\u0003Roberto E. Troncoso,1Wolfgang Belzig,2and Arne Brataas1,y\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany\nAbstract\nWe present a systematic phenomenological description of Gilbert damping in two-sublattice mag-\nnets. Our theory covers the full range of materials from ferro- via ferri- to antiferromagnets. Fol-\nlowing a Rayleigh dissipation functional approach within a Lagrangian classical \feld formulation,\nthe theory captures intra- as well as cross-sublattice terms in the Gilbert damping, parameterized\nby a 2\u00022 matrix. When spin-pumping into an adjacent conductor causes dissipation, we obtain\nthe corresponding Gilbert damping matrix in terms of the interfacial spin-mixing conductances.\nOur model reproduces the experimentally observed enhancement of the ferromagnetic resonance\nlinewidth in a ferrimagnet close to its compensation temperature without requiring an increased\nGilbert parameter. It also predicts new contributions to damping in an antiferromagnet and sug-\ngests the resonance linewidths as a direct probe of the sublattice asymmetry, which may stem from\nboundary or bulk.\n1arXiv:1808.04385v2  [cond-mat.mtrl-sci]  6 Nov 2018I. INTRODUCTION\nThe fundamental connection1between magnetic moment and spin angular momentum\nunderlies the important role for magnets in nearly all spin-based concepts. An applied mag-\nnetic \feld provides the means to manipulate the state of a ferromagnet (FM), and thus the\nassociated spin. Conversely, a spin-polarized current absorbed by the FM a\u000bects its mag-\nnetization2{5. Exploiting a related phenomenon, switching the state of an antiferromagnet\n(AFM) has also been achieved6. Emboldened by this newly gained control, there has been\nan upsurge of interest in AFMs7{10, which o\u000ber several advantages over FMs. These include\nthe absence of stray \felds and a larger anisotropy-induced gap in the magnon spectrum. The\ntwo-sublattice nature of the AFMs further lends itself to phenomena distinct from FMs11.\nConcurrently, ferrimagnets (FiMs) have been manifesting their niche in a wide range of\nphenomena such as ultrafast switching12{14and low-dissipation spin transport15{22. A class of\nFiMs exhibits the so-called compensation temperature23{28, at which the net magnetization\nvanishes, similar to the case of AFMs. Despite a vanishing magnetization in the compensated\nstate, most properties remain distinct from that of AFMs29. Thus, these materials can be\ntuned to mimic FMs and AFMs via the temperature. In conjunction with the possibility of a\nseparate angular-momentum compensation, when the magnetization does not vanish but the\ntotal spin does, FiMs provide a remarkably rich platform for physics and applications. An\nincreased complexity in the theoretical description29,30hence accompanies these structurally\ncomplicated materials, and may be held responsible for comparatively fewer theoretical\nstudies. Nevertheless, a two-sublattice model with distinct parameters for each sublattice\nqualitatively captures all the phenomena mentioned above.\nDissipation strongly in\ruences the response of a magnet to a stimulus and is thus cen-\ntral to the study of magnetic phenomena such as switching, domain wall motion and spin\ntransport. Nevertheless, magnetic damping has conventionally been investigated via the\nferromagnetic resonance (FMR) linewidth. It is accounted for phenomenologically in the\nLandau-Lifshitz description of the magnetization dynamics via the so-called Gilbert damp-\ning term31, which produces a good agreement with experiments for a wide range of systems.\nThe Gilbert damping represents the viscous contribution and may be `derived' within a\nLagrangian formulation of classical \feld theory by including the Rayleigh dissipation func-\ntional31. While the magnetic damping for FMs has been studied in great detail29,31{35,\n2from phenomenological descriptions to microscopic models, a systematic development of an\nanalogous description for ferri- and antiferromagnets has been lacking in literature. Further-\nmore, recent theoretical results on spin pumping in two-sublattice magnets36and damping\nin AFMs37suggest an important role for the previously disregarded29cross-sublattice terms\nin Gilbert damping, and thus set the stage for the present study. Yuan and co-workers have\nrecently presented a step in this direction focussing on spin torques in AFMs38.\nHere, we formulate the magnetization dynamics equations in a general two-sublattice\nmagnet following the classical Lagrangian approach that has previously been employed for\nFMs31. The Gilbert damping is included phenomenologically via a Rayleigh dissipation\nfunctional appropriately generalized to the two-sublattice system, which motivates intra-\nas well as cross-sublattice terms. The Gilbert damping parameter thus becomes a 2 \u00022\nmatrix, in contrast with its scalar form for a single-sublattice FM. Solving the system of\nequations for spatially homogeneous modes in a collinear ground state, we obtain the decay\nrates of the two eigenmodes \fnding direct pathways towards probing the dissipation mech-\nanism and asymmetries in the system. Consistent with recent experiments28,39, we \fnd an\nenhancement in the decay rates39close to the magnetization compensation in a FiM with\nan unaltered damping matrix28. The general description is found to be consistent with the\nspin pumping mediated damping in the magnet34{36, and allows for relating the Gilbert\ndamping matrix with the interfacial spin-mixing conductances. Focusing on AFMs, we ex-\npress the magnetization dynamics in terms of the Neel variable thus clarifying the origin\nof the di\u000berent damping terms in the corresponding dynamical equations38,40. Apart from\nthe usually considered terms, we \fnd additional contributions for the case when sublattice-\nsymmetry is broken in the AFM36,41{45. Thus, FMR linewidth measurements o\u000ber a direct,\nparameter-free means of probing the sublattice asymmetry in AFMs, complementary to the\nspin pumping shot noise36.\nThis paper is organized as follows. We derive the Landau-Lifshitz-Gilbert (LLG) equa-\ntions for the two-sublattice model in Sec. II. The ensuing equations are solved for the\nresonance frequencies and decay rates of the uniform modes in a collinear magnet in Sec.\nIII. Section IV presents the application of the phenomenology to describe a compensated\nferrimagnet and spin pumping mediated Gilbert damping. The case of AFMs is discussed in\nSec. V. We comment on the validity and possible generalizations of the theory in Sec. VI.\nThe paper is concluded with a summary in Sec. VII. The discussion of a generalized Rayleigh\n3dissipation functional and properties of the damping matrix is deferred to the appendix.\nII. MAGNETIZATION DYNAMICS AND GILBERT DAMPING\nWe consider a two-sublattice magnet described by classical magnetization \felds MMMA\u0011\nMMMA(rrr;t) andMMMB\u0011MMMB(rrr;t) corresponding to the sublattices AandB. The system is\ncharacterized by a magnetic free energy F[MMMA;MMMB] with the magnetization \felds assumed\nto be of constant magnitudes MA0andMB0. Here, the notation F[ ] is employed to emphasize\nthat the free energy is a functional over the magnetization \felds, i.e. an integration of the\nfree energy density over space.\nThe undamped magnetization dynamics is described by equating the time derivative of\nthe spin angular momentum associated with the magnetization to the torque experienced\nby it. The resulting Landau-Lifshitz equations for the two \felds may be written as:\nd\ndt\u0012MMMA;B\n\u0000j\rA;Bj\u0013\n=\u0000_MMMA;B\nj\rA;Bj=MMMA;B\u0002\u00160HHHA;B; (1)\nwhere\rA;B(<0) are the gyromagnetic ratios for the two sublattices, and HHHA;Bare the\ne\u000bective magnetic \felds experienced by the respective magnetizations. This expression of\nangular momentum \row may be derived systematically within the Lagrangian classical \feld\ntheory31. The same formalism also allows to account for a restricted form of damping via\nthe so-called dissipation functional R[_MMMA;_MMMB] in the generalized equations of motion:\nd\ndt\u000eL[\u0001]\n\u000e_MMMA;B\u0000\u000eL[\u0001]\n\u000eMMMA;B=\u0000\u000eR[_MMMA;_MMMB]\n\u000e_MMMA;B; (2)\nwhereL[\u0001]\u0011 L [MMMA;MMMB;_MMMA;_MMMB] is the Lagrangian of the magnetic system. Here,\n\u000eL[\u0001]=\u000eMMMArepresents the functional derivative of the Lagrangian with respect to the var-\nious components of MMMA, and so on. The left hand side of Eq. (2) above represents the\nconservative dynamics of the magnet and reproduces Eq. (1) with31\n\u00160HHHA;B=\u0000\u000eF[MMMA;MMMB]\n\u000eMMMA;B; (3)\nwhile the right hand side accounts for the damping.\nThe Gilbert damping is captured by a viscous Rayleigh dissipation functional parame-\nterized by a symmetric matrix \u0011ijwithfi;jg=fA;Bg:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (4)\n4whereVis the volume of the magnet. The above form of the functional assumes the damping\nto be spatially homogeneous, isotropic, and independent of the equilibrium con\fguration.\nA more general form with a lower symmetry is discussed in appendix A. Including the\ndissipation functional via Eq. (2) leads to the following replacements in the equations of\nmotion (1):\n\u00160HHHA!\u00160HHHA\u0000\u0011AA_MMMA\u0000\u0011AB_MMMB; (5)\n\u00160HHHB!\u00160HHHB\u0000\u0011BB_MMMB\u0000\u0011AB_MMMA: (6)\nHence, the LLG equations for the two-sublattice magnet become:\n_MMMA=\u0000j\rAj(MMMA\u0002\u00160HHHA) +j\rAj\u0011AA\u0010\nMMMA\u0002_MMMA\u0011\n+j\rAj\u0011AB\u0010\nMMMA\u0002_MMMB\u0011\n; (7)\n_MMMB=\u0000j\rBj(MMMB\u0002\u00160HHHB) +j\rBj\u0011AB\u0010\nMMMB\u0002_MMMA\u0011\n+j\rBj\u0011BB\u0010\nMMMB\u0002_MMMB\u0011\n:(8)\nThese can further be expressed in terms of the unit vectors ^mmmA;B=MMMA;B=MA0;B0:\n_^mmmA=\u0000j\rAj(^mmmA\u0002\u00160HHHA) +\u000bAA\u0010\nmmmA\u0002_^mmmA\u0011\n+\u000bAB\u0010\n^mmmA\u0002_^mmmB\u0011\n; (9)\n_^mmmB=\u0000j\rBj(^mmmB\u0002\u00160HHHB) +\u000bBA\u0010\n^mmmB\u0002_^mmmA\u0011\n+\u000bBB\u0010\n^mmmB\u0002_^mmmB\u0011\n; (10)\nthereby introducing the Gilbert damping matrix ~ \u000bfor a two-sublattice system:\n~\u000b=0\n@\u000bAA\u000bAB\n\u000bBA\u000bBB1\nA=0\n@j\rAj\u0011AAMA0j\rAj\u0011ABMB0\nj\rBj\u0011ABMA0j\rBj\u0011BBMB01\nA; (11)\n\u000bAB\n\u000bBA=j\rAjMB0\nj\rBjMA0: (12)\nAs elaborated in appendix B, the positivity of the dissipation functional implies that the\neigenvalues and the determinant of ~ \u000bmust be non-negative, which is equivalent to the\nfollowing conditions:\n\u0011AA;\u0011BB\u00150; \u0011 AA\u0011BB\u0015\u00112\nAB=)\u000bAA;\u000bBB\u00150; \u000b AA\u000bBB\u0015\u000bAB\u000bBA: (13)\nThus, Eqs. (9) and (10) constitute the main result of this section, and introduce the damping\nmatrix [Eq. (11)] along with the constraints imposed on it [Eq. (12) and (13)] by the\nunderlying formalism.\n5III. UNIFORM MODES IN COLLINEAR GROUND STATE\nIn this section, we employ the phenomenology introduced above to evaluate the resonance\nfrequencies and the decay rates of the spatially homogeneous modes that can be probed in a\ntypical FMR experiment. We thus work in the macrospin approximation, i.e. magnetizations\nare assumed to be spatially invariant. Considering an antiferromagnetic coupling J(>0)\nbetween the two sublattices and parameterizing uniaxial easy-axis anisotropies via KA;B(>\n0), the free energy assumes the form:\nF[MMMA;MMMB] =Z\nVd3r\u0002\n\u0000\u00160H0(MAz+MBz)\u0000KAM2\nAz\u0000KBM2\nBz+JMMMA\u0001MMMB\u0003\n;(14)\nwhereH0^zzzis the applied magnetic \feld. The magnet is assumed to be in a collinear ground\nstate:MMMA=MA0^zzzandMMMB=\u0000MB0^zzzwithMA0>M B0. Employing Eq. (3) to evaluate the\ne\u000bective \felds, the magnetization dynamics is expressed via the LLG equations (9) and (10).\nConsidering MMMA=MAx^xxx+MAy^yyy+MA0^zzz,MMMB=MBx^xxx+MBy^yyy\u0000MB0^zzzwithjMAx;Ayj\u001c\nMA0,jMBx;Byj \u001cMB0, we linearize the resulting dynamical equations. Converting to\nFourier space via MAx=MAxexp (i!t) etc. and switching to circular basis via MA\u0006(B\u0006)=\nMAx(Bx)\u0006iMAy(By), we obtain two sets of coupled equations expressed succinctly as:\n0\n@\u0006!\u0000\nA\u0000i!\u000b AA\u0000\u0010\nj\rAjJMA0+i!\u000b ABMA0\nMB0\u0011\n\u0010\nj\rBjJMB0+i!\u000b BAMB0\nMA0\u0011\n\u0006!+ \n B+i!\u000b BB1\nA0\n@MA\u0006\nMB\u00061\nA=0\n@0\n01\nA; (15)\nwhere we de\fne \n A\u0011j\rAj(JMB0+ 2KAMA0+\u00160H0) and \n B\u0011j\rBj(JMA0+ 2KBMB0\u0000\n\u00160H0). Substituting !=!r\u0006+i!i\u0006into the ensuing secular equation, we obtain the\nresonance frequencies !r\u0006to the zeroth order and the corresponding decay rates !i\u0006to the\n\frst order in the damping matrix elements:\n!r\u0006=\u0006(\nA\u0000\nB) +p\n(\nA+ \n B)2\u00004J2j\rAjj\rBjMA0MB0\n2; (16)\n!i\u0006\n!r\u0006=\u0006!r\u0006(\u000bAA\u0000\u000bBB) +\u000bAA\nB+\u000bBB\nA\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000: (17)\nIn the expression above, Eq. (16) and Eq. (17), we have chosen the positive solutions of\nthe secular equations for the resonance frequencies. The negative solutions are equal in\nmagnitude to the positive ones and physically represent the same two modes. The positive-\npolarized mode in our notation corresponds to the typical ferromagnetic resonance mode,\nwhile the negative-polarized solution is sometimes termed `antiferromagnetic resonance'25.\n6020406080100120\n0 0.2 0.4 0.6 0.8 10.040.050.060.070.080.090.1FIG. 1. Resonance frequencies and normalized decay rates vs. the applied \feld for a quasi-\nferromagnet ( MA0= 5MB0).j\rAj=j\rBj= 1;1:5;0:5 correspond to solid, dashed and dash-dotted\nlines respectively. The curves in blue and red respectively depict the + and \u0000modes. The damping\nparameters employed are \u000bAA= 0:06,\u000bBB= 0:04 and\u000bAB= 0.\nIn order to avoid confusion with the ferromagnetic or antiferromagnetic nature of the un-\nderlying material, we call the two resonances as positive- and negative-polarized. The decay\nrates can further be expressed in the following form:\n!i\u0006\n!r\u0006=\u0016\u000b(\nA+ \n B)\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000\u0006\u0001\u0016\u000b; (18)\nwith \u0016\u000b\u0011(\u000bAA+\u000bBB)=2 and \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2. Eq. (18) constitutes the main result\nof this section and demonstrates that (i) asymmetric damping in the two sublattices is\nmanifested directly in the normalized decay rates of the two modes (Figs. 1 and 2), and\n(ii) o\u000b-diagonal components of the damping matrix may reduce the decay rates (Fig. 2).\nFurthermore, it is consistent with and reproduces the mode-dependence of the decay rates\nobserved in the numerical studies of some metallic AFMs37.\nTo gain further insight into the results presented in Eqs. (16) and (18), we plot the\n7resonance frequencies and the normalized decay rates vs. the applied magnetic \feld for a\ntypical quasi-ferromagnet, such as yttrium iron garnet, in Fig. 1. The parameters employed\nin the plot arej\rBj= 1:8\u00021011,MB0= 105,KA=KB= 10\u00007, andJ= 10\u00005in SI units,\nand have been chosen to represent the typical order of magnitude without pertaining to a\nspeci\fc material. The plus-polarized mode is lower in energy and is raised with an increasing\napplied magnetic \feld. The reverse is true for the minus-polarized mode whose relatively\nlarge frequency makes it inaccessible to typical ferromagnetic resonance experiments. As\nanticipated from Eq. (18), the normalized decay rates for the two modes di\u000ber when \u000bAA6=\n\u000bBB. Furthermore, the normalized decay rates are independent of the applied \feld for\nsymmetric gyromagnetic ratios for the two sublattices. Alternately, a measurement of the\nnormalized decay rate for the plus-polarized mode is able to probe the sublattice asymmetry\nin the gyromagnetic ratios. Thus it provides essential information about the sublattices\nwithout requiring the measurement of the large frequency minus-polarized mode.\nIV. SPECIFIC APPLICATIONS\nWe now examine two cases of interest: (i) the mode decay rate in a ferrimagnet close to\nits compensation temperature, and (ii) the Gilbert damping matrix due to spin pumping\ninto an adjacent conductor.\nA. Compensated ferrimagnets\nFMR experiments carried out on gadolinium iron garnet23,39\fnd an enhancement in the\nlinewidth, and hence the mode decay rate, as the temperature approaches the compensation\ncondition, i.e. when the two e\u000bective46sublattices have equal saturation magnetizations.\nThese experiments have conventionally been interpreted in terms of an e\u000bective single-\nsublattice model thereby ascribing the enhancement in the decay rate to an increase in the\nscalar Gilbert damping constant allowed within the single-sublattice model24. In contrast,\nexperiments probing the Gilbert parameter in a di\u000berent FiM via domain wall velocity\n\fnd it to be essentially unchanged around compensation28. Here, we analyze FMR in a\ncompensated FiM using the two-sublattice phenomenology developed above and thus address\nthis apparent inconsistency.\n8020406080100120\n1 2 3 4 500.050.10.150.20.250.3FIG. 2. Resonance frequencies and normalized decay rates vs. relative saturation magnetizations\nof the sublattices. The curves which are not labeled as + or \u0000represent the common normalized\ndecay rates for both modes. The parameters employed are the same as for Fig. 1 with \rA=\rB.\nThe compensation behavior of a FiM may be captured within our model by allowing\nMA0to vary while keeping MB0\fxed. The mode frequencies and normalized decay rates\nare examined with respect to the saturation magnetization variation in Fig. 2. We \fnd an\nenhancement in the normalized decay rate, consistent with the FMR experiments23,39, for a\n\fxed Gilbert damping matrix. The single-sublattice interpretation ascribes this change to a\nmodi\fcation of the e\u000bective Gilbert damping parameter24, which is equal to the normalized\ndecay rate within that model. In contrast, the latter is given by Eq. (18) within the\ntwo-sublattice model and evolves with the magnetization without requiring a modi\fcation\nin the Gilbert damping matrix. Speci\fcally, the enhancement in decay rate observed at\nthe compensation point is analogous to the so-called exchange enhancement of damping in\nAFMs47. Close to compensation, the FiM mimics an AFM to some extent.\nWe note that while the spherical samples employed in Ref. 23 are captured well by our\nsimple free energy expression [Eq. (14)], the interfacial and shape anisotropies of the thin\n9\flm sample employed in Ref. 39 may result in additional contributions to decay rates. The\nsimilarity of the observed linewidth trends for the two kinds of samples suggests that these\nadditional anisotropy e\u000bects may not underlie the observed damping enhancement. Quan-\ntitatively accounting for these thin \flm e\u000bects requires a numerical analysis, as discussed\nin Sec VI below, and is beyond the scope of the present work. Furthermore, domain forma-\ntion may result in additional damping contributions not captured within our single-domain\nmodel.\nB. Spin pumping mediated Gilbert damping\nSpin pumping34from a FM into an adjacent conductor has been studied in great detail35\nand has emerged as a key method for injecting pure spin currents into conductors48. The\nangular momentum thus lost into the conductor results in a contribution to the magnetic\ndamping on top of the intrinsic dissipation in the bulk of the magnet. A variant of spin\npumping has also been found to be the dominant cause of dissipation in metallic magnets37.\nThus, we evaluate the Gilbert damping matrix arising due to spin pumping from a two-\nsublattice magnet36into an adjacent conductor acting as an ideal spin sink.\nWithin the macrospin approximation, the total spin contained by the magnet is given by:\nSSS=\u0000MA0V^mmmA\nj\rAj\u0000MB0V^mmmB\nj\rBj: (19)\nThe spin pumping current emitted by the two-sublattice magnet has the following general\nform36:\nIIIs=~\neX\ni;j=fA;BgGij\u0010\n^mmmi\u0002_^mmmj\u0011\n; (20)\nwithGAB=GBA, where the spin-mixing conductances Gijmay be evaluated within di\u000berent\nmicroscopic models36,49{51. Equating the spin pumping current to \u0000_SSSand employing Eqs.\n(9) and (10), the spin pumping contribution to the Gilbert damping matrix becomes:\n\u000b0\nij=~Gijj\rij\neMi0V; (21)\nwhich in turn implies\n\u00110\nij=~Gij\neMi0Mj0V; (22)\n10for the corresponding dissipation functional. The resulting Gilbert damping matrix is found\nto be consistent with its general form and constraints formulated in Sec. II. Thus, employing\nthe phenomenology developed above, we are able to directly relate the magnetic damping in\na two-sublattice magnet to the spin-mixing conductance of its interface with a conductor.\nV. ANTIFERROMAGNETS\nDue to their special place with high symmetry in the two-sublattice model as well as the\nrecent upsurge of interest7{10,52{54, we devote the present section to a focused discussion on\nAFMs in the context of the general results obtained above. It is often convenient to describe\nthe AFM in terms of a di\u000berent set of variables:\nmmm=^mmmA+^mmmB\n2; nnn=^mmmA\u0000^mmmB\n2: (23)\nIn contrast with ^mmmAand ^mmmB,mmmandnnnare not unit vectors in general. The dynamical\nequations for mmmandnnnmay be formulated by developing the entire \feld theory, starting with\nthe free energy functional, in terms of mmmandnnn. Such a formulation, including damping,\nhas been accomplished by Hals and coworkers40. Here, we circumvent such a repetition and\ndirectly express the corresponding dynamical equations by employing Eqs. (9) and (10) into\nEq. (23):\n_mmm=\u0000(mmm\u0002\rm\u00160HHHm)\u0000(nnn\u0002\rn\u00160HHHn) +X\np;q=fm;ng\u000bm\npq(ppp\u0002_qqq); (24)\n_nnn=\u0000(mmm\u0002\rn\u00160HHHn)\u0000(nnn\u0002\rm\u00160HHHm) +X\np;q=fm;ng\u000bn\npq(ppp\u0002_qqq); (25)\nwith\n\rm\u00160HHHm\u0011j\rAj\u00160HHHA+j\rBj\u00160HHHB\n2; (26)\n\rn\u00160HHHn\u0011j\rAj\u00160HHHA\u0000j\rBj\u00160HHHB\n2; (27)\n\u000bm\nmm=\u000bn\nnm=\u000bAA+\u000bBB+\u000bAB+\u000bBA\n2; (28)\n\u000bm\nmn=\u000bn\nnn=\u000bAA\u0000\u000bBB\u0000\u000bAB+\u000bBA\n2; (29)\n\u000bm\nnn=\u000bn\nmn=\u000bAA+\u000bBB\u0000\u000bAB\u0000\u000bBA\n2; (30)\n\u000bm\nnm=\u000bn\nmm=\u000bAA\u0000\u000bBB+\u000bAB\u0000\u000bBA\n2: (31)\n11A general physical signi\fcance, analogous to \rA;B, may not be associated with \rm;nwhich\nmerely serve the purpose of notation here. The equations obtained above manifest new\ndamping terms in addition to the ones that are typically considered in the description\nof AFMs. Accounting for the sublattice symmetry of the antiferromagnetic bulk while\nallowing for the damping to be asymmetric, we may assume \rA=\rBandMA0=MB0, with\n\u0016\u000b\u0011(\u000bAA+\u000bBB)=2, \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2, and\u000bAB=\u000bBA\u0011\u000bod. Thus, the damping\nparameters simplify to\n\u000bm\nmm=\u000bn\nnm=\u0016\u000b+\u000bod; (32)\n\u000bm\nmn=\u000bn\nnn=\u0001\u0016\u000b; (33)\n\u000bm\nnn=\u000bn\nmn=\u0016\u000b\u0000\u000bod; (34)\n\u000bm\nnm=\u000bn\nmm=\u0001\u0016\u000b; (35)\nthereby eliminating the \\new\" terms in the damping when \u000bAA=\u000bBB. However, the sublat-\ntice symmetry may not be applicable to AFMs, such as FeMn, with non-identical sublattices.\nFurthermore, the sublattice symmetry of the AFM may be broken at the interface41{43via,\nfor example, spin mixing conductances36,45,55resulting in \u000bAA6=\u000bBB.\nThe resonance frequencies and normalized decay rates [Eqs. (16) and (18)] take a simpler\nform for AFMs. Substituting KA=KB\u0011K,\rA=\rB\u0011\r, andMA0=MB0\u0011M0:\n!r\u0006=\u0006j\rj\u00160H0+ 2j\rjM0p\n(J+K)K; (36)\n!i\u0006\n!r\u0006=J(\u0016\u000b\u0000\u000bod) + 2K\u0016\u000b\n2p\n(J+K)K\u0006\u0001\u0016\u000b\u0019(\u0016\u000b\u0000\u000bod)\n2r\nJ\nK+ \u0016\u000br\nK\nJ\u0006\u0001\u0016\u000b; (37)\nwhere we have employed J\u001dKin the \fnal simpli\fcation. The term /p\nK=J has typically\nbeen disregarded on the grounds K\u001cJ. However, recent numerical studies of damping in\nseveral AFMs37\fnd \u0016\u000b\u001d\u0016\u000b\u0000\u000bod>0 thus suggesting that this term should be comparable\nto the one proportional top\nJ=K and hence may not be disregarded. The expression above\nalso suggests measurement of the normalized decay rates as a means of detecting the sublat-\ntice asymmetry in damping. For AFMs symmetrical in the bulk, such an asymmetry may\narise due to the corresponding asymmetry in the interfacial spin-mixing conductance36,45,55.\nThus, decay rate measurements o\u000ber a method to detect and quantify such interfacial e\u000bects\ncomplementary to the spin pumping shot noise measurements suggested earlier36.\n12VI. DISCUSSION\nWe have presented a phenomenological description of Gilbert damping in two-sublattice\nmagnets and demonstrated how it can be exploited to describe and characterize the system\ne\u000bectively. We now comment on the limitations and possible generalizations of the formal-\nism presented herein. To begin with, the two-sublattice model is the simplest description of\nferri- and antiferromagnets. It has been successful in capturing a wide range of phenomenon.\nHowever, recent measurements of magnetization dynamics in nickel oxide could only be ex-\nplained using an eight-sublattice model56. The temperature dependence of the spin Seebeck\ne\u000bect in yttrium iron garnet also required accounting for more than two magnon bands57.\nA generalization of our formalism to a N-sublattice model is straightforward and can be\nachieved via a Rayleigh dissipation functional with N2terms, counting \u0011ijand\u0011jias sepa-\nrate terms. The ensuing Gilbert damping matrix will be N \u0002N while obeying the positive\ndeterminant constraint analogous to Eq. (13).\nIn our description of the collinear magnet [Eq. (14)], we have disregarded contributions\nto the free energy which break the uniaxial symmetry of the system about the z-axis. Such\nterms arise due to spin-nonconserving interactions58, such as dipolar \felds and magnetocrys-\ntalline anisotropies, and lead to a mixing between the plus- and minus-polarized modes30.\nIncluding these contributions converts the two uncoupled 2 \u00022 matrix equations [(15)] into\na single 4\u00024 matrix equation rendering the solution analytically intractable. A detailed\nanalysis of these contributions30shows that their e\u000bect is most prominent when the two\nmodes are quasi-degenerate, and may be disregarded in a \frst approximation.\nIn evaluating the resonance frequencies and the decay rates [Eqs. (16) and (18)], we\nhave assumed the elements of the damping matrix to be small. A precise statement of the\nassumption employed is !i\u001c!r, which simply translates to \u000b\u001c1 for a single-sublattice\nferromagnet. In contrast, the constraint imposed on the damping matrix within the two-\nsublattice model by the assumption of small normalized decay rate is more stringent [Eq.\n(18)]. For example, this assumption for an AFM with \u000bAB= \u0001\u0016\u000b= 0 requires \u0016 \u000b\u001c\np\nK=J\u001c1. This stringent constraint may not be satis\fed in most AFMs37, thereby\nbringing the simple Lorentzian shape description of the FMR into question. It can also be\nseen from Fig. 2 that the assumption of a small normalized decay rate is not very good for\nthe chosen parameters.\n13VII. SUMMARY\nWe have developed a systematic phenomenological description of the Gilbert damping\nin a two-sublattice magnet via inclusion of a Rayleigh dissipation functional within the La-\ngrangian formulation of the magnetization dynamics. Employing general expressions based\non symmetry, we \fnd cross-sublattice Gilbert damping terms in the LLG equations in con-\nsistence with other recent \fndings36{38. Exploiting the phenomenology, we explain the en-\nhancement of damping23,39in a compensated ferrimagnet without requiring an increase in\nthe damping parameters28. We also demonstrate approaches to probe the various forms\nof sublattice asymmetries. Our work provides a uni\fed description of ferro- via ferri- to\nantiferromagnets and allows for understanding a broad range of materials and experiments\nthat have emerged into focus in the recent years.\nACKNOWLEDGMENTS\nA. K. thanks Hannes Maier-Flaig and Kathrin Ganzhorn for valuable discussions. We\nacknowledge \fnancial support from the Alexander von Humboldt Foundation, the Research\nCouncil of Norway through its Centers of Excellence funding scheme, project 262633, \\QuS-\npin\", and the DFG through SFB 767 and SPP 1538.\nAppendix A: Generalized Rayleigh dissipation functional\nAs compared to the considerations in Sec. II, a more general approach to parameterizing\nthe dissipation functional is given by:\nR[_MMMA;_MMMB] =1\n2Z\nVZ\nVd3r0d3rX\np;q=fA;BgX\ni;j=fx;y;zg_Mpi(rrr)\u0011ij\npq(rrr;rrr0)_Mqj(rrr0): (A1)\nThis form allows to capture the damping in an environment with a reduced symmetry.\nHowever, the larger number of parameters also makes it di\u000ecult to extract them reliably\nvia typical experiments. The above general form reduces to the case considered in Sec. II\nwhen\u0011ij\npq(rrr;rrr0) =\u0011pq\u000eij\u000e(rrr\u0000rrr0) and\u0011pq=\u0011qp. Furthermore, the coe\u000ecients \u0011ij\npqmay depend\nuponMMMA(rrr) andMMMB(rrr) as has been found in recent numerical studies of Gilbert damping\nin AFMs37.\n14Appendix B: Damping matrix\nThe Rayleigh dissipation functional considered in the main text is given by:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (B1)\nwhich may be brought into the following concise form with the notation~_MMM\u0011[_MMMA_MMMB]|:\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_MMM|~\u0011~_MMM; (B2)\nwhere ~\u0011is the appropriate matrix given by:\n~\u0011=0\n@\u0011AA\u0011AB\n\u0011AB\u0011BB1\nA: (B3)\nConsidering an orthogonal transformation~_MMM=~Q~_M, the dissipation functional can be\nbrought to a diagonal form\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_M|~Q|~\u0011~Q~_M; (B4)\nwhere ~Q|~\u0011~Qis assumed to be diagonal. 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Various\nmodels to deal with spin fluctuations are discussed concerni ng their impact on the resulting tem-\nperature dependent magnetic moment, longitudinal conduct ivity and Gilbert damping parameter.\nIt is demonstrated that using the Monte Carlo (MC) spin config uration as an input, the alloy ana-\nlogy model is capable to reproduce results of MC simulations on the average magnetic moment\nwithin all spin fluctuation models under discussion. On the o ther hand, response quantities are\nmuch more sensitive to the spin fluctuation model. Separate c alculations accounting for either the\nthermal effect due to lattice vibrations or spin fluctuations show their comparable contributions\nto the electrical conductivity and Gilbert damping. Howeve r, comparison to results accounting for\nboth thermal effects demonstrate violation of Matthiessen’ s rule, showing the non-additive effect of\nlattice vibrations and spin fluctuations. The results obtai ned for bcc Fe and fcc Ni are compared\nwith theexperimental data, showing rather good agreement f or thetemperature dependentelectrical\nconductivity and Gilbert damping parameter.\nI. INTRODUCTION\nFinite temperature has often a very crucial influence\non the response properties of a solid. A prominent ex-\nample for this is the electrical resistivity of perfect non-\nmagnetic metals and ordered compounds that only take\na non-zero value with a characteristic temperature ( T)\ndependence due to thermal lattice vibrations. While the\nHolstein transport equation1,2provides a sound basis for\ncorresponding calculations numerical work in this field\nhas been done so far either on a model level or for sim-\nplified situations.3–6In practice often the Boltzmann-\nformalism is adopted using the constant relaxation time\n(τ) approximation. This is a very popular approach in\nparticular when dealing with the Seebeck effect, as in\nthis case τdrops out.7,8The constant relaxation time\napproximation has also been used extensively when deal-\ning with the Gilbert damping parameter α.9–11Within\nthe description of Kambersky10,12the conductivity- and\nresistivity-like intra- and inter-band contributions to α\nshow a different dependency on τleading typically to\na minimum for α(τ) or equivalently for α(T).10,11A\nscheme to deal with the temperature dependent resistiv-\nity that is formally much more satisfying than the con-\nstant relaxation time approximation is achieved by com-\nbining the Boltzmann-formalism with a detailed calcula-\ntion of the phonon properties. As was shown by various\nauthors,13–16this parameter-free approach leads for non-\nmagneticmetalsingeneraltoaverygoodagreementwith\nexperimental data.\nAs an alternative to this approach, thermal lattice\nvibrations have also been accounted for within various\nstudies by quasi-static lattice displacements leading to\nthermallyinducedstructuraldisorderinthesystem. This\npoint of view provides the basis for the use of the al-\nloy analogy, i.e. for the use of techniques to deal withsubstitutional chemical disorder also when dealing with\ntemperature dependent quasi-static random lattice dis-\nplacements. An example for this are investigations on\nthe temperature dependence of the resistivity and the\nGilbert parameter αbased on the scattering matrix ap-\nproach applied to layered systems.17The necessary aver-\nageovermanyconfigurationsoflatticedisplacementswas\ntakenbymeansofthe supercelltechnique. Incontrastto\nthistheconfigurationalaveragewasdeterminedusingthe\nCoherent Potential Approximation (CPA) within invest-\nigations using a Kubo-Greenwood-like linear expression\nforα.18The same approach to deal with the lattice dis-\nplacements was also used recently within calculations of\nangle-resolved photo emission spectra (ARPES) on the\nbasis of the one-step model of photo emission.19\nAnother important contribution to the resistivity in\nthe case of magnetically ordered solids are thermally in-\nduced spin fluctuations.20Again, the alloy analogy has\nbeen exploited extensively in the past when dealing with\ntheimpactofspinfluctuationsonvariousresponsequant-\nities. Representing a frozen spin configuration by means\nof super cell calculations has been applied for calcula-\ntions of the Gilbert parameter for α17as well as the\nresistivity or conductivity, respectively.17,21,22Also, the\nCPA has been used for calculations of α23as well as the\nresistivity.20,24A crucial point in this context is obvi-\nously the modeling of the temperature dependent spin\nconfigurations. Concerning this, rather simple models\nhave been used,23but also quite sophisticated schemes.\nHere one should mention the transfer of data from Monte\nCarlo simulations based on exchange parameters calcu-\nlated in an ab-initio way25as well as work based on the\ndisordered local moment (DLM) method.24,26Although,\nthe standard DLM does not account for transversal spin\ncomponents it nevertheless allows to represent the para-\nmagnetic regime with no net magnetization in a rigor-2\nous way.Also, for the magnetically ordered regime below\nthe Curie-temperature it could be demonstrated that the\nuncompensated DLM (uDLM) leads for many situations\nstill to goodagreementwith experimentaldata on the so-\ncalled spin disorder contribution to the resistivity.20,24\nIn the following we present technical details and exten-\nsionsofaschemethatwasalreadyused beforewhendeal-\ning with the temperature dependence of response quant-\nities on the basis of Kubo’s response formalism. Various\napplications will be presented for the conductivity and\nGilbert damping parameter accounting simultaneously\nfor various types of disorder.\nII. THEORETICAL FRAMEWORK\nA. Configurational average for linear response\nfunctions\nMany important quantities in spintronics can be\nformulated by making use of linear response formal-\nism. Important examples for this are the electrical\nconductivity,27,28the spin conductivity29or the Gilbert\ndamping parameter.18,30Restricting here for the sake of\nbrevity to the symmetric part of the corresponding re-\nsponse tensor χµνthis can be expressed by a correlation\nfunction of the form:\nχµν∝Tr/angbracketleftbigˆAµℑG+ˆAνℑG+/angbracketrightbig\nc. (1)\nIt should be stressed that this not a real restriction as\nthe scheme described below has been used successfully\nwhen dealing with the impact of finite temperatures on\nthe anomalous Hall conductivity of Ni.31In this case the\nmore complex Kubo-Stˇ reda- or Kubo-Bastin formulation\nfor the full response tensor has to be used.32\nThe vector operator ˆAµin Eq. (1) stands for example\nin case of the electrical conductivity σµνfor the cur-\nrent density operator ˆjµ28while in case of the Gilbert\ndamping parameter αµνit stands for the torque oper-\natorˆTµ.9,18Within the Kubo-Greenwood-like equation\n(1) the electronic structure of the investigated system\nis represented in terms of its retarded Green function\nG+(r,r′,E). Within multiple scattering theory or the\nKKR (Korringa-Kohn-Rostoker)formalism, G+(r,r′,E)\ncan be written as:33–35\nG+(r,r′,E) =/summationdisplay\nΛΛ′Zm\nΛ(r,E)τmn\nΛΛ′(E)Zn×\nΛ′(r′,E)(2)\n−δmn/summationdisplay\nΛZn\nΛ(r,E)Jn×\nΛ′(r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(r,E)Zn×\nΛ′(r′,E)Θ(rn−r′\nn).\nHerer,r′refer to points within atomic volumes around\nsitesRm,Rn, respectively, with Zn\nΛ(r,E) =ZΛ(rn,E) =\nZΛ(r−Rn,E) being a function centered at site Rn. Ad-\nopting a fully relativistic formulation34,35for Eq. (2) one\ngets in a natural way access to all spin-orbit inducedproperties as for example the anomalous and spin Hall\nconductivity29,32,36or Gilbert damping parameter.18In\nthis case, the functions Zn\nΛandJn\nΛstand for the reg-\nular and irregular, respectively, solutions to the single-\nsite Dirac equation for site nwith the associated single-\nsite scattering t-matrix tn\nΛΛ′. The corresponding scat-\ntering path operator τnn′\nΛΛ′accounts for all scattering\nevents connecting the sites nandn′. Using a suitable\nspinor representation for the basis functions the com-\nbined quantum number Λ = ( κ,µ) stands for the relativ-\nistic spin-orbit and magnetic quantum numbers κandµ,\nrespectively.34,35,37\nAs was demonstrated by various authors27,28,38rep-\nresenting the electronic structure in terms of the Green\nfunction G+(r,r′,E) allows to account for chemical dis-\norder in a random alloy by making use of a suitable al-\nloy theory. In this case ∝an}bracketle{t...∝an}bracketri}htcstands for the configura-\ntional average for a substitutional alloy concerning the\nsite occupation. Corresponding expressions for the con-\nductivity tensor have been worked out by Velick´ y27and\nButler28usingthe single-siteCoherentPotentialApprox-\nimation (CPA) that include in particular the so-called\nvertex corrections.\nThe CPA can be used to deal with chemical but also\nwith any other type of disorder. In fact, making use of\nthe different time scales connected with the electronic\npropagation and spin fluctuations the alloy analogy is\nexploited when dealing with finite temperature magnet-\nism on the basis of the disordered local moment (DLM)\nmodel.26,39Obviously, the same approach can be used\nwhen dealing with response tensors at finite temperat-\nures. In connection with the conductivity this is often\ncalled adiabatic approximation.40Following this philo-\nsophy, the CPA has been used recently also when calcu-\nlating response tensors using Eq. ( 1) with disorder in the\nsystem caused by thermal lattice vibrations18,31as well\nas spin fluctuations.20,41\nB. Treatment of thermal lattice displacement\nA way to account for the impact of the thermal dis-\nplacement of atoms from their equilibrium positions, i.e.\nfor thermal lattice vibrations, on the electronic struc-\nture is to set up a representative displacement configura-\ntion for the atoms within an enlargedunit cell (super-cell\ntechnique). In this case one has to use either a very large\nsuper-cell or to take the average over a set of super-cells.\nAlternatively, one may make use of the alloy analogy for\nthe averaging problem. This allows in particular to re-\nstrict to the standard unit cell. Neglecting the correla-\ntion between the thermal displacements of neighboring\natoms from their equilibrium positions the properties of\nthe thermal averaged system can be deduced by making\nuse of the single-site CPA. This basic idea is illustrated\nby Fig.1. To make use of this scheme a discrete set\nofNvdisplacement vectors ∆ Rq\nv(T) with probability xq\nv\n(v= 1,..,Nv) is constructed for each basis atom qwithin3\nFigure 1. Configurational averaging for thermal lattice dis -\nplacements: the continuous distribution P(∆Rn(T)) for the\natomic displacement vectors is replaced by a discrete set of\nvectors ∆ Rv(T) occurring with the probability xv. The con-\nfigurational average for this discrete set of displacements is\nmade using the CPA leading to a periodic effective medium.\nthe standard unit cell that is conform with the local sym-\nmetry and the temperature dependent root mean square\ndisplacement ( ∝an}bracketle{tu2∝an}bracketri}htT)1/2according to:\n1\nNvNv/summationdisplay\nv=1|∆Rq\nv(T)|2=∝an}bracketle{tu2\nq∝an}bracketri}htT. (3)\nIn the general case, the mean square displacement along\nthe direction µ(µ=x,y,z) of the atom ican be either\ntaken from experimental data or represented by the ex-\npression based on the phonon calculations42\n∝an}bracketle{tu2\ni,µ∝an}bracketri}htT=3/planckover2pi1\n2Mi/integraldisplay∞\n0dωgi,µ(ω)1\nωcoth/planckover2pi1ω\n2kBT,(4)\nwhereh= 2π/planckover2pi1the Planck constant, kBthe Boltzmann\nconstant, gi,µ(ω) is a partial phonon density of states.42\nOn the other hand, a rather good estimate for the root\nmean square displacement can be obtained using Debye’s\ntheory. In this case, for systems with one atom per unit\ncell, Eq. ( 4) can be reduced to the expression:\n∝an}bracketle{tu2∝an}bracketri}htT=1\n43h2\nπ2MkBΘD/bracketleftbiggΦ(ΘD/T)\nΘD/T+1\n4/bracketrightbigg\n(5)\nwith Φ(Θ D/T) the Debye function and Θ Dthe Debye\ntemperature43. Ignoring the zero temperature term 1 /4\nand assuming a frozen potential for the atoms, the situ-\nationcanbe dealt with in full analogytothe treatmentof\ndisorderedalloysonthebasisoftheCPA.Theprobability\nxvfor a specific displacement vmay normally be chosen\nas 1/Nv. The Debye temperature Θ Dused in Eq. ( 5) can\nbe either taken fromexperimental data orcalculated rep-\nresenting it in terms of the elastic constants44. In general\nthe latter approach should give more reliable results in\nthe case of multicomponent systems.\nTo simplify notation we restrict in the following to sys-\ntems with one atom per unit cell. The index qnumbering\nsites in the unit cell can therefore be dropped, while the\nindexnnumbers the lattice sites.\nAssuming a rigid displacement of the atomic potential\nin the spirit of the rigid muffin-tin approximation45,46\nthe correspondingsingle-site t-matrix tlocwith respect to\nthe local frame of reference connected with the displaced\natomic position is unchanged. With respect to the globalframe of reference connected with the equilibrium atomic\npositions Rn, however, the corresponding t-matrix tis\ngiven by the transformation:\nt=U(∆R)tlocU(∆R)−1. (6)\nThe so-called U-transformation matrix U(s) is given in\nits non-relativistic form by:45,46\nULL′(s) = 4π/summationdisplay\nL′′il+l′′−l′CLL′L′′jl′′(|s|k)YL′′(ˆs).(7)\nHereL= (l,m) represents the non-relativistic angu-\nlar momentum quantum numbers, jl(x) is a spherical\nBesselfunction, YL(ˆr) a realsphericalharmonics, CLL′L′′\na corresponding Gaunt number and k=√\nEis the\nelectronic wave vector. The relativistic version of the\nU-matrix is obtained by a standard Clebsch-Gordan\ntransformation.37\nThe various displacement vectors ∆ Rv(T) can be used\nto determine the properties of a pseudo-component of a\npseudo alloy. Each of the Nvpseudo-components with\n|∆Rv(T)|=∝an}bracketle{tu2∝an}bracketri}ht1/2\nTis characterized by a corresponding\nU-matrix Uvand t-matrix tv. As for a substitutional\nalloy the configurational average can be determined by\nsolving the multi-component CPA equations within the\nglobal frame of reference:\nτnn\nCPA=Nv/summationdisplay\nv=1xvτnn\nv (8)\nτnn\nv=/bracketleftbig\n(tv)−1−(tCPA)−1+(τnn\nCPA)−1/bracketrightbig−1(9)\nτnn\nCPA=1\nΩBZ/integraldisplay\nΩBZd3k/bracketleftbig\n(tCPA)−1−G(k,E)/bracketrightbig−1,(10)\nwhere the underline indicates matrices with respect to\nthe combined index Λ. As it was pointed out in the pre-\nvious work41, the cutoff for the angular momentum ex-\npansionin these calculations should be taken l≥lmax+1\nwith the lmaxvalue used in the calculations for the non-\ndistorted lattice.\nThe first of these CPA equations represents the re-\nquirement for the mean-field CPA medium that embed-\nding of a component vshould lead in the average to no\nadditional scattering. Eq. ( 9) gives the scattering path\noperator for the embedding of the component vinto the\nCPA medium while Eq. ( 10) gives the CPA scattering\npath operator in terms of a Brillouin zone integral with\nG(k,E) the so-called KKR structure constants.\nHaving solved the CPA equations the linear response\nquantity of interest may be calculated using Eq. ( 1)\nas for an ordinary substitutional alloy.27,28This im-\nplies that one also have to deal with the so-called ver-\ntex corrections27,28that take into account that one\nhas to deal with a configuration average of the type\n∝an}bracketle{tˆAµℑG+ˆAνℑG+∝an}bracketri}htcthat in general will differ from the\nsimpler product ∝an}bracketle{tˆAµℑG+∝an}bracketri}htc∝an}bracketle{tˆAνℑG+∝an}bracketri}htc.4\nC. Treatment of thermal spin fluctuations\nAs for the disorder connected with thermal displace-\nments the impact of disorder due to thermal spin fluc-\ntuations may be accounted for by use of the super-cell\ntechnique. Alternatively one may again use the alloy\nanalogy and determine the necessary configurational av-\nerage by means of the CPA as indicated in Fig. 2. As\nFigure 2. Configurational averaging for thermal spin fluc-\ntuations: the continuous distribution P(ˆen) for the orienta-\ntion of the magnetic moments is replaced by a discrete set of\norientation vectors ˆ efoccurring with a probability xf. The\nconfigurational average for this discrete set of orientatio ns is\nmade using the CPA leading to a periodic effective medium.\nfor the thermal displacements in a first step a set of rep-\nresentative orientation vectors ˆ ef(withf= 1,...,Nf) for\nthelocalmagneticmomentisintroduced(seebelow). Us-\ning the rigid spin approximation the spin-dependent part\nBxcoftheexchange-correlationpotentialdoesnotchange\nfor the local frame of reference fixed to the magnetic mo-\nment when the moment is oriented along an orientation\nvector ˆef. This implies that the single-site t-matrix tloc\nf\nin the local frame is the same for all orientation vectors.\nWith respect to the common global frame that is used\nto deal with the multiple scattering (see Eq. ( 10)) the\nt-matrix for a given orientation vector is determined by:\nt=R(ˆe)tlocR(ˆe)−1. (11)\nHere the transformation from the local to the global\nframe of reference is expressed by the rotation matrices\nR(ˆe) that are determined by the vectors ˆ eor correspond-\ning Euler angles.37\nAgain the configurational average for the pseudo-alloy\ncan be obtained by setting up and solvingCPAequations\nin analogy to Eqs. ( 8) to (10).\nD. Models of spin disorder\nThe central problem with the scheme described above\nis obviously to construct a realistic and representative\nset of orientation vectors ˆ efand probabilities xffor each\ntemperature T. A rather appealing approach is to cal-\nculate the exchange-coupling parameters Jijof a sys-\ntem in an ab-initio way25,47,48and to use them in sub-\nsequent Monte Carlo simulations. Fig. 3(top) shows\nresults for the temperature dependent average reduced\nmagnetic moment of corresponding simulations for bcc-\nFe obtained for a periodic cell with 4096 atom sites. The0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC*\nKKR (MC*)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)\nMC\nMF-fit to MC (wMC(T))\nMF-fit to MC (w=const)\nExpt\nMF-fit to Expt (wExpt(T))\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC\nKKR (MC)\nKKR (DLM)\nFigure 3. Averaged reduced magnetic moment M(T) =\n/angbracketleftmz/angbracketrightT/|/angbracketleftm/angbracketrightT=0|along the z-axis as a function of the tem-\nperature T. Top: results of Monte Carlo simulations using\nscheme MC* (full squares) compared with results of sub-\nsequent KKR-calculations (open squares). Middle: results\nof Monte Carlo simulations using scheme MC (full squares)\ncompared with results using a mean-field fit with a constant\nWeiss field wMC(TC) (open diamonds) and a temperature de-\npendent Weiss field wMC(T) (open squares). In addition ex-\nperimental data (full circles) together with a correspondi ng\nmean-field fit obtained for a temperature dependent Weiss\nfieldwexp(T). Bottom: results of Monte Carlo simulations\nusing scheme MC (full squares) compared with results sub-\nsequent KKR-calculations using the MC (triangles up) and\na corresponding DLM (triangle down) spin configuration, re-\nspectively.\nfull line gives the value for the reduced magnetic mo-\nmentMMC∗(T) =∝an}bracketle{tmz∝an}bracketri}htT/m0projected on the z-axis for\nthe lastMonteCarlostep (ˆ zis the orientationofthetotal\nmoment, i.e. ∝an}bracketle{tm∝an}bracketri}htT∝bardblˆz; the saturated magnetic moment at\nT= 0 K is m0=|∝an}bracketle{tm∝an}bracketri}htT=0|). This scheme is called MC∗\nin the following. In spite of the rather large number of\nsites (4096) the curve is rather noisy in particular when\napproaching the Curie temperature. Nevertheless, the5\nspin configuration of the last MC step was used as an\ninput for subsequent SPR-KKR-CPA calculations using\ntheorientationvectors ˆ efwiththeprobability xf= 1/Nf\nwithNf= 4096. As Fig. 3(top) shows, the temperature\ndependent reducedmagnetic moment MKKR(MC∗)(T) de-\nduced from the electronic structure calculations follows\none-to-one the Monte Carlo data MMC∗(T). This is a\nvery encouraging result for further applications (see be-\nlow) as it demonstrates that the CPA although being a\nmean-field method and used here in its single-site formu-\nlation is nevertheless capable to reproduce results of MC\nsimulations that go well beyond the mean-field level.\nHowever, using the set of vectors ˆ efof scheme MC*\nalso for calculations of the Gilbert damping parameters\nαas a function of temperature led to extremely noisy\nand unreliable curves for α(T). For that reason an av-\nerage has been taken over many MC steps (scheme MC)\nleading to a much smoother curve for MMC(T) as can\nbe seen from Fig. 3(middle) with a Curie temperature\nTMC\nC= 1082 K. As this enlarged set of vectors ˆ efgot\ntoo large to be used directly in subsequent SPR-KKR-\nCPA calculations, a scheme was worked out to get a set\nof vectors ˆ efand probabilities xfthat is not too large\nbut nevertheless leads to smooth curves for M(T).\nThe first attempt was to use the Curie temperature\nTMC\nCtodeduceacorrespondingtemperatureindependent\nWeiss-field w(TC) on the basis of the standard mean-field\nrelation:\nw(TC) =3kBTC\nm2\n0. (12)\nThis leads to a reduced magnetic moment curve MMF(T)\nthat shows by construction the same Curie temperature\nas the MC simulations. For temperatures between T=\n0 K and TC, however, the mean-field reduced magnetic\nmoment MMF(T) is well below the MC curve (see Fig. 3\n(middle) ).\nAs an alternative to this simple approach we intro-\nduced a temperature dependent Weiss field w(T). This\nallows to describe the temperature dependent magnetic\nproperties using the results obtained beyond the mean-\nfield approximation. At the same time the calculation\nof the statistical average can be performed treating the\nmodel Hamiltonian in termsofthe mean field theory. For\nthis reason the reduced magnetic moment M(T), being\na solution of equation (see e.g.49)\nM(T) =L/parenleftbiggwm2\n0M(T)\nkBT/parenrightbigg\n, (13)\nwas fitted to that obtained from MC simulations\nMMC(T)withtheWeissfield w(T)asafittingparameter,\nsuch that\nlim\nw→w(T)M(T) =MMC(T), (14)\nwithL(x) the Langevin function.\nThe corresponding temperature dependent probability\nx(ˆe) for an atomic magnetic moment to be oriented alongˆeis proportionalto exp( −w(T)ˆz·ˆe/kBT) (see, e.g.49). To\ncalculate this value we used NθandNφpoints for a reg-\nular grid for the spherical angles θandφcorresponding\nto the vector ˆ ef:\nxf=exp(−w(T)ˆz·ˆef/kBT)/summationtext\nf′exp(−w(T)ˆz·ˆef′/kBT).(15)\nFig.4shows for three different temperatures the θ-\ndependent behavior of x(ˆe). As one notes, the MF-fit\n0 30 60 90 120 150 180\nθ00.050.10.150.20.250.3P(θ)MC\nMF-fit to MC (wMC(T))T = 200 K\n0 30 60 90 120 150 180\nθ00.050.10.150.2P(θ)MC\nMF-fit to MC (wMC(T))T = 400 K\n0 30 60 90 120 150 180\nθ00.050.1P(θ)MC\nMF-fit to MC (wMC(T))T = 800 K\nFigure 4. Angular distribution P(θ) of the atomic magnetic\nmoment mobtained from Monte Carlo simulations (MC) for\nthe temperature T= 200, 400, and 800 K compared with field\nmean-field (MF) data, xf, (full line) obtained by fitting using\na temperature dependent Weiss field w(T) (Eq.13).\nto the MC-results perfectly reproduces these data for all\ntemperatures. This applies of course not only for the\nangular resolved distribution of the magnetic moments\nshown in Fig. 4but also for the average reduced mag-\nnetic moment recalculated using Eq.( 13), shown in Fig.\n3. Obviously, the MF-curve MMF(MC)(T) obtained using\nthe temperature dependent Weiss field parameter w(T)\nperfectly reproduces the original MMC(T) curve. The\ngreat advantage of this fitting procedure is that it al-\nlows to replace the MC data set with a large number6\nNMC\nfof orientation vectors ˆ ef(pointing in principle into\nany direction) with equal probability xf= 1/NMC\nfby a\nmuch smaller data set with Nf=NθNφwithxfgiven\nby Eq. (15).\nAccordingly, the reduced data set can straight for-\nwardly be used for subsequent electronic structure cal-\nculations. Fig. 3(bottom) shows that the calcu-\nlated temperature dependent reduced magnetic moment\nMKKR−MF(MC)(T) agrees perfectly with the reduced\nmagnetic moment MMC(T) given by the underlying MC\nsimulations.\nThe DLM method has the appealing feature that it\ncombines ab-initio calculations and thermodynamics in\na coherent way. Using a non-relativistic formulation, it\nwas shown that the corresponding averaging over all ori-\nentations of the individual atomic reduced magnetic mo-\nments can be mapped onto a binary pseudo-alloy with\none pseudo-component having up- and downward orient-\nation of the spin moment with concentrations x↑and\nx↓, respectively.24,50For a fully relativistic formulation,\nwith spin-orbitcoupling included, this simplificationcan-\nnot be justified anymore and a proper average has to be\ntaken over all orientations.51As we do not perform DLM\ncalculationsbut use hereonly the DLM picture to repres-\nent MC data, this complication is ignored in the follow-\ning. Having the set of orientation vectors ˆ efdetermined\nby MC simulations the corresponding concentrations x↑\nandx↓can straight forwardly be fixed for each temper-\nature by the requirement:\n1\nNfNf/summationdisplay\nf=1ˆef=x↑ˆz+x↓(−ˆz), (16)\nwithx↑+x↓= 1. Using this simple scheme electronic\nstructure calculations have been performed for a binary\nalloy having collinear magnetization. The resulting re-\nduced magnetic moment MKKR−DLM(MC) (T) is shown in\nFig.3(bottom). As one notes, again the original MC\nresults are perfectly reproduced. This implies that when\ncalculating the projected reduced magnetic moment Mz\nthat is determined by the averaged Green function ∝an}bracketle{tG∝an}bracketri}ht\nthe transversal magnetization has hardly any impact.\nFig.3(middle) gives also experimental data for\ntheM(T).52While the experimental Curie-temperature\nTexp\nC= 1044 K52is rather well reproduced by the MC\nsimulations TMC\nC= 1082 K one notes that the MC-curve\nMMC(T) is well below the experimental curve. In partic-\nular,MMC(T) drops too fast with increasing Tin the\nlow temperature regime and does not show the T3/2-\nbehavior. The reason for this is that the MC simulations\ndo not properly account for the low-energy long-ranged\nspinwaveexcitationsresponsibleforthelow-temperature\nmagnetization variation. Performing ab-initio calcula-\ntions for the spin wave energies and using these data for\nthe calculation of M(T) much better agreement with ex-\nperiment can indeed be obtained in the low-temperature\nregime than with MC simulations.53\nAs the fitting scheme sketched above needs only thetemperature reduced magnetic moment M(T) as input\nit can be applied not only to MC data but also to ex-\nperimental data. Fig. 3shows that the mean field fit\nMMF(exp)(T) again perfectly fits the experimental re-\nduced magnetic moment curve Mexp(T). Based on this\ngood agreement this corresponding data set {ˆef,xf}has\nalso been used for the calculation of responsetensors (see\nbelow).\nAn additional much simpler scheme to simulate the\nexperimental Mexp(T) curve is to assume the individual\natomic moments to be distributed on a cone, i.e. with\nNθ= 1 and Nφ>>1.23In this case the opening angle\nθ(T) of the cone is chosen such to reproduce M(T). In\ncontrasttothestandardDLMpicture,thissimplescheme\nallows already to account for transversal components of\nthe magnetization. Corresponding results for response\ntensor calculations will be shown below.\nFinally, it should be stressed here that the various spin\nconfiguration models discussed above assume a rigid spin\nmoment, i.e. its magnitude does not change with temper-\nature nor with orientation. In contrast to this Ruban et\nal.54usealongitudinalspinfluctuation Hamiltonianwith\nthe corresponding parameters derived from ab-initio cal-\nculations. As a consequence, subsequent Monte Carlo\nsimulations based on this Hamiltonian account in par-\nticular for longitudinal fluctuations of the spin moments.\nA similar approach has been used by Drchal et al.55,56\nleading to good agreement with the results of Ruban et\nal. However, the scheme used in these calculations does\nnot supply in a straightforward manner the necessary\ninput for temperature dependent transport calculations.\nThis is different from the work of Staunton et al.57who\nperformed self-consistent relativistic DLM calculations\nwithout the restriction to a collinear spin configuration.\nThis approach in particular accounts in a self-consistent\nway for longitudinal spin fluctuations.\nE. Combined chemical and thermally induced\ndisorder\nThe various types of disorder discussed above may be\ncombined with each other as well as with chemical i.e.\nsubstitution disorder. In the most general case a pseudo-\ncomponent ( vft) is characterized by its chemical atomic\ntypet, the spin fluctuation fand lattice displacement\nv. Using the rigid muffin-tin and rigid spin approxim-\nations, the single-site t-matrix tloc\ntin the local frame is\nindependent from the orientation vector ˆ efand displace-\nment vector ∆ Rv, and coincides with ttfor the atomic\ntypet. With respect to the common global frame one\nhas accordingly the t-matrix:\ntvft=U(∆Rv)R(ˆef)ttR(ˆef)−1U(∆Rv)−1.(17)\nWith this the corresponding CPA equations are identical\nto Eqs. ( 8) to (10) with the index vreplaced by\nthe combined index ( vft). The corresponding pseudo-\nconcentration xvftcombines the concentration xtof the7\natomic type twith the probability for the orientation\nvector ˆefand displacement vector ∆ Rv.\nIII. COMPUTATIONAL DETAILS\nThe electronic structure of the investigated ferro-\nmagnets bcc-Fe and fcc-Ni, has been calculated self-\nconsistently using the spin-polarized relativistic KKR\n(SPR-KKR) band structure method.58,59For the ex-\nchangecorrelationpotential the parametrizationas given\nby Vosko et al.60has been used. The angular-momentum\ncutoff of lmax= 3 was used in the KKR multiple scatter-\ning expansion. The lattice parameters have been set to\nthe experimental values.\nIn a second step the exchange-coupling parameters\nJijhave been calculated using the so-called Lichten-\nstein formula.25Although the SCF-calculations have\nbeen done on a fully-relativistic level the anisotropy of\nthe exchange coupling due to the spin-orbit coupling has\nbeen neglected here. Also, the small influence of the\nmagneto-crystallineanisotropyfor the subsequent Monte\nCarlo (MC) simulations has been ignored, i.e. these have\nbeen based on a classical Heisenberg Hamiltonian. The\nMC simulations were done in a standard way using the\nMetropolis algorithm and periodic boundary conditions.\nThe theoretical Curie temperature TMC\nChas been de-\nduced from the maximum of the magnetic susceptibility.\nThe temperature dependent spin configuration ob-\ntained during a MC simulation has been used to con-\nstruct a set of orientations ˆ efand probabilities xfac-\ncording to the schemes MC* and MC described in sec-\ntionIIDto be used within subsequent SPR-KKR-CPA\ncalculations (see above). For the corresponding calcu-\nlation of the reduced magnetic moment the potential\nobtained from the SCF-calculation for the perfect fer-\nromagnetic state ( T= 0K) has been used. The calcu-\nlation for the electrical conductivity as well as the Gil-\nbertdampingparameterhasbeenperformedasdescribed\nelsewhere.41,61\nIV. RESULTS AND DISCUSSION\nA. Temperature dependent conductivity\nEq. (1) has been used together with the various\nschemes described above to calculate the temperature\ndependent longitudinal resistivity ρ(T) of the pure fer-\nromagnets Fe, Co and Ni. In this case obviously disorder\ndue to thermal displacements of the atoms as well as spin\nfluctuations contribute to the resistivity.\nTo give an impression on the impact of the thermal\ndisplacementsaloneFig. 5givesthe temperaturedepend-\nent resistivity ρ(T) of pure Cu (Θ Debye= 315 K) that\nis found in very good agreement with corresponding ex-\nperimental data.62This implies that the alloy analogy\nmodel that ignores any inelastic scattering events should0 100 200 300 400 500\nTemperature (K)01234ρxx (10-6Ω⋅cm)Expt\nTheory - alloy analogy\nTheory - LOVA\nCu\nFigure 5. Temperature dependent longitudinal resistivity of\nfcc-Cuρ(T) obtained by accounted for thermal vibrations as\ndescribed in section IIBcompared with corresponding ex-\nperimental data.62In addition results are shown based on\nthe LOVA (lowest order variational approximation) to the\nBoltzmann formalism.14\nin general lead to rather reliable results for the resistivity\ninduced by thermal displacements. Accordingly, com-\nparison with experiment should allow for magnetically\nordered systems to find out the most appropriate model\nfor spin fluctuations.\nFig.6(top) shows theoretical results for ρ(T) of bcc-\nFe due to thermal displacements ρv(T), spin fluctuations\ndescribed by the scheme MC ρMC(T) as well as the com-\nbination of the two influences ( ρv,MC(T)). First of all\none notes that ρv(T) is not influenced within the adop-\ntedmodelbytheCurietemperature TCbutisdetermined\nonly by the Debye temperature. ρMC(T), on the other\nhand, reaches saturation for TCas the spin disorder does\nnot increase anymore with increasing temperature in the\nparamagnetic regime. Fig. 6also shows that ρv(T) and\nρMC(T)arecomparableforlowtemperaturesbut ρMC(T)\nexceedsρv(T) more and more for higher temperatures.\nMost interestingly, however, the resistivity for the com-\nbined influence of thermal displacements and spin fluctu-\nationsρv,MC(T) does not coincide with the sum of ρv(T)\nandρMC(T) but exceeds the sum for low temperatures\nand lies below the sum when approaching TC.\nFig.6(bottom) shows the results of three differ-\nent calculations including the effect of spin fluctuations\nas a function of the temperature. The curve ρMC(T)\nis identical with that given in Fig. 6(top) based on\nMonte Carlo simulations. The curves ρDLM(MC) (T) and\nρcone(MC)(T) are based on a DLM- and cone-like repres-\nentation of the MC-results, respectively. For all three\ncases results are given including as well as ignoring the\nvertex corrections. As one notes the vertex corrections\nplay a negligible role for all three spin disorder models.\nThis is fully in line with the experience for the longitud-\ninal resistivity of disordered transition metal alloys: as\nlong as the the states at the Fermi level have domin-\nantly d-character the vertex corrections can be neglected\nin general. On the other hand, if the sp-character dom-8\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)vib\nfluct (MC)\nvib + fluct (MC)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)MC (VC)\nMC (NVC)\nDLM (VC)\nDLM (NVC)\ncone (VC)\ncone (NVC)\nFigure 6. Temperature dependent longitudinal resistivity of\nbcc-Feρ(T) obtained by accounted for thermal vibrations\nand spin fluctuations as described in section IIB. Top: ac-\ncounting for vibrations (vib, diamonds), spin fluctuations us-\ning scheme MC (fluct, squares) and both (vib+fluct, circles).\nBottom: accounting for spin fluctuations ˆ ef= ˆe(θf,φf) us-\ning the schemes: MC (squares) with 0 ≤θf≤π;0≤φf≤\n2π, DLM(MC) (triangles up) with θf1= 0,θf2=π, and\ncone(MC) (triangles down) θf=/angbracketleftθf/angbracketrightT;0≤φf≤2π. The\nfull and open symbols represent the results obtained with th e\nvertex corrections included (VC) and excluded (NV), respec t-\nively.\ninates inclusion of vertex corrections may alter the result\nin the order of 10 %.63,64\nComparing the DLM-result ρDLM(MC) (T) with\nρMC(T) one notes in contrast to the results for M(T)\nshown above (see Fig. 3(bottom)) quite an appreciable\ndeviation. This implies that the restricted collinear\nrepresentation of the spin configuration implied by the\nDLM-model introduces errors for the configurational\naverage that seem in general to be unacceptable, For\nthe Curie temperature and beyond in the paramagnetic\nregimeρDLM(MC) (T) andρMC(T) coincide, as it was\nshown formally before.20\nComparing finally ρcone(MC)(T) based on the conical\nrepresentationofthe MCspin configurationwith ρMC(T)\none notes that also this simplification leads to quite\nstrong deviations from the more reliable result. Never-\ntheless, one notes that ρDLM(MC) (T) agrees with ρMC(T)\nfor the Curie temperature and also accounts to some ex-\ntent for the impact of the transversal components of themagnetization.\nThe theoretical results for bcc-Fe (Θ Debye= 420 K)\nbased on the combined inclusion of the effects of thermal\ndisplacementsandspinfluctuationsusingtheMCscheme\n(ρv,MC(T)) are compared in Fig. 7(top) with experi-\nmental data ( ρexp(T)). For the Curie temperature ob-\n0 0.2 0.4 0.6 0.8 1 1.2 1.4\nT/TC020406080100120ρxx  (10-6Ω⋅cm)Expt: J. Bass and K.H. Fischer \nvib + fluct (MC)\nvib + fluct (exp)\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8\nT/TC01020304050ρxx  (10-6Ω⋅cm)Expt.: C.Y. Ho et al. (1983)\nvib\nvib (PM)\nfluct\nvib + fluct\nFigure 7. Top: Temperature dependent longitudinal res-\nistivity of bcc-Fe ρ(T) obtained by accounted for thermal\nvibrations and spin fluctuations using the scheme MC\n(vib+fluct(MC), squares) and a mean-field fit to the experi-\nmental temperature magnetic moment Mexp(vib+fluct(exp),\ndiamonds) compared with experimental data (circles).62Bot-\ntom: corresponding results for fcc-Ni. In addition results are\nshown accounting for thermal displacements (vib) only for\nthe ferromagnetic (FM) as well paramagnetic (PM) regime.\nExperimental data have been taken from Ref. 65.\nviously a very good agreement with experiment is found\nwhile for lower temperatures ρv,MC(T) exceeds ρexp(T).\nThis behavior correlates well with that of the temperat-\nure dependent reduced magnetic moment M(T) shown\nin Fig.3(middle). The too rapid decrease of MMC(T)\ncompared with experiment implies an essentially overes-\ntimated spin disorder at any temperature leading in turn\nto a too large resistivity ρv,MC(T). On the other hand,\nusing the temperature dependence of the experimental\nreducedmagneticmoment Mexp(T)tosetup thetemper-\nature dependent spin configuration as described above a\nvery satisfying agreement is found with the experimental\nresistivity data ρexp(T). Note also that above TCthe\ncalculated resistivity riches the saturation in contrast to\nthe experimental data where the continuing increase of9\nρexp(T) can be attributed to the longitudinal spin fluctu-\nations leading to a temperature dependent distribution\nof local magnetic moments on Fe atoms.54However, this\ncontribution was not taken into account because of re-\nstriction in present calculations using fixed value for the\nlocal reduced magnetic moments.\nFig.7(bottom) shows corresponding results for the\ntemperature dependent resistivity of fcc-Ni (Θ Debye=\n375 K). For the ferromagnetic (FM) regime that the\ntheoretical results are comparable in magnitude when\nonly thermal displacements ( ρv(T)) or spin fluctuations\n(ρMF(T)) are accounted for. In the later case the mean\nfieldw(T) has been fitted to the experimental M(T)-\ncurve. Taking both into account leads to a resistivity\n(ρv,MF(T)) that are well above the sum of the individual\ntermsρv(T) andρMF(T). Comparing ρv,MF(T) with ex-\nperimentaldata ρexp(T)ourfindingshowsthatthetheor-\netical results overshoots the experimental one the closer\none comes to the critical temperature. This is a clear\nindication that the assumption of a rigid spin moment\nis quite questionable as the resulting contribution to the\nresistivity due to spin fluctuations as much too small.\nIn fact the simulations of Ruban et al.54on the basis of\na longitudinal spin fluctuation Hamiltonian led on the\ncase of fcc-Ni to a strong diminishing of the averagelocal\nmagnetic moment when the critical temperature is ap-\nproachedfrom below (about 20% comparedto T= 0K).\nFor bcc-Fe, the change is much smaller (about 3 %) justi-\nfying on the case the assumption of a rigid spin moment.\nTaking the extreme point of view that the spin moment\nvanishescompletely abovethe criticaltemperature orthe\nparamagnetic (PM) regime only thermal displacements\nhave to be considered as a source for a finite resistivity.\nCorresponding results are shown in Fig. 7(bottom) to-\ngether with corresponding experimental data. The very\ngood agreementbetween both obviouslysuggeststhat re-\nmaining spin fluctuations above the critical temperature\nare of minor importance for the resistivity of fcc-Ni.\nB. Temperature dependent Gilbert damping\nparameter\nFig.8shows results for Gilbert damping parameter α\nof bcc-Fe obtained using different models for the spin\nfluctuations. All curves show the typical conductivity-\nlike behaviorfor low temperatures and the resistivity-like\nbehavior at high temperatures reflecting the change from\ndominating intra- to inter-band transitions.66The curve\ndenoted expt isbasedon aspin configurationtoted tothe\nexperimental Mexpt(T) data. Using the conical model to\nfitMexpt(T) as basis for the calculation of α(T) leads\nobviously to a rather good agreement with αM(expt)(T).\nHaving instead a DLM-like representation of Mexpt(T),\non the other hand, transverse spin components are sup-\npressed and noteworthy deviations from αM(expt)(T) are\nfound for the low temperature regime. Nevertheless, the\ndeviations are less pronounced than in the case of the0 200 400 600 800\nTemperature (K)02468α × 103fluct (MC)\nfluct (Expt)\nfluct (DLM)\nfluct (cone)\nFigure 8. Temperature dependent Gilbert damping α(T) for\nbcc-Fe, obtainedbyaccountedfor thermal vibrations andsp in\nfluctuations accounting for spin fluctuations using scheme\nMC (squares), DLM(MC) (triangles up), cone(MC) (triangles\ndown) and a MF fit to the experimental temperature reduced\nmagnetic moment (circles).\nlongitudinal resistivity (see Fig. 6(bottom)), where cor-\nresponding results are shown based on MMC(T) as a ref-\nerence. Obviously, the damping parameter αseems to\nbe less sensitive to the specific spin fluctuation model\nused than the resistivity. Finally, using the spin con-\nfiguration deduced from Monte Carlo simulations, i.e.\nbased on MMC(T) quite strong deviations for the result-\ningαM(MC)(T) fromαM(expt)(T) are found. As for the\nresistivity (see Fig. 6(bottom)) this seems to reflect the\ntoo fast drop of the reduced magnetic moment MMC(T)\nwith temperature in the low temperature regime com-\npared with temperature (see Fig. 3). As found before18\naccountingonly for thermal vibrations α(T) (Fig.6(bot-\ntom)) is found comparableto the casewhen only thermal\nspan fluctuations are allowed. Combing both thermal ef-\nfects does not lead to a curve that is just the sum of the\ntwoα(T) curves. As found for the conductivity (Fig. 6\n(top)) obviously the two thermal effects are not simply\nadditive. As Fig. 9(top) shows, the resulting damping\nparameter α(T) for bcc-Fe that accounts for thermal vi-\nbrationsaswellasspinfluctuationsisfoundinreasonable\ngood agreement with experimental data.18\nFig.9shows also corresponding results for the Gilbert\ndampingoffcc-Niasafunctionoftemperature. Account-\ning only for thermal spin fluctuations on the basis of the\nexperimental M(T)-curveleadsinthis casetocompletely\nunrealistic results while accounting only for thermal dis-\nplacements leads to results already in rather good agree-\nment with experiment. Taking finally both sources of\ndisorder into account again no simple additive behavior\nis found but the results are nearly unchanged compared\nto those based on the thermal displacements alone. This\nimplies that results for the Gilbert damping parameter\nof fcc-Ni hardly depend on the specific spin configura-\ntion model used but are much more governed by thermal\ndisplacements.10\n0 200 400 600 800\nTemperature (K)0246810α × 103vib\nvib + fluct (Expt)\nExpt 1\nExpt 2\n0 100 200 300 400 500\nTemperature (K)00.050.10.150.2αvib\nfluct (Expt)\nvib + fluct (Expt)\nExpt\nFigure 9. Top: Temperature dependent Gilbert damping\nα(T) for bcc-Fe, obtained byaccounted for thermal vibrations\nand spin fluctuations accounting for lattice vibrations onl y\n(circles) and lattice vibrations and spin fluctuations base d on\nmean-field fit to the experimental temperature reduced mag-\nnetic moment Mexpt(diamonds) compared with experimental\ndata (dashed and full lines).67,68Bottom: corresponding res-\nults for fcc-Ni. Experimental data have been taken from Ref.\n67.\nV. SUMMARY\nVarious schemes based on the alloy analogy that al-\nlow to include thermal effects when calculating responseproperties relevant in spintronics have been presented\nand discussed. Technical details of an implementation\nwithin the framework of the spin-polarized relativistic\nKKR-CPA band structure method have been outlined\nthat allow to deal with thermal vibrations as well as spin\nfluctuations. Various models to represent spin fluctu-\nations have been compared with each other concerning\ncorresponding results for the temperature dependence\nof the reduced magnetic moment M(T) as well as re-\nsponse quantities. It was found that response quantities\nare much more sensitive to the spin fluctuation model as\nthe reduced magnetic moment M(T). 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Grenoble Alpes INAC-SPINTEC F-38000 Grenoble France \n2CNRS INAC-SPINTEC F-38000 Grenoble France \n3CEA INAC-SPINTEC F-38000 Grenoble France \n4CNRS Institut Néel Grenoble France \n5King Abdullah University of Science and Technology (KAUST), Physical Science and \nEngineering Division, Thuwal 23955-6900, Saudi Arab ia \n  \n \n \n* To whom correspondence should be addressed.  \nE-mail: mihai.miron@cea.fr \n† Currently at NIST Boulder, Colorado, US  2 Structural symmetry breaking in magnetic materials is responsible for a variety of \noutstanding physical phenomena. Examples range from  the existence of multiferroics 1, to \ncurrent induced spin orbit torques (SOT) 2-7 and the formation of topological magnetic \nstructures 8-11 . In this letter we bring into light a novel effect  of the structural inversion \nasymmetry (SIA): a chiral damping mechanism. This p henomenon is evidenced by \nmeasuring the field driven domain wall (DW) motion in perpendicularly magnetized \nasymmetric Pt/Co/Pt trilayers. The difficulty in ev idencing the chiral damping is that the \nensuing DW dynamics exhibit identical spatial symme try to those expected from the \nDzyaloshinskii-Moriya interaction (DMI) 12-18. Despite this fundamental resemblance, the \ntwo scenarios are differentiated by their time reve rsal properties: while DMI is a \nconservative effect that can be modeled by an effec tive field, the chiral damping is purely \ndissipative and has no influence on the equilibrium  magnetic texture. When the DW \nmotion is modulated by an in-plane magnetic field, it reveals the structure of the internal \nfields experienced by the DWs, allowing to distingu ish the physical mechanism. The \nobservation of the chiral damping, not only enriche s the spectrum of physical phenomena \nengendered by the SIA, but since it can coexists wi th DMI it is essential for conceiving \nDW and skyrmion devices 19 .  \nThe role of the SIA on the magnetic properties of t he metallic ferromagnets has been \nthe focus of extensive research. Whether the SIA oc curs at an interface or in the bulk of a \nmaterial, it may induce non-collinear exchange inte ractions 8 (DMI). This gives rise to a rich \nvariety of magnetic phases, such as spin helices 9,10  and skyrmions 11  or to magnetic frustration 23. \nRecently, it was proposed that the DMI promotes the  existence of Néel DWs in materials where \nthe configuration otherwise favored would be Bloch 12 . In the context of current induced DW \nmotion 20-22 this is an important idea, as it may solve a long- standing issue of DWs moving in \nthe direction of the electric current and not into that of the electron flow, as one would expect  3 from spin transfer torque 12-14 . Moreover, the observation that the field-driven D W motion in \nPt/Co based multilayers is made asymmetric by in-pl ane magnetic fields was also linked to the \npossible occurrence of DMI 15,16 . \nUntil now, DMI was the only well established intera ction known to link the DW \nmagnetic texture to the SIA of the layer structure.  In this letter we propose a new mechanism, \nwhere the magnetic texture can sense the SIA as a d amping whose intensity depends on the DW \nchirality.  \nDWs in materials with perpendicular magnetic anisot ropy (PMA) are the perfect test-\nground for such phenomena: they are chiral objects and their motion is relatively easy to detect. \nFrom the perspective of the magnetization dynamics,  all relevant physical phenomena translate \ninto torques that control the magnetization accordi ng to the phenomenological Landau–\nLifshitz–Gilbert (LLG) equation 24. Their time reversal properties separate them in t wo \ncategories. They can, either derive from a form of energy and be invariant to time reversal \n(anisotropy, exchange), or they may be dissipative and break time reversal symmetry \n(damping). These two scenarios are exclusive, in th e sense that there is no other possibility. \nSimilarly, the physical phenomenon linking the DW m otion asymmetry to the material’s SIA \nmust fall into one of these categories. It can only  derive from chiral energy (DMI) or be a form \nof chiral dissipation.  \nThe effect of chiral energy has already been eviden ced experimentally 15 . The DMI \nimplies the existence of an internal field H DMI  setting the DW chirality (Figure 1b). As a \nconsequence, the core magnetization of up/down ( ↑↓) and down/up ( ↓↑) DWs will point in \nopposite directions. The field driven DW velocity c an sense this interaction if an external in-\nplane field (H ip ) is applied simultaneously with the perpendicular field (H z). ↑↓ and ↓↑ DWs \nmoving along H ip  will experience different effective fields (H ip -HDMI  and H ip +H DMI ) and \ntherefore will have different velocity. As long as the applied field does not alter the equilibrium  4 DW profile (by significantly tilting the domains) t he DW dynamics are correctly described by \na collective coordinate model. Since in this case H ip  and H DMI  enter the DW equation of motion \non equal footing 15 , DMI will bias the velocity dependence on H ip  by +H DMI  and -H DMI  \nrespectively (Figure 1b), making the motion asymmet ric.  \nThe chiral damping mechanism, although allowed by s ymmetry has never been \nobserved experimentally. Its chief attribute is tha t, unlike DMI, it causes asymmetric \ndisplacements without biasing the magnetic configur ation (Figure 1d). \nWe use wide field Kerr microscopy to probe the DW m otion in asymmetric \nPt 30Å/Co 6Å/Pt xÅ trilayers subjected simultaneously to H z and H ip . A pulse of H z nucleates a \nreversed domain.  A second pulse of H z accompanied by H ip  further propagates the circular \nDW. Figure 2a shows typical magnetic differential i mages corresponding to the DW \ndisplacement produced by H z and H ip . The reversed magnetization produces bright (or da rk) \ncontrast. Without H ip  the DWs move over equal distances in all direction s. The direction and \nthe magnitude of the asymmetry are controlled by H ip . This is qualitatively consistent with the \nexistence of the H DMI : depending on the type of DW, ( ↑↓) or (↓↑), Hip  either adds-up or \nsubtracts to H DMI , leading to different effective field for the two DWs. To verify this possibility, \nwe measured the DW velocity ( vDW ) vs. H ip  for a fixed H z. While H ip  can be applied continuously, \nas it provokes no DW motion, H z is applied in pulses of 400 ms. vDW  is extracted by dividing the \ndisplacement observed on the Kerr image by the puls e length.  \nFigure 2 shows the effect of H ip  on vDW . Besides the chiral effects emerging from SIA, \nHip  may also affect vDW  through phenomena unrelated to SIA (such as the va riations of the \neffective anisotropy). To separate the chiral effec ts we decompose vDW  into a symmetric ( S) and \nanti-symmetric ( A) component with respect to H ip : ) 21 (ASv+=↑↓  and ) 21 (ASv−=↓↑ , \nwhere ()2↓↑ ↑↓ −=vvS  and ()()↓↑ ↑↓ ↓↑ ↑↓ + −⋅= vvvvA2 . While S can be affected by non-SIA \nphenomena, A is a robust indicator of chiral effects, as by sym metry, only SIA related  5 phenomena can create it. Since A dictates the ratio \nAAvv−+=↓↑ ↑↓ 22, it is also an indicator of the \nasymmetry of DW motion the along the direction of H ip  (Figure 2a). \nThe variation of A vs. H ip   has two distinctive features: it varies continuou sly up to a \nsaturation field (H sat =40mT), and then saturates (Figure 2a,2b), meaning that below H sat  the \nvelocities of the two DWs vary differently, while a bove H sat  they vary identically to each other. \nThis indicates that the parameter responsible for ↓↑ ↑↓ ≠vv must also saturate at H sat . In our \nsamples, the only micromagnetic parameter that may vary monotonically up to 40 mT and then \nsaturate, is the DW core magnetization. Due to the large uniaxial anisotropy (H k=500 mT) 25, \nthe domain tilting affecting the DW internal struct ure and the effective DW width, becomes \nsignificant at much higher fields. \n Hsat  must then designate the field value for which both  DWs become magnetized along \nHip , implying that H DMI  is smaller than H sat . If DMI imposes homochiral Néel DWs, H ip  acts \nonly on the magnetization of one of the two DWs, as  the other one is already aligned  (Figure \n1a). Therefore, vDW  should display a feature at H sat  for only one of the DWs (Figure 1c). On the \ncontrary, we observe (Figure 2c,S5) that vDW  vs. Hip  changes slope close to H sat  for both DWs. \nThe fact that H sat  is physically relevant for both DWs excludes the e xistence of homochiral Néel \nDWs in our samples. It also sets a ceiling value fo r HDMI  within the precision of H sat  \ndetermination (±10mT). \nAs explained earlier, if A is produced by a small DMI (that we cannot rule ou t \ncompletely), the velocity curves for the two DWs sh ould have the same shape and only differ \nthrough a horizontal offset given by the value of t he H DMI  bias field (Figure 1b). By shifting the \ntwo curves in opposite directions by H DMI , they should coincide. Clearly, the measured vDW  do \nnot follow this scenario: it is impossible to overl ap the curves by any amount of lateral shifting \n(Figure 2c).  6 Based on the variations of vDW  induced by H ip  at constant H z, we established that our \nsamples support Bloch DWs and that the asymmetry me chanism evidenced in samples with \nNéel DWs 15,16  does not apply here. Now we switch perspectives an d study the DW motion as a \nfunction of H z at fixed H ip . To separate the effect of H ip  on the chiral energy and chiral damping, \nwe rely on the well-known creep scaling law, descri bing the DW velocity at very small driving \nfields:  \n\n\n\n\n⋅ −⋅=−4 / 14 / 1\n0exp z\nBpc\ncreep HTkHUvv         (1) \n cU - height of the pinning barrier; pH - pinning field;  \nSince the prefactor 31 \n⋅⋅=TkUCf dv\nBcexp 0 0 0 (1≈C is an empirical constant) also \ndepends exponentially on the pinning barrier, it is  more convenient to write the creep scaling \nlaw 15,26 as \n( ) \n−⋅−⋅=− − 4 / 1 4 / 1\n0 0exp c z\nBDW HHT kEf d v         (2) \nand group all the terms related to energy into a si ngle exponential (4 / 1\npcHUE= ). 4CHHp\nc=is \nthe critical field that determines the limit of the  creep regime 31. Since the dissipation does not \ncontribute to the exponent, the damping can only af fect the prefactor 25-26 \n0 0f d31 (d0 – disorder \ncorrelation length; f0 – the attempt frequency). \nAs long as the creep scaling law applies, the plot of 4 / 1.)ln( −\nz DW Hvs v allows to analyze \nseparately the effect of H ip  on the prefactor and the exponent, as they appear as the intercept \nand the slope of the linear dependence (Figure S2).  Figure 3a,b shows typical results for two \nsamples with different capping thickness (Pt 15.6Å and Pt 30Å). The linear dependence specific for  7 the creep regime is observed for the two samples an d both DW polarities. To evidence the \nsymmetric and anti-symmetric effects of H ip  we consider  \n ()()↓↑ ↑↓ − =mT mT \ncreep v v A80 80 ln ln        (3) \n()() [ ]()mT mT mT \ncreep v v v S0 80 80 ln ln ln 21− + =↓↑ ↑↓       (4) \n According to (2) they can be written \n( )\n\n\n −− −\n\n\n=↓↑ ↑↓ \n− −\n↓↑ ↑↓ \nT kE EH H\nf df dA\nBmT mT \nc z mT mT \ncreep 80 80 \n4 / 1 4 / 1\n80 \n0 080 \n0 0ln      (5) \n( )( )\n\n\n\n − +− −\n\n\n⋅=↓↑ ↑↓ \n− −↓↑ ↑↓ \nTkE E EHHf df d f dS\nBmT mT mT \nc z mT mT mT \ncreep 0 80 80 \n21\n4 / 1 4 / 1\n0\n0 080 \n0 080 \n0 0ln \n (6) \nRemarkably, Acreep  shows negligible dependence on 4 / 1−\nzH (inset of Figure 3a,b), \nimplying that the bubble asymmetry is not due to ch iral energy ( ↓↑ ↑↓ =mT mT E E80 80 ) but to chiral \ndissipation (↓↑ ↑↓ ≠mT mT fdfd80 \n0 080 \n0 0 ). At the same time, Screep  decreases linearly with 4 / 1−\nzH, \nsignaling the existence of non-chiral energy contri butions related to H ip  (\nmT mT mT E E E0 80 80 ≠ =↓↑ ↑↓ ).  \nThough it has been predicted 27  that friction affects the creep of an elastic memb rane via \nthe velocity prefactor, to our knowledge, in the ca se of DWs no microscopic mechanism has \nyet been proposed. For this particular case, the ph ysical parameter most likely to depend on \ndamping is the attempt frequency f0. We verify this possibility using micromagnetic si mulations \nby calculating the DW depinning time from a single defect for different damping values (Figure \n3c,d). Indeed, we observe a strong dependence of th e release time on damping. Since the barrier \nheight is kept constant in the simulations, it is o nly f0 that can be responsible for this variation. \nWe find that f0 is proportional to the inverse of the damping (DW m obility). This trend (\nα1=DW v ) is further confirmed for a wider range of applied  fields and damping values using a \n1D model 25.  8 The above mechanism can account for the chiral DW d ynamics if the damping \ncoefficient depends on the orientation of the DW’s core with respect to the magnetization \ngradient: ()Zip Cmm∇⋅∝rαα . However, this kind of damping cannot exist by its elf in a real \nphysical system since the dissipation may become ne gative if the chiral contribution exceeds \nthe intrinsic damping. To prevent this unphysical b ehavior, the chiral damping must include at \nleast a second component, which always offsets its value in the positive range. For example, its \nmathematical expression could be ()Zip Cmm∇⋅+∝rααα0 , where Cαα>0. Based on this \nexpression, the amplitude of the chiral damping con tribution can be extracted from the value of \nA. Above H sat , where m ip  saturates, \ncc\nvv\nαααα\n−+=\n↓↑ ↑↓ \n00 and 112 /+−=\n↓↑ ↑↓ ↓↑ ↑↓ \nvvvvA  we obtain \n02 /ααcA= \nFor the sample exhibiting the largest asymmetry (Fi gure S1), the amplitude of the chiral \ndamping reaches approximately 50% of the constant p art. \nTo illustrate the effect of the chiral damping on t he DW dynamics we use a numerical  \ncollective coordinate model. Since such models do n ot include explicitly the magnetization \ngradients, we write the chiral damping using a simp lified form:  φααα cos 0C+= . Here the \nvalue of the magnetization gradient is considered c onstant and is implicitly included in the \nvalues of α0 and αC. The orientation of the DW magnetization is descri bed by the azimuthal \nangle φ. Figure 2e shows the computed vDW  as a function of H ip  at 0K and without pinning. The \nsalient feature is that the velocities of both DWs saturate simultaneously, at the field value \nwhere their core magnetization saturates. In order to compare the experimental results with the \nsimulations, the symmetric component S of the measured velocity must be removed, as it is  not \nproduced by the chiral damping. For this purpose we  normalize v↓↑ and  v↑↓ to the average \nvelocity along x and y,  the in-plane directions. The normalized DW velocit y (Figure 2d, Figure \nS5) saturates at H sat , in perfect agreement with the simulations.  9 At this stage, we do not understand why, contrary t o previous studies, the effect of the \nchiral damping in our samples is larger than the ef fect of DMI. We infer that these two \nphenomena should co-exist, but their relative stren gth may be strongly dependent on the precise \ncrystallographic structure. An independent hint of the co-existence of chiral damping and DMI \nwas provided by the magnon dynamics in Fe/W bi-laye rs, where it was observed that the not \nonly the dispersion relation was shifted due to DMI , but also that magnon lifetime depends on \nthe chirality 28 . \nWhile theoretical work is definitely required to un derstand the origin of the chiral \ndamping, we would like to speculate and point towar ds a plausible explanation: both \ntheoretically and experimentally it was shown that the spin orbit torques have two components: \na field like 2,3  (conservative) part and a damping like (dissipativ e) part 4,5,6 . Theoretically the \nDMI was linked to the conservative (field like) par t 29,30 , as they share a common microscopic \norigin. In a similar way, the chiral damping report ed here, might be reminiscent of its dissipative \ncomponent. \nFrom a practical perspective, the DW motion asymmet ry can be useful for memory 22 or \nlogic devices 32 requiring unidirectional DW movement. The applicat ion of an alternating H z \ncauses the domain to enlarge and then shrink to its  initial position. When H ip  is applied along \nwith H z, the DW motion will become asymmetric. If H ip  changes sign together with H z, the \nasymmetry of the domain growth will be opposite to that of its contraction. Consequently, the \ndomain will not come back to its initial position b ut will be shifted along the axis of the in-\nplane field 15 . The relative sign of the two field components con trols the direction. Compared to \ncurrent induced DW motion where trains of DWs are p ushed along 1D wires, the phenomenon \nthat we describe affords a distinctive ability of s hifting 2D magnetic domains in any directions \nof a full sheet magnetic layer (Figure 4c; suppleme ntary movie).  10  In conclusion, the quantitative analysis of field i nduced DW motion has revealed the \nexistence of a chiral damping term whose effects on  the DW dynamics have the same spatial \nsymmetry as those expected from the DMI. Understand ing and controlling the chiral damping \nis essential for the designing spintronics devices that rely on chiral magnetic textures, such as \nDW or skyrmion racetracks.   11   \nFigure 1.  Graphic illustration of the asymmetric DW dynamics  in PMA materials. a. A DW \n(dotted line) separates two oppositely magnetized d omains (grey and white). The DMI can be \nmodeled by an effective field that only exists insi de the DW and that always rotates the DW \nmagnetization to point from the domain pointing up to the one pointing down. If H ip  is applied \nalong x, the two DWs standing orthogonal to its direction will react differently: while one \nremains magnetized in the same direction (red), the  other will reverse its chirality (blue).  \nb Without SIA and hence with no DMI to differentiate  the two DWs, H ip  will have the same \neffect on the field driven velocity of the two DWs that move along x (black dashed lines). If \npresent, the H DMI  bias field will shift laterally the velocity curve s for the two DWs (red and blue \nlines). An important consequence is that any featur e of the initial velocity curve allowing to \nidentify the DW transformation, such as a change of  slope when its magnetization saturates, \nwill translate laterally, and always manifest at th e same velocity ( vsat ). c.  If no DMI is present, \nthe DWs have Bloch structure; the structural change s produced by H ip  will be identical for the \ntwo DWs. d. The chiral damping does not shift laterally the in itial “non-SIA” velocity curve \n(black dotted line), but creates an asymmetric vert ical correction.  In this case, the DW \ntransformation from Bloch to Néel, will not occur a t the same velocity, but at the same field \nvalue (H sat ). \n 12   \nFigure 2. DW velocity modulation by H ip  in Pt/Co/Pt layers. a.  Kerr microscopy images of \nfield induced DW displacements. Dark contrast corre sponds to the growth of a “down” domain; \nBright contrast corresponds to an “up” domain. With out in-plane field the DW motion is \nisotropic. The in-plane field (white arrows) makes the bubble growth asymmetric: in the field \ndirection for an “up” domain; opposite to the field  direction for a “down” domain. It appears \nthat the asymmetry depends on the direction of the in-plane component of the DW’s \nmagnetization ( ip mr) with respect to the gradient of its vertical comp onent ( Zm∇), defining a \nchirality. b. The anti-symmetric component of the DW velocity (P t 30 /Co 6/Pt 15.6 sample) varies \nmonotonically and then saturates at a characteristi c field value (H sat ). Since any deviations from \nthe condition: v↓↑ (H ip ) = v↑↓ (–Hip ), are caused by noise or measurement artifacts, to  improve \nthe signal/noise ratio we plot A = ½ ( Araw (H ip ) – Araw (–Hip )), instead of Araw  = 2( v↓↑ – v↑↓)/( v↓↑ \n+ v↑↓) (Figure S5) c. The measured DW velocity exhibits changes in the s lope at H sat  for both \nDWs (red and blue) and both directions of H ip . Note that despite the different overall shape of \nthe DW velocity dependence for the two samples, bot h measurements exhibit a change of slope \n 13  at H sat  d. the velocity of the left and right moving DWs norm alized to the average velocity along \nthe in-plane directions ( x and y) show that while one of the DWs is accelerated the  other slows \ndown. Both these relative variations saturate at H sat . e. DW velocity calculated by a collective \ncoordinate model using a chiral damping defined as XCm⋅+∝ααα0 . The red and black \ncurves correspond to the ↑↓ and the ↓↑ DWs. The values were used in the calculation are:  \nα0=0.6 αc=0.3, ΔDW=5nm, H dip =35mT, H z=0.01mT.  14   \nFigure 3. Effect of chiral damping on thermally activated D W motion a,b.  Creep scaling law \nfor v↓↑ and v↑↓ for the Pt 30 Co 6Pt 15.6 and Pt 30 Co 6Pt 30  samples. The measurements were performed \nat Hip =0 (black) and H ip =80 mT (red and blue). The grey horizontal line is the limit of the creep \nregime, where we expect the transition to the flow (Figure S2). The rectangular black square at \nthe intercept of the linear fit with this limiting region is an approximate indicator of H c-1/4 . In \nthe inset we plot Screep  (empty symbols) and Acreep . (full symbols) c. Design of the \nmicromagnetic model system. The first picture from left to right is a differential Kerr image of \na DW propagated by H z showing the typical DW shape. The second one is a schematic \nrepresentation of the localized pinning that create s the rippled DW shape evidenced by Kerr \nimaging. The third picture is a micromagnetic confi guration depicting a pinned DW in a 100 \nnm wide nanowire pinned to a localized defect; the dark rectangle designates an area with 50% \nsmaller anisotropy that attracts the DW d. The simulated depinning probability (60 independen t \nevents) for two damping values (0.5 and 0.25) at Hz =18mT. The de-pinning field at 0K is 30mT. \nThe inset shows that the curves overlap when the ti mescale is normalized by the damping \nconstant. This indicates that f0 is proportional to the inverse of the damping.  \n 15   \nFigure 4 . Bubble shifting by magnetic fields. Sequence of d irect Kerr images showing the \nbubble shifting in the Pt 30 Co 6Pt 18,7  sample by combined in-plane and perpendicular fiel ds               \n(H ip = 40mT and H z = 11mT) The relative direction of the in-plane and perpendicular magnetic \nfield is indicated on the images. The same experime ntal parameters are used in the \nsupplementary movie. \n \n 16  References: \n1.  Khomskii, D. Classifying multiferroics: Mechanisms and effects Physics  2, 20 (2009)   \n2.  Chernyshov, A. et al . Evidence for reversible control of magnetization in a ferromagnetic \nmaterial by means of spin–orbit magnetic field. Nature Phys.  5, 656–659 (2009). \n3.  Miron, I. M. et al . 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Creep and depinning in disordered media Phys. \nRev. B 62 , 6241 (2000) \n28.  Zakeri, K., Zhang, Y., Chuang, T. H., & Kirschner, J.. “Magnon lifetimes on the Fe (110) \nsurface: The role of spin-orbit coupling” Physical review letters,  108(19), 197205. (2012) \n29.  Freimuth F., Blugel S., & Mokrousov Y. Berry phase theory of Dzyaloshinskii-Moriya \ninteraction and spin-orbit torques  arXiv:1308.5983  \n30.  Kim, K. W., Lee, H. W., Lee, K. J., & Stiles, M. D.  Chirality from interfacial spin-orbit \ncoupling effects in magnetic bilayers. Phys. Rev. Lett.  111 (21), 216601 (2013) \n31.  Gorchon, J., et al. \"Pinning-Dependent Field-D riven Domain Wall Dynamics and Thermal \nScaling in an Ultrathin Pt/Co/Pt Magnetic Film.\" Phys. Rev. Lett. 113.2  027205 (2014) \n32. Allwood, D. A. et al . Magnetic domain-wall logic. Science  309 , 1688–1692 (2005). \n \nAcknowledgements \nWe thank H. W. Lee, K-J Lee and T. Giamarchi for he lpful discussions as well as A Thiaville, S Pizzini , and J \nVogel for critically reading the manuscript and dis cussing the results. This work was partially suppor ted by the \nANR (11 BS10 008, ESPERADO) project and European Co mmission under the Seventh Framework Programme \n(GA 318144, SPOT) and (GA-2012-322369, Sport for Me mory). AM has been supported by King Abdullah \nUniversity of Science and Technology. \n \nCompeting financial interests \n      The authors declare no competing financial in terests \n   19  SUPPLEMENTARY INFORMATION \n \nS1. Thickness dependence of DW motion asymmetry and  interface anisotropy \nS2. Prefactor of the DW creep scaling law \nS3. Influence of damping on the DW velocity in the thermally activated regime \na)  Chiral damping and thermal fluctuations \nb)  Depinning time vs. Gilbert damping \nc)  Velocity dependence on damping in the thermally act ivated regime \nS4. Asymmetry dependence on DW velocity.   \nS5. Link between the asymmetry and m ip  variation \nS1. Thickness dependence of DW motion asymmetry and  interface \nanisotropy.  \nTo probe the dependence of this phenomenon on the m aterial properties, we have \ncompared the asymmetries measured on several sample s having different thickness of the Pt \nupper layer (15Å–70Å). We chose this approach, to e nsure that neither the magnetic properties \nof the Co layer nor the interface asymmetry 1,2,3 , known to be essential for the DMI, are affected. \nAll the samples exhibit the same qualitative behavi or: A varies up to H sat , and fully saturates \nabove. Surprisingly, A decreases as the top Pt is made thicker. The possi bility to experimentally \nturn off A affords another independent way of probing its ori gin. The DMI creates an \nasymmetry by inducing a bias field that shifts the velocity curves towards positive H ip  for one \nDW and towards negative H ip  for the other (Figure 1c). Consequently, the satur ation value of \nA should be proportional to the bias field (H DMI ). And since H sat  is the experimental indicator \nof H DMI , the amplitude of A should be related to H sat . Experimentally we do not observe such a \nconnection: the variations of A are not associated to changes of H sat  (Figure S1), thus showing \nthat A does not originate from H DMI .  \nThe uniaxial anisotropy fields (H k) were determined from measurements of anomalous \nHall effect for all the samples. The results plotte d in figure S1 shows a reduction of the \nanisotropy fields from the sample with the thinnest  (15.6Å) Pt capping to the thickest (70Å).  20  The variation of the anisotropy field reflects a ch ange of the interfacial anisotropy. While H k \nvaries by 30% (0.55 T to 0.38 T) after correcting f or the layer’s demagnetizing field (µ 0Ms=1.3 \nT), we calculate a maximum variation of the anisotr opy constant of 10%.   \nThough the relative change of the interfacial aniso tropy is small, it is nevertheless \nsurprising to observe such a variation for Pt cappi ng layers exceeding several nm. A plausible \nhypothesis is that the heating of the sample during  deposition may influence its crystallographic \nstructure in the vicinity to the interface. It is w ell known that high temperature annealing \nenhances the Pt and Co intermixing. \n \nFigure S1 . a.  Asymmetry curves extracted from Kerr microscopy im ages for Pt \nthickness ranging from 15 Å to 70 Å. b.  When normalized to 1 all the curves overlap, \nindicating that H sat  is the same for all the samples. The inset shows t he dependence of \nthe normalizing factor (saturation value of the asy mmetry) with the thickness of the Pt \ncapping. c.  Measurement of the anisotropy fields of all the sa mples used in the \nexperiment. d.  Asymmetry of the annealed sample compared to the a s deposited ones. If \nthe asymmetry value would be correlated to the anis otropy field the asymmetry of the \nannealed sample (red) would be comprised in-between  the asymmetries measured for the \nsamples with 30Å and 56Å of Pt capping. \n 21  In order to corroborate this hypothesis we have gro wn several series of the following \nsamples: Pt 30 Co 6Pt 20, Pt 30 Co 6Pt 60 and Pt 30 Co 6Pt 3520 . While the Pt 60  layer of the second was \ndeposited continuously, the Pt capping of the third  was deposited in three steps consisting of \nPt 20  separated by 30 seconds pauses, to allow cooling. N.B.  The changes of H k correspond to \nsmall variations of the interfacial anisotropy cons tant, and therefore are sensitive to small \nvariations of the growth conditions. \nDespite the variations from series to series, we co nsistently find that             \nHk(Pt 30 Co 6Pt 20) > Hk(Pt 30 Co 6Pt 3520 )+50mT > H k(Pt 30 Co 6Pt 60)+100mT, in agreement with the \ndifferent heating that the samples might have suffe red. \n To verify experimentally that the heat-induced cha nges in the anisotropy do not affect \nthe asymmetry we annealed the tPt =18.7Å sample at 200°C in vacuum for 30 minutes, an d \nmeasured again its H k as well as the DW motion asymmetry. Indeed, we obs erve that the \nannealed sample has a smaller anisotropy field (the  same as the sample with tPt =30Å), consistent \nwith the Pt-Co inter-diffusing, while the asymmetry  remains almost unchanged (Figure S1 d). \nThe only consequence of the anisotropy variation is  that the DW width might vary from sample \nto sample. However, since it depends on the square root of the anisotropy field, its variations \nwill not exceed 10%. Consequently, the DW demagneti zing field stabilizing the Bloch DW \nstructure will vary by the same ratio. In the absen ce of DMI, H sat  is given by the DW \ndemagnetizing field. The fact that we do not detect  any significant sample-to-sample variation \nof Hsat  is consistent with this scenario. \nS2. The prefactor of the DW creep scaling law \nRecently, Gorchon et al. 7 evidenced experimentally an exponential dependence  of the \ncreep velocity prefactor on temperature. They found  that: \n \n⋅⋅=TkUCfdv\nBcexp 0 0 0  \n0 0f d, is the product of the pinning potential correlati on length ( d0) and the depinning \nattempt frequency ( f0). 1≈C is an empirical constant. Inserting this formula in to the creep \nscaling law leads to: \n  ( )\n\n\n\n\n−⋅ −⋅=− − 4 / 1 4 / 14 / 1\n0 0exp c z\nBpc\nDW HHTkHUfdv , where  4CHHp\nc=  \nBy rewriting the creep law under this form all the energy contributions are removed \nfrom the velocity prefactor; they are included in H c-1/4 , which is an offset of the abscissa.   22  Figure S2 represents schematically the changes of t he creep scaling law produced by \nvariations of the energy barrier and of the attempt  frequency. Changes of the pinning barrier \nhave two effects. On one hand, the slope is modifie d, but the curve is also shifted laterally, due \nto the variation of the pinning field (related to t he energy barrier). On the contrary, the attempt \nfrequency does not produce any variation of the slo pe; it only shifts the curve vertically. Our \nexperimental data (Figure 3 in the main text) does not exhibit any significant change of slope \nwhen H ip  is reversed (from -80mT to 80mT); we only observe a shift. We deduce that this must \nbe a vertical shift (associated to f0), since a lateral shift (associated to the pinning  field) would \nbe accompanied by a change of slope. The counter ex ample can be seen when comparing the \ncurve at H ip =0mT to the curves obtained at 80mT. Their differen ce in slope is associated to a \nchange of the pinning field (Figure 3 in the main t ext). \n \nFigure S2 . Influence of energy barrier and attempt frequency  on the DW creep scaling \nlaw. a.  The variations of the pinning barrier lead to a ch ange of slope (red dotted line), \nbut also to a lateral shift due to a change of the pinning field. The horizontal gray line \nrepresents the approximate transition from creep to  flow. The black square is the \napproximate H c-1/4 b. The variation of the attempt frequency leads only to a vertical shift \nof the curve. c.  Hc-1/4  and d0 f0 can be roughly estimated from the intercept of the  creep \nvelocity curves corresponding to different pinning barriers. \n \nOrder of magnitude of the attempt frequency.   \nBesides the measurements of Gorchon et al. 7 the order of magnitude of d0 f0 can be \nextracted also from previous measurements of DW cre ep in Pt/Co/Pt samples with varying Co \nthickness 6. In this case it is the change of thickness that l eads to changes of the effective \nparameters (exchange, anisotropy, pinning…) affecti ng the energy landscape. Figure 2d of \nsupplementary reference 6, exhibiting the DW veloci ty in the 4 / 1.)ln( −\nz DW Hvs v format, shows \nthat all the curves corresponding to the different samples converge close to a small area. This \n 23  area allows estimating the range of H c-1/4  and d0 f0 values for the series of samples. Since our \nsamples are nominally identical, and are made under  the same conditions in the same sputtering \nchamber as these previous studies 6,7 , we use their estimate for the order of magnitude of 0 0f d. \nIndependent estimates from these two studies coinci de to smfd /10 0 0≈. Considering this \nvalue, we estimate that the deviations from the cre ep regime should occur when 5 . 2)ln( ≈DW v . \nBy considering that the disorder correlation length  should be larger than the typical grain size \n(20 nm), the order of magnitude of the attempt freq uency could be anywhere between 10MHz \nand 1GHz.   \nS3. Influence of damping on the DW velocity in the thermally activated \nregime \nThermally activated DW motion consists of relativel y fast DW displacements \ninterrupted by pinning and de-pinning events. If th e driving field is sufficiently small compared \nto the pinning field, the DW spends most of the tim e trapped at pinning sites, trying to de-pin. \nWhen the temperature fluctuations finally free it, the DW moves to the next pinning center. In \nthis motion regime the experimentally determined DW  velocity is much smaller than the steady \nDW velocity (which is effective in-between pinning events); it is merely a measure of the \naverage de-pinning time. For this reason, instead o f trying to assess the complete influence of \ndamping on the thermally activated DW velocity we w ill first focus on the DW depinning. \nThe thermally activated de-pinning of a DW depends on two ingredients: the usually \ndominant one is the exponential dependence on the h eight of the energy barrier in units of \nthermal energy. The other, often neglected, is the pre-factor of this exponential, given by the \nattempt frequency ( \u0001=\u0003\u0004∙exp\t\n∆\f/\u000e\u000f\u0010\u0011). While the energy barrier does not depend on the \ndamping value, the attempt frequency may be altered  by variations of the damping. We analyze \nthis possibility numerically. \n \na) Chiral damping and thermal fluctuations \nThe effect of temperature along with the chiral dam ping proves to be impossible to include in \nthe numerical modeling. We find that the temperatur e alone induces DW displacements. This \nis a direct consequence of a combination of two fac tors:  \n1) the random thermal field is created using a pseu do random number generator;  \n2) the chiral damping makes the DW behave as a ratc het.  24   A ratchet is an object having an asymmetric respon se to excitations 4,5 . When subjected \nto periodic excitations a ratchet produces a net di rectional motion (rotation, displacement, etc). \nIn the case of DW motion, the asymmetric response i s given by the chiral damping. And since \nthe temperature fluctuations are generated using a pseudo random number generator whose \nautocorrelation is not null, it includes periodic c omponents that inevitably create an artificial \ndisplacement.  \nDespite this difficulty, it is still possible to te st the effect of damping in the thermally \nactivated DW motion, using an alternative approach.  As long as m ip  is established by the \ncompetition between H ip  and the DW’s demagnetizing field, the only variabl e determining the \ndamping value is H ip . Therefore, instead of computing the velocity depe ndence on H ip  we can \nuse the damping value as the free parameter. We use  a constant damping (independent of m ip ) \nand vary its value. These two methods are equivalen t as long as H z is sufficiently small, not to \ninduce any supplementary DW deformation and m ip  is determined only by the balance between \nHip  and the DW’s demagnetizing field. This approximati on is valid in the case of very slow \nthermally activated DW motion (corresponding to our  experimental conditions).  \n \nb)  Depinning time vs. Gilbert damping \nIn the micromagnetic simulations (Figure 3d of the main text) we observed a net \ndependence of the depinning time on the damping coe fficient.  Since the energy barrier for the \ntwo simulations is the same, this difference may on ly originate in the DW’s attempt rate. \nFurthermore, the fact that the depinning time is pr oportional to the damping value shows that \nthe DW mobility plays a significant role in the mot ion of the DW inside the pinning center. The \nDW depinning is a probabilistic process whose full characterization requires averaging over a \nstatistically significant amount of events. Consequ ently, micromagnetic modeling, being \nrelatively time consuming, is not the best-suited t ool for the extensive study of the thermally \nactivated DW motion. In order to further analyze th e influence of the damping coefficient on \nthe DW depinning we use a numerical collective coor dinate model. \n \nc) Velocity dependence on damping in the thermally activated regime \nWhile 2D micromagnetics allow including realistical ly the partial pinning of the DW, \nby definition in the 1D models the entire DW is enc losed in the pinning potential. As a \nconsequence, when projecting the 2D DW dynamics ont o a single dimension, the effective \npinning potential needs to be adapted accordingly. The simplest way is to use effective pinning  25  centers that are much wider than the DW, such that the DW will have to shift far from the \nenergy minimum before actually applying pressure on  the pinning potential. \n \nFigure S3 . DW velocity in the 1D model for 3 different dampi ng values. Each velocity \npoint is obtained after averaging 30 displacements produced by 500ns long pulses a.  DW \nvelocity when the width of the pinning potential is  10 times larger than the size of the DW \n(5nm). The inset shows the velocity curves normaliz ed by the inverse of the damping \nvalue. b.  The velocity calculated for a potential half the w idth of the DW (5nm). In this \ncase, the DW velocity is independent of the damping  value.  \nIn order to probe the influence of the damping vari ation on the thermally activated DW \nmotion, we calculate the velocity (Figure S3) as a function of the perpendicular magnetic field, \nfor several damping values. The DW parameters used in the simulations are ΔDW=5nm, \nHdip =40mT. The effective pinning field is modeled by a sinusoidal function with a period of \n50nm, (10 times larger than the DW width) with ampl itude of 3  mT. This form of pinning is \nchosen only for ease of implementing in the numeric al model. Though it is not realistic for the \nquantitative calculation of the DW velocity, since it allows modeling the repeated pinning and \nde-pinning of the DW, it is sufficient to evidence the impact of damping on the DW velocity. \nWe observe that the DW velocity depends strongly on  the damping value (0.2, 0.4, 0.6). \nMoreover, by normalizing all the velocity curves by  the inverse of the damping (proportional \nto the DW mobility) they overlap (figure S2a). This  indicates that the DW velocity depends on \nthe intrinsic DW mobility. On the contrary, when tr apping the DW in a narrow potential well \nwith the same barrier height (period of 2.5 nm and amplitude of pinning field of 60 mT) the \nvelocity curves completely loose the damping depend ence (figure S2b). \n \n d) Attempt frequency in the 1D model \nTo identify the physical parameter responsible for the damping dependence of the DW \nvelocity, we analyze the motion of the DW driven on ly by temperature (without H z) inside the \n 26  pinning potential. To estimate the DW attempt frequ ency we compute the frequency spectrum \n(by FFT analysis) of the DW position. We observe th at the attempt frequency is not described \nby a well-defined peak, but rather by a cut-off val ue.  \n \n \n \n \n \n \n \n \nFigure S4 . Fourier analysis of a pinned DW with Hz=0. The FF Ts were obtained by \naveraging 100 independent evolutions with a duratio n of 500 ns each a.  The case of a \nwide pinning potential. The inset shows that by nor malizing the frequency by the domain \nwall mobility (inverse of the damping value), the c urves overlap.  b.  For a narrow pinning \npotential the FFT exhibits a peak whose position is  almost independent of damping.  \nWhen normalizing the different FFTs to the inverse of the damping (DW mobility) the \nspectra overlap. This proves that the characteristi c attempt frequency of the DW (cut-off \nfrequency) changes proportionally to its mobility. In the case where the pinning potential is not \nsufficiently wide (of the order of the DW width or smaller) the FFT shows a relatively well-\ndefined resonance peak (corresponding to the DW’s f erromagnetic resonance) whose position \ndoes not depend on damping. The fact that the attem pt frequency (in this case given by the \nposition of the peak) loses its damping dependence,  explains why the DW velocity in the case \nof narrow pinning potential does not depend on damp ing. \nS4. Asymmetry dependence on DW velocity.   \nThe numerical analysis of the DW motion reveals an important feature: since the \nvelocity variation induced by the damping does not depend of the value of H z (Figure S2a), the \nmeasured asymmetry should also be independent of H z.   \n 27   \nFigure S5 . Asymmetry dependence on H z. a.  Velocity curves for different values of H z. \nb.  The corresponding asymmetries A raw  = 2( v↓↑ – v↑↓)/( v↓↑ + v↑↓). c. The normalized \nvelocities.  \n \nTo verify this prediction, we repeated the measurem ents for different values of H z. \nIndeed we find that, though the velocity changes by  three orders of magnitude, both H sat  and A \nremain the same. On the contrary, the symmetric com ponent varies significantly, changing the \noverall shape of the curves. The uncorrelated behav ior of the symmetric and asymmetric \ncomponents of the velocity points to their differen t origin: while the asymmetry is given by the \nchiral damping through the attempt frequency, the s ymmetric part is linked to changes in the \npinning potential induced by H ip .  \nS5. Link between the asymmetry and m ip  variation  \nThe DW velocity measurements correspond to the cree p regime, where the DW velocity \ndepends linearly on the attempt frequency and expon entially on the DW energy and pinning \nstrength. For simplicity, we merge all energy contr ibutions under a same parameter (E)  \n()Efvexp 0⋅≈  \n 28  We can write explicitly for the two DWs: ()↑↓ ↑↓ ↑↓ ⋅≈ E fv exp 0  and ()↓↑ ↓↑ ↓↑ ⋅≈ E fv exp 0  \nWe observe that below H sat  the two DW velocities vary differently, while abov e, their ratio \nstays constant. \nSince   ( )↓↑ ↑↓ \n↓↑ ↑↓ \n↓↑ ↑↓ −⋅= EEff\nvvexp \n00 \ncst vv=↓↑ ↑↓  implies that both cst ff=↓↑ ↑↓ 00  and cst EE =−↓↑ ↑↓ , or that the attempt \nfrequency and energy variations coincidently compen sate each other.  This scenario can be \nexcluded based on the robustness of the observation : the asymmetry saturation is observed for \na wide range of H ip  for all our samples. There are two possibilities l eft: \n \nCase 1) Damping induced asymmetry \nThis mechanism links directly the two DW velocities  to m ip . Therefore, as indicated in the \nmanuscript the asymmetry saturation coincides with the saturation of the DWs magnetization. \n \nCase 2) energy induced asymmetry \nIn thin films with vertical SIA and isotropic in th e plane, the chiral energy for the two DWs is: \n ∫\nΔ∝\nDW dx xE )cos( ϕ∂∂θ   \nThe three parameters that define this energy and th at can be in principle affected by H ip  are: \nthe magnetization gradient, the DW width and the m ip  orientation. Since the variation and the \nsaturation of the asymmetry occur at low fields, wh ile the magnetization gradient varies \nsignificantly at much larger fields, comparable wit h the uniaxial anisotropy, we deduce that \nthe chiral energy variation is essentially due to a  variation of m ip . \n∫ ∫\nΔ Δ↓↑ ↑↓ − ∝−\nDW DW dx Hxdx HxEEip ip )(cos )(cos 2 1 ϕ∂∂θ ϕ∂∂θ  \nSince x∂∂θ  has opposite sign for up/down and down/up DWs, and cos( ϕ) variations induced by \nHip  have the same sign, any variation of m ip  for any one of the DWs will amplify ↓↑ ↑↓ −EE . \nAs a consequence, cst EE =−↓↑ ↑↓  is verified only if m ip  is constant for both DWs.  \n \nTherefore, independently of the mechanism that crea tes the asymmetry, its saturation at low \nfields, is a robust indicator of the m ip  saturation.  29   \nSupplementary References: \n31.  Bode, M. et al . Chiral magnetic order at surfaces driven by inver sion asymmetry. Nature  \n447 , 190–193 (2007). \n32.  Je, S-G et al. Asymmetric Magnetic Domain-Wall Moti on by the Dzyaloshinskii-Moriya \nInteraction Physical Review B 88, 214401 (2013) \n33.  Thiaville, A., Rohart, S., Jué, É., Cros, V., & Fer t, A. Dynamics of Dzyaloshinskii domain \nwalls in ultrathin magnetic films. EPL (Europhysics Letters) , 100 (5), 57002. (2012). \n34.  Franken, J. H., Swagten, H. J. M., & Koopmans, B. ( 2012). Shift registers based on \nmagnetic domain wall ratchets with perpendicular an isotropy. Nature Nanotechnology , \n7(8), 499-503  \n35.  Pérez-Junquera, A. et al.,  Crossed-ratchet effects for magnetic domain wall m otion. \nPhysical review letters , 100 (3), 037203 (2008) \n36.  Metaxas, P. J., et al. \"Creep and flow regimes of m agnetic domain-wall motion in ultrathin \nPt/Co/Pt films with perpendicular anisotropy.\" Phys . Rev. Lett. 99.21 (2007): 217208. \n37.  Gorchon, J., et al. \"Pinning-Dependent Field-Driven  Domain Wall Dynamics and Thermal \nScaling in an Ultrathin Pt/Co/Pt Magnetic Film.\" Phys. Rev. Lett. 113.2  027205 (2014)  \n "    },    {        "title": "1609.06883v1.Damping_of_nonlinear_standing_kink_oscillations__a_numerical_study.pdf",        "content": "Astronomy &Astrophysics manuscript no. 29010_jn c\rESO 2022\nMay 24, 2022\nDamping of nonlinear standing kink oscillations: a numerical study\nN. Magyar and T. Van Doorsselaere\nCentre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, bus 2400, 3001 Leu-\nven, Belgium\nMay 24, 2022\nABSTRACT\nAims. We aim to study the standing fundamental kink mode of coronal loops in the nonlinear regime, investigating the changes in\nenergy evolution in the cross-section and oscillation amplitude of the loop which are related to nonlinear e \u000bects, in particular to the\ndevelopment of the Kelvin-Helmholtz instability (KHI).\nMethods. We run ideal, high-resolution three-dimensional (3D) magnetohdydrodynamics (MHD) simulations, studying the influence\nof the initial velocity amplitude and the inhomogeneous layer thickness. We model the coronal loop as a straight, homogeneous\nmagnetic flux tube with an outer inhomogeneous layer, embedded in a straight, homogeneous magnetic field.\nResults. We find that, for low amplitudes which do not allow for the KHI to develop during the simulated time, the damping time\nagrees with the theory of resonant absorption. However, for higher amplitudes, the presence of KHI around the oscillating loop can\nalter the loop’s evolution, resulting in a significantly faster damping than predicted by the linear theory in some cases. This questions\nthe accuracy of seismological methods applied to observed damping profiles, based on linear theory.\nKey words. Sun: corona, Sun: oscillations, Magnetohydrodynamics (MHD)\n1. Introduction\nWith the first observations of transverse oscillations in coro-\nnal loops by TRACE (Aschwanden et al. 1999; Nakariakov\net al. 1999) a new era has begun in the exploration and under-\nstanding of the solar corona. These observations of waves al-\nlow the inference of di \u000berent physical parameters which were\npreviously largely unknown, a method called coronal seismol-\nogy. It was quickly noted that the observed coronal loop oscilla-\ntions were strongly damped, and if such a high dissipation was\ndue to viscous or resistive damping, then the associated trans-\nport coe \u000ecients must be orders of magnitude higher than pre-\ndicted by the classical theory. Nearly all of the observed, high-\namplitude, lower coronal eruption-related coronal loop oscilla-\ntions (Zimovets & Nakariakov 2015) are strongly damped, with\nrare but nevertheless very intriguing exceptions (see, e.g. As-\nchwanden et al. (2002); Aschwanden & Schrijver (2011); Wang\net al. (2012)). Recently, small amplitude decayless kink oscil-\nlations were observed in coronal loops (Nisticò et al. 2013; An-\nfinogentov et al. 2013), and are thought to be ubiquitous in active\nregions (Anfinogentov et al. 2015). These oscillations appear to\nbe constantly driven, and do not originate from an impulsive or\neruptive event like the high amplitude oscillations. It is gener-\nally accepted that the mechanism responsible for the fast damp-\ning of transverse oscillations is the extensively studied resonant\nabsorption or mode coupling (e.g. Ionson 1978; Hollweg 1984;\nRuderman & Roberts 2002; Goossens et al. 2002). For resonant\nabsorption to damp the oscillations, the existence of a surface\nwithin the limits of the loop, with Alfvén speed matching the\nglobal kink phase speed, is required. This is usually portrayed as\na smooth layer around a homogeneous flux tube, but resonance\nhas been proven to exist regardless of the geometrical shape of\nsuch a layer (Terradas et al. 2008b; Pascoe et al. 2011). The lin-\near theory of kink oscillations is well studied (for a review, see\nRuderman & Erdélyi 2009). The damping profile of oscillations,in the presence of a driver (or initial perturbation in the case\nof standing modes) has been shown to deviate from a purely\nexponential decay, being described rather by an initially Gaus-\nsian damping profile followed by an exponential damping pro-\nfile (Pascoe et al. 2012; Hood et al. 2013; Pascoe et al. 2013,\n2016b,a). Observed average periods, amplitudes and exponen-\ntial damping times of kink oscillations in coronal loops were re-\nported by, Aschwanden et al. (2002), for example. In their anal-\nysis of 26 loop oscillation events, they find average oscillation\nperiods of 321\u0006140 s, oscillation amplitudes of 2200 \u00062800 km\nand damping times of 580 \u0006385 s. The measured amplitudes cor-\nrespond to relative amplitudes (to the loop length L) of approx-\nimately 1-5%. Arregui et al. (2007) and Goossens et al. (2008)\nused the analytical formula for the damping time (Ruderman &\nRoberts 2002) to construct a seismological method for inferring\nthe loop parameters from the observed damping time and pe-\nriod.\nWaves in the solar atmosphere often have high enough ampli-\ntudes to be considered nonlinear. The study of nonlinear waves\nin the solar atmosphere has been carried out especially in the\ncontext of chromospheric and coronal heating. For a review of\nearlier theoretical work on the subject, see Ruderman (2006). In\nparticular, the analytical theory of nonlinear kink oscillations has\nbeen studied, both for propagating (Ruderman et al. 2010), and\nstanding waves (Ruderman & Goossens 2014). In these studies\nit was shown that damping of propagating kink waves can be\nenhanced by the nonlinearity of an m-mode resonance, where\nenergy from the m=1 kink mode is transferred to m\u00152\nfluting modes, which, by resonant absorption, can damp faster\ndue to shorter wavelengths. It was noted that the m-mode reso-\nnance can damp the kink wave even in the absence of a resonant\nlayer. However, in an axially inhomogeneous flux tube, due to\ndensity stratification, for example, the m-modes no longer have\nthe same phase speed, thus the m-mode resonance and enhanced\nArticle number, page 1 of 10arXiv:1609.06883v1  [astro-ph.SR]  22 Sep 2016A&A proofs: manuscript no. 29010_jn\ndamping disappears. These calculations were valid for weakly\nnonlinear oscillations, and the authors anticipated that the am-\nplitude would be a \u000bected in the fully nonlinear case. In general,\nthe study of fully nonlinear problems is only possible numeri-\ncally. Numerical studies of nonlinear kink oscillations of coro-\nnal loops have been carried out by, Terradas & Ofman (2004);\nOfman (2005); Terradas et al. (2008a); Ofman (2009b); Antolin\net al. (2014, 2015); Magyar et al. (2015); Magyar & Van Doors-\nselaere (2016), for example. See also Ofman (2009a) for a re-\nview. Of particular and renewed interest in nonlinear evolution\nof transverse waves, even for low amplitudes, is the suscepti-\nbility of the resonant layer, due to its high velocity shear, to\nthe Kelvin-Helmoltz instability (KHI) (Heyvaerts & Priest 1983;\nBrowning & Priest 1984; Karpen et al. 1994; Ofman et al. 1994;\nPoedts et al. 1997). This can enhance the wave energy dissipa-\ntion via local turbulence, and thus is also relevant in the coronal\nheating problem. Direct observational evidence is lacking of the\nKHI in coronal loops. However, it has been observed in coronal\nmass ejections and quiescent prominences (Berger et al. 2010;\nRyutova et al. 2010; Foullon et al. 2011; Ofman & Thompson\n2011), and it was suggested that, in fact, the KHI may appear as\nstrands of coronal loops as seen in EUV (Antolin et al. 2014).\nThe previous statement is strengthened by the first observational\nevidence of resonant absorption in prominences (Okamoto et al.\n2015; Antolin et al. 2015). Recently, the idea of observing the\ndamping profile of kink oscillations of coronal loops in order to\ninfer various local parameters such as density ratios and inhomo-\ngeneous layer widths in the context of coronal seismology was\nput forward. (Pascoe et al. 2016a). Therefore, it is essential to\nknow the e \u000bects that nonlinearity might have on damping pro-\nfiles. New observational evidence suggests that the damping of\nthe transverse oscillations of coronal loops is dependent on the\namplitude (Goddard & Nakariakov 2016).\nIn this paper we investigate the nonlinear standing kink oscilla-\ntion of coronal loops, for which the main e \u000bect of nonlinearity\nis the development of the KHI at the loop edges, altering the en-\nergy distribution in the loop cross section and ultimately leading\nto a change in the damping profile of the loop displacement.\n2. Numerical Model\n2.1. Initial conditions\nWe model the coronal loop as a straight, density-enhanced mag-\nnetic flux tube. The loop consists of a homogeneous inner core,\nand an inhomogeneous annulus (transitional layer), in which the\ndensity varies sinusoidally from the core ( \u001ai=2:5\u000110\u000012kg m\u00003)\nto the background density value ( \u001ae=0:5\u000110\u000012kg m\u00003), where\nthe subscripts i;estand for internal and external, respectively.\nThe numerical domain is permeated by a straight, homogeneous\nmagnetic field, of 12 :5 G strength. The distance between the\ncentral axis of the flux tube and the midpoint of the inhomo-\ngeneous layer defines its radius, R=1:5 Mm. The thickness of\nthe inhomogeneous layer is denoted by l. We neglect the e \u000bects\nof curvature, gravity (i.e. no stratification), energy sources and\nsinks (heating, thermal conduction, radiative processes), and we\nlack a realistic lower solar atmosphere (i.e. photosphere, chro-\nmosphere). The plasma- \fis constant throughout the domain,\n0:06, corresponding to Te=4:5 MK and Ti=0:9 MK. The\nresulting temperature profile is most probably not realistic. This\nsimplistic profile was chosen in order to start with an initial equi-\nlibrium, that is, constant total pressure throughout the domain.\nTests with isothermal models ( Te=Ti=0:9 MK) show no\nsignificant di \u000berence in comparison with the present model, asexpected (period, growth rates and damping rates not depending\non the temperature profile).\nInitially, we impose a perturbation in a component of the veloc-\nity transverse to the loop axis, of the form\nVy=8>><>>:A V A;icos\u0010\u0019z\nL\u0011\n;x2+y2\u0014\u0010\nR+l\n2\u00112\n0; otherwise(1)\nVA;iis the Alfvén speed inside the loop ( VA;i\u00190:7 Mm s\u00001),\nL=120 Mm is the total loop length, and Ais the perturbation\namplitude. Thus the perturbation acts only inside the loop, in-\ncluding the inhomogeneous layer. This excites the fundamental\nkink mode of the loop, and we stop the simulation at tf=1500\ns.\n2.2. Boundary conditions\nIn order to excite a standing transverse oscillation, we fix the\nfootpoint of the loop ( z=60 Mm), by setting the transverse\nvelocities to antisymmetric, while the other variables are set to\ncontinuous (Neumann-type, zero-gradient boundary condition)\nat this boundary. Exploiting the symmetric properties of the fun-\ndamental kink mode, we model only half of the loop in both the z\nandxdirections. This reduces the computational time four-fold.\nThe boundary conditions for these mirroring boundaries are de-\nscribed in Magyar et al. (2015). At the other, lateral boundaries\nwe let any waves leave the domain freely by imposing an outflow\n(zero-gradient) condition on all variables.\n2.3. Numerical method and mesh\nTo solve the ideal 3D MHD problem, we use the MPI-AMRVAC\ncode (Keppens et al. 2012; Porth et al. 2014). We use the imple-\nmented second-order ‘onestep’ TVD method with the Roe solver\nand ‘Woodward’ slope limiter. The constraint on the magnetic\nfield divergence is maintained using Powell’s scheme. The nu-\nmerical domain has dimensions of (0 ;6)\u0002(\u000010;10)\u0002(0;60)\nMm. The base resolution is 24 \u000280\u000232, and we use 4 levels of\nrefinement, fully refining the region around the loop, resulting in\nanx\u0000yplane resolution of 31.2 km per cell, or 0 :02R, where Ris\nthe radius of the loop. The fully-refined resolution in the zdirec-\ntion is 234 km per cell. Comparison with simulations with one\nmore level of refinement shows no important di \u000berences in the\ndynamics, even though the KHI instability (and later turbulence)\nchanges quantitatively.\n3. Results and discussion\nWe ran a series of simulations which explored the parameter\nspace, varying the initial velocity perturbation and thickness\nof the inhomogeneous layer of the loop. The chosen values\nare the following: A=f0:005;0:01;0:02;0:035;0:05gandl=\nf\u00190:0R;0:1R;0:33R;0:5Rg, for a total of 20 di \u000berent runs (Fig-\nure 1). The first value for ldenotes an initially step profile which,\nafter the start of the simulation, evolves into an inhomogeneous\nlayer due to numerical di \u000busion. This layer is thus the thinnest\npossible layer in our simulation, corresponding to approximately\n0:08R. The initial displacements of the loop are, in increasing\norder with the amplitude, f0:079R;0:16R;0:32R;0:55R;0:78Rg.\nObtained oscillation periods are discussed later in the text. The\nnonlinearity parameter in the case of kink oscillations of a flux\ntube is given approximately by Ruderman & Goossens (2014):\n\u0017'AL\nR: (2)\nArticle number, page 2 of 10N. Magyar and T. Van Doorsselaere: Damping of nonlinear kink oscillations\nFig. 1. Plots showing the density in the anti-node cross-section of the loop ( z=0) at t=540 s for the di \u000berent parameters used ( A=0:005 not\nshown for brevity, looking almost identical to A=0:01). Di \u000berent columns have di \u000berent initial inhomogeneous layer widths, shown at the top,\nand the di \u000berent rows are for di \u000berent amplitudes, shown in the left margin. Axis units are in Mm and density in units of 10\u000012kg m\u00003.Article number, page 3 of 10A&A proofs: manuscript no. 29010_jn\nThe oscillations are weakly nonlinear for \u0017\u001c1 and strongly\nnonlinear for \u0017\u00151. For the amplitudes chosen, the nonlinear-\nity parameter varies between 0.4 and 4. Thus, all the simulations\nrepresent nonlinear and strongly nonlinear regimes of oscilla-\ntions. As the simulations with A=0:005 and A=0:01 show\nalmost similar evolution, lower amplitudes were not essential for\nthis study. In Figure 2, the evolution of the oscillation in the anti-\nnode cross-section can be followed, for a specific set of param-\neters. In this sequence, we can identify the principal signatures\nof nonlinear kink oscillations previously described in detail in\nnumerical simulations by Terradas et al. (2008a); Antolin et al.\n(2014, 2015); Magyar et al. (2015). These works however, lack\na description of one aspect which plays a role in the nonlinearly\nenhanced damping of the oscillations, namely the resonance be-\ntween the kink and m\u00152 or fluting modes, which extracts en-\nergy from the kink mode, leading to damping of the transverse\noscillation. This e \u000bect was studied analytically by Ruderman\net al. (2010), and is present only in the absence of longitudi-\nnal inhomogeneities (e.g. gravitational stratification), which de-\nstroys the resonance between the modes (Ruderman & Goossens\n2014). This e \u000bect can be seen clearly in the second snapshot of\nFigure 2 ( t=135 s, approximately half-period), before the de-\nvelopment of the Kelvin-Helmholtz instability (KHI), as a defor-\nmation of the initially circular cross-section in a manner resem-\nbling the m=2 fluting mode. There are three main mechanisms\nacting to damp the transverse oscillation (aside from a very small\nnumerical dissipation): resonant absorption or mode conversion,\nthe previously mentioned m\u00152 resonance, and the development\nof the KHI (in the case shown in Figure 2, in less than a period).\nThese mechanisms act concomitantly and are coupled, resulting\nin the significance and magnitude of each mechanism varying in\ntime, which is di \u000ecult to estimate. The kink mode presents a ve-\nlocity shear near the lateral boundary of the loop, which is prone\nto the KHI. Resonant absorption, acting immediately after t=0,\nfurther enhances the velocity shear, thus shortening the time af-\nter which the instability sets in. Note that, even in the absence\nof an inhomogeneous layer (which, due to numerical di \u000busivity,\nis not possible in our simulations), the KHI can still develop. In\nthis sense, in the nonlinear regime, transverse oscillations will\nalways be damped.\nAs previously stated, estimating the relative importance of the\ninteracting damping mechanisms is not an easy task. By looking\nat the transverse velocity ( vy), in Figure 3, we can appreciate the\ninteraction of two mechanisms, resonant absorption and KHI.\nInitially, the analytical kink eigenfunction for velocity develops\nafter the perturbation at t=0, representing a uniform speed\nwithin the boundaries of the loop and a dipolar field around it.\nAs noted earlier, a velocity shear is present in this configuration,\nwhich, as shown in the t=135splot, is later further enhanced\nby resonant absorption. Later on, the resonant layer is disrupted,\nits KE contributing to the development of the KHI. Due to the\nrobust nature of resonant absorption (Terradas et al. 2008b), it\nis still present in patches, that is, on the surface of the roll-ups,\nbut ceases to exist in the form which was present at t=135s.\nIt is di \u000ecult to estimate the relative e \u000bectiveness and the con-\ntribution to the damping of the patchy resonant absorption com-\npared to the pre-disruption phase. Ultimately, the roll-ups break\nup, resulting in a turbulent inhomogeneous layer, where most\nof the remaining wave energy is situated. The impact of the m-\nmode resonance on the damping time is not clear. Magyar et al.\n(2015) studied nonlinear fundamental standing kink oscillations\nin a stratified atmosphere, in which the resonance is still notice-\nable, despite the longitudinal inhomogeneity. In Figure 2 of Ru-\nderman et al. (2010) the enhanced damping as a function of anonlinearity parameter N is shown, which is roughly equal to the\ndriving amplitude divided by the width of the inhomogeneous\nlayer in units of R. For our parameters, this value is at most 0.5.\nThis value corresponds to a very small e \u000bect on the oscillation\namplitude. In this sense, we can be somewhat confident that this\ndamping mechanism does not have a substantial impact on the\nresulting damping times, and in what follows we focus on the\nother damping mechanisms.\nWe studied the energy evolution in the anti-node cross-section\nof the loop for the di \u000berent parameters. For the case in Figure 2,\nthe average values of relevant variables are plotted as a function\nof simulation time in Figure 4. Note that, in the anti-node of the\nfundamental kink, all of the kink mode’s energy is in the form of\nKE. The oscillations present in this plot, with the double period\nof the kink mode, represent the exchange between KE and mag-\nnetic energy (ME), analogous to the exchange between kinetic\nand elastic potential energy in a classical harmonic oscillator (if\nwe account only for the perturbation to the ME). To proceed\nfurther, we make the distinction between the core of the loop,\ndefined as the region of the cross-section where \u001a\u00150:98\u001aiand\nthe ME is negligible (see Goossens et al. (2014) for example),\nand the inhomogeneous layer, defined as the region of the cross-\nsection where 0 :98\u001ai\u0015\u001a\u00151:1\u001ae. Inspecting the figures, we\nobserve that the average KE density in the core of the loop has a\ndecreasing peak energy. At the same time the energy contained\nin the inhomogeneous layer is undergoing a di \u000berent evolution:\nthe peak values are decreasing, but its minimal values show a\nstrong initial increase, which can be attributed to the increase\nin the energy of localized Alfvén waves, through resonant ab-\nsorption (see, e.g. Poedts & Kerner (1991)). Simultaneously, the\nKE in the layer is dissipated, resulting in an increase of internal\nenergy (IE). The average IE density shows a long-period oscil-\nlation, triggered by the perturbation at t=0, corresponding to\na longitudinal slow mode. The slow mode is present both in the\ncore and the layer, however the background trend of average IE\ndensity shows a steady increase in the layer, in accordance with\nthe KE loss. The IE density plot shows the average perturba-\ntion to its equilibrium value over time, and the rise due to the\ndissipation of the KE represents a \u00190:36% increase in IE den-\nsity. This cannot account for the apparent heating of the loop\n(core with layer), from an average temperature of 1 :17 MK to\n1:47 MK: this would require around 50 times more energy con-\nverted into IE. Inspecting the change in the area of the core and\nthe surrounding corona (outside the loop, area of the numerical\ndomain), we notice that both shrink, resulting in an enlargement\nof the inhomogeneous layer. However, we also notice that the\ncore area diminishes to a lesser extent than the corona. This im-\nplies that the higher average temperature of the loop is due to\nthe enlarging inhomogeneous layer, mixing with the hot and rar-\nefied plasma surrounding the loop, driven by the KHI. This e \u000bect\nis present only because of the chosen initial conditions, that is,\nwith a corona five times hotter surrounding the loop. Note that\nthe total IE in the layer (not shown here) presents a considerable\nincrease (\u0019150%), but this is almost entirely (i.e. except the\nminute increase converted from KE) a result of the enlargement\nof the layer, caused by the KHI.\nIn the plots of Figure 5 the evolution of the average KE density\nin the inhomogeneous layer for di \u000berent initial amplitudes and\nlayer thickness can be compared. These plots are normalized to\nthe initial energy density contained in the layer, (i.e. of the ini-\ntial perturbation), so that a comparison for di \u000berent amplitudes is\nstraightforward. We note that the relative increase of KE density\nin the layer is higher for smaller amplitudes, and it requires more\ntime to dissipate. This demonstrates that the presence of KHI in\nArticle number, page 4 of 10N. Magyar and T. Van Doorsselaere: Damping of nonlinear kink oscillations\nFig. 2. Plots of density in the anti-node cross-section of the loop ( z=0) at di \u000berent times (in steps of approximately one half-period, shown in the\nupper part of each plot) for A=0:035 and l=0:33R. Axis units are in Mm and density in units of 10\u000012kg m\u00003.\nFig. 3. Plots of the y-component of the velocity ( vy) at di \u000berent times (selected times corresponding to plots in Figure 2, shown in the upper part\nof each plot), for A=0:035 and l=0:33R. Axis units are in Mm and velocity in units of km s\u00001.\nthe higher amplitude oscillations acts to enhance the conversion\nof KE to IE. This e \u000bect is more pronounced for wider initial in-\nhomogeneous layers.\nIn the following, we analyze the evolution of the kink oscilla-\ntions triggered by the perturbation and their associated damping.\nFor this, we track the centre of mass for the loop in the funda-\nmental kink anti-node cross-section where the displacement is\nmaximum. The damping profile of impulsively triggered stand-\ning kink modes in a flux tube is not purely exponential (Hoodet al. 2013; Pascoe et al. 2013, 2016a). Instead, initially the\ndamping is approximated by a Gaussian profile, and after a time\ntsby an exponential profile. This profile has the form:\nA(t)=8>>><>>>:A0exp\u0012\n\u0000t2\n2\u001c2g\u0013\nt\u0014ts\nA(ts) exp\u0010\n\u0000t\u0000ts\n\u001cd\u0011\nt>ts; (3)\nArticle number, page 5 of 10A&A proofs: manuscript no. 29010_jn\n0 200 400 600 800 1000 1200 1400\nTime (s)02000400060008000\nLayer\nCore<KE> (erg m−3)\n0 200 400 600 800 1000 1200 1400\nTime (s)−4000−20000200040006000<IE> (erg m−3)Layer\nCore\n0 200 400 600 800 1000 1200 1400\nTime (s)1.1×1061.2×1061.3×1061.4×1061.5×106Temperature (K)\n0 200 400 600 800 1000 1200 1400\nTime (s)−2.0−1.5−1.0−0.50.00.5Area (Mm2)Corona\nCore\nFig. 4. Plots showing the time evolution of di \u000berent quantities in the anti-node cross-section of the loop ( z=0) with A=0:035 and l=0:33R.\nTop-left : Average kinetic energy (KE) density in the core and layer region of the loop. Top-right : Average internal energy density in the core and\nlayer regions of the loop. Bottom-left : Average temperature of the loop (average over both the core and layer). Bottom-right : Change in the area\nrelative to the initial value, for the core region of the loop and the numerical domain outside the loop (outside both core and layer).\nand the switch time tsis defined as:\nts=\u001c2\ng\n\u001cd=\u001a0=\u001ae+1\n\u001a0=\u001ae\u00001P: (4)\nThus, by fitting observed standing kink oscillations of coronal\nloops to a function\nA(t) sin(!t+\u001e); (5)\none could seismologically estimate the density ratio of the loop\nand inhomogeneous layer thickness (Pascoe et al. 2016a). We\nfit Eq. 5, using mpfitfun.pro (Markwardt 2009), to the simu-\nlated damping profiles to obtain four parameters, A0;\u001cg;\u001cd, and\n!=2\u0019\nP. Note that tsis a constrained parameter (Eq. 4) using the\n‘.TIED ’ entry of the PARINFO structure, and \u001eis zero as it results\nfrom the initial condition. We find best-fittings for the period be-\ntween 253.5 and 265 :1 s; thicker inhomogeneous layers result-\ning in the shorter periods. The period for the simulation closest\nto the linear case ( A=0:005;l\u00190:0R) is 265.1 s. The theo-\nretical fundamental kink period for a step-density flux tube with\nthe same parameters is 263 :6 s (Edwin & Roberts 1983). Some\nresults from fitting to the simulated damping profiles are shown\nin Figure 6. The amplitudes were normalized to the value of themaximal initial displacements. Note that for the low-amplitude\ncase ( A=0:01, upper plots), the fit is much more accurate\n(lower\u001f2) than for the high-amplitude counterpart ( A=0:05,\nlower plots). Furthermore, we find that, in the runs which de-\nvelop KHI ( A\u00150:02), the last periods of oscillation are practi-\ncally undamped. This can also be seen in Figure 4 (top left), in\nthe core average KE, which remains practically constant in max-\nimal value after\u00191000 s. We interpret this as a consequence\nof the KHI: some of the kinetic energy percolates into the core\nregion. Although a small e \u000bect, this unfortunately makes the fit-\nting of Eq. 5 meaningless (i.e. a Gaussian profile will always fit\nbetter to the oscillations, for any value of ts). Overcoming this\nproblem by truncating the dataset where the constant KE period\nbegins (e.g. 800 s) is unreliable as tsvaries greatly even for small\nchanges in the truncation. Thus, great care should be taken when\nfitting Eq. 5 to observed oscillations for seismology. The oscil-\nlations resulting from A\u00150:02 are best described by a Gaussian\nprofile. Therefore, we encourage the fitting of Gaussian damping\nprofiles alongside the traditional exponential damping profile in\nfuture studies of transverse coronal loop oscillations. For our re-\nsults to be readily comparable to available studies, we quantify\nthe exponential damping time, and to obtain tsfor the high am-\nArticle number, page 6 of 10N. Magyar and T. Van Doorsselaere: Damping of nonlinear kink oscillations\n0 200 400 600 800 1000 1200 1400\nTime (s)0.00.20.40.60.81.0<KE> (Normalized)0.01\n0.02\n0.035\n0.05\n0 200 400 600 800 1000 1200 1400\nTime (s)0.00.20.40.60.81.0<KE> (Normalized)0.01\n0.02\n0.035\n0.05\nFig. 5. Plots of the average kinetic energy density over time in the inhomogeneous layer of the anti-node cross-section of the loop ( z=0).Left:\nl=0:1R.Right :l=0:5R. The di \u000berent lines are for di \u000berent initial amplitudes, from 0 :01 to 0:05.\nplitude oscillations, we fit an exponentially damped sine;\nA(t)=A0exp \n\u0000t\n\u001cd!\nsin(!t); (6)\nfor the whole dataset. The exponential damping times obtained\nin this way are plotted in Figure 7. Note that for ‘well-behaving’\namplitude profiles (for A\u00140:01) we show \u001cdobtained from\nEq. 5. We also plotted the theoretical damping times due to res-\nonant absorption for di \u000berent inhomogeneous layer widths, for\nboth linear and sinusoidal density profiles in the layer (Ruder-\nman & Roberts 2002; Goossens et al. 2002). We find that, for\nthe low amplitude cases ( A\u00140:01) the simulated damping times\nare in between the predicted ones: the density profile in the in-\nhomogeneous layer can greatly influence the resulting theoreti-\ncal damping time, especially for thin layers (Soler et al. 2013).\nWe would like to highlight three important characteristics of the\nnonlinear damping of kink oscillations from Figure 7:\n\u000fThe damping time is no longer generally independent of am-\nplitude, as in the linear case. There is a strong damping, even\nwith initially thin inhomogeneous layers of the kink oscilla-\ntions, for high enough amplitudes. In these cases, the damp-\ning time can be less than a third of the theoretically predicted\none. Furthermore, the damping time appears to be saturated\nfor amplitudes larger than some threshold value.\n\u000fFor thick enough inhomogeneous layers (in our case, for l=\n0:5R), the damping time is independent of the amplitude, and\nthus coincidentally well approximated by the theory.\n\u000fIn the saturated high amplitude regime (here for A\u00150:035),\nthe damping is only weakly dependent on the initial thick-\nness of the inhomogeneous layer, and can thus not be used\nfor seismological purposes.\nHaving in mind the aforementioned characteristics, we can state\nthat high amplitude oscillations of coronal loops (as induced by\na flare or Low Coronal Eruption (LCE), see Zimovets & Nakari-\nakov (2015)) are constrained to damp with specific damping\ntimes, only weakly dependent on the amplitude and the initial in-\nhomogeneous layer width. This behaviour is associated with the\npresence of the KHI in the high amplitude oscillations. Thus, the\ngrowth rate of the KHI determines the switch between the lin-\nearly well-approximated damping and high amplitude damping\nregimes. For an analytical estimate of the growth rate of KHI forstanding transverse oscillations, see Zaqarashvili et al. (2015).\nIn our case, this switch is around A=0:02, which can be seen\nas a transitional regime: the KHI is developing but not fully, and\nis not disrupting the classical m=1 resonant absorption (as at\nt=135 s in Figure 3).\nFor our initial conditions and measured oscillation periods, the\ntime of transition between Gaussian and exponential damping\nprofiles (Eq. 4) is at ts\u0019400 s. In Figure 8 we plotted the val-\nues for tsobtained through the fitted Gaussian and exponential\ndamping times. As described above, we obtain \u001cdand\u001cgthrough\nfitting Eq. 5 for simulations with A\u00140:01. For the higher am-\nplitude simulations, we fit the Gaussian of Eq. 5 to obtain \u001cg\n(\u001cs\u0015tf) and Eq. 6 for \u001cd. We can see that we recover the theo-\nretical value for tswell (within 10%) for A\u00140:01, but for higher\namplitudes there are deviations, especially for the transitional\namplitude of A=0:02. Once again, this shows that we have to\nbe careful when applying seismology on high amplitude, nonlin-\near oscillations.\nVery recently, observational evidence has been produced show-\ning that the damping time of transverse oscillations of coronal\nloops is a function of the initial displacement amplitude (God-\ndard & Nakariakov 2016) . Unfortunately, a quantitative com-\nparison of our results to observations is not possible given the\nlarge parameter space which would need to be covered. On the\nother hand, we can state that there are encouraging qualitative\nagreements between our Figure 7 and Figure 2 of Goddard &\nNakariakov (2016): Firstly, the damping time is decreasing as\nthe initial amplitude of the oscillations increases, as is the case\nfor our simulations with thin inhomogeneous layers. Secondly,\nthis dependence weakens as the amplitude increases. The ob-\nservational data also indicates that there is a considerable pop-\nulation between the values 1 and 2 of the damping time to the\noscillation period ratio, which seem to be weakly dependent on\nthe amplitude, similarly to our case of the loops with thick in-\nhomogeneous layers. It is interesting to note that the transitional\nregime (defined earlier) appears to occur for much higher ampli-\ntudes for the observed oscillations than in our simulations. This\ncould possibly imply a lower growth rate of the KHI for the ob-\nserved loops.\nArticle number, page 7 of 10A&A proofs: manuscript no. 29010_jn\n0 200 400 600 800 1000 1200 1400\nTime (s)−1.0−0.50.00.51.01.5Amplitude (Normalized)Exp\nGauss\n0 200 400 600 800 1000 1200 1400\nTime (s)−1.0−0.50.00.51.01.52.0Amplitude (Normalized)Exp\nGauss\n0 200 400 600 800 1000 1200 1400\nTime (s)−1.0−0.50.00.51.01.52.0Amplitude (Normalized)Exp\nGauss\n0 200 400 600 800 1000 1200 1400\nTime (s)−1.0−0.50.00.51.01.52.0Amplitude (Normalized)Exp\nGauss\nFig. 6. Plots of the displacement amplitude over time in the loop anti-node ( z=0). The values are normalized to the initial displacement. For the\ntop plots ( A=0:01,Left:l=0:1R,Right :l=0:5R), the Gaussian and exponential profiles from Eq. 3 obtained by fitting Eq. 5 are plotted. For the\nbottom plots ( A=0:05,Left:l=0:1R,Right :l=0:5R), the Gaussian of Eq. 3 is fitted for the whole data and plotted, as well as the exponential\nresulting from fitting Eq. 6.\n4. Conclusions\nIn this study, we investigated the standing kink oscillations of\na straight flux tube, aiming to model transverse oscillations of\ncoronal loops. We ran 3D ideal MHD simulations, exploring\nthe parameter space of initial amplitudes and inhomogeneous\nlayer thicknesses. The chosen amplitudes cover the weakly lin-\near and fully nonlinear regimes. The resulting oscillation peri-\nods are well described by the linear theory. Furthermore, for low\namplitudes, we find that the resulting damping of the oscillation\nis close to the damping times computed by linear theory. How-\never, for higher amplitudes, nonlinear e \u000bects have a definitive\nimpact on the resulting oscillation characteristics, especially the\ndevelopment of the KHI around the loop edges where the ve-\nlocity shear is highest, coinciding with the layer where resonant\nabsorption is taking place. The development of KHI (and ulti-\nmately the threshold amplitude at which its e \u000bects become im-\nportant) is dictated by its growth rate, which in turn depends on\nthe ratio of loop radius to length, oscillation amplitude, Alfvén\nspeeds, inhomogeneous layer thickness and numerical dissipa-\ntion. We show that even for initially thin inhomogeneous layers,\nthe oscillations undergo rapid damping due to the presence of\nKHI. For high amplitudes (with KHI developing in under one os-\ncillation period), the damping time is almost independent of theinitial thickness of the inhomogeneous layer. On the other hand,\nfor thick inhomogeneous layers, the damping time does not seem\nto depend on the initial amplitude of the perturbation. To put\nthings in perspective, the highest amplitude perturbation used in\nour simulations initially displaces the loop by less than its radius,\nand is thus at the lower boundary of observed flare-related coro-\nnal loop oscillation displacements (Aschwanden et al. 2002).\nStudying the energy distribution in the anti-node cross-section\nof the loop, we arrive at the conclusion that the kinetic energy in\nthe inhomogeneous layer is converted to plasma internal energy\nmore quickly in the presence of KHI. The increase in average\ninternal energy is less than one percent, thus the energy budget\nof the wave is not enough to cause any significant temperature\nchange in the 0.9 MK plasma. However, this heating might be\nsignificant for prominences (Antolin et al. 2015). Even if the dis-\nsipated wave energy is not enough to cause significant heating,\nwe show that if the loop is surrounded by hotter plasma, mixing\ninduced by the KHI can increase the average temperature of the\nloop. In the presence of KHI, the peak value of average kinetic\nenergy deposited in the inhomogeneous layer is lower than in\nsimulations without KHI. This is a consequence of the acceler-\nated conversion of kinetic to internal energy in the presence of\nKHI, which cascades energy to smaller scales where it can be\nArticle number, page 8 of 10N. Magyar and T. Van Doorsselaere: Damping of nonlinear kink oscillations\n0.00 0.01 0.02 0.03 0.04 0.05\nA050010001500200025003000d(s)0.1R0.0R\n0.33R\n0.5R\nFig. 7. Plot showing the exponential damping times for the di \u000berent\ninitial amplitudes, ( A), and inhomogeneous layer widths, represented by\ndiamond (\u0005), triangle (4), square ( 2), and pentagon ( D), in increasing\norder. The stars ( ?) of the same colour as the symbols, represent the\ntheoretically predicted damping time for the specific layer width and\nsinusoidal layer density profile. Analogously, the asterisks ( E) represent\nthe predicted damping for a linear layer density profile (Goossens et al.\n2002).\n0.00 0.01 0.02 0.03 0.04 0.05\nA200300400500600700\n≈  0             .0    R\n0 . 1R\n0.33R\n0.5Rτ\ng2/τ\nd(s)\nFig. 8. Plot showing the switch time ts(Eq. 4) between the Gaussian\nand exponential damping, obtained through fitting as described in the\narticle body. The symbols used for di \u000berent layer widths are the same\nas in Figure 7. The dashed line at 400 s is the theoretical value of ts, for\nour parameters.\ndissipated (by numerical dissipation) more e \u000eciently. The dis-\nruption of the resonant layer may also contribute to the reduction\nin the peak value of average kinetic energy, though it is unclear\nhow e \u000bective the resulting ‘patchy’ resonant absorption is.\nAccording to the present study, it becomes uncertain whether\nseismology schemes based on the linear theory for the damp-\ning rates of coronal loops are valid for high-amplitude, non-\nlinear transverse oscillations. Using the observed switch between\nGaussian and exponential damping profiles of transverse coronal\nloop oscillations for coronal seismology has recently been sug-\ngested. However, because of the nature of nonlinear kink oscilla-\ntions it is questionable how accurately one can infer parameters\nsuch as inhomogeneous layer thickness and density ratio from\nthe observed damping profiles, given that the non-linear dampingtimes are nearly insensitive to them. However, it is important that\nfuture studies also include Gaussian damping profiles in their\nanalyses, as this profile seems to better describe the damping of\nnonlinear transverse oscillations of coronal loops. The new ob-\nservations of amplitude-dependent damping times qualitatively\nsupport our conclusions.\nAcknowledgements. The authors would like to thank the referee for valuable\ncomments which helped to improve the manuscript. N.M. acknowledges the\nFund for Scientific Research-Flanders (FWO-VLaanderen). T.V .D. was sup-\nported by an Odysseus grant, the Belspo IAP P7 /08 CHARM network and the\nGOA-2015-014 (KU Leuven). Inspiration for this research was found during\nISSI and ISSI-BJ workshops. Visualization was done with the help of VisIt soft-\nware (Childs et al. (2012)).\nReferences\nAnfinogentov, S., Nisticò, G., & Nakariakov, V . M. 2013, A&A, 560, A107\nAnfinogentov, S. A., Nakariakov, V . M., & Nisticò, G. 2015, A&A, 583, A136\nAntolin, P., Okamoto, T. 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Chantrell,2and\nYongbing Xu1, 3,b)\n1)York-Nanjing Joint Center (YNJC) for Spintronics and Nano-engineering, School\nof Electronic Science and Engineering, Nanjing University, Nanjing, 210093,\nChina\n2)Department of Physics, University of York, York, YO10 5DD, UK\n3)Spintronics and Nanodevice Laboratory, Department of Electronic Engineering, University of York,\nYork YO10 5DD, UK\n(Dated: 25 June 2021)\nWe have investigated the magnetic damping of precessional spin dynamics in defect-controlled epitaxial grown\nFe3O4(111)/Yttria-stabilized Zirconia (YSZ) nanoscale \flms by all-optical pump-probe measurements. The\nintrinsic damping constant of the defect-free Fe 3O4\flm is found to be strikingly larger than that of the as-\ngrown Fe 3O4\flm with structural defects. We demonstrate that the population of the \frst-order perpendicular\nstanding spin wave (PSSW) mode, which is exclusively observed in the defect-free \flm under su\u000eciently high\nexternal magnetic \felds, leads to the enhancement of the magnetic damping of the uniform precession (Kittel)\nmode. We propose a physical picture in which the PSSW mode acts as an additional channel for the extra\nenergy dissipation of the Kittel mode. The energy transfer from Kittel mode to PSSW mode increases as\nin-plane magnetization precession becomes more uniform, resulting in the unique intrinsic magnetic damping\nenhancement in the defect-free Fe 3O4\flm.\nThe photo-induced precessional spin dynamics in vari-\nous magnetic materials has attracted signi\fcant attention\nsince the observation of the uniform magnetic precession\n(Kittel mode) and the corresponding \frst-order perpen-\ndicular standing spin wave (PSSW mode) in Ni \flms by\nthe all-optical pump-probe technique.1,2After excitation\nby a femtosecond laser pulse, besides the uniform Kit-\ntel mode, di\u000berent spin wave modes can be stimulated\nincluding \frst-order PSSW and Damon-Eshbach dipo-\nlar surface spin waves (DE modes).3At the same time,\nall-optical pump-probe measurements allow determina-\ntion of the magnetic Gilbert damping \u000b,4,5which is a\nkey parameter for magnetic data recording and the next-\ngeneration spintronic memory devices such as Magne-\ntoresistive Random Access Memory (MRAM)6,7. There-\nfore, understanding and controlling the magnetic damp-\ning is of crucial importance. Among many factors af-\nfecting the magnetic damping, structural defects are cru-\ncial because they are generally inevitable when preparing\n\flms or devices. It was proposed theoretically that de-\nfects scatter the Kittel mode into short wavelength spin\nwaves via two magnon scattering, producing an extrinsic\ncontribution to magnetic damping.8This extrinsic mech-\nanism was veri\fed by the fact that in thin NiFe \flms\nthe FMR linewidth increases with decreasing thickness.9\nAlso, the magnetic damping was found to signi\fcantly\nincrease due to interfacial defects in ultrathin Fe/Cr\nlayers.10It is reported that the antisite disorder can in-\ncreases the intrinsic Gilbert damping in L10FePt \flms in\na)Email: jing.wu@york.ac.uk\nb)Email: yongbing.xu@york.ac.ukwhich the spin\rip scattering plays dominant role.11Thus,\nit can be expected that the magnetic damping should be-\ncome smaller with reduced structural defects. Although\nthe damping coe\u000ecient is inversely proportional to the\nCo layer thickness in Co/Pd multilayers,12a thickness-\ndependent study of magnetic damping in Fe 3O4=MgO\n\flms has shown a strong increase of e\u000bective damping\nfrom 0:037 up to 0 :2 with increasing \flm thickness from\n5 nm up to 100 nm,13and the mechanisms of this ob-\nservation is unknown. As a half metal, Fe 3O4is one\nof the promising materials for spintronics due to nearly\n100% spin polarization and highly e\u000ecient spin injec-\ntion and transport. Fe 3O4contains defects including an-\ntiphase boundaries (APBs) and twin defects.14,15APBs\nare structural defects formed during the \flm growth\noriginating from the mismatch between Fe 3O4\flms and\nsubstrates.16The low formation energy of APBs means\nthat they are commonly observed in thin \flm Fe 3O4\nsamples.17,18In addition to APBs, it was demonstrated\nthat as-grown \flms can also have signi\fcant numbers of\ntwin defects.19The in\ruence of the defects in Fe 3O4thin\n\flms has not been investigated and very few studies on\nthe magnetization dynamics and magnetic damping of\nFe3O4thin \flms have been reported.20,21\nTo explore the e\u000bects of the structure on magnetic\ndamping, the defect-controlled epitaxial Fe 3O4(111)\nnanoscale \flms on Yttria-stabilized Zirconia (YSZ) sub-\nstrate were prepared by PLD growth and annealing.\nUsing time-resolved magneto-optical Kerr e\u000bect (TR-\nMOKE), we \fnd that the PSSW mode only exists in\nthe annealed defect-free Fe 3O4\flm, and numerical sim-\nulations support this \fnding. More importantly, in con-\ntrast to the general belief, the magnetic damping of thearXiv:1710.10938v2  [cond-mat.mtrl-sci]  17 May 20192\nFIG. 1. (a)(b) TEM images of (a) the as-grown\nFe3O4/YSZ(111) interface and (b) the annealed\nFe3O4/YSZ(111) interface. (c)(d) Schematic pump-probe\nTR-MOKE setup and sample geometry. (e)(f) Measured\ntransient Kerr signal of as-grown Fe 3O4(e) and annealed\nFe3O4(f). Kittel mode are observed in both \flms while\nthe PSSW mode is exclusively observed in annealed Fe 3O4.\nThe red solid line in (e) and (f) show the best \ftting using\nsingle and double damped sinusoidal formulas, respectively.\nApplied \feld range is from 3616 Oe to 5404 Oe.\ndefect-free Fe 3O4\flm is found to be signi\fcantly larger\nthan that of the as-grown \flm with defects. We \fnd that\nthe non-uniform PSSW mode is an extra energy dissi-\npation channel in addition to the uniform Kittel mode\nin the defect-free Fe 3O4\flm. The PSSW mode is only\npopulated when the in-plane magnetization precession is\nuniform under high magnetic \feld. Our work demon-\nstrates that the PSSW mode draws energy from the Kit-\ntel mode and then leads to the enhancement of the mag-\nnetic damping of the uniform magnetic precession.\nTo compare the defect conditions between the two\nas-grown and annealed \flms, transmission electron mi-\ncroscopy (TEM) observations have been performed. We\n\fnd that the annealed Fe 3O4\flm contains a lower density\nof defects, including APBs and twin defects, compared to\nthe as-grown \flm. The cross-sectional TEM images of as-\ngrown and annealed Fe 3O4\flms are shown in Fig. 1(a)\nand Fig. 1(b), respectively(details in supplemental ma-\nterials). The thickness Lof as-grown \flm and annealed\n\flms are about 80 nm and 50 nm, respectively.\nThe schematic pump-probe TR-MOKE setup and the\nsample geometry in the external \feld are illustrated\nin Fig. 1(c) and (d). TR-MOKE results for both\nas-grown and annealed Fe 3O4samples are presented\nin Fig. 1(e) and (f) under a pump beam \ruence of\n5:09 mJ cm\u00002. The photo-induced magnetic precession\nis described using a phenomenological formula:\u0001 \u0012K/P2\ni=1Aiexp (\u0000t=\u001ci) sin (2\u0019fit+'i) +B(t) where\u001ciand\nfiare the relaxation time and the precession frequency,\nrespectively. 'iis the initial phase of magnetization pre-\ncession and B(t) represents the background accounting\nfor the slow recovery of the magnetization, which is very\nweak in our measurements due to the low pump beam\nFIG. 2. Fourier spectrum of as-grown (a) and annealed (b)\nFe3O4. Kittel mode is recognized as the main peak. In an-\nnealed \flm, the second mode (red dashed line) originate from\nthe PSSW. The \feld independent high-frequency third mode\nis CAP mode (blue dashed line) from the Fe 3O4=YSZ inter-\nface. (c)f1of Kittel mode of both as-grown \flm (blue trian-\ngles) and annealed \flm (red circles). f2of the PSSW mode\nobserved in the annealed \flm is also plotted (pink squares).\nThe solid lines represent the \ftting curves. Dashed grey line\nrepresents the estimated PSSW mode in as-grown \flm which\nis not observed. (d) Numerical simulation of magnetization\ndynamics in as-grown (blue) and anneled (red) Fe 3O4thin\n\flm. Inset is the corresponding Fourier spectrums.\n\ruence. A beating can be clearly seen from the Kerr ro-\ntation oscillations of the annealed \flm in Fig. 1(f)(open\ncircles). Therefore, a sum of two damped sinusoidal func-\ntions is used to give the best \ft of the data as shown in\nsolid lines in Fig. 1(f). In the case of the as-grown sam-\nple, only a single damped sinusoidal function ( A2= 0)\nis needed in order to give the best \ft (solid lines) to the\nresults (open circles) as shown in Fig. 1(e). The best\n\ft curves demonstrate the \ftting function a good repre-\nsentation of the experimental results. The \frst dominat-\ning mode (i= 1) observed in both \flms is the magnetic\nuniform precession mode obeying the Landau-Lifshitz-\nGilbert (LLG) equation while the second mode ( i= 2),\nwhich only presents in the annealed \flm, is the \frst-order\nPSSW. We note that the possibility of a DE mode is ex-\ncluded as discussed in supplemental materials.\nFourier analysis is used to explore the observed\nmultiple-mode oscillations which are presented in Fig. 2.\nFig. 2(a) presents the Fourier spectrum for the as-grown\n\flm showing a single precession mode, the Kittel mode.\nFig. 2(b) presents the Fourier spectrum of the annealed\n\flm. In addition to the main peak denoting the Kittel\nmode (1st mode), two extra high-frequency modes are\nobserved. The second mode (red dashed line), which\nonly becomes profound at high external \felds, is the\nPSSW mode. The amplitude of the Fourier spectrum3\nof the PSSW mode increases as the external magnetic\n\feld increases, indicating that the excitation of PSSW\nmodes gets stronger as the sample magnetization be-\ncomes more uniform. The frequency of the Kittel mode\nin the as-grown \flm is plotted as a function of the ex-\nternal \feld strength in blue triangles in Fig. 2(c). The\nfrequencies of the Kittel mode and PSSW mode in the\nannealed \flm are plotted vs the \feld strength in red cir-\ncles and pink squares in Fig. 2(c), respectively. The third\nmode (blue dashed line in Fig. 2(b)) with a frequency\nof 95:8 GHz shows no magnetic-\feld-dependence, there-\nfore is not of a magnetic origin. This third mode is a\ncoherent acoustic phonon (CAP) mode from the inter-\nface between the Fe 3O4\flm and substrat. The non-\nappearance of the CAP mode in the as-grown \flm may\nbe due to phonon di\u000braction caused by defects. The\nKittel and the PSSW frequencies in Fig. 2(c) are \ft-\nted using the Eq. (S8) and Eq. (S10) derived from the\nLLG equation. The \ftting processes are detailed in the\nsupplemental material. The \fts are in agreement with\nthe values (data points) extracted from damped sinu-\nsoidal \ftting and plotted in solid lines in Fig. 2(c). The\nbest \ftted values of the perpendicular anisotropy con-\nstantKuand cubic anisotropy constant Kcare obtained\nfor both samples and are comparable with other stud-\nies on Fe 3O4thin \flms.21,22The extracted value of the\nexchange sti\u000bness constant Aexfrom the PSSW mode \ft-\nting is (1:47\u00060:15)µerg=cm which is reasonably close to\nthe reported value of Aex= 1:19µerg=cm estimated for\nexchange coupling constant of Fe 3O4.23If we substitute\nKu,Kc,LandMsof as-grown \flm as well as the ex-\ntractedAexfrom annealed \flm into Equation (S10), the\n\feld dependence of the frequency of the PSSW mode in\nthe as-grown \flm is estimated and plotted as the dashed\nline in Fig. 2(c). The di\u000berence in the PSSW mode be-\ntween two samples comes from the di\u000berent \flm thickness\nbetween them. However, this PSSW mode is not ob-\nserved, neither in the TR-MOKE results (Fig. 1(e)) nor\nthe corresponding Fourier spectrum (Fig. 2(a)). There-\nfore, the PSSW mode is only observed in the annealed\nFe3O4\flm and is not present in the as-grown \flm due\nto magnon scattering induced by a high density of de-\nfects in this sample. The fact that the PSSW mode is\nnot observed in the low \feld region can be attributed to\nmagnon scattering due to the magnetic inhomogeneities.\nIn support of the experimental data a model based on\nthe Landau-Lifshitz-Bloch (LLB)24{27equation has been\nused to provide further insight into the cause of the ad-\nditional peaks seen in the Fourier spectrum.\nThe relaxation time \u001c1of the Kittel mode for both\nas-grown and annealed samples is obtained from the\nbest \ft of Equation (1) as a function of external \feld\nstrengths and presented as blue triangles and red circles\nin Fig. 3(a), respectively. The relaxation time \u001c2of the\nsecond mode, the PSSW mode, observed in the annealed\nsample is also obtained and presented as pink squares\nin Fig. 3(a). The PSSW mode emerges in the annealed\n\flm only when the applied \feld is su\u000eciently large. InFig. 3(a), the relaxation time of the Kittel mode, \u001c1, in-\ncreases with the \feld strengths in the as-grown sample.\nThis is as expected since the relaxation time is nega-\ntively correlated to the energy dissipation rate of each\nspin wave modes, and the energy dissipation rate de-\ncreases when the magnetization becomes more uniform.\nIn the annealed \flm, however, the relaxation time of the\nKittel mode decreases with the \feld strengths. This is in\ncontrast with the observation in the as-grown \flm. There\nare less lattice defects in annealed \flm, which means that\nthe contribution of magnon scattering on the energy dis-\nsipation rate is not as strong as that in the as-grown\n\flm. However, one still expects that as the external \feld\nincreases, the magnetization precession becomes more\nuniform and leads its energy dissipation rate decreasing\nrather than increasing. The fact that the precessional\nenergy dissipates quicker as the increased \feld strength\nindicates that there must be an additional channel. This\nadditional channel is responsible for the extra energy dis-\nsipation from the Kittel mode, and this channel draws\nmore energy from the Kittel mode as the external \feld\nstrength increases. On the other hand, PSSW modes are\nobserved only in the annealed sample. The relaxation\ntime,\u001c2, of the PSSW mode increases dramatically with\nthe external \feld strength while \u001c1of the Kittel mode\ndecreases. This suggests that the PSSW mode acts as\nthe additional channel responsible for the extra energy\ndissipation of the Kittel mode. In this case, the energy\nis transferred from the Kittel mode to the PSSW mode\nduring magnetization precession in the annealed sam-\nple. This energy transfer between two modes increases\nas the in-plane magnetization precession becomes more\nuniform. It should be noted that the relaxation time of\nKittel model decreases with a larger \feld for the annealed\nsample may also result from that the e\u000bective damping\nreaches the intrinsic damping value as shown in the fol-\nlowing discussion. Even though, the PSSW mode acting\nas an additional energy dissipation channel is clari\fed.\nThe e\u000bective damping constants \u000be\u000bare derived from\nthe Kittel mode using28,29\u000be\u000b= 1=2\u0019f1\u001c1, wheref1\nand\u001c1are the frequency and relaxation time of Kittel\nmode, respectively. The variations of the damping with\nthe external \feld strength for the as-grown and annealed\n\flms are shown in Fig. 3(d). The e\u000bective damping con-\nstant\u000be\u000bin both \flms decreases signi\fcantly with in-\ncreasing applied \feld. This \feld dependence of the ef-\nfective damping is commonly observed and explained.\nThe e\u000bective damping constant consists of intrinsic and\nextrinsic components and the extrinsic damping mainly\ncomes from magnetic inhomogeneity or two magnon scat-\ntering. In the low magnetic \feld region, the e\u000bective\n\feld is dominated by the spatially \ructuating anisotropy\n\feld which leads to increased damping, while in the high\n\feld region the e\u000bect of anisotropy \feld becomes weak\nsince the external \feld dominates and hence the e\u000bect\nof the magnetic inhomogeneity decreases along with the\ndamping constant.30{32In our measurement, the e\u000bec-\ntive damping constant of the as-grown \flm is larger than4\nFIG. 3. Extracted \u001c1of Kittel mode from as-grown Fe 3O4\n\flm (blue triangles) and annealed Fe 3O4\flm (red circles), \u001c2\nof PSSW mode from the annealed \flm (pink squares). Solid\nlines are guide to eyes. (b)(c) Normalized time-domain \ftting\ncurves of PSSW mode (c) and Kittel mode (d), dashed lines\nare the envelop curves to monitor the amplitude decay. (d)\n\u000be\u000bof as-grown (blue) and annealed (red) Fe 3O4\flms. From\nthe single exponential decay \ftting (solid lines), the intrinsic\ndamping\u000b0of both \flms are estimated.\nthat of the annealed Fe 3O4\flm at small external \feld\nstrengths. This is consistent with the larger damping\nassociated with lattice defects introduced by APBs or\nother defects present in the as-grown \flm. However, an\nintriguing \fnding is that as the external \feld strength be-\ncomes larger, \u000be\u000bof the as-grown \flm becomes smaller\nthan\u000be\u000bof the annealed \flm. In light of the fact that\nthe measured e\u000bective damping parameter usually de-\ncreases dramatically with the increasing of the applied\n\feld and eventually reaches a constant value,30,31,33,34\na phenomenological \ftting using single exponential de-\ncay is applied to describe the \feld dependence of \u000be\u000bas\nshown in\u000be\u000b=\u000b0+\u000bextexp (\u0000\fH), where\u000b0denotes\nthe intrinsic damping. The second term \u000bextexp (\u0000\fH)\nrepresents the extrinsic damping term which is depen-\ndent to the applied \feld. The best \fttings as shown in\nFig. 3(d) give the value \u000b0= 0:039\u00060:004 for the as-\ngrown and\u000b0= 0:063\u00060:010 for the annealed \flm. The\nintrinsic damping constant \u000b0of the defect-free annealed\nFe3O4\flm is thus 62% higher than that of the as-grown\n\flm with high density of defects.\nAs discussed, the energy is transferred from the Kit-\ntel mode to the PSSW mode at large external \felds.\nTo visualise this energy transfer, the time sequences of\nthe Kittel and PSSW modes under the highest external\n\feld strengths are simulated using the extracted parame-\nters (relaxation time, frequencies and initial phases), and\nplotted in Fig. 3(c) and (d), respectively. Comparing the\ntwo sets of time sequences from bottom to top as the\n\feld increases from 3616 Oe to 5401 Oe in Fig. 3(c) and\n(d), the decay of the PSSW mode slows down while the\ndecay of the Kittel mode gradually speeds up. With-\nout the PSSW mode, the decay of the Kittel mode is\nexpected to decrease with the increasing of the external\n\feld strength, as in the case of the as-grown \flm. The\nexistence of the PSSW mode has reversed this trend by\nFIG. 4. Left panel: Without defect-induced magnon scatter-\ning, the PSSW is observed in annealed Fe 3O4\flm along with\ncoherent Kittel magnetic precession. The Gilbert damping\nis enhanced due to the additional energy dissipation chan-\nnel from Kittel mode to PSSW mode. Right panel: only the\nKittel mode is observed because of the existence of magnetic\ninhomogeneity (red symbols). The energy dissipation channel\nfrom Kittel mode to PSSW mode no longer exists leading to\na smaller damping compared to the defect-free \flm.\nproviding an extra energy dissipation channel, which con-\ntributed to the intrinsic damping in the annealed Fe 3O4\n\flm. We therefore propose that this unexpected enhance-\nment of magnetic damping in the annealed \flm is related\nto the emerging of the \frst order PSSW mode, which\ndraws the energy from the Kittel mode and speeds its\nrelaxation process. This energy transfer process may be\nattributed to two potential mechanisms: nonlinear spin-\nwaves transition35and two-magnon scattering36. The\nphysical picture to explain the di\u000berence in magnetic\ndamping of these two Fe 3O4\flms is illustrated in Fig. 4.\nIn the as-grown \flm, only the Kittel mode exists and\nPSSW modes isnt populated even in high \feld region\ndue to the magnetization inhomogeneity caused by de-\nfects. The magnetic damping follows the usual trend and\napproaches to its intrinsic value as the extrinsic contribu-\ntion reduces in the high \feld region. In the annealed \flm\nwith no structural defects, a \frst order PSSW mode is\npopulated at the high \feld region. The amplitude of this\nPSSW mode increases and its relaxation time decreases\nwith the external \feld strength, which indicates the en-\nergy drawn into the PSSW mode increases. This sug-\ngests that the uniformity of in-plane magnetization is the\nkey to populate higher energy magnon, PSSW modes,37\nacross the \flm thickness. The more uniform the mag-\nnetization is, the more energy the PSSW mode draws\nfrom the Kittel mode, which leads to a larger intrinsic\ndamping of the Kittel mode in the defect-free \flms.\nIn summary, the dynamic damping properties of the\nepitaxial grown Fe 3O4thin \flms with controlled atomic5\nscale structures have been studied by an all-optical TR-\nMOKE technique. In addition to the dominate uni-\nform magnetization precession Kittel mode, the \frst-\norder PSSW mode is observed exclusively in the annealed\n\flm with low density of defects when the applied \feld is\nsu\u000eciently strong. The amplitude of the PSSW mode in-\ncreases and its relaxation time decreases as the external\nmagnetic \feld increases. Furthermore, the extracted in-\ntrinsic magnetic damping constant of the defect-free \flm\nis much larger than that of the \flm with defects. This\nenhancement is attributed to the PSSW mode, which\nprovides the additional channel for the extra energy dis-\nsipation of the Kittel mode. During the magnetization\nprecession of the defect-free Fe 3O4\flm, energy is trans-\nferred from the Kittel mode to the PSSW mode and this\nenergy transfer between two modes increase as in-plane\nmagnetization precession become more uniform result-\ning in the enhanced intrinsic magnetic damping. Our\nwork demonstrates that the defect-free \flm structure is\nessential for the population of the PSSW modes and at\nthe same time the existence of these PSSW modes pro-\nvides an additional channel for the energy dissipation of\nthe Kittel mode, which leads to the enhanced intrinsic\nmagnetic damping in the epitaxial defect-free Fe 3O4thin\n\flms. This result o\u000bers new insights into engineering the\nmagnetic damping for potential spintronics applications\nemploying Fe 3O4as a function material.\nSee supplemental material for details of sample prepa-\nration, TR-MOKE setup, TEM images, data \ftting pro-\ncess, exchlusion of DE mode, numeriacal simulations,\nhysterisis loops and CAP mode.\nACKNOWLEDGMENTS\nThis work was supported by the National Basic Re-\nsearch Program of China (No. 2014CB921101), National\nNatural Science Foundation of China (No. 61274102,\nNo. 61427812 and No. 11574137), The National\nKey Research and Development Program of China (No.\n2016YFA0300803), Jiangsu NSF (BK20140054), Jiangsu\nShuangchuan Programme. RFLE acknowledges the \f-\nnancial support of the Engineering and Physical Sciences\nResearch Council (Grant No. EPSRC EP/P022006/1).\n1M. Van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae,\nW. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88,\n227201 (2002).\n2M. J ackl, V. I. 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B 73, 144424 (2006)."    },    {        "title": "1503.01478v2.Critical_current_destabilizing_perpendicular_magnetization_by_the_spin_Hall_effect.pdf",        "content": "arXiv:1503.01478v2  [cond-mat.mes-hall]  1 Aug 2015Critical current destabilizing perpendicular magnetizat ion by the spin Hall effect\nTomohiro Taniguchi1, Seiji Mitani2, and Masamitsu Hayashi2\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba 305-8568, Japan\n2National Institute for Materials Science, Tsukuba 305-004 7, Japan\n(Dated: July 5, 2018)\nThe critical current needed to destabilize the magnetizati on of a perpendicular ferromagnet via\nthe spin Hall effect is studied. Both the dampinglike and field like torques associated with the spin\ncurrent generated by the spin Hall effect is included in the La ndau-Lifshitz-Gilbert equation to\nmodel the system. In the absence of the fieldlike torque, the c ritical current is independent of the\ndamping constant and is much larger than that of conventiona l spin torque switching of collinear\nmagnetic systems, as in magnetic tunnel junctions. With the fieldlike torque included, we find that\nthe critical current scales with the damping constant as α0(i.e., damping independent), α, and\nα1/2depending on the sign of the fieldlike torque and other parame ters such as the external field.\nNumerical and analytical results show that the critical cur rent can be significantly reduced when\nthe fieldlike torque possesses the appropriate sign, i.e. wh en the effective field associated with the\nfieldlike torque is pointing opposite to the spin direction o f the incoming electrons. These results\nprovideapathwaytoreducingthecurrentneededtoswitch ma gnetization usingthespin Hall effect.\nPACS numbers: 75.78.-n, 75.70.Tj, 75.76.+j, 75.40.Mg\nI. INTRODUCTION\nThe spin Hall effect1–3(SHE) in a nonmagnetic heavy\nmetal generates pure spin current flowing along the di-\nrection perpendicular to an electric current. The spin\ncurrent excites magnetization dynamics in a ferromagnet\nattached to the nonmagnetic heavy metal by the spin-\ntransfer effect4,5. There have been a number of exper-\nimental reports on magnetization switching and steady\nprecession induced by the spin Hall effect6–9. These dy-\nnamics have attracted great attention recently from the\nviewpoints ofboth fundamental physicsand practicalap-\nplications.\nAn important issue to be solved on the magnetization\ndynamics triggered by the spin Hall effect is the reduc-\ntion of the critical current density needed to destabilize\nthe magnetization from its equilibrium direction, which\ndetermines the current needed to switch the magneti-\nzation direction or to induce magnetization oscillation.\nThe reported critical current density for switching8,10–13\nor precession9is relatively high, typically larger than 107\nA/cm2. One of the reasons behind this may be related\nto the recently predicted damping constant independent\ncritical current when SHE is used14,15. This is in con-\ntrast to spin-transfer-induced magnetization switching in\na typical giant magnetoresistance (GMR) or magnetic\ntunnel junction (MTJ) device where the critical current\nis expected to be proportional to the Gilbert damping\nconstant α. Here the magnetization dynamics is excited\nas a result of the competition between the spin torque\nand the damping torque16. Since the damping constant\nfor typical ferromagnet in GMR or MTJ devices is rela-\ntively small ( α∼10−2−10−3)17,18, it can explain why\nthe critical current is larger for the SHE driven systems.\nThus in particular for device application purposes, it is\ncrucial to find experimental conditions in which the mag-netization dynamics can be excited with lower current.\nAnother factor that might contribute to the reduc-\ntion of the critical current is the presence of the field\nlike torque19. In the GMR/MTJ systems, both the con-\nventional spin torque, often referred to as the damp-\ninglike torque, and the fieldlike torque arise from the\nspin transfer between the conduction electrons and the\nmagnetization4,19–23. Due to the short relaxation length\nof the transverse spin of the conduction electrons24,25,\nthe damping like torque is typically larger than the field-\nlike torque. Indeed, the magnitude of the field like\ntorque experimentally found in GMR/MTJ systems has\nbeen reported to be much smaller than the damping like\ntorque26–29. Because of its smallness, the fieldlike torque\nhad notbeen consideredin estimatingthe criticalcurrent\nintheGMR/MTJsystems16,30–32,althoughitdoesplaya\nkeyrolein particularsystems33,34. In contrast, recentex-\nperiments found that the fieldlike torque associated with\nthe SHE is larger than the damping like torque35–40.\nThe physical origin of the large SHE-induced field-\nlike torque still remains unclear. Other possible sources\ncan be the Rashba effect36,41–44, bulk effect45, and the\nout of plane spin orbit torque46. Interestingly, the field\nlike torque has been reported to show a large angu-\nlar dependence36,37,47(the angle between the current\nand the magnetization), which cannot be explained by\nthe conventional formalism of spin-transfer torque in\nGMR/MTJsystems. Thefieldliketorqueactsasatorque\nduetoanexternalfieldandmodifiestheenergylandscape\nof the magnetization. As a result, a large fieldlike torque\ncan significantly influence the critical current. However,\nthe fieldlike torque had not been taken into account in\nconsidering the current needed to destabilize the magne-\ntization from its equilibrium direction and thus its role\nis still unclear.\nIn this paper, we study the critical current needed to2\ndestabilize a perpendicular ferromagnet by the spin Hall\neffect. The Landau-Lifshitz-Gilbert(LLG) equationwith\nthe dampinglike and fieldlike torques associated with the\nspin Hall effect is solved both numerically and analyti-\ncally. Wefindthatthecriticalcurrentcanbesignificantly\nreduced when the fieldlike torque possesses the appropri-\nate sign with respect to the dampinglike torque. With\nthe fieldlike torque included, the critical current scales\nwith the damping constant as α0(i.e., damping indepen-\ndent),α, andα1/2, depending on the sign of the field-\nlike torque and other parameters. Analytical formulas\nof such damping-dependent critical current are derived\n[Eqs. (19)-(21)], and they show good agreement with the\nnumerical calculations. From these results, we find con-\nditions in which the critical current can be significantly\nreduced compared to the damping-independent thresh-\nold, i.e., systems without the fieldlike torque.\nThe paper is organized as follows. In Sec. II, we\nschematically describe the system under consideration.\nWe discuss the definition of the critical current in Sec.\nIII. Section IV summarizes the dependences of the crit-\nical current on the direction of the damping constant,\nthe in-plane field, and the fieldlike torque obtained by\nthe numerical simulation. The analytical formulas of the\ncritical current and their comparison to the numerical\nsimulations are discussed in Sec. V. The condition at\nwhich damping-dependent critical current occurs is also\ndiscussed in this section. The conclusion follows in Sec.\nVI.\nII. SYSTEM DESCRIPTION\nThe system we consider is schematically shown in Fig.\n1, where an electric current flowing along the x-direction\ninjects a spin current into the ferromagnet by the spin\nHall effect. The magnetization dynamics in the ferro-\nmagnet is described by the LLG equation,\ndm\ndt=−γm×H+αm×dm\ndt\n−γHsm×(ey×m)−γβHsm×ey,(1)\nwhereγandαarethe gyromagneticratioandtheGilbert\ndamping constant, respectively. We assume that the\nmagnetization of the ferromagnet points along the film\nnormal (i.e., along the zaxis), and an external in-plane\nmagnetic field is applied along the xoryaxis. The total\nmagnetic field His given by\nH=HapplnH+HKmzez, (2)\nwhereHapplis the external field directed along the xor\nyaxis and HKis the uniaxial anisotropy field along the\nzaxis.nHandeiare unit vectors that dictate the di-\nrection of the uniaxial anisotropy field and the iaxis,\nrespectively. Here we call the external field along the x\nandydirections the longitudinal and transverse fields,\nrespectively. The third and fourth terms on the right-\nhand side of Eq. (1) are the damping like and fieldlikeHappl // y\nHappl // x\nm\ncurrentxz\ny\nFIG. 1. Schematic view of the spin-Hall system. The x\naxis is parallel to current, whereas the zaxis is normal to the\nfilm plane. The spin direction of the electrons entering the\nmagnetic layer via the spin Hall effect points along the + yor\n−ydirection.\ntorques associated with the spin Hall effect, respectively.\nThetorquestrength Hscanbeexpressedwiththecurrent\ndensityj, the spin Hall angle ϑ, the saturation magneti-\nzationM, and the thickness of the ferromagnet d, i.e.,\nHs=/planckover2pi1ϑj\n2eMd. (3)\nThe ratio of the fieldlike torque to the damping like\ntorque is represented by β. Recent experiments found\nthatβis positive and is larger than 135–40.\nThe magnetization dynamics described by the LLG\nequation can be regarded as a motion of a point particle\non a two-dimensional energy landscape. In the presence\nof the fieldlike torque, the energy map is determined by\nthe energy density given by34\nE=−M/integraldisplay\ndm·H−βMHsm·ey.(4)\nThen, the external field torque and the fieldlike torque,\nwhich are the first and fourth terms on the right-hand-\nside of Eq. (1), can be expressed as −γm×B, where the\neffective field Bis\nB=−∂E\n∂Mm. (5)\nThe initial state of the numerical simulation is chosen to\nbe the direction corresponding to the minimum of the\neffective energy density E. The explicit forms of the ini-\ntial state for the longitudinal and the transverse external\nfields are shown in Appendix A.\nWe emphasize for the latter discussion in Sec. V that,\nusing Eqs. (1), (4), and (5), the time change of the effec-\ntive energy density is described as\ndE\ndt=dEs\ndt+dEα\ndt. (6)3\nHere the first and second terms on the right-hand side\nare the rates of the work done by the spin Hall torque\nand the dissipation due to damping, respectively, which\nare explicitly given by\ndEs\ndt=γMHs[ey·B−(m·ey)(m·B)],(7)\ndEα\ndt=−αγM/bracketleftBig\nB2−(m·B)2/bracketrightBig\n. (8)\nThe sign of Eq. (7) depends on the current direction\nand the effective magnetic field, while that of Eq. (8) is\nalways negative.\nThe magnetic parameters used in this paper mimic the\nconditions achieved in CoFeB/MgO heterostructures48;\nM= 1500 emu/c.c., HK= 540 Oe, ϑ= 0.1,γ=\n1.76×107rad/(Oe s), and d= 1.0 nm. The value of\nβis varied from −2, 0, to 2. Note that we have used a\nreducedHK(Refs.8,49) in ordertoobtain criticalcurrents\nthat are the same order of magnitude with that obtained\nexperimentally. We confirmed that the following discus-\nsions are applicable for a large value of HK(∼1T).\nIII. DEFINITION OF CRITICAL CURRENT\nIn this section, we describe how we determine the crit-\nical current from the numerical simulations. In exper-\niments, the critical current is determined from the ob-\nservation of the magnetization reversal8,12,41,46,48–50. As\nmentioned in Sec. II, in this paper, the initial state for\ncalculation is chosen to be the minimum of the effective\nenergy density. Usually, there are two minimum points\nabove and below the xyplane because of the symmetry.\nThroughout this paper, the initial state is chosen to be\nthe minimum point above the xyplane, i.e., mz(0)>0,\nfor convention.” It should be noted that, once the mag-\nnetization arrives at the xyplane during the current ap-\nplication, it can move to the other hemisphere after the\ncurrent is turned off due to, for example, thermal fluc-\ntuation. Therefore, here we define the critical current as\nthe minimum current satisfying the condition\nlim\nt→∞mz(t)< ǫ, (9)\nwhere a small positive real number ǫis chosen to be\n0.001. The duration of the simulations is fixed to 5 µs,\nlong enough such that all the transient effects due to the\ncurrent application are relaxed. Figures 2(a) and 2(b)\nshow examples of the magnetization dynamics close to\nthe critical current, which are obtained from the numer-\nical simulation of Eq. (1). As shown, the magnetization\nstays near the initial state for j= 3.1×106A/cm2, while\nit moves to the xyplane for j= 3.2×106A/cm2. Thus,\nthe critical current is determined as 3 .2×106A/cm2in\nthis case.\nWe note that the choice of the definition of the criti-\ncal current has some arbitrariness. For comparison, weFIG. 2. Time evolution ofthe zcomponentof themagnetiza-\ntionmzin the presence of the transverse field of Happl= 200\nwith (a) j= 3.1×106A/cm2and (b)j= 3.2×106A/cm2.\nThe value of βis zero.\nshow numerically evaluated critical current with a differ-\nent definition in Appendix B. The main results of this\npaper, e.g., the dependence of the critical current on the\ndamping constant, are not affected by the definition.\nWe also point out that the critical current defined by\nEq. (9) focuses on the instability threshold, and does\nnot guarantee a deterministic reversal. For example,\nin the case of Fig. 2(b), the reversal becomes prob-\nabilistic because the magnetization, starting along + z,\nstops its dynamics at the xyplane and can move back\nto its original direction or rotate to a point along −z\nresulting in magnetization reversal. Such probabilistic\nreversal can be measured experimentally using transport\nmeasurements8,12,41,46,49,50or by studying nucleation of\nmagnetic domains via magnetic imaging48. On the other\nhand, it hasbeen reportedthat deterministicreversalcan\ntake place when a longitudinal in-plane field is applied\nalongside the current41,49. It is difficult to determine the\ncritical current analytically for the deterministic switch-\ning for all conditions since, as in the case of Fig. 2(b),\nthe magnetization often stops at the xyplane during the\ncurrent application. This occurs especially in the pres-\nence of the transverse magnetic field because all torques\nbecome zero at m=±eyand the dynamics stops. Here\nwe thus focus on the probabilistic reversal.4\nFIG. 3. Numerically evaluated mzatt= 5µs for (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse ( nH=ey)\nfields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The damping constant is α= 0.005. The color scale\nindicates the zcomponent of the magnetization ( mz) att= 5µs. The red/white boundary indicates the critical current fo r\nprobabilistic switching, whereas the red/blue boundary gi ves the critical current for deterministic switching.\nIV. NUMERICALLY ESTIMATED CRITICAL\nCURRENT\nIn this section, we show numerically evaluated critical\ncurrent for different conditions. We solve Eq. (1) and\napply Eq. (9) to determine the critical current. Figure\n3 shows the value of mzatt= 5µs in the presence of\n(a)-(c) the longitudinal ( nH=ex) and (d)-(f) the trans-\nverse (nH=ey) fields. The value of βis 0 for Figs.\n3(a) and 3(d), 2 .0 for Figs. 3(b) and 3(e), and −2.0 for\nFigs. 3(c) and 3(f), respectively. The damping constant\nisα= 0.005. The red/white boundary indicates the crit-\nical current for the probabilistic switching, whereas the\nred and blue ( mz=−1) boundary gives the critical cur-\nrent for the deterministic switching. Using these results\nand the definition of the critical current given by Eq. (9),\nand performing similar calculations for different values of\nα, wesummarizethedependenceofthecriticalcurrenton\nthe longitudinal and transverse magnetic fields in Fig. 4.\nThe damping constant is varied as the following in each\nplot:α= 0.005, 0.01, and 0 .02. The solid lines in Fig. 4\nrepresent the analytical formula derived in Sec. V.A. In the presence of longitudinal field\nIn the case of the longitudinal field and β= 0\nshown in Fig. 4(a), the critical current is damping-\nindependent. Such damping-independent critical current\nhas been reported previously for deterministic magneti-\nzation switching14,15. Similarly, in the case of the longi-\ntudinal field and negative β(β=−2.0) shown in Fig.\n4(c), the critical current is damping-independent. In\nthese cases, the magnitude of the critical current is rel-\natively high. In particular, near zero field, the critical\ncurrent exceeds ∼108A/cm2, which is close to the limit\nof experimentally accessible value. These results indicate\nthat the useofthe longitudinal field with zeroornegative\nβis ineffective for the reduction of the critical current.\nOn the other hand, when βis positive, the critical cur-\nrent depends on the damping constant, as shown in Fig.\n4(b). Note that positive βis reported for the torques\nassociated with the spin Hall effect or Rashba effect in\nthe heterostructures studied experimentally35–37,39. The\nmagnitude of the critical current, ∼10×106A/cm2, is\nrelatively small compared with the cases of zero or neg-\nativeβ. In this case, the use of a low damping material\nis effective to reduce the critical current. Interestingly,\nthe critical current is not proportional to the damping\nconstant, while that previously calculated for a GMR or\nMTJ system16is proportional to α. For example, the5\nlongitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-6040 60 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-5040 50 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 \ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=-2.0\n-50\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)\n(d) (e) (f)longitudinal magnetic field (Oe)0 50 100 150 200β=-2.0\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02critical current density (10 6 A/cm 2)\n0\n-10050 \n-150100150\n-50\nFIG. 4. Numerically evaluated critical currents in the pres ence of (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse\n(nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0, respectively. The solid lines are analytically\nestimated critical current in Sec. V.\ncritical current at zero longitudinal field in Fig. 4(b) is\n12.3,17.2, and24 .0×106A/cm2forα= 0.005,0.01, and\n0.02, respectively. These values indicate that the critical\ncurrent is proportional to α1/2. In fact, the analytical\nformula derived in Sec. V shows that the critical current\nis proportional to α1/2for positive β[see Eq. (19)].\nTo summarize the case of the longitudinal field, the\nuse of a heterostructure with positive β, which is found\nexperimentally, has the possibility to reduce the critical\ncurrent if a ferromagnet with low damping constant is\nused. In this case, the critical current is proportional to\nα1/2, which has not been found in previous works.\nB. In the presence of transverse field\nIn the presence of the transverse field with β= 0,\nthe critical current shows a complex dependence on the\ndamping constant α, as shown in Fig. 4(d). When the\ncurrent and the transversefield areboth positive (or neg-\native), the critical current is proportional to the damping\nconstant αexcept near zero field. The numerically cal-\nculated critical current matches well with the analytical\nresult, Eq. (20), shown by the solid lines. In this case,\nthe use of the low damping material results in the reduc-\ntion of the critical current. On the other hand, when the\ncurrent and the transversefield possessthe opposite sign,\nthecriticalcurrentisdampingindependent. Moreover,in\nthis case, thecriticalcurrentisofthe orderof108A/cm2.\nThus, it is preferable to use the current and field having\nthe same sign for the reduction of the critical current. Itshould be noted that, in our definition, the same sign of\ncurrent and field corresponds to the case when the direc-\ntion ofincoming electrons’spin (due to the SHE) and the\ntransverse field are opposite to each other. The reason\nwhy the critical current becomes damping dependent in\nthis situation will be explained in Sec. V.\nWhenβis positive the critical current depends on the\ndamping constant for the whole range of the transverse\nfield, as shown in Fig. 4(e). The critical current is\nroughly proportional to α1/2, in particular, close to zero\nfield. The solid lines display the analytical formula, Eq.\n(21), and showgood agreementwith the numericalcalcu-\nlations. The damping dependence of the critical current\nbecomes complex when the magnitude of the transverse\nfield is increased [see Eq. (21)]. We note that the critical\ncurrentfor the positive βin Fig. 4(e) is smallerthan that\nforβ= 0 in Fig. 4(d) for the whole range of Happl.\nOn the other hand, when βis negative, the critical\ncurrent is almost independent of α, especially near zero\nfield. However, when the transverse field is increased,\nthere is a regime where the critical current depends on\nthe damping constant. Such transition of the critical\ncurrent with the transverse field is also predicted by the\nanalytical solution, Eq. (21).\nTosummarizethe caseofthe transversefield, the αde-\npendence of the critical current can be categorized into\nthe following: α0(damping independent), α,α1/2, or\nother complex behavior. As with the case of the longi-\ntudinal field, the use of a heterostructure with positive β\nallowsreductionofthe criticalcurrentwhen lowdamping\nferromagnet is used. Overall, the most efficient condition6\nto reduce the critical current is to use the transverse field\nwith heterostructures that possess low αand positive β.\nIn this case, the critical current is reduced to the order\nof 106A/cm2.\nV. ANALYTICAL FORMULA OF CRITICAL\nCURRENT\nIn this section, we derive the analytical formula of the\ncritical current from the linearized LLG equation51. The\ncomplex dependences ofthe critical currentonthe damp-\ning constant αdiscussed in Sec. IV are well explained by\nthe analytical formula. We also discuss the physical in-\nsight obtained from the analytical formulas.\nA. Derivation of the critical current\nTo derive the critical current, we consider the stable\ncondition of the magnetization near its equilibrium. It is\nconvenient to introduce a new coordinate XYZin which\ntheZaxis is parallel to the equilibrium direction. The\nrotationfromthe xyz-coordinatetothe XYZcoordinate\nis performed by the rotation matrix\nR=\ncosθ0−sinθ\n0 1 0\nsinθ0 cosθ\n\ncosϕsinϕ0\n−sinϕcosϕ0\n0 0 1\n,(10)\nwhere (θ,ϕ) are the polar and azimuth angles of the\nmagnetization at equilibrium. The equilibrium magne-\ntization direction under the longitudinal and transverse\nmagnetic field is given by Eqs. (A1) and (A2), respec-\ntively. Since we are interested in small excitation of the\nmagnetization around its equilibrium, we assume that\nthe components of the magnetization in the XYZcoor-\ndinate satisfy mZ≃1 and|mX|,|mY| ≪1. Then, the\nLLG equation is linearized as\n1\nγd\ndt/parenleftbigg\nmX\nmY/parenrightbigg\n+M/parenleftbigg\nmX\nmY/parenrightbigg\n=−Hs/parenleftbigg\ncosθsinϕ\ncosϕ/parenrightbigg\n,(11)\nwhere the components of the 2 ×2 matrix Mare\nM1,1=αBX−Hssinθsinϕ, (12)\nM1,2=BY, (13)\nM2,1=BX (14)\nM2,2=αBY−Hssinθsinϕ. (15)\nHere,BXandBYare defined as\nBX=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,\n(16)BY=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,\n(17)\nwhereϕHrepresents the direction of the external field\nwithin the xyplane:ϕH= 0 for the longitudinal field\nandπ/2 for the transverse field.\nThe solution of Eq. (11) is mX,mY∝\nexp{γ[±i/radicalbig\ndet[M]−(Tr[M]/2)2−Tr[M]/2]t}, where\ndet[M] and Tr[ M] are the determinant and trace of\nthe matrix M, respectively. The imaginary part of the\nexponent determines the oscillation frequency around\ntheZaxis, whereas the real part determines the time\nevolution of the oscillation amplitude. The critical\ncurrent is defined as the current at which the real part\nof the exponent is zero. Then, the condition Tr[ M] = 0\ngives\nα(BX+BY)−2Hssinθsinϕ= 0,(18)\nFor the longitudinal field, Eq. (18) gives\njLONG\nc=±2e√αMd\n/planckover2pi1ϑ/radicalBig\n2H2\nK−H2\nappl/radicalbig\nβ(2+αβ),(19)\nindicating that the critical current is roughly propor-\ntional to α1/2. This formula works for positive βonly52\nif we assume 0 <2+αβ≃2, which is satisfied for typical\nferromagnets. The critical current when the transverse\nfield is applied reads\njTRANS\nc=2αeMd\n/planckover2pi1ϑ(Happl/HK)HK/bracketleftBigg\n1−1\n2/parenleftbiggHappl\nHK/parenrightbigg2/bracketrightBigg\n,(20)\nwhenβ= 0, indicating that the critical current is pro-\nportional to α. The critical current for finite βis\njTRANS\nc=2eMd\n/planckover2pi1ϑ\n×−(1+αβ)Happl±/radicalBig\nH2\nappl+2αβ(2+αβ)H2\nK\nβ(2+αβ).\n(21)\nEquation (21) works for the whole range of |Happl|(<\nHK) for positive β, while it only works when |Happl|>\n2αβ(2 +αβ)HKfor negative β. For example, when\nβ=−2.0, this condition is satisfied when |Happl|>108\nOe forα= 0.005 and |Happl|>152 Oe for α= 0.01.\nHowever the condition is not satisfied for the present\nrange of Happlforα= 0.02. The solid lines in Fig. 4(f)\nshow where Equation (21) is applicable. The zero-field\nlimits of Eqs. (19) and (21) become identical,\nlim\nHappl→0jc=±2e√αMd\n/planckover2pi1ϑ√\n2HK/radicalbig\nβ(2+αβ),(22)\nindicating that the critical current near zero field is pro-\nportional to α1/2whenβ >0.7\nFIG. 5. Magnetization dynamics under the conditions of (a)\nnH=ey,Happl= 50 Oe, β= 0,α= 0.005, and j= 13.2×106\nA/cm2, and (b) nH=ex,Happl= 50 Oe, β= 0,α= 0.005,\nandj= 90×106A/cm2.\nB. Discussions\nThe solid lines in Fig. 4(b), 4(d), 4(e), and 4(f) show\nthe analytical formulas, Eqs. (19), (20), and (21). As\nevident, these formulas agree well with the numerical re-\nsults in the regions where the critical currents depend on\nthe dampingconstant. In this section, we discussthe rea-\nson why the critical current becomes damping dependent\nor damping independent depending on the field direction\nand the sign of β.\nIt is useful for the following discussion to first study\ntypical magnetization dynamics found in the numerical\ncalculations. Figure 5 shows the time evolution of the\nx,yandzcomponents of the magnetization when the\ncritical current depends on [Fig. 5(a)] or is independent\nof [Fig. 5(b)] the damping constant. For the former,\nthe instability is accompanied with a precession of the\nmagnetization. On the other hand, the latter shows that\nthe instability takes place without the precession.\nWe start with the case when the critical current be-\ncomes damping dependent. To provide an intuitive pic-\nture, we schematically show in Fig. 6(a) the torques ex-\nerted on the magnetization during one precession period\nwhen current is applied. The condition is the same with\nthat described in Fig. 5(a), i.e., the transverse magnetic\nfield is applied with β= 0. In Fig. 6(a), magnetization\nis shown by the large black arrow, while the directions\nof the spin Hall torque, the damping torque and the ex-ternal field torque are represented by the solid, dotted\nand dashed lines, respectively (the external field torque\nis tangent to the precession trajectory). As evident in\nFig. 5(a), the precession trajectory is tilted to the posi-\ntiveydirection due to the transversefield. Depending on\nthe direction of the magnetization the spin Hall torque\nhas a component parallel, antiparallel, or normal to the\ndamping torque. This means that the work done by the\nspin Hall torque, denoted by ∆ Esin Fig. 6 (a), is pos-\nitive, negative, or zero at these positions. This can be\nconfirmed numerically when we calculate the work done\nby the spin Hall torque using Eq. (7). For an infinites-\nimal time ∆ t, the work done by the spin Hall torque\nis equal to the rate of its work ( dEs/dt), given in Eq.\n(7), times ∆ t, i.e. ∆Es= (dEs/dt)∆t. The solid line\nin Fig. 6(b) shows an example of the calculated rate of\nthe work done by the spin Hall torque (solid line), dEs/dt\nin Eq. (7). As shown, dEs/dtis positive, negative, and\nzero, when the magnetization undergoes one precession\nperiod. Similarly, the energy dissipated by the damping\ntorque,dEα/dt, can be calculated using Eq. (8) and is\nshown by the dotted line in Fig. 6(b). The calculated\ndissipation due to damping over a precession period is\nalways negative. Details of how the rates, shown in Fig.\n6, are calculated are summarized in Appendix C.\nNote that the strength of the spin Hall torque for\n∆Es>0 is larger than that for ∆ Es<0 due to the an-\ngular dependence of the spin Hall torque, |m×(ey×m)|.\nAlthough it is difficult to see, thesolid line in Fig. 6(b) is\nslightly shifted upward. Thus the total energy supplied\nby the spin Hall torque during one precession, given by/contintegraltext\ndt(dEs/dt), does not average to zero and becomes posi-\ntive. When the current magnitude, |j|, is larger than |jc|\nin Eq. (20), the energy supplied by the spin Hall torque\novercomes the dissipation due to the damping and con-\nsequently the precession amplitude grows, which leads to\nthe magnetization instability shown in Fig. 5(a). The\nsame picture is applicable when both directions of field\nand current are reversed. For this condition, the insta-\nbility of the magnetization is induced by the competition\nbetween the spin Hall torque and the damping torque.\nTherefore, the critical current depends on the damping\nconstant α. When only the current direction is reversed\nin Figs. 6(a) and 6(b) (i.e., the sign of the magnetic field\nand current is opposite to each other), the sign of ∆ Esis\nreversed and thus the total energy supplied by the spin\nHall torque becomes negative. This means that the spin\nHall torque cannot overcome the damping torque to in-\nduce instability. Therefore, the critical current shown in\nEq. (20) only applies to the case when the sign of the\nfield and current is the same. As described in Sec. IV,\nthe same sign of the current and field in our definition\nmeans that the incoming electrons’ spin direction, due\nto the spin Hall effect, is opposite to the transverse field\ndirection.\nNext, we consider the case when the critical current is\ndamping independent. Figure 6 (c) schematically shows\nthe precession trajectory when the applied field points to8\nFIG. 6. (a) A schematic view of the precession trajectory\nin the presence of the applied field in the positive y-direction.\nThe solid and dotted arrows indicate the directions of the\nspin Hall torque and the damping torque, respectively. The\ndashed line, which is the tangent line to the precession tra-\njectory, shows the field torque. The damping torque always\ndissipates energy from the ferromagnet. On the other hand,\nthe spin Hall torque supplies energy (∆ Es>0) when its di-\nrection is anti-parallel to the damping torque, and dissipa tes\nenergy (∆ Es<0) when the direction is parallel to the damp-\ning torque. When the direction of the spin Hall torque is\northogonal to the damping torque, the spin Hall torque does\nnot change the energy (∆ Es= 0). (b) Typical temporal vari-\nation of the rates of the work done by the spin Hall torque,\nEq. (7), (solid) and the dissipation due to damping, Eq. (8)\n(dotted) in the presence of the transverse field. The time is\nnormalized by the period given by Eq. (C7). (c), (d) Similar\nfigures with the longitudinal field.\nthexdirection and β= 0. The corresponding rate of\nwork done by the spin Hall torque and the dissipation\nrate due to the damping torque are shown in Fig. 6 (d).\nSimilar to the previous case, ∆ Escan be positive, nega-\ntive, or zero during one precession period. However, the\ntotal workdoneby the spin Hall torque,/contintegraltext\ndt(dEs/dt), be-\ncomes zero in this case due to the symmetry of angular\ndependence of the spin Hall torque. This means that the\nspin Hall torque cannot compensate the damping torque,\nand thus, a steady precession assumed in the linearized\nLLG equation is not excited. This is evident in the nu-\nmerically calculated magnetization trajectory shown in\nFig. 5(b). For this case, the linearized LLG equation\ngives|jc| → ∞, indicating that the spin Hall torque can-\nnot destabilize the magnetization. The same picture is\nalsoapplicable, forexample, in the absenceofthe applied\nfield and β= 0.\nHowever, an alternative mechanism can cause destabi-\nlization of the magnetization. As schematically shown in\nFigs. 6(a) and 6(c), there is a component of the damping\nlike spin Hall torque that is orthogonal to the damping\ntorque when ∆ Es= 0. The spin Hall torque at this pointis parallel or antiparallel to the field torque depending on\nthe position of the magnetization. When the spin Hall\ntorqueissufficientlylargerthanthefieldtorque,themag-\nnetization moves from its equilibrium position even if the\ntotal energy supplied by the spin Hall torque is zero or\nnegative. This leads to an instability that occurs before\none precession finishes. In this case, it is expected that\nthe critical current is damping-independent because the\ninstability is induced as a competition between the spin\nHall torque and the field torque, not the damping torque.\nThe time evolution of the magnetization shown in Fig.\n5 (b) represents such instability. The work reported in\nRefs.14,49discusses a similar instability condition.\nThe above physical picture is also applicable in the\npresence of the fieldlike torque. The fieldlike torque,\nwhich acts like a torque due to the transversefield, modi-\nfies the equilibrium direction ofthe ferromagnetand thus\nthe precession trajectory. Consequently, the amount of\nenergy supplied by the spin Hall torque and the dissipa-\ntion due to damping is changed when the fieldlike torque\nis present. Depending on the sign of β, the amount of the\nwork done by the spin Hall torque increases or decreases\ncompared to the case with β= 0. In our definition, posi-\ntiveβcontributes to the increase of the supplied energy,\nresulting in the reduction of the critical current. The\ncomplex dependence of the critical current on αarises\nwhen the fieldlike torque is present.\nTo summarize the discussion, the critical current be-\ncomes damping dependent when the energy supplied by\nthe spin Hall torque during a precession around the equi-\nlibrium is positive. The condition that meets this criteria\ndepends on the relative direction of the spin Hall torque\nand the damping torque, as briefly discussed above. To\nderive an analytical formula that describes the condition\natwhichthe criticalcurrentbecomesdamping dependent\nis not an easy task except for some limited cases53.\nVI. CONCLUSION\nIn summary, we have studied the critical current\nneeded to destabilize a perpendicularly magnetized fer-\nromagnet by the spin Hall effect. The Landau-Lifshitz-\nGilbert (LLG) equation that includes both the damping-\nlike and fieldlike torques associated with the spin Hall\neffect is solved numerically and analytically. The criti-\ncal current is found to have different dependence on the\ndamping constant, i.e., the critical current scales with α0\n(damping-independent), α, andα1/2depending on the\nsign of the fieldlike torque. The analytical formulas of\nthe damping-dependent critical current, Eqs. (19), (20),\nand (21), are derived from the linearized LLG equation,\nwhich explain well the numerical results. We find that\nsystems with fieldlike torque having the appropriate sign\n(β >0 in our definition) are the most efficient way to re-\nduce the criticalcurrent. Fortypicalmaterialparameters\nfound in experiment, the critical current can be reduced\nto the order of 106A/cm2when ferromagnets with rea-9\nsonable parameters are used.\nACKNOWLEDGMENTS\nThe authorsacknowledgeT. Yorozu, Y. Shiota, and H.\nKubota in AIST for valuable discussion sthey had with\nus. This workwassupported by JSPS KAKENHIGrant-\nin-AidforYoungScientists(B),GrantNo. 25790044,and\nMEXT R & D Next-Generation Information Technology.\nAppendix A: Initial state of the numerical\nsimulation\nWe assume that the magnetization in the absence of\nthe applied field points to the positive zdirection. In\nthe presence of the field, the equilibrium direction moves\nfrom the zaxis to the xyplane. Let us denote the zenith\nand azimuth angles of the initial state m(t= 0) asθand\nϕ, i.e.,m(t= 0) = (sin θcosϕ,sinθsinϕ,cosθ). When\nthe applied field points to the x-direction ( nH=ex), the\ninitial state is\n/parenleftbigg\nθ\nϕ/parenrightbigg\nnH=ex=/parenleftBigg\nsin−1[/radicalBig\nH2\nappl+(βHs)2/HK]\ntan−1(βHs/Happl)/parenrightBigg\n,(A1)\nwhere the value of ϕis 0< ϕ < π/ 2 forHappl>0 and\nβHs>0,π/2< ϕ < π forHappl<0 andβHs>0,π <\nϕ <3π/2forHappl<0andβHs<0,and3π/2< ϕ <2π\nforHappl>0 andβHs<0. On the other hand, when\nthe applied field points to the y-direction ( nH=ey), the\ninitial state is\n/parenleftbigg\nθ\nϕ/parenrightbigg\nnH=ey=/parenleftbigg\nsin−1[(Happl+βHs)/HK]\nπ/2/parenrightbigg\n,(A2)\nwhere the range of the inverse sine function is −π/2≤\nsin−1x≤π/2. We note that the choice of the initial\nstate does not affect the evaluation of the critical cur-\nrent significantly, especially in the small field and current\nregimes.\nAppendix B: Numerically evaluated critical current\nwith different definition\nAs mentioned in Sec. III, the definition of the critical\ncurrent has arbitrariness. As an example, we show the\ntime evolution of mzunder the conditions of nH=ex,\nHappl=−30 Oe,β= 0, and j= 110×106A/cm2in\nFig. 7. In this case, the magnetization initially starts at\nmz= cos[sin−1(Happl/HK)]≃0.99, and finally moves to\na pointmz→0.12. Since the final state does not satisfy\nEq. (9), this current, j= 110×106A/cm2, should be\nregarded as the current smaller than the critical current\nin Sec. IV. However, from the analytical point of view,\nthis current can be regarded as the current larger than\nmagnetization 01\n-1 -0.50.5\nj=110×10 6 A/cm2\ntime (ns)0 2 4 6 8 10 Happl=-30 Oe\nFIG. 7. Time evolution of the zcomponent of the mag-\nnetization mzin the presence of the longitudinal field with\nHappl=−30 Oe,β= 0, and j= 110×106A/cm2. The\ndotted line is a guide showing mz= 0.\nthe critical current because the final state of the magne-\ntization is far away from the initial equilibrium.\nRegarding this point, we show the numerically eval-\nuated critical current with a different definition. The\nmagnetic state can be regarded as unstable when it fi-\nnally arrives at a point far away from the initial state54.\nThus, for example, one can define the critical current as\na minimum current satisfying\nlim\nt→∞|mz(t)−mz(0)|> δ, (B1)\nwhere a small positive real number δis chosen to be\n0.1 here. Figure 8 summarizes the numerically evalu-\nated critical current with the definition of Eq. (B1). The\nanalytical formulas, Eqs. (19)-(21), still fit well with the\nnumerical results. The absolute values of the damping-\ndependent critical current are slightly changed when the\ndefinition of the critical current is changed. This is be-\ncause Eq. (B1) is more easily satisfied than Eq. (9),\nand thus the critical current in Fig. 8 is smaller than\nthat shown in Fig. 4. However, the main results of this\npaper, such as the damping dependence of the critical\ncurrent, are not changed by changing the definition of\nthe critical current in the numerical simulations.\nAppendix C: Energy change during a precession\nAs described in Sec. V, the linearized LLG equation\nassumes a steady precession of the magnetization due to\nthe field torque when the current magnitude is close to\nthe critical current. This is because the spin Hall torque\ncompensates with the damping torque. Thus, Figs. 6(b)\nand 6(d) are obtained by substituting the solution of m\nprecessing a constant energy curve of Einto Eqs. (7) and\n(8).\nWhen the transverse field is applied and β= 0, i.e.,\nE=E, whereE=−M/integraltext\ndm·H, the precession trajec-\ntory on the constant energy curve of Eis given by55\nmx(E) = (r2−r3)sn(u,k)cn(u,k),(C1)10\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-6040 60 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=-2.0\n-50\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)\n(d) (e) (f)longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-5040 50 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 \nlongitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\nlongitudinal magnetic field (Oe)0 50 100 150 200β=-2.0\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ncritical current density (10 6 A/cm 2)\n0\n-10050 \n-150100150\n-50\nFIG. 8. Numerically evaluated critical currents with a diffe rent definition, Eq. (B1), in the presence of (a)-(c) the long itudinal\n(nH=ex) and (d)-(f) the transverse ( nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The\nsolid lines are the analytically estimated critical curren t described in Sec. V.\nmy(E) =r3+(r2−r3)sn2(u,k),(C2)\nmz(E) =/radicalBig\n1−r2\n3−(r2\n2−r2\n3)sn2(u,k),(C3)\nwhereu=γ/radicalbig\nHtHK/2√r1−r3t, andrℓare given by\nr1(E) =−E\nMHappl, (C4)\nr2(E) =Happl\nHK+/radicalBigg\n1+/parenleftbiggHappl\nHK/parenrightbigg2\n+2E\nMHK,(C5)\nr3(E) =Happl\nHK−/radicalBigg\n1+/parenleftbiggHappl\nHK/parenrightbigg2\n+2E\nMHK.(C6)The modulus of Jacobi elliptic functions is k=/radicalbig\n(r2−r3)/(r1−r3). The precession period is\nτ(E) =2K(k)\nγ/radicalbig\nHapplHK/2√r1−r3,(C7)\nwhereK(k) is the first kind of complete elliptic inte-\ngral. The initial state is chosen to be my(0) =r3. Fig-\nure 6(b) is obtained by substituting Eqs. 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Serpico, Nonlinear mag-\nnetization Dynamics in Nanosystems (Elsevier, Oxford,\n2009).\n54S. Wiggins, “Introduction to applied nonlinear dynamical\nsystems and chaos,” (Springer, 2003) Chap. 1.\n55T. Taniguchi, “Nonlinear analysis of magnetization dy-\nnamics excited by spin Hall effect,” Phys. Rev. B 91,\n104406 (2015)."    },    {        "title": "2107.00982v3.Anomalous_Gilbert_Damping_and_Duffing_Features_of_the_SFS___boldmath___varphi_0___Josephson_Junction.pdf",        "content": "arXiv:2107.00982v3  [cond-mat.supr-con]  25 Aug 2021Anomalous Gilbert Damping and Duffing Features of the SFS ϕ0Josephson Junction\nYu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan\n4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran\n(Dated: August 26, 2021)\nWe demonstrate unusual features of phase dynamics, IV-char acteristics and magnetization dy-\nnamics of the ϕ0Josephson junction at small values of spin-orbit interacti on, ratio of Josephson to\nmagnetic energy and Gilbert damping. In particular, an anom alous shift of the ferromagnetic reso-\nnance frequency with an increase of Gilbert damping is found . The ferromagnetic resonance curves\nshow the Duffing oscillator behaviour, reflecting the nonline ar nature of Landau-Lifshitz-Gilbert\n(LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an\napproximated equation demonstrated both damping effect and Duffing oscillator features. The re-\nsulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative\nterm and the restoring force. A resonance method for the dete rmination of spin-orbit interaction in\nnoncentrosymmetric materials which play the role of barrie r inϕ0junctions is proposed.\nIntroduction. The Josephson junctions (JJ) with the\ncurrent-phaserelation I=Icsin(ϕ−ϕ0), wherethephase\nshiftϕ0is proportional to the magnetic moment of ferro-\nmagneticlayerdetermined bythe parameterofspin-orbit\ninteraction, demonstratea number ofunique featuresim-\nportant for superconducting spintronics, and modern in-\nformation technology [1–6]. The phase shift allows one\nto manipulate the internal magnetic moment using the\nJosephson current, and the reverse phenomenon which\nleads to the appearance of the DC component in the su-\nperconducting current [7–9].\nInteractive fields can bring nonlinear phenomena of\nboth classical, and quantum nature. A basic example\nis the magnons strongly interacting with microwave pho-\ntons [10]. As a result we could name Bose-Einstein con-\ndensation of such quasiparticles, i.e. magnons [11, 12],\nand synchronization of spin torque nano-oscillators as\nthey coherently emit microwave signals in response to\nd.c. current [13]. It is interesting that (semi)classical an-\nharmonic effects in the magnetodynamics described by\nthe Landau-Lifshitz-Gilbert (LLG) model in thin films or\nheterostructures [14, 15], and the quantum anharmonic-\nity in the cavity mangnonics [16] can well be modeled\nby so simple a nonlinear oscillator as Duffing. The cor-\nresponding Duffing equation contains a cubic term and\ndescribesthe oscillationsofthe variousnonlinearsystems\n[17].\nDespite the fact that nonlinear features of LLG are\nstudied often during a long time and in different systems,\nmanifestation of the Duffing oscillator behavior in the\nframeworkofthisequationisstill notcompletelystudied.\nCloser to our present investigation, in the study of the\ndynamics of antiferromagnetic bimeron under an alter-\nnatingcurrent,Duffingequationformsagoodmodel, and\nthis has applications in weak signal detection [14, 18, 19].\nAs another application with Duffing oscillator at work,\nwe can mention the ultra thin Co 20Fe60B20layer, andits largeangle magnetizationprecessionunder microwave\nvoltage. There are also ‘foldover’ features, characteris-\ntic of the Duffing spring, in the magnetization dynamics\nof the Co/Ni multilayer excited by a microwave current\n[15, 20, 21]. But nonlinear features of ϕ0Josephson junc-\ntions have not been carefully studied yet. In this Letter,\nwe show that the Duffing oscillator helps in the under-\nstanding of the nonlinear features of ϕ0Josephson junc-\ntions at small values of system parameters.\nCoupling of superconducting current and magnetiza-\ntion and its manifestation in the IV-characteristics and\nmagnetizationdynamicsopensthedoorfortheresonance\nmethod determination of spin-orbit intensity in noncen-\ntrosymmetric materials playing the role of barrier in ϕ0\njunctions. As it is well known, the spin-orbit interaction\nplays an important role in modern physics, so any novel\nmethod for its determination in real materials would be\nvery important. There are a series of recent experiments\ndemonstrating the modification of Gilbert damping by\nthe superconducting correlations (see Ref.[22] and cita-\ntionstherein). Inparticular, the pronouncedpeaksin the\ntemperature dependence of Gilbert damping have been\nobserved for the ferromagnetic insulator/superconductor\nmultilayers [23] which might be explained by the pres-\nence of spin relaxation mechanisms like the spin-orbit\nscattering [22]. Here, we use the noncentrosymmetric\nferromagnetic material as a weak link in ϕ0junctions.\nThe suitable candidates may be MnSi or FeGe, where\nthe lack of inversion center comes from the crystalline\nstructure [8].\nThe Gilbert damping determines the magnetization\ndynamics in ferromagnetic materials but its origin is not\nwell understood yet. Effect of nonlinearity on damp-\ning in the system is very important for application of\nthese materials in fast switching spintronics devices. Our\nstudy clarifies such effects. In Ref.[24] the authors dis-\ncuss the experimental study of temperature-dependent2\nGilbert damping in permalloy (Py) thin films of varying\nthicknesses by ferromagnetic resonance, and provide an\nimportant insight into the physical origin of the Gilbert\ndamping in ultrathin magnetic films.\nIn this Letter we demonstrate an anomalous depen-\ndence of the ferromagnetic resonance frequency with an\nincrease of the Gilbert damping. We find that the reso-\nnance curves demonstrate features of Duffing oscillator,\nreflecting the nonlinear nature of LLG equation. The\ndamped precession of the magnetic moment is dynami-\ncally driven by the Josephson supercurrent, and the res-\nonance behavior is given by the dynamics of the Duffing\nspring. The resonance methods for the determination of\nspin-orbit interaction in the ϕ0junction are proposed.\nModel and Methods. In the considered SFS ϕ0junc-\ntion (see Fig.1) the superconducting phase difference ϕ\nand magnetization Mof the F layer are two coupled dy-\nnamical variables. Based on the LLG equation for the\nFigure 1: Schematic view of SFS ϕ0Josephson junction. The\nexternal current applied along xdirection, ferromagnetic easy\naxis is along zdirection.\nmagnetic moment Mwith effective magnetic field Heff,\nresistively capacitively shunted junction (RCSJ) model,\nand Josephson relation for the phase difference ϕ, we de-\nscribe dynamics of the SFS ϕ0junction by the system of\nequations in normalized variables\ndm\ndt=ωFheff×m+α/parenleftbigg\nm×dm\ndt/parenrightbigg\n,\nheff=Grsin(ϕ−rmy)/hatwidey+mz/hatwidez, (1)\ndV\ndt=1\nβc[I−V+rdmy\ndt−sin(ϕ−rmy)],\ndϕ\ndt=V,\nwheremis vector of magnetization with components\nmx,y,z, normalized to the M0=/bardblM/bardbland and satisfy-\ning the constraint/summationtext\ni=x,y,zm2\ni(t) = 1,ωF= ΩF/ωc,\nΩF=γK/νis ferromagnetic resonance frequency, γis\nthe gyromagnetic ratio, Kis an anisotropic constant, ν\nis the volume of the ferromagnetic F layer, αis the phe-\nnomenologicaldamping constant(Gilbert damping), heff\nis the vector of effective magnetic field, normalized to\ntheK/M0(heff=HeffM0/K),G=EJ/(Kν) relation\nof Josephson energy to magnetic one, ris a parameter\nof spin-orbit coupling, ϕis phase difference of JJ, Vis\nvoltage normalized to the Vc=IcR,Iccritical current\nof JJ,Rresistance of JJ, βc= 2eIcCR2//planckover2pi1is McCumberparameter, Cis capacitance of JJ, Iis bias current nor-\nmalized to the Ic. In this system of equation time tis\nnormalized to the ω−1\nc, whereωc= 2eIcR//planckover2pi1is character-\nistic frequency. In the chosen normalization, the average\nvoltage corresponds to the Josephson frequency ωJ.\nFerromagnetic resonance in ϕ0junction. The ferro-\nmagnetic resonance features are demonstrated by aver-\nage voltage dependence of the maximal amplitude of the\nmycomponent ( mmax\ny), taken at each value of bias cur-\nrent. To stress novelty and importance of our finding,\nwe first present the analytical results for average volt-\nage dependence of mmax\nyalong IV-characteristics in the\nferromagnetic resonance region. As it was discussed in\nRefs.[8, 25, 26], in case Gr≪1,mz≈1, and neglecting\nquadratic terms mxandmy, we get\n/braceleftBigg\n˙mx=ξ[−my+GrsinωJt−αmx]\n˙my=ξ[mx−αmy],(2)\nwhereξ=ωF/(1 +α2). This system of equations can\nbe written as the second order differential equation with\nrespect to the my\n¨my=−2αξ˙my−ξ2(1+α2)my+ξ2GrsinωJt.(3)\nCorresponding solution for myhas the form\nmy(t) =ω+−ω−\nrsinωJt−α++α−\nrcosωJt,(4)\nwhere\nω±=Gr2ωF\n2ωJ±ωF\n((ωJ±ωF)2+(αωJ)2),(5)\nand\nα±=Gr2ωF\n2αωJ\n((ωJ±ωF)2+(αωJ)2).(6)\nSo,mydemonstrates resonance with dissipation when\nJosephson frequency is approaching the ferromagnetic\none (ωJ→ωF). The maximal amplitude mmax\nyas a\nfunction of voltage (i.e., Josephson frequency ωJ) at dif-\nferentα, calculated using (4), is presented in Fig.2 (a).\nWe see the usual characteristicvariation of the resonance\ncurve with an increase in dissipation parameter when the\nmaximal amplitude and position of resonance pick cor-\nresponds to the damped resonance. We note that the\nanalytical result (4) were obtained in the case Gr≪1.\nPresented in Fig.2(b) results of numerical simulations\nmmax\ny(V) dependence at different values of dissipation\nparameter αdemonstrate the essential differences with\nthe results followedfrom the analytical consideration(4).\nWe note also that the strong coupling of the supercon-\nducting phase difference ϕand magnetization Mof the\nF layermanifests itself by appearanceof subharmonics of\nthe resonance at ω= 1/2,1/3,1/4 demonstrated in the\ninset to Fig.2(b).3\nFigure 2: (a) Analytical results for maximal amplitude mmax\ny\nin the ferromagnetic resonance region for different α; (b)\nNumerical results for maximal amplitude of magnetization\nmy−component at each values of bias current and voltage\nalong IV-characteristics of the ϕ0junction in the ferromag-\nnetic resonance region for various α. Inset shows the man-\nifestation of the resonance subharmonics. Parameters are:\nβc= 25, G=0.05, r=0.05, ωF= 0.5.\nWe stress two important features followed from the\npresented results. First, the ferromagnetic resonance\ncurves show the foldover effect, i.e., the features of Duff-\ning oscillator. Different from a linear oscillator, the non-\nlinear Duffing demonstrates a bistability under external\nperiodic force [27]. Second, the ferromagnetic resonance\ncurves demonstrate an unusual dependence of the reso-\nnance frequency as a function of Gilbert damping α. As\nshown in Fig. 3(a), an increase in damping leads to a\nnonuniform change in the resonant frequency, i.e., with\nan increase in damping the resonance maximum shifts\ntoωFat small α, but then moves to the opposite side,\ndemonstrating the usual damped resonance. So, with\nan increase in α, unusual dependence of the resonance\nvoltage transforms to the usual one. For the parameters\nchosen, the critical value of this transformation is around\nα= 0.02−0.03. We call this unusual behaviour of the\nresonance maximum of mmax\nyas an “α-effect”. Both the\nα−effect and Duffing features in our system appear due\nto the nonlinear features of the system dynamics at small\nFigure 3: (a) α-dependence of the resonance curve mmax\ny(V)\npeak presented in Fig.2 in the damping parameter interval\n[0.006 – 0.2]. Dashed line indicates ferromagnetic resonan ce\nposition; (b) Comparison of the resonance curves mmax\ny(V)\ncalculated by full LLG equation (1) and the approximate\nequation (8).\nG,r,α≪1. To prove it, we have carried out the nu-\nmerical analysis of each term of LLG full equation (first\ntwo equations in (1)) for the set of model parameters\nG= 0.05,r= 0.05α= 0.005. After neglecting the\nterms of order 10−6, we have\n˙mx\nξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2\nz,\n˙my\nξ=mxmz−αmym2\nz, (7)\n˙mz\nξ=−Grmxsin(ϕ−rmy)+αmz(m2\nx+m2\ny),\nInthisapproximationweobserveboththe“ α–effect”and\nDuffing oscillator features. Neglecting here the last term\nαmz(m2\nx+m2\ny) in third equationfor ˙ mz, which is orderof\n10−4, leadstothe losingoftheDuffing oscillatorfeatures,\nbut still keeps alpha-effect. We note that equation (7)\nkeeps the time invariance of the magnetic moment, so\nthat term plays an important role for manifestation of\nDuffing oscillator features by LLG equation.\nThe generalized Duffing equation for ϕ0junction.\nThe LLG is a nonlinear equation and in case of simple\neffective field it can be transformed to the Duffing equa-\ntion [14, 17]. Such transformation was used in Ref.[17]\nto demonstrate the nonlinear dynamics of the magnetic\nvortex state in a circular nanodisk under a perpendicular\nalternating magnetic field that excites the radial modes\nof the magnetic resonance. They showed Duffing-type\nnonlinear resonance and built a theoretical model corre-\nsponding tothe Duffing oscillatorfromthe LLG equation\nto explore the physics of the magnetic vortex core polar-\nity switching for magnetic storage devices.\nThe approximated LLG system of equations (7)\ndemonstrates both α-effect and features of Duffing os-\ncillator. As demonstrated in the Supplemental Materials\n[28], the generalizedDuffing equation forthe ϕ0junction,\n¨my+2ξα˙my+ξ2(1+α2)my\n−ξ2(1+α2)m3\ny=ξ2GrsinωJt.(8)4\ncan be obtained directly from the LLG system of equa-\ntions.\nAs we see, for small enough Gandr, it is only the\ndimensionless damping parameter αin LLG that plays a\nrole in the dynamics of the system. We can think of a\nharmonic spring with a constant that is hardened or soft-\nened by the nonlinear term. For a usual Duffing spring,\nwith independent coefficients of the various terms, the\nresonancepeak relative to the harmonic (linear) resonant\nfrequency folds over to the smaller (softening) or larger\n(hardening) frequencies. In the frequency response, the\ninterplay of the specific dependence of each coefficient on\nαplays an important role and as Fig.3(a) shows, there is\na particular αthat brings the resonant frequency closest\nto ferromagnetic resonance.\nSimulations of the mydynamics in the framework of\nDuffing equation can explain observed foldover effect in\nthe frequency dependence of mmax\ny. Comparison the re-\nsults followed from analytical approximate equation (8)\nand results of full equation (1) for maximal amplitude of\nmmax\nyin the ferromagnetic resonance region is presented\nin Fig.3(b). So, the magnetization dynamics in the SFS\nϕ0-junction due to the voltage oscillations can effectively\nbe described by a scalar Duffing oscillator, synchronizing\nthe precession of the magnetic moment with the Joseph-\nson oscillations.\nEffect of spin-orbit interactions. As we mentioned\nabove, the spin-orbit interaction plays an important role\nin different fields of modern physics. Here we have sug-\ngested a novel method for its determination in real non-\ncentrosymmetric ferromagnetic materials like MnSi or\nFeGe, where the lack of inversion center comes from\nthe crystalline structure Ref.[8] and which play role a\nweak link in ϕ0junctions. Based on the obtained re-\nsults, presented in Fig.4, we propose different versions of\nthe resonance method for the determination of spin-orbit\ninteraction in these materials. Particularly, in Fig.4(a)\nwe present the simulation results of maximal amplitude\nmmax\nybased on (1) at G= 0.05,α= 0.01 at different\nvalues of spin-orbit parameter rin the ferromagnetic res-\nonance region. This case corresponds to the nonlinear\napproximation leading to the Duffing equation (8). The\nsame characteristics calculated by equation (1) for larger\nvalueα= 0.1, i.e. corresponding to the linear approxi-\nmation (3) are presented in Fig.4(b). As it was expected,\nin caseα= 0.01 the foldover effect is more distinct.\nIn Fig.4(c) the r-dependence of the resonancepeak po-\nsition, obtained from the simulation results of full equa-\ntion atα= 0.01 andα= 0.1 for the same set of model\nand simulation parameters is demonstrated. We stress\nhere that nonlinear features of LLG equation leading to\nthe Duffing’s shift of the mmax\nypeak of main harmonic\nwith r presented in Fig.4(c) show the manifestation of\nnonlinearity.\nDespite the noted differences between results for α=\n0.01 andα= 0.1 , we see in both cases a monotonic\nFigure 4: (a) Voltage dependence of mmax\nyin the ferromag-\nnetic resonance region at different values of spin-orbit int er-\naction based on (1) at G= 0.05,α= 0.01. Inset enlarges\nthe main harmonic; (b) The same as in (a) for α= 0.1; (c)\nShift ofmmax\nypeak as a function of spin-orbit interaction at\ntwo values of Gilbert damping; (d) r-dependence of the main\nharmonic and subharmonics peaks in case (a); (e) The same\nas in (d) for the case (b).\nlinear increase of mmax\nypeak of main harmonic and sub-\nharmonics with rdemonstrated in Fig.4(d) and Fig.4(e).\nSuch lineardependence canbe noted fromEq. (6) ofRef.\n[14], but the authors did not discuss it. This dependence\nmight serve as a calibrated curve for spin-orbit interac-\ntion intensity, thus creating the resonance methods for r\ndetermination.\nConclusions. Based on the reported features of the\nϕ0Josephson junction at small values of spin-orbit in-\nteraction, ratio of Josephson to magnetic energy and\nGilbert damping, we have demonstrated that the cou-\npled superconducting current and the magnetic moments\nin theϕ0-junction result in the current phase relation in-5\ntertwining with the ferromagnetic LLG dynamics. The\nferromagnetic resonance clearly shows this interplay. In\nparticular, an anomalous shift of the ferromagnetic res-\nonance frequency with an increase of Gilbert damping\nis found. The ferromagnetic resonance curves demon-\nstrate features of Duffing oscillator, reflecting the nonlin-\near nature of LLG equation. The obtained approximated\nequation demonstrates both damping effect and Duffing\noscillator features. We have shown that due to the non-\nlinearity, asmodeledbythe generalizedDuffing equation,\nthe parameters of the system can compensate each other\nresulting in unusual response. The position of the maxi-\nmum can shift towards and then away from the expected\nresonant frequency, as the damping is decreased. There\nare also foldover effects that was explained by the pro-\nposed model. A resonance method for the determination\nof spin-orbit interaction in noncentrosymmetric materi-\nals which play the role of barrier in ϕ0junctions was\nproposed.\nThe experimental testing of our results would in-\nvolve SFS structures with ferromagnetic material having\nenough small value of Gilbert damping. Potential candi-\ndate for experimental realization could be ferromagnetic\nmetals or insulators which have small values of damping\nparameter ( α∼10−3−10−4). In Ref.[29] the authors\nreport on a binary alloy of cobalt and iron that exhibits\na damping parameterapproaching10−4, which is compa-\nrable to values reported only for ferrimagnetic insulators\n[30, 31]. Using superconductor-ferromagnetic insulator-\nsuperconductor on a 3D topological insulator might be\na way to have strong spin-orbit coupling needed for ϕ0\nJJ and small Gilbert dissipation for α-effect [5]. We note\nin this connection that the yttrium iron garnet YIG is\nespecially interesting because of its small Gilbert damp-\ning (α∼10−5). The interaction between the Joseph-\nson current and magnetization is determined by the ra-\ntio of the Josephson to the magnetic anisotropy energy\nG=EJ/(Kν) and spin-orbit interaction r. The value of\nthe Rashba-type parameter rin a permalloy doped with\nPt[32] and in the ferromagnets without inversion sym-\nmetry, like MnSi or FeGe, is usually estimated to be in\nthe range 0 .1−1. The value of the product Grin the ma-\nterialwith weakmagneticanisotropy K∼4×10−5KA−3\n[33], and a junction with a relatively high critical current\ndensity of (3 ×105−5×106)A/cm2[34] is in the range\n1−100. It givesthe set offerromagneticlayerparameters\nand junction geometry that make it possible to reach the\nvalues used in our numerical calculations for the possible\nexperimental observation of the predicted effect.\nNumerical simulations were funded by the project 18-\n71-10095oftheRussianScientificFund. A.J.andM.R.K.\nare grateful to IASBS for financial support.[1] Jacob Linder and W. A. Jason Robinson, Nature Physics\n11, 307 (2015).\n[2] Yu. M. Shukrinov, Accepted for UFN.\nDOI:https://doi.org/10.3367/UFNe.2020.11.038894\n[3] A.A. Mazanik, I.R. Rahmonov, A.E. Botha, and Yu.M.\nShukrinov, Phys. Rev. Applied 14, 014003 (2020).\n[4] M. Nashaat and Yu. M. Shukrinov, Physics of Particles\nand Nuclei Letters, 17, 79. (2020).\n[5] I. V. Bobkova , A. M. Bobkov, I. R. Rahmonov, A. 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Cros, and A. Fert, Appl. Phys.\nLett.103, 082408 (2013).[31] M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M.\nKl¨ aui, A. V. Chumak, B. Hillebrands, and C. A. Ross,\nAPL Mater. 2, 106102 (2014).\n[32] A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Aren-\nholz, A. T. N’Diaye, R. L. Stamps, and C. H. Marrows,\nPhys. Rev. B 93, 014432 (2016).\n[33] A. Yu. Rusanov, M. Hesselberth, J. Aarts, and A. I.\nBuzdin, Phys. Rev. Lett. 93, 057002 (2004).\n[34] J. W. A. Robinson, F. Chiodi, M. Egilmez, G. B. Hal´ asz\nand M. G. Blamire, Scientific Report 2, 699 (2012).arXiv:2107.00982v3  [cond-mat.supr-con]  25 Aug 2021Supplemental Material to “Anomalous Gilbert Damping and Du ffing Features of the\nSFSϕ0Josephson Junction”\nYu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan\n4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran\n(Dated: August 26, 2021)\nHere, we demonstrate by numerical methods that a generalize d Duffing equation can be obtained\ndirectly from LLG system of equations, for small system para meters of S/F/S junction.\nBoth the α−effect and Duffing features obtained by\nLLG system of equations appear due to the nonlinear\nfeatures of its dynamics at small G,r,α≪1. To proveit,\nwe have carried out the numerical analysis of each term\nof LLG full equation (first two equations in the equation\n(1) of the main text) for the set of model parameters\nG= 0.05,r= 0.05α= 0.005. After neglecting the\nterms of order 10−6, we have\n˙mx\nξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2\nz,\n˙my\nξ=mxmz−αmym2\nz, (1)\n˙mz\nξ=−Grmxsin(ϕ−rmy)+αmz(m2\nx+m2\ny),\nThe procedure is as follows. Expanding mn\nzin a series\nwith the degree of ( mz−1) we can find\nmn\nz=nmz−(n−1). (2)\nFrom expression m2\nx+m2\ny+m2\nz= 1 and (2), we obtain\nmz=2−m2\ny\n2. (3)\nUsing approximation sin( ϕ−rmy) = sin(ωJt) in (1),\ndifferentiatingsecondequationofthe system(1) andsub-\nstituting ˙ mx,mxand ˙mzfrom first second and third\nequations of the system (1), respectively and using the\nexpression (2), (3) and assuming mz= 1 only in denom-\ninators, we come to a second order differential equation\nwith respect to my\n¨my=a1˙m3\ny+a2my˙m2\ny+a3m4\ny˙my+a4m2\ny˙my+a5˙my\n+a6m5\ny+a7m3\ny+a8my−c1˙m2\nysinωJt (4)\n+c2m4\nysinωJt+c3m2\nysinωJt+AsinωJt.The numerical calculation for the used set of model\nparameters allows us to estimate each of the terms in the\nequation, as presented in Table I.\nNow, if we neglect those terms smaller than 10−4, the\nequation (4) takes on the form of Duffing equation with\nTable I: Numerical analysis of equation (4) terms.\na1α\nξa1˙m3\ny∼1.76×10−5\na2 α2a2my˙m2\ny∼3.4×10−8\na3ξα3a3m4\ny˙my∼7.7×10−12\na4ξ(3α−α3)a4m2\ny˙my∼2×10−5\na52ξα a5˙my∼6×10−4\na6ξ2(α2+2α4)a6m5\ny∼5.56×10−9\na7ξ2(1+α2−α4)a7m3\ny∼3.7×10−3\na8ξ2(1+α2)a8my∼6.1×10−2\nc1 Gr c1˙m2\nysinϕ∼3.6×10−5\nc22ξ2α2Grc2m4\nysinϕ∼5.3×10−11\nc3ξ2Gr(α2−2)c3m2\nysinϕ∼4.5×10−5\nAξ2Gr AsinωJt∼6.25×10−4\ndamping dependent coefficients, i.e., we have a general-\nization of the Duffing equation\n¨my+2ξα˙my+ξ2(1+α2)my\n−ξ2(1+α2)m3\ny=ξ2GrsinωJt.(5)"    },    {        "title": "2306.04617v2.Helicity_dependent_optical_control_of_the_magnetization_state_emerging_from_the_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "1  Helicity-dependent optical control of the magnetization state emerging from the Landau-Lifshitz-Gilbert equation  Benjamin Assouline, Amir Capua*  Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel   *e-mail: amir.capua@mail.huji.ac.il  Abstract:   It is well known that the Gilbert relaxation time of a magnetic moment scales inversely with the magnitude of the externally applied field, 𝑯, and the Gilbert damping, 𝜶. Therefore, in ultrashort optical pulses, where 𝑯 can temporarily be extremely large, the Gilbert relaxation time can momentarily be extremely short, reaching even picosecond timescales. Here we show that for typical ultrashort pulses, the optical control of the magnetization emerges by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. Surprisingly, when circularly polarized optical pulses are introduced to the LLG equation, an optically induced helicity-dependent torque results. We find that the strength of the interaction is determined by 𝜼=𝜶𝜸𝑯/𝒇𝒐𝒑𝒕, where 𝒇𝒐𝒑𝒕 and 𝜸 are the optical frequency and gyromagnetic ratio. Our results illustrate the generality of the LLG equation to the optical limit and the pivotal role of the Gilbert damping in the general interaction between optical magnetic fields and spins in solids.  2  The ability to control the magnetization order parameter using ultrashort circularly polarized (CP) optical pulses has attracted a great deal of attention since the early experiments of the all-optical helicity dependent switching (AO-HDS) [1-4]. This interaction was found intriguing since it appears to have all the necessary ingredients to be explained by a coherent transfer of angular momentum, yet it occurs at photon energies of 1\t−\t2\t𝑒𝑉, very far from the typical resonant transitions in metals. The technological applications and fundamental scientific aspects steered much debate and discussion [5,6], and the experiments that followed found dependencies on a variety of parameters including material composition [7-9], magnetic structure [10-12], and laser parameters [1,3,13], that were often experiment-specific [4]. Consequently, a multitude of mechanisms that entangle photons [14,15], spins [16,17], and phonons [18,19] have been discovered. References [4,20] provide a state of the art review of the theoretical and experimental works of the field. Ferromagnetic resonance (FMR) experiments are usually carried out at the 𝐺𝐻𝑧 range. In contrast, optical fields oscillate much faster, at ~\t400−800\t𝑇𝐻𝑧. Therefore, it seems unlikely that such fast-oscillating fields may interact with magnetic moments. However, the amplitude of the magnetic field in ultrashort optical pulses can, temporarily, be very large such that the magnetization may respond extremely fast. For example, in typical experiments having 40\t𝑓𝑠−1\t𝑝𝑠 pulses at 800\t𝑛𝑚, with energy of 0.5\t𝑚𝐽 that are focused to a spot size of ~0.5\t𝑚𝑚$, the peak magnetic flux density can be as high as ~\t5\t𝑇, for which the corresponding Gilbert relaxation time reduces to tens of picoseconds in typical ferromagnets. Here we show that ultrashort optical pulses may control the magnetization state by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. We find that the strength of the interaction is determined by 𝜂=𝛼𝛾𝐻/𝑓%&', where 𝑓%&' and 𝛼 are the angular optical frequency and the Gilbert damping, respectively, and 𝛾 is the gyromagnetic ratio. Moreover, we show that for circularly polarized (CP) pulses, the polarity of the optically induced torque is determined by the optical helicity. From a quantitative analysis, we find that a sizable effective out-of-plane field is generated which is comparable to that measured experimentally in ferromagnet/heavy-metal (FM/HM) material systems. 3  The LLG equation is typically not applied in the optical limit, and hence requires an alternative mathematical framework whose principles we adopt from the Bloch equations for semiconductor lasers [21,22]. We exploit the analogy between the magnetization state and the Bloch vector of a two-level system (TLS) [23,24] by transforming the LLG equation under a time-varying magnetic field excitation to the dynamical Maxwell-Bloch (MB) equations in the presence of an electrical carrier injection. In this transformation, the reversal of the magnetization is described in terms of population transfer between the states. The paper is organized as follows: We begin by transforming the LLG equation to the density matrix equations of a TLS. We then identify the mathematical form of a time-dependent magnetic field in the LLG equation, 𝐻AA⃗&()&↓↑, that is mapped to a time-independent carrier injection rate into the TLS. Such excitation induces a population transfer that varies linearly in time and accordingly to a magnetization switching profile that is also linear in time. The mathematical 𝐻AA⃗&()&↓↑ field emerges naturally as a temporal impulse-like excitation. We then show that when 𝛼 is sizable, 𝐻AA⃗&()&↓↑ acquires a CP component whose handedness is determined by the direction of the switching. By substituting 𝐻AA⃗&()&↓↑ for an experimentally realistic picosecond CP Gaussian optical magnetic pulse, we show that it can also exert a net torque on the magnetization. In this case as well, the helicity determines the polarity of the torque. Finally, we present a quantitative analysis that is based on experimental data. The LLG equation describing the dynamics of the magnetization, 𝑀AA⃗, where the losses are introduced in the Landau–Lifshitz form is given by [25]: 𝑑𝑀AA⃗𝑑𝑡=\t−𝛾1+𝛼$𝑀AA⃗×𝐻AA⃗−𝛾𝛼1+𝛼$1𝑀,𝑀AA⃗×𝑀AA⃗×𝐻AA⃗.(1) Here 𝑀, and 𝐻AA⃗ are the magnetization saturation and the time dependent externally applied magnetic field, respectively. We define 𝐻AA⃗-.. by: 𝐻AA⃗-..≜K𝐻AA⃗−\t𝛼𝑀,𝐻AA⃗×𝑀AA⃗\tL,(2) and in addition, 𝜅≜/012!O𝐻-..\t4−𝑗𝐻-..\t5Q/2 and 𝜅6\t≜/012!𝐻-..\t7, where 𝜅 and 𝜅6 can be regarded as effective AC and DC magnetic fields acting on 𝑀AA⃗, respectively. We 4  transform 𝑀AA⃗ to the density matrix elements of the Bloch state in the TLS picture and compare it to the Bloch equations describing a semiconductor laser that is electrically pumped [26]:  ⎩⎪⎨⎪⎧𝜌̇00=𝛬0−𝛾0𝜌00+𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇$$=𝛬$−𝛾$𝜌$$−𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇0$=\t−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗(𝜌00−𝜌$$)𝑉0$\t.\t\t(3) In this reference model, 𝛬0 and 𝛬$ are injection rates of carriers to the ground and excited states of the TLS, respectively. They are assumed to be time independent and represent a constant injection of carriers from an undepleted reservoir [27]. 𝛾0 and 𝛾$ are the relaxation rates of the ground and excited states, and 𝛾;<= is the decoherence rate due to an inhomogeneous broadening. 𝑉0$ is the interaction term and 𝜔89: is the resonance frequency of the TLS. Figure 1(a) illustrates schematically the analogy between the magnetization dynamics and the electrically pumped TLS. We find the connection between the LLG equation expressed in the density matrix form and the model of the electrically pumped TLS: ]𝛬0−𝛾0𝜌00+[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t−𝑗𝜅𝜌$0+𝑐.𝑐.𝛬$−𝛾$𝜌$$−[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t𝑗𝜅𝜌$0+𝑐.𝑐.−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗𝑀7𝑉0$=\t−𝑗𝜅6𝜌0$\t+𝑗𝜅𝑀7.(4) The pumping of the excited and ground states by the constant 𝛬0 and 𝛬$ rates implies that the reversal of the magnetization along the ∓\t𝑧̂ direction is linear in time. Using Eq. (4) we find 𝜅, and hence a field 𝐻AA⃗, that produces such 𝛬0 and 𝛬$. We define this field as 𝐻AA⃗&()&↓↑: 𝐻AA⃗&()&↓↑=\t±𝛬&𝑀,$−𝑀7$f𝑀5−\t𝑀40g.(5) 𝐻AA⃗&()&↓↑ depends on the temporal state of 𝑀AA⃗ while 𝛬&=𝛾𝛬0/(1+𝛼$) is the effective field strength parameter. 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ induce a linear transition of 𝑀AA⃗ towards the –𝑧̂ and +𝑧̂ direction, respectively. 5  Figure 1(b) presents the outcome of the application of 𝐻AA⃗&()&↓↑ by numerically integrating the LLG equation. The Figure illustrates 𝐻AA⃗(𝑡), 𝑀7(𝑡), and the 𝑧̂ torque, O−𝑀AA⃗×𝐻AA⃗Q7, for alternating 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ that switch 𝑀AA⃗ between ∓𝑀,𝑧̂. The magnitude of 𝛬& determines the switching time, 𝛥𝜏↓↑, chosen here to describe a femtosecond regime. Equation (4) yields 𝛥𝜏↓↑=(1+𝛼$)𝑀,/(\t𝛾𝛬&)≈𝑀,/𝛾𝛬& in which 𝑀7 is driven from 𝑀7=0 to 𝑀7≅±𝑀, (for derivation, see Supplemental Material Note 1). It is seen that O−𝑀AA⃗×𝐻AA⃗Q7 is constant when 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑ are applied so that the switching profile of 𝑀7 is linear in time. It is also seen that 𝐻AA⃗&()&↓↑ requires that m𝐻AA⃗m diverge as 𝑀7 approaches ±𝑀,, which is not experimentally feasible. To account for a more realistic excitation, in Fig. 1(c) we simulated a pulse whose trailing edge was taken as a reflection in time of 𝐻AA⃗&()&↓↑, and that is shorter by an order of magnitude as compared to the leading edge. In this case 𝑀AA⃗ remains in its final state when 𝐻AA⃗ is eventually turned off. The polarization state of 𝐻AA⃗&()&↓↑ is determined from the polarization state of the transverse components of 𝑀AA⃗. Next, we show that for larger 𝛼, 𝑀5(𝑡) becomes appreciable such that 𝐻AA⃗&()&↓↑ acquires an additional CP component. This result emerges naturally from the Bloch picture: we recall that the transverse components of 𝑀AA⃗ are expressed by the off-diagonal density matrix element. According to Eq. (3), 𝜌0$ oscillates at 𝜔89: and decays at the rate 𝛾;<=, whereas the sign of 𝜔89: determines the handedness of the transverse components of 𝑀AA⃗. Namely, the ratio between 𝜔89: and 𝛾;<= determines the magnitude of the circular component in the n𝑀4(𝑡),𝑀5(𝑡)o trajectory. Under the application of 𝐻AA⃗&()&↓↑, Eq. (4) yields 𝜔89:=±𝛾𝛬&𝛼𝑀,/[(𝑀,$−𝑀7$)(1+𝛼$)] and 𝛾;<==∓𝛾𝛬&𝑀7/[(𝑀,$−𝑀7$)(1+𝛼$)] readily showing that |𝜔89:/𝛾;<=|=𝛼𝑀,/𝑀7 increases with 𝛼, so that 𝐻AA⃗&()&↓↑ acquires an additional CP component (see Supplemental Note 2 for full derivation). Figure 2 illustrates these results. Panel (a) presents the components of 𝑀AA⃗(𝑡) for the same simulation in Fig. 1(b). It is seen that 𝑀5(𝑡) is negligible and thus 𝐻AA⃗&()&↓↑ remains linearly polarized. When 𝛼 is increased, an elliptical trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane emerges, while the constant transition rate of 𝑀7 persists as illustrated in Fig. 2(b). In this case, 6  𝐻AA⃗&()&↓↑ acquires a right-CP (RCP) or left-CP (LCP) component depending on the choice of 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑. The coupling between the handedness and reversal direction in a femtosecond excitation is reminiscent of the switching reported in AO-HDS experiments and emerges naturally in our model. These results call to examine the interaction of the CP magnetic field of a short optical pulse with 𝑀AA⃗. Figure 3(a) presents the calculation for experimental conditions [4]. The results are shown for an 800\t𝑛𝑚 optical magnetic field of an RCP Gaussian optical pulse 𝐻AA⃗%&'(𝑡). The pulse has a duration determined by 𝜏&, an angular frequency 𝜔%&', and a peak amplitude 𝐻&->? that is reached at 𝑡=𝑡&->?. In our simulations 𝜏&=3\t𝑝𝑠 and 𝑡&->?=10\t𝑝𝑠. The pulse energy was ~\t5\t𝑚𝐽 and assumed to be focused to a spot size of ~\t100\t𝜇𝑚$, for which 𝐻&->?=8⋅10@\t𝐴/𝑚. Here we take 𝛼=0.035 [28,29]. For such conditions, the Gilbert relaxation time corresponding to 𝐻&->? is 𝜏2=02/A\"#$%≈16\t𝑝𝑠 [30]. It is readily seen that for such 𝜏2 the magnetization responds within the duration of the optical pulse indicating that the interaction between the optical pulse and 𝑀AA⃗ becomes possible by the LLG equation. Following the interaction, 𝑀7=−5×10BC⋅𝑀:, namely a sizable net longitudinal torque results. In agreement with the prediction of the TLS model, pulses of the opposite helicity induce an opposite transition as shown in Fig. 3(b). The results are compared to the measured data discussed in Supplemental Material Note 3.  To this end we simulate the same conditions of the measurements including optical intensity and sample parameters. Accordingly, we find from our calculations an effective field which is of the same order of magnitude as measured.   For a given pulse duration, we define the interaction strength parameter 𝜂=2𝜋𝛼𝛾𝐻&->?/𝜔%&', which expresses the ratio between 𝜏2 and the optical cycle and is 2.5⋅10BC in Fig. 3(a). The principles of the interaction can be better understood at the limit where 𝜂→1 and for which the interaction can be described analytically. To this end, we set 𝜂=1. The higher optical magnetic fields required for this limit are achievable using conventional amplified femtosecond lasers, for example by focusing a ~\t5\t𝑚𝐽 pulse into a spot size of ~\t1\t𝜇𝑚$. Figure 3(c) illustrates the results for an RCP 𝐻AA⃗%&' pulse of a duration of 20\t𝑓𝑠 determined by the full width at half-maximum of the 7  intensity. The Figure reveals the different stages of the interaction. During the leading edge, for 𝑡<~\t40\t𝑓𝑠, the relative phase between 𝐻AA⃗%&' and 𝑀AA⃗ seems arbitrary. As 𝑡&->? is reached, the Gilbert relaxation time becomes as short as the optical cycle allowing 𝑀AA⃗ to follow 𝐻AA⃗%&' until it is entirely locked to 𝐻AA⃗%&'. In this case, 𝑀AA⃗ undergoes a right-circular trajectory about 𝑧̂. The switching of 𝑀AA⃗ takes place at the final stage of the interaction: During the trailing edge of the pulse, the amplitude of 𝐻AA⃗%&' reduces and 𝜏2 extends, thereby releasing the locking between 𝑀AA⃗ and 𝐻AA⃗%&'. In this case, the switching profile of 𝑀7 is monotonic linear-like in time, closely resembling the transition stemming from a constant carrier injection rate in the Bloch picture. The optically induced transition can be described analytically following the calculation presented in Supplemental Note 4, from which we find the transition rate:  𝛤/𝑀,=∓32√2𝑙𝑛K43L1𝜏&}~𝑙𝑛K𝐻&->?0.27𝐻'=L−~𝑙𝑛𝐻&->?𝐻'=/√2,(6) where 𝐻'==D&\"'$E/2 is the value of 𝐻&->? at 𝜂=1. The rate 𝛤/𝑀, is plotted as well in Fig. 3(c) and reproduces the numerical calculation. 𝛤 depends on the ratio between 𝐻&->? and 𝐻'= and is only weekly dependent on 𝐻&->?. Namely, when 𝐻&->?≫𝐻'=, the circular trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane persists longer after 𝑡&->?, but as the amplitude of the pulse decays below 𝐻'=/√2, 𝑀AA⃗ is driven out of the 𝑥−𝑦 plane and the reversal takes place (see Supplemental Material Note 5). This analysis also holds for LCP pulses, which result in an opposite reversal of 𝑀AA⃗, as shown in Fig. 3(d).  To summarize, in this work we demonstrated that the control of the magnetization by an optical field arises from first principles by introducing the magnetic part of the optical radiation to the LLG equation. This was seen from the comparison between the case where 𝜂≪1 and the case of 𝜂=1. Using the TLS model, we demonstrated the coupling between the optical helicity state and the polarity of the longitudinal torque. A quantitative analysis of the optically induced torque revealed that it can be comparable to that observed in experiments.   8  Figure 1 \n       Fig. 1. (a) Left panel: Illustration of 𝑴AAA⃗ on the Bloch sphere. Right panel: Illustration of the electrically pumped TLS. (b) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ of Eq. (5). The Figure illustrates the temporal plots of 𝑴𝒛/𝑴𝒔, 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑,𝒚 and O−𝑴AAA⃗×𝑯AAA⃗Q𝒛 normalized to unity. (c) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ and a more realistic trailing edge, for the same conditions in (b). Full lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and dashed lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.    \n9  Figure 2 \n Fig. 2. Temporal evolution of the components of 𝑴AAA⃗ under the influence of alternating 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑ for (a) small and (b) large damping. Black dashed lines indicate the alternation between 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.   \n10   \n11  Fig. 3. (a) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟐.𝟓⋅𝟏𝟎B𝟒. Top and middle panels depict the temporal evolution of the 𝒙 and 𝒚 components of 𝑴AAA⃗ and 𝑯AAA⃗𝒐𝒑𝒕\tin normalized units. Bottom panel depicts 𝑴𝒛/𝑴𝒔. (b) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. (c) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟏. Top panel presents the temporal behavior of m𝑯AAA⃗𝒐𝒑𝒕m and 𝝉𝜶, where 𝑯𝒄𝒓𝒊𝒕=𝑯𝒕𝒉/√𝟐 and 𝑯𝟏/𝟐=𝟎.𝟐𝟕𝑯𝒕𝒉. (d) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. In (c) and (d), black solid lines represent the analytical solution of 𝜞/𝑴𝒔.   References  [1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, \"All-Optical Magnetic Recording with Circularly Polarized Light\", Physical Review Letters 99, 047601 (2007). [2] J. Hohlfeld, C. D. Stanciu, and A. Rebei, \"Athermal all-optical femtosecond magnetization reversal in GdFeCo\", Applied Physics Letters 94, 152504 (2009). [3] D. Steil, S. Alebrand, A. Hassdenteufel, M. Cinchetti, and M. 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Bigot, \"Distinguishing the ultrafast dynamics of spin and orbital moments in solids\", Nature 465, 458 (2010). [17] B. Y. Mueller, T. Roth, M. Cinchetti, M. Aeschlimann, and B. Rethfeld, \"Driving force of ultrafast magnetization dynamics\", New Journal of Physics 13, 123010 (2011). [18] B. Koopmans, J. J. M. Ruigrok, F. D. Longa, and W. J. M. de Jonge, \"Unifying Ultrafast Magnetization Dynamics\", Physical Review Letters 95, 267207 (2005). [19] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fahnle, T. Roth, M. Cinchetti, and M. Aeschlimann, \"Explaining the paradoxical diversity of ultrafast laser-induced demagnetization\", Nature Materials 9, 259 (2010). [20] C. Wang and Y. Liu, \"Ultrafast optical manipulation of magnetic order in ferromagnetic materials\", Nano Convergence 7, 35 (2020). [21] A. Capua, O. Karni, G. Eisenstein, V. Sichkovskyi, V. Ivanov, and J. P. Reithmaier, \"Coherent control in a semiconductor optical amplifier operating at room temperature\", Nature Communications 5, 5025 (2014). [22] A. Capua, O. Karni, G. Eisenstein, and J. P. Reithmaier, \"Rabi oscillations in a room-temperature quantum dash semiconductor optical amplifier\", Physical Review B 90, 045305 (2014). [23] R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, \"Geometrical Representation of the Schrödinger Equation for Solving Maser Problems\", Journal of Applied Physics 28, 49 (1957). [24] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997). [25] Alexander G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC Press, Boca Raton Florida, 1996). [26] L. Allen and J. Eberly, Optical Resonance and Two Level Atoms (Dover Publications, New York, 1987). [27] J. Yao, G. P. Agrawal, P. Gallion, and C. M. Bowden, \"Semiconductor laser dynamics beyond the rate-equation approximation\", Optics Communications 119, 246 (1995). [28] A. Capua, C. Rettner, S.-H. Yang, T. Phung, and S. S. P. Parkin, \"Ensemble-averaged Rabi oscillations in a ferromagnetic CoFeB film\", Nature Commun. 8, 16004 (2017). [29] N. Fujita, N. Inaba, F. Kirino, S. Igarashi, K. Koike, and H. Kato, \"Damping constant of Co/Pt multilayer thin-film media\", Journal of Magnetism and Magnetic Materials 320, 3019 (2008). [30] A. H. Morrish, The Physical Principles of Magnetism (Wiley-IEEE Press, 2001).  "    },    {        "title": "1210.0868v2.Coherence_and_Stimulated_Emission_in_the_Tavis_Cummings_Model__A_Quantum_Description_of_the_Free_Induction_Signal_and_Radiation_Damping_in_Magnetic_Resonance.pdf",        "content": " \n 1       A  Quantum Description of Radiation Damping and  the Free Induction Signal in Magnetic Resonance†   James Tropp General Electric Healthcare Technologies 47697 Westinghouse Drive   Fremont CA, 94539 james.tropp@med.ge.com      † Preliminary accounts of this work were given at annual meeting of the International Society for Magnetic Resonance in Medicine (ISMRM),  and at the Experimental NMR Conference (ENC), both in 2011. \n 2 Abstract  We apply the methods of cavity quantum electrodynamics (CQED), to obtain a microscopic and fully quantum-mechanical picture of radiation damping in magnetic resonance, and the nascent formation of the free induction signal.   Numerical solution of the Tavis-Cummings model --i.e. multiple spins 1/2 coupled to a lossless single-mode cavity --shows in fine detail the transfer of Zeeman energy,  via spin coherence,  to excite the cavity -- here represented by a quantized LC resonator.  The case of a single spin is also solved analytically.  Although the motion of the Bloch vector is non-classical,  we nonetheless show that the quantum mechanical Rabi nutation frequency (as enhanced by cavity coupling and stimulated emission) gives realistic estimates of macroscopic signal strength and the radiation damping constant in NMR.  We also show how to introduce dissipation: cavity losses by means of a master equation, and relaxation by the phenomenological method of Bloch.  The failure to obtain the full Bloch equations (unless semi-classical conditions  are imposed on the cavity) is discussed in light of similar issues arising in CQED (and in earlier work in magnetic resonance as well), as are certain problems relative to quantization of the electromagnetic near-field. \n 3   Introduction  Despite longstanding connections between quantum optics and nuclear magnetic resonance [1], NMR theoreticians –excepting those working in force microscopy [2-4] -- have paid scant attention to the Jaynes-Cummings (J-C) model for a two level atom (or a spin ½)  coupled to a quantized cavity [5]; much less to that model’s extension, by Tavis and Cummings (T-C), to accommodate multiple atoms or spins [6].  Given the centrality of NMR as a tool in modern chemistry, physics, biomedicine, and clinical medicine, it is to be wondered that a fully quantum mechanical theory has not been given, using these comparatively simple tools from cavity quantum electrodynamics (CQED) [7].  This may be due in part to the fact that CQED does not directly yield the Bloch equations, without the assumption of semi-classical conditions, such as a cavity Glauber state [8], or the replacement of quantum mechanical operators by their expectation values.  However, earlier approaches [9, 10] to a quantum theory of NMR have not avoided  equivalent assumptions; and CQED, without losing rigor, sidesteps other difficulties-- particularly those arising in the application to NMR of near-field quantum electrodynamics, such as quantization of the longitudinal field. Here we apply the Tavis-Cummings model to NMR.  Using the quantized LC circuit [11] as a stand-in for the cavity, we model the time course of Rabi oscillations, for several spins – coherently excited or simply inverted-- which drive an NMR probe initially in its ground state.  This allows us to demonstrate -- in a quantum calculation comprising both spins  \n 4 and cavity—not only radiation damping [12-14], but also the nascent formation of the free induction signal.  For simplicity we begin with a lossless cavity, and sketch later the inclusion of dissipative processes: cavity losses and spin relaxation.  The theory will be developed along lines which reflect typical experimental practice in CQED, which proceeds in two stages [15] -- first for preparation and then for subsequent evolution and observation. For preparation, the spins are excited by a semi-classical field, which nutates but does not entangle with them; the prepared spins then undergo fully quantum-mechanical evolution, in a low-temperature cavity prepared in a well-defined quantum state of low occupation number, leading to an outcome with spins and cavity entangled.  We will assume a preparatory density matrix with spins and cavity all in their ground states, and then apply, deus ex machina, a rotation to the spins alone, to produce the desired initial state.  The cavity therefore will always start the evolution period in its ground Fock state.  The outcomes are assessed by calculating the reduced density matrices for spins and cavity.     NMR practitioners will find some of the results intuitive, and others less so.  For example, despite exhibiting the transduction of Zeeman energy into cavity energy-- a prototypical form of radiation damping, even in the lossless cavity, (cf the discussion below under Cavity Damping) , -- the signal exhibits quantum collapse and revival, i.e. decay and recovery through interference of isochromats of incommensurable frequencies [16, 17]. (This is a confounding effect, but unavoidable in these calculations.)  Also, driving the cavity with inverted spins gives Rabi oscillation purely of the longitudinal magnetization, with no transverse component developing, i.e. no detectable NMR signal.  This reflects the fact that neither the T-C nor  \n 5 J-C Hamiltonian induces a pure rotation of the spins, which, in a loss-free system, would conserve the length of the Bloch vector.    This non-rotational behavior of the quantum Bloch vector is central to the present work, but is not surprising, being pre-figured in the work of J-C [5], and also in the theory of micromasers [18-20]. The classical Bloch equations are not obeyed;  nor (equivalently) is the familiar pendulum model, [21-24] which posits that the Bloch angle tracks the displacement angle of a classical pendulum.  This relates to another counter-intuitive result, namely that the longitudinal and transverse magnetizations evolve at different frequencies, offset by a factor of two, (with the transverse the slower.)  In fact, what is conventionally called the ‘Rabi frequency’ is that for nutation of the longitudinal moment; and we will adhere to this usage.  As is well known, the optical Bloch equations, which do generate rotation of the spins (more properly, of the atomic dipole moment), may be derived from the J-C or T-C Hamiltonian, on the assumption of a cavity excited by a classical oscillatory field, or alternatively, a cavity Glauber (i.e. coherent) state [8, 25-26].  But, while driving the cavity, initially quiescent, with a small number of spin coherences does produce coherences of the field, a fully formed Glauber state is not quick to appear; nor have we seen it in any of our calculations.   We will begin by presenting some elementary theory, and then move on to examples of the microscopic evolution of magnetizations and cavity fields, following suitable excitations.    We next introduce dissipative effects: cavity losses and spin relaxation.  Then, to trace a path, from a microscopic theory of NMR transduction, to a realistic estimate of the macroscopic signal strength, we calculate the cavity enhancement of Rabi oscillation, using the coupled oscillator model of J-C [5].   This allows us to  \n 6 connect the Rabi nutation frequency with the usual (classical) radiation damping constant, and moves us towards a quantum description of the experimentally observed NMR signal, which has to date remained a work in progress [9-10, 27-31].    Theory  The usual presentation of quantum electrodynamics [32] is largely concerned with the radiation field, at the expense of the near-field,  for example, that in the vicinity of a nano-scale dipole [33]. The near field need does not radiate; also, it comprises longitudinal as well as transverse components; and its quantization is correspondingly complicated [34-35].  While earlier workers in NMR have sought to address the quantum near-field directly [9], we have chosen here to neglect it, and to focus instead on the cavity operators, which describe the excitation of a resonant LC circuit, or equivalently, a single mode of a tuned cavity.  This places our approach squarely in the mainstream of CQED, while allowing a direct focus upon the most consequential process in NMR reception: the transfer of photons from spins to cavity.  Due to cavity enhancement of emission -- the cornerstone of CQED [36-37]--  the rate of this process far exceeds that of spontaneous emission by spins into free space.  To write the Hamiltonian we follow the basic procedure of J-C [5], beginning with the details of the NMR antenna, considered as a quantum oscillator, with inductance and capacitance replacing mass and spring constant.  We modify the field operators defined by Louisell [11] (with canonical variables electric charge and magnetic flux) by multiplying with ±i (5), to write the flux (i.e. canonical momentum) in terms of the  \n 7 inductance and Larmor frequency as ϕ=ω0L2[ˆa+ˆa†].  Postulating (for definiteness) that the inductor comprises a singly wound Helmholtz pair, we approximate the laboratory frame radiofrequency field B1 as ϕ/2a, where a is the window aperture in meters-squared.  This leads directly to the Rabi fundamental frequency, i.e.  Ω0=γω0L22a where γ is the gyromagnetic ratio, and the factor of ½, required for the correct nutation rate, is implicit in the partition of field operators according to the rotating wave  approximation.   Then for the coupling of several spins to a single cavity mode we write the Tavis-Cummings (i.e. extended Jaynes-Cummings) Hamiltonian in the interaction picture:   H=−Ω02{ˆa†ˆI+(j)+ˆaˆI−(j)}j=1∑    [1] where the Rabi fundamental Ω0 is effectively the coupling constant between spins and cavity, and the frequency for the transition connecting the Fock states n and n+1 is Ωn=(n+1)Ω0; (for large nwe will approximate n+1with n.)   The sum is over spins; the operator pairings are for a spin ½ with positive gyromagnetic ratio.  The Zeeman interaction is implied by the presence of the Larmor frequency (above), but a direct treatment of the static polarizing field is unnecessary. For the case of a single spin (the J-C model), several informative results are obtainable by elementary analytic calculation.  Starting from the preparatory density matrix ρ(prep)=0α0α, and applying (as described  \n 8 earlier) a spin rotation of π2 , (about the y axis of a rotating reference frame) we arrive the density matrix describing our initial conditions:                                        ρ(init)=121−10−110000⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥   ,   [2]  where the product basis elements 0α, 0β, 1α give the occupation numbers and spin projections.  Then with the abbreviations c=cos12Ω0t ands=sin12Ω0t, the density matrix, at time t, evolves, to:                                   ρ(t)=121−cis−cc2−isc−isiscs2⎤⎦⎥⎥⎥⎡⎣⎢⎢⎢   .   [3]                                The reduced [38]  spin and photon density matrices are: ρ(spin)(t)=121+s2−c−cc2⎤⎦⎥⎥⎡⎣⎢⎢ .   [4]  and                          ρ(cavity)(t)=121+c2is−iss2⎤⎦⎥⎥⎡⎣⎢⎢        .      [5]   From these expressions it is easily seen that the longitudinal magnetization oscillates at the Rabi fundamental frequency, and the transverse at one half  \n 9 that value; also the cavity one-quantum coherence and transverse magnetization are in time quadrature in the laboratory frame (the two are related like the real and imaginary components of the complex NMR signal.)  The case of an initial nutation of π  is easily worked out, and shows a perfect absence of transverse magnetization.  These effects will be borne out in the numerical examples presented below.  The Case of Two Spins   As above, the basis set comprises simple product kets,  e.g. nαβ〉, giving the Fock state indices (0, 1, and 2) and the spin projections.   The eigenvalues, in units of Ω0 , are ±32 , ±1 , ±12, and 0 (multiplicity 6).   The twelve members of the basis set include seven elements, which would together comprise a pair invariant subspaces of constant excitation 1 and 2 [39]; but a redundant basis is easily adaptable to the case of more spins and is therefore preferred.    Excited states of the spins are generated, as noted, by applying spin rotations (π2 or π)  to the preparatory density matrixρ(prep)=0αα〉〈0αα; also as noted, the rotation is performed by a classical field; the subsequent time evolution of the spins, and their prompt entanglement with the cavity, is then determined by solution of the Liouville equation, based upon the Hamiltonian of Eq [1].  The form of  ρ(prep) is appropriate for small numbers of spins, and also ensures a trace of unity.  Since the cavity is presumed lossless, and the NMR linewidth perfectly homogeneous, excitation along  \n 10 the rotating axes x or y ensures that all off-diagonal elements of the reduced spin density matrix, ρ(spin),  are either pure real or pure imaginary.  Figure 1A shows the non-classical time evolution of the longitudinal and transverse magnetizations (for two periods of the Rabi fundamental),  starting from an initial condition with both spins tipped by π2  and the cavity is in its ground state.  The subsequent time course is calculated by evolving the propagator in the eigenbasis, following numerical diagonalization of the Hamiltonian [40].  The magnetizations  are plotted at baseband, i.e. with the harmonic time dependence at the Larmor (carrier) frequency having been removed by demodulation.  All possible spin coherences are excited, of orders zero, one, and two, although only the latter two are visualized; the dotted trace shows the total number of excitations, constant and equal to 1.  The longitudinal magnetization (blue trace) oscillates at the Rabi frequency.  The transverse magnetization (a single-quantum coherence)  oscillates at half the Rabi frequency, 12Ω0 , between the negative and positive x axes of the rotating coordinate frame [41].  Since the two magnetizations evolve at different frequencies, they cannot be said to be in time quadrature; nonetheless, the extrema of one occur at or near the zeros of the other, as expected for the time evolution of the corresponding classical signals.  The incipient damping, of both magnetizations, signals the onset of quantum collapse [17]; revival is easily demonstrated in a time course of longer duration. The red trace (shown at doubled amplitude for better visualization) is the two-quantum coherence (a Schrödinger’s cat state),  It oscillates (roughly sinusoidally) at the highest eigenfrequency, 32Ω0.  \n 11 Figure 1B shows the time evolution of the cavity coherences (also at baseband.)  There are two single quantum coherences (blue and green traces) with complicated time dependences, representing the summed cavity reduced density matrix elementsρ12(cavity)±ρ21(cavity)   and ρ23(cavity)±ρ32(cavity), where  the index 1 denotes the ground Fock state.  The red trace  undergoes approximately sinusoidal oscillation at32Ω0, and represents the two- quantum coherence ρ13(cavity)±ρ31(cavity); its time evolution tracks that of the corresponding spin coherence.  The phase of the one quantum coherences depends upon the choice of axis for the initial excitation of the spins; for initial rotation about y (leading to purely real spin coherences, i.e.  entirely expressed in terms of operator products, Ix(i)and Iz(j)),  the single-quantum cavity coherences are pure imaginary, of which more below.   Figure 2A illustrates our central result: that is, the transfer of Zeeman energy (via spin coherence) from the precessing spins to the tuned cavity, with the concomitant appearance of an induced field.   The transverse magnetization (cf  Fig. 1A)  is shown in green for reference.     Then summation of the pair of one-quantum cavity coherences (cf Fig. 1B) yields the quasi-sinusoidal waveform shown in dashed blue. Fourier analyses (not shown) demonstrate minor differences in frequency content between the transverse magnetization and the summed cavity coherences, despite which the two are approximately in time quadrature at baseband, as is expected [13] in a conventional pulsed  NMR experiment.   The dashed black trace gives the expectation value at baseband of magnetic flux (normalized to the square root of the occupation number).  Since the off-diagonal terms of the cavity reduced matrix are here purely imaginary, its trace with the summed field operators vanishes when the  \n 12 operator time dependence (at the Larmor frequency) is omitted.  The summed operators may nonetheless be evaluated at Larmor, yet plotted at baseband, i.e. the phasor amplitude plotted, as we have done. Voltage and current are in phase for a perfectly tuned oscillator excited at resonance, and the net amplitude of flux – or field-- scales directly with current.  The time course of flux directly tracks that of the induced magnetic field.  Since the transverse moment oscillates at about one half the Rabi frequency, and since our cavity is assumed lossless, the zero of transverse moment coincides with the maximum field, in contrast to a classical model with losses (which is discussed later.)  Overall, the figure illustrates the dynamics of NMR transduction, which may be described qualitatively as the conversion of spin coherence to cavity coherence.  It may also be viewed as the primordial form of radiation damping -- even in the absence of cavity losses-- since Zeeman energy is transduced to create an oscillatory field.  (We return to this point below, in the section on cavity enhancement.)   For reference, Figure 2B shows the phenomena of collapse and revival –here of longitudinal magnetization -- observed over several periods of the Rabi fundamental, following preparation with a nutation pulse of π2( above).   After a π pulse (below),  pure Rabi oscillation is observed without quantum collapse.   The Case of Multiple Spins (N > 2)   We show in this section some details of dynamic behavior for larger numbers of spins; we restrict consideration to an initial density matrix  \n 13 ρ(prep)=0α....α〉〈0α....α.   Figure 3A shows the time evolution of transverse magnetization, over a single period of the Rabi fundamental, for spin clusters with N ranging from two to seven, and the vertical scale normalized correspondingly.   The curves show the expected behavior, inasmuch as the decay rate of transverse magnetization (which essentially equals the emission rate in the semi-classical regime) grows with the number of emitted photons, and therefore demonstrates stimulated emission.   This is clearly seen in the figure by tracing the position of the first zero crossing, for progressively larger N.  The longitudinal magnetization (not shown) displays similar behavior, although not as pronounced.   Typical cavity dynamics are complex, and are shown in Figure 3B, which gives, over the same time interval, (in solid lines) the time courses of the individual one-quantum cavity coherences, for five spins, – each calculated as above by summing conjugate elements), plus (in dashed blue) their net resultant, which exhibits the expected quasi-sinusoidal shape (cf Figure 2A), also with evidence of quantum collapse.  The net magnetic flux (dashed black, calculated as describe earlier) follows closely the total summed coherences.   The rapid initial growth of cavity one-quantum coherence is explained by the early dominance of the 1,2element (plus conjugate) in the cavity density matrix,  which involves the cavity ground state – fully populated at the outset.  The other tributaries (which we write in abbreviated form)  -- e.g. ρ23(cavity), ρ34(cavity), ρ45(cavity) – show the expected delayed growth characteristic of stimulated emission, as higher cavity levels are progressively populated.  Fig. 4A shows, for five spins in a single period of the Rabi fundamental, the interchange of the net photon population (dotted blue  \n 14 trace)  with transverse (green) and longitudinal (blue) magnetizations.  For  comparison, we show in Figure 4B the analytic results (starting from a negative transverse moment) for a single spin (above) for two periods of the Rabi fundamental. Note the absence of quantum collapse and resulting high symmetry.  Otherwise these are similar to the results of Figure 4A, and show usefulness as a qualitative guide.  Introduction of Cavity Losses and Relaxation   Since a complete theory of radiation damping will include dissipation, we sketch the introduction of cavity losses and spin relaxation.  The examples are illustrative, not definitive.  Earlier attempts at a quantum theory of the NMR signal treated cavity damping artificially [9-10, 42], e.g. by simple escape of photons from an active volume, or by  a transmission line dashpot , (i.e. a non reflective termination.)  Here we employ the master equation from the theory of micromasers [18-20],  which derives from coupling to a thermal bath, and gives the damping contribution for ρ:    ρ=(γ2){(n+1)(2ˆaρˆa†−ˆa†ˆaρ−ρˆa†ˆa)+n(2ˆa†ρˆa−ˆaˆa†ρ−ρˆaˆa†)},   [6] where nis the mean photon occupation number, and γis now the photon damping rate.  For micromasers, ρis usually the reduced photon density matrix [18-19], but here it is the combined spin-photon matrix.  The dissipative term in Eq. 4 is added to the coherent Liouville equation, to give the master equation with damping.  To this we then superadd the effects of relaxation, following the phenomenological Bloch equations, with separate  \n 15 decay constants for the longitudinal and transverse moments; for simplicity both are assumed to damp to zero.  We consider a single spin, since multispin longitudinal order associated with high polarization [43] requires additional magnetization modes and decay constants [44].  The full master equation is then solved numerically by iteration, using the Euler method (45).  Figure 5 shows that the effects of cavity losses and spin relaxation can be adjusted empirically to mimic the results of classical radiation damping in NMR,  that is, to produce substantial damping of Rabi oscillation in half a cycle.  Figure 5A gives the damped signal from a single spin; figure 5B shows a realistic calculation based upon the classical Bloch-Kirchhoff [12, 46] equations,  for a sample of water protons at 14 tesla, using coil and sample parameters derived from an experimental study [14].  Of particular interest is the peak damping current, which reaches a value of 7 milliamps.  The calculated damping linewidth here -- based upon the measured coil efficiency, and excluding the (unmeasureable) filling factor--  is 55 Hz, compared with a measured value of  65 Hz.  Also, compare particularly Fig. 5A with Figure 4B (one spin, no dissipation, no quantum collapse), to judge the rapidity of signal decay with losses.  The magnetizations have similar trajectories in both figures 5A and 5B; also the oscillator current in 5B resembles the photon population in 5A, when account is taken of the early time behavior (inset) .    Cavity Enhancement and Stimulated Emission in NMR: the Rabi Frequency and the Radiation Damping Constant   \n 16 We first numerically estimate the cavity-enhanced Rabi fundamental; we then establish a relationship between the Rabi frequency and the radiation damping constant; we also discuss the importance of stimulated emission in setting the signal power in NMR (i.e. the rate of energy transfer from spins to cavity).   Throughout, we follow the coupled oscillator model of J-C [5], i.e. for a radiating atom coupled to a single cavity mode, or in our case, a quantum oscillator.  For a reasonable Helmholtz coil, (round windows of inside diameter 0.75 cm, separated by the diameter, inductance of 58 nH) the Rabi fundamental (cf Theory) takes the value  Ω0=8.13×10−5sec−1 for a proton in a polarizing field of 14.1 tesla, i.e. with  Larmor frequency ω02π of 600 MHz.  This corresponds to an emission time (12 of the Rabi period) of 6.47×10−6sec.  By contrast, the inverse lifetime for spontaneous emission at 600 MHz,  by a single excited proton in free space, is: µ0γ2ω3πc3=6×10−21sec−1.  The staggering difference (~15 orders of magnitude) is due to the high concentration of magnetic flux created per photon, inside the coil, relative to that in free space.  J-C estimate a cavity enhancement of ~107for spontaneous emission in the ammonia beam maser [5].  Then the rotating frame B1 field for our model NMR coil carrying unit current is justL/4a. Writing the current in terms of the oscillator mean occupation number n as2nω0L, (according to12LI2=nω0), and multiplying by field per current and the gyromagnetic ratio, yieldsγnω0L/22a, which , for large n, is just the Rabi nutation frequency connecting the ntt and n+1st  Fock states.  This is therefore the product of a nutation rate per unit current, and a current.  Incidentally, this  \n 17 can also be written in terms of the occupation number n as Ωn=Ω0n,  which shows the importance of stimulated emission in setting the overall emission rate in NMR; we shall return to this point below.   The classical radiation damping constant [12] also factors as the product of a nutation rate per current, and a current.  We start by writing it in terms of the fill factorη, as k=µ0γηM0Q2 (in SI units), where Q is the source-loaded tuned-circuit quality factor, M0 is the equilibrium magnetization, γis the gyromagnetic ratio and other symbols have their usual meanings.  Using reciprocity theory [14, 47-48], this may be rewritten as k=γω0VM0ζ24 with the transceiver efficiency defined as ζ=B1(I)/(IR),  where B1(I) is the laboratory frame radiofrequency  field at coil current I, and R is the coil resistance, (without source loading); V the sample volume.  Thus the damping constant is a product of two factors: γB1(I)2I and ω0VM0B1(I)2IR: the first a nutation rate per unit current, and the second (per reciprocity theory) a current, i.e. voltage over resistance.  This establishes (if it were doubted) the close relationship of Rabi nutation to radiation damping.  Even for a lossless cavity, we have used the term 'radiation damping' to describe the diminution of the transverse moment, and concomitant growth of the longitudinal, (as the spins emit), by analogy with the Wigner-Weisskopf  theory of spectroscopic linewidth [11], in which the fundamental event is the emission of a photon, whose subsequent fate (e.g absorption by a black body) is of small interest.  Finally we return to the question of stimulated emission, specifically for the water protons in a realistic NMR probe  as given elsewhere, [14] and described in the legend of Fig. 5B.  From the familiar equation for Zeeman  \n 18 energy balance in terms of nutation angle (E=M0VB0ϑsinϑ) we calculate, for ϑ=Ω0, a net power due to coherent spontaneous emission of  11 pW , that is, for nutation of the net moment at the Rabi fundamental, without stimulated emission. This is far below the rms power of 25µW, calculated classically from Bloch-Kirchhoff with a peak oscillator current of 7 mA (cf Figure 5B above), and the given resistance (with source loading) of 1.0 ohm [14].  However, the oscillator occupation number for 7 mA is 3.57×1012 , yielding an increase in the Rabi frequency of a factorn=1.9×106.  The resultant emitted power is now increased from 11 pW to 21µW, which lies within 20% of the 25 µWcalculated classically.  (In reckoning the classical power, the amplitude of the current must be treated as AC, and its rms value used.  This is verified by a direct calculation in terms of the Zeeman energy, as gotten from the longitudinal moment.) The numerical agreement between the two values of power (21 µW and 25µW) is perhaps fortuitous, given the approximate nature of the calculations-- but  it is nonetheless consistent with the view that the power in radiation damping has a substantial contribution, amounting to several orders of magnitude, from stimulated emission.  This result is also is also consistent with the increase of the Rabi frequency measured in experiments on large populations of excited Rydberg atoms in a tuned cavity [49], as well for single atoms in the presence of larger injected fields [50].   Discussion   \n 19 Cavity quantum electrodynamics originated with the observation by Purcell, that spontaneous emission inside a tuned cavity is enhanced by the increased the density of states [36-37].  Bloembergen and Pound [12] then argued that the observed signal power in NMR results from the twinned factors of cavity enhancement and coherent spontaneous emission; and this viewpoint has been accepted for decades [30].  Our own calculations suggest that these two factors do not suffice to explain the strength of the FID signal, but that stimulated emission makes an essential contribution, amounting to several orders of magnitude. The treatment of the enhanced radiation density inside the cavity then becomes important, as we follow here not Bloembergen-Pound, but Jaynes-Cummings [5]. That is, B-P, -- and others [51-52] -- typically write the enhancement factor (i.e. the density of states) in terms of the cavity Q, as a measure of the sharpness of the cavity resonance. This approach blurs the distinction between the atom-field coupling constant and the cavity dissipation rate, as set out in the bad cavity limit [53], and also as manifest in the master equation, which clearly separates the coherent atom-field interaction from the incoherent cavity damping.    J-C, in contrast to B-P, write the Hamiltonian directly in terms of the atom-field coupling constant, without reference to cavity dissipation.  This also comports with the theory of reciprocity [47-48], inasmuch as the emf in nuclear magnetism depends only upon the radiofrequency B field per unit oscillator current,  which determines the nutation rate (i.e. Rabi frequency) given the actual current, and serves in the classical theory as an analog to the coupling constant between spins and cavity.  The J-C treatment is also intrinsically a theory of large nutations, in contrast to that of B-P, which  \n 20 (despite solving the classical damping equation for large excursions of the Bloch vector) relies nonetheless on small perturbations for the quantum treatment of cavity-enhanced emission, with the density of states entering through the Fermi Golden Rule. On a different tack, we have noted earlier that CQED does not lead directly to the Bloch equations, without strong assumptions amounting to the imposition of semi-classical behavior of the cavity; this path has been chosen by other workers in NMR [9-10] including an instance in which  the validity of Bloch equations is taken to define the zero order condition in a perturbation scheme [9].  In a larger sense, the question of the transition to semi-classical behavior remains open in CQED [21, 54-56].  The disappearance of quantum collapse and revival has been proposed as a marker for the arrival of classical or semi-classical behavior [54] ; but in practical NMR,  where low resonator quality factors  (~50 to 500) enforce the bad cavity limit [53], collapse and revival have never been  (and likely never will be) observed.  We therefore propose that the transition to classical behavior in NMR comes with the onset of rotation of the Bloch vector, which we take to coincide with the appearance of a cavity Glauber state.  That is, if the Liouville equation for a spin ½ coupled to a cavity be written for the initial cavity state a Glauber state, then the Liouville matrix elements are diagonal in photon variables, and pure rotations of the Bloch vector can occur.  On the other hand, if the cavity starts in a Fock state, the Hamiltonian disallows pure rotations of the spins, and non-classical effects ensue-- an issue which has not been addressed by prior workers in magnetic resonance [9-10, 27-31]. The theory given here, although incomplete, demonstrates the gap between the pure quantum and semi-classical regimes, with the latter corresponding to the customary world of NMR observations.   \n 21 Bridging the gap between the two will probably require calculations including both dissipation and many more spins.  Acknowledgement This work was supported by General Electric Healthcare Technologies.   Figure Legends  Figure 1.  Time evolution of magnetizations and cavity coherences.   For two periods of the Rabi fundamental, following a π2 pulse of two perfectly polarized spins. Spin and cavity coherences are offset in time by one quarter Larmor period.  A) Longitudinal magnetization (blue), transverse magnetization (green), two quantum coherence (red.).  Vertical axis normalized to number of spins.  The dotted black line is the number of excitations, constant at 1.0.   B) Individual one-quantum cavity coherences: ρ12(cavity)±ρ21(cavity) (blue) and ρ23(cavity)±ρ32(cavity) (green); and  also two quantum cavity coherence ρ13(cavity)±ρ31(cavity) (red).     Figure 2.  Energy transfer from spins to cavity, and Rabi oscillation of longitudinal moment.  A), Time course of transverse magnetization (green), and  summed one quantum cavity coherences (dotted blue.)  Also, (dotted black) the normalized magnetic flux (per square root photon). B)  Time  \n 22 course of longitudinal magnetization following preparatory nutation of π2 (above, note quantum collapse and revival) and π (below), for 10 periods of the Rabi fundamental.  Note the division of the vertical scale for the upper and lower traces.   Figure 3.  A) Illustration of stimulated emission over one period of the Rabi fundamental: time courses of transverse magnetization for increasing numbers of spins, from two – (solid blue trace) through seven (dotted red trace.)  (Number of spins increases in color sequence blue, green, red, and from solid to dotted trace) The damping (emission) rate is indicated by the first zero crossing, which arrives progressively sooner with each additional spin, as expected from stimulated emission.  Vertical axis normalized to number of spins.  B) One quantum cavity coherences (solid traces) for five spins, and their summations, to give approximately sinusoidal resultant (dashed navy trace); also,  weighted sum (dashed black trace) gives net flux. Color sequence of one quantum coherences (solid lines) starting from ρ12(cavity) is navy, red, green, violet, cerulean.  Note the acceleration relative to two spins shown in Figure 2. Refer to text for details.   Figure 4.  Comparison of numerical and analytic results.  A) Signal formation with five spins, following nutation of π2, for one period of the Rabi fundamental.  Longitudinal moment blue, transverse green, total photons dashed blue, total excitations dashed black.  B) Analytical solution of Jaynes-Cummings model, single spin in cavity tuned at Larmor  \n 23 frequency, following nutation of π2, for two periods of the Rabi fundamental.  Colors as in  3A.     Figure 5.  Effects of dissipation: cavity losses and spin relaxation. A)  J-C model (one spin) with cavity damping and spin relaxation, to simulate radiation damping in conventional NMR.  Strong cavity damping (γ=5τ0),  short T2, (τ04) long T1 (τ00.1), with τ0 the Rabi fundamental period.  Longitudinal and transverse magnetization  solid blue and green respectively, total photons dotted blue, total excitations dotted black.  B) Classical radiation damping per the Bloch Kirchhoff equations.  Magnetizations as in 5A, magnitude current  in dotted blue (milliamp scale).  Inset: early time current, 1 µs duration, 8 mA excursion starting at zero; compare zero initial photons in 4A.  Details of the sample and the measurement of the coil efficiency match experimental conditions [14].  Starting magnetic moment:  9. 73×10−9 amp-meter2 , corresponds to neat water at T = 298 K  in a 5 mm NMR tube, with vertical probe window of 1.6 cm.   Measured RF coil efficiency: 2.64×10−4teslawatt. Measured quality factor of Q=220, plus assumed coil resistance of  R=1.0 ohm (both Q and resistance are source loaded), gives coil inductance L=58 nH.       \n 24 References  1. N. Bloembergen, Nonlinear Optics, 4th Edn. (World Scientific Publishing Singapore 1996).  2.  J. A. Sidles, Phys. Rev. Lett. 68, 1124 (1991).  3. I. Bargatin and M. L. Roukes, Phys. Rev. Lett. 91, 138302 (2003)  4.  M. C. Butler, V. A. Norton, D. P. Weitekamp, Phys. Rev. Lett. 105, 177601 (2010).  5. .  E. T. Jaynes and F. W. 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Lett.  105, 163601, (2010).  \n 30                            00.511.52−0.8−0.6−0.4−0.200.20.40.60.81.0\n00.511.52−2−1.5−1−0.500.511.52ATropp Figure 1B\n0246810−2.0−1.001.001.02.0\n00.511 . 5 2−2.0−1.5−1.0−0.500.51.51.02.0BATropp Figure 2\n00.20.40.60.81−8−6−4−202468AB\n00.20.40.60.81−1.5−1.0−0.500.51.01.52.0Tropp Figure 3 \n 31                            00.20.40.60.81−5−4−3−2−1012345\n00.40.81.21.62−1−0.8−0.6−0.4−0.200.20.40.60.81ABTropp Figure 4\n00.511.52−1−0.8−0.6−0.4−0.200.20.40.60.81BATropp Figure 5\n00.0050.010.0150.020.025−1−0.8−0.6−0.4−0.200.20.40.60.810246810x 10−8amp-m2\nmilliamp secondB"    },    {        "title": "1607.04983v3.Magnetic_Skyrmion_Transport_in_a_Nanotrack_With_Spatially_Varying_Damping_and_Non_adiabatic_Torque.pdf",        "content": "1\nMagnetic Skyrmion Transport in a Nanotrack With Spatially\nVarying Damping and Non-adiabatic Torque\nXichao Zhang1,2, Jing Xia1, G. P. Zhao3, Xiaoxi Liu4, and Yan Zhou1\n1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China\n2School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China\n3College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n4Department of Information Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan\nReliable transport of magnetic skyrmions is required for any future skyrmion-based information processing devices. Here we\npresent a micromagnetic study of the in-plane current-driven motion of a skyrmion in a ferromagnetic nanotrack with spatially\nsinusoidally varying Gilbert damping and/or non-adiabatic spin-transfer torque coefficients. It is found that the skyrmion moves in\na sinusoidal pattern as a result of the spatially varying Gilbert damping and/or non-adiabatic spin-transfer torque in the nanotrack,\nwhich could prevent the destruction of the skyrmion caused by the skyrmion Hall effect. The results provide a guide for designing\nand developing the skyrmion transport channel in skyrmion-based spintronic applications.\nIndex Terms —magnetic skyrmions, racetrack memories, micromagnetics, spintronics.\nI. I NTRODUCTION\nMagnetic skyrmions are quasiparticle-like domain-wall\nstructures with typical sizes in the sub-micrometer regime [1]–\n[7]. They are theoretically predicted to exist in magnetic metals\nhaving antisymmetric exchange interactions [8], and confirmed\nby experiments [9], [10] just after the turn of the twenty-\nfirst century. Isolated skyrmions are expected to be used to\nencode information into bits [11], which might lead to the\ndevelopment of novel spintronic applications, such as the\nracetrack memories [12]–[19], storage devices [20]–[22], and\nlogic computing devices [23].\nThe write-in and read-out processes of skyrmions in thin\nfilms are realizable and controllable at low temperatures [24]–\n[26]. A recent experiment has realized the current-induced\ncreation and motion of skyrmions in Ta/CoFeB/TaO trilayers\nat room temperature [27]. Experimental investigations have\nalso demonstrated the increased stability of skyrmions in mul-\ntilayers [28]–[30], which makes skyrmions more applicable to\npractical room-temperature applications.\nHowever, the skyrmion experiences the skyrmion Hall effect\n(SkHE) [31], [32], which drives it away from the longitudinal\ndirection when it moves in a narrow nanotrack. As a con-\nsequence, in the high-speed operation, the transverse motion\nof a skyrmion may result in its destruction at the nanotrack\nedges [18], [33]–[36]. Theoretical and numerical works have\nproposed several intriguing methods to reduce or eliminate the\ndetrimental transverse motion caused by the SkHE. For ex-\nample, one could straightforwardly enhance the perpendicular\nmagnetic anisotropy near the nanotrack edges to better confine\nthe skyrmion motion [33]. An alternative solution is to trans-\nport skyrmions on periodic substrates [37]–[40], where the\nskyrmion trajectory can be effectively controlled. Moreover, by\nconstructing antiferromagnetic skyrmions [34], [35] and anti-\nThe first two authors contributed equally to this work. Corre-\nsponding authors: X. Liu (email: liu@cs.shinshu-u.ac.jp) and Y . Zhou\n(email: zhouyan@cuhk.edu.cn).ferromagnetically exchange-coupled bilayer skyrmions [18],\n[36], the SkHE can be completely suppressed. Recently, it is\nalso found that the skyrmionium can perfectly move along\nthe driving force direction due to its spin texture with a zero\nskyrmion number [41], [42].\nIn this paper, we propose and demonstrate that a skyrmion\nguide with spatially sinusoidally varying Gilbert damping\nand/or non-adiabatic spin-transfer torque (STT) coefficients\ncan be designed for transporting skyrmions in a sinusoidal\nmanner, which is inspired by a recent study on the magnetic\nvortex guide [43], where the vortex core motion is controlled\nvia spatially varying Gilbert damping coefficient. The results\nprovide a guide for designing and developing the skyrmion\ntransport channel in future spintronic devices based on the\nmanipulation of skyrmions.\nII. M ETHODS\nOur simulation model is an ultra-thin ferromagnetic nan-\notrack with the length land the width w, where the thick-\nness is fixed at 1nm. We perform the simulation using\nthe standard micromagnetic simulator, i.e., the 1.2 alpha 5\nrelease of the Object Oriented MicroMagnetic Framework\n(OOMMF) [44]. The simulation is accomplished by a set of\nbuilt-in OOMMF extensible solver (OXS) objects. We employ\nthe OXS extension module for modeling the interface-induced\nantisymmetric exchange interaction, i.e., the Dzyaloshinskii-\nMoriya interaction (DMI) [45]. In addition, we use the updated\nOXS extension module for simulating the in-plane current-\ninduced STTs [46]. The in-plane current-driven magnetization\ndynamics is governed by the Landau-Lifshitz-Gilbert (LLG)\nequation augmented with the adiabatic and non-adiabatic\nSTTs [44], [47]\ndM\ndt=\u0000\r0M\u0002Heff+\u000b\nMS(M\u0002dM\ndt) (1)\n+u\nM2\nS(M\u0002@M\n@x\u0002M)\u0000\fu\nMS(M\u0002@M\n@x);arXiv:1607.04983v3  [cond-mat.mes-hall]  15 Dec 20162\nFig. 1. (a) The magnetic damping coefficient \u000b(x)and non-adiabatic STT\ncoefficient\f(x)as functions of xin the nanotrack. (b) Trajectories of current-\ndriven skyrmions with \f=\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6.\nDot denotes the skyrmion center. Red cross indicates the skyrmion destruction.\n(c) Skyrmion Hall angle \bas a function of xfor skyrmion motion with \f=\n\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6. The dashed lines indicate\n\b =\u000614\u000e. (e) Real-space top-views of skyrmion motion with \f=\u000b=2 =\n0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6.wandvdenote the nanotrack\nwidth and velocity direction, respectively. The dashed line indicates the central\nline of the nanotrack. The skyrmion is destroyed at t= 870 ps when\f=\n2\u000b= 0:6. The out-of-plane magnetization component is represented by the\nred (\u0000z)-white ( 0)-green ( +z) color scale.\nwhere Mis the magnetization, MSis the saturation magne-\ntization,tis the time, \r0is the Gilbert gyromagnetic ratio,\n\u000bis the Gilbert damping coefficient, and \fis the strength of\nthe non-adiabatic STT. The adiabatic STT coefficient is given\nbyu, i.e., the conduction electron velocity. The effective field\nHeffis expressed as\nHeff=\u0000\u0016\u00001\n0@E\n@M; (2)\nwhere\u00160is the vacuum permeability constant. The average\nenergy density Econtains the exchange, anisotropy, demag-\nnetization, and DMI energies, which is given as\nE=A[r(M\nMS)]2\u0000K(n\u0001M)2\nM2\nS\u0000\u00160\n2M\u0001Hd(M) (3)\n+D\nM2\nS(Mz@Mx\n@x+Mz@My\n@y\u0000Mx@Mz\n@x\u0000My@Mz\n@y);\nwhereA,K, andDare the exchange, anisotropy, and DMI\nenergy constants, respectively. nis the unit surface normal\nvector, and Hd(M)is the demagnetization field. Mx,My\nandMzare the three Cartesian components of M.\nThe model is discretized into tetragonal volume elements\nwith the size of 2nm\u00022nm\u00021nm, which ensures a\ngood compromise between the computational accuracy and ef-\nficiency. The magnetic parameters are adopted from Refs. [14],\nFig. 2. (a)vx, (b)vy, and (c) \bas functions of \u000band\fgiven by Eq. (11)\nand Eq. (12), respectively. vxandvyare reduced by u.\n[23]:\r0= 2:211\u0002105m/(A\u0001s),A= 15 pJ/m,D= 3mJ/m2,\nK= 0:8MJ/m3,MS= 580 kA/m. In all simulations, we\nassumeu= 100 m/s andw= 50 nm. The skyrmion is initially\nlocated at the position of x= 100 nm,y= 25 nm.\nThe Gilbert damping coefficient \u000bis defined as a function\nof the longitudinal coordinate xas follows [Fig. 1(a)]\n\u000b(x) =\u000bamp\u0001f1 + sin [2\u0019(x=\u0015\u000b)]g+\u000bmin; (4)\nwhere\u000bamp= (\u000bmax\u0000\u000bmin)=2is the amplitude of the \u000b\nfunction.\u000bmaxand\u000bminstand for the maximum and mini-\nmum values of the \u000bfunction, respectively. \u0015\u000bdenotes the\nwavelength of the \u000bfunction. It is worth mentioning that the\nspatially varying \u000bcan be achieved by gradient doping of\nlanthanides impurities in ferromagnets [43], [48], [49]. Exper-\niments have found that \u000bis dependent on the interface [50].\nThus it is also realistic to construct the varying \u000bby techniques\nsuch as interface engineering. Indeed, as shown in Ref. [51],\nlocal control of \u000bin a ferromagnetic/non-magnetic thin-film\nbilayer has been experimentally demonstrated by interfacial\nintermixing induced by focused ion-beam irradiation.\nIn a similar way, the non-adiabatic STT coefficient \fis\nalso defined as a function of the longitudinal coordinate xas\nfollows [Fig. 1(a)]\n\f(x) =\famp\u0001f1 + sin [2\u0019(x=\u0015\f)\u0000']g+\fmin;(5)\nwhere\famp= (\fmax\u0000\fmin)=2is the amplitude of the \f\nfunction.\fmaxand\fminstand for the maximum and minimum\nvalues of the \ffunction, respectively. \u0015\fand'denote the\nwavelength and phase of the \ffunction, respectively. Since\nthe value of \fdepends on the material properties [52], it is\nexpected to realize the spatial varying \fby constructing a\nsuperlattice nanotrack using different materials, similar to the\nmodel given in Ref. [43]. Note that the effect of varying \f\nhas also been studied in spin torque oscillators [53].\nIII. R ESULTS\nA. Nanotrack with spatially uniform \u000band\f\nWe first recapitulate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially uniform \u000band\f. As\nshown in Fig. 1(b), the skyrmion moves along the central line\nof the nanotrack when \f=\u000b= 0:3. However, due to the\nSkHE, it shows a transverse shift toward the upper and lower\nedges when \f= 2\u000b= 0:6and\f=\u000b=2 = 0:15, respectively.3\nThe skyrmion is destroyed by touching the upper edge when\n\f= 2\u000b= 0:6att= 870 ps.\nThe skyrmion Hall angle \b, which characterizes the trans-\nverse motion of the skyrmion caused by the SkHE, is defined\nas\n\b = tan\u00001(vy=vx): (6)\nFigure 1(c) shows \bas a function of xfor the skyrmion motion\nwith\f=\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6. It\ncan be seen that \b = 0\u000ewhen\f=\u000b= 0:3, indicating\nthe moving skyrmion has no transverse motion [Fig. 1(d)].\nWhen\f=\u000b=2 = 0:15,\bincreases from\u000015\u000eto0\u000e,\nindicating the moving skyrmion has a transverse shift toward\nthe lower edge which is balanced by the transverse force due to\nthe SkHE and the edge-skyrmion repulsive force [Fig. 1(d)].\nWhen\f= 2\u000b= 0:6,\bdecreases from 15\u000eto3\u000ewithin\n870 ps, indicating the moving skyrmion shows a transverse\nmotion toward the upper edge. At t= 870 ps, the skyrmion\nis destroyed as it touches the upper edge of the nanotrack\n[Fig. 1(d)]. It should be noted that the skyrmion profile is\nrigid before it touches the nanotrack edge. In order to better\nunderstand the transverse motion caused by the SkHE, we also\nanalyze the in-plane current-driven skyrmion motion using the\nThiele equation [54]–[57] by assuming the skyrmion moves in\nan infinite film, which is expressed as\nG\u0002(v\u0000u) +D(\fu\u0000\u000bv) =0; (7)\nwhere G= (0;0;\u00004\u0019Q)is the gyromagnetic coupling vector\nwith the skyrmion number\nQ=1\n4\u0019Z\nm\u0001\u0012@m\n@x\u0002@m\n@y\u0013\ndxdy: (8)\nm=M=M Sis the reduced magnetization and Dis the\ndissipative tensor\nD= 4\u0019\u0012DxxDxy\nDyxDyy\u0013\n: (9)\nu= (u;0)is the conduction electron velocity, and vis the\nskyrmion velocity. For the nanoscale skyrmion studied here,\nwe have\nQ=\u00001;Dxx=Dyy= 1;Dxy=Dyx= 0: (10)\nHence, the skyrmion velocity is given as\nvx=u(\u000b\f+ 1)\n\u000b2+ 1; vy=u(\f\u0000\u000b)\n\u000b2+ 1: (11)\nThe skyrmion Hall angle \bis thus given as\n\b = tan\u00001(vy=vx) = tan\u00001\u0012\f\u0000\u000b\n\u000b\f+ 1\u0013\n: (12)\nBy calculating Eq. (11), we show vxas functions of \u000band\f\nin Fig. 2(a). vxranges between 0:5uand1:21u, indicating the\nskyrmion always moves in the +xdirection. When \u000b= 0:42\nand\f= 1,vxcan reach the maximum value of vx= 1:21u.\nSimilarly, we show vyas functions of \u000band\fin Fig. 2(b).\nvyranges between\u00000:5uandu, indicating the skyrmion can\nmove in both the \u0006ydirections. When \u000b < \f ,vy>0, the\nskyrmion shows a positive transverse motion, while when \u000b>\n\f,vy<0, the skyrmion shows a negative transverse motion.\nFig. 3. (a) Trajectories of current-driven skyrmions with \u000bamp =\n0:315;0:225;0:215.\u0015\u000b= 2wand\f= 0:3. (b) \bas a function of x\nfor skyrmion motion with \u000bamp= 0:315;0:225;0:215.\u0015\u000b= 2wand\n\f= 0:3. (c) Trajectories of current-driven skyrmions with \u0015\u000b=w;2w;4w.\n\u000bamp= 0:225 and\f= 0:3. (d)\bas a function of xfor skyrmion motion\nwith\u0015\u000b=w;2w;4w.\u000bamp= 0:225 and\f= 0:3.\nFig. 4. (a) Trajectories of current-driven skyrmions with \famp =\n0:315;0:225;0:215.\u0015\f= 2w,'= 0, and\u000b= 0:3. (b) \bas a function\nofxfor skyrmion motion with \famp= 0:315;0:225;0:215.\u0015\f= 2w,\n'= 0 , and\u000b= 0:3. (c) Trajectories of current-driven skyrmions with\n\u0015\f=w;2w;4w.\famp= 0:225,'= 0, and\u000b= 0:3. (d)\bas a function\nofxfor skyrmion motion with \u0015\f=w;2w;4w.\famp= 0:225,'= 0, and\n\u000b= 0:3.\nBy calculating Eq. (12), we also show \bas functions of \u000b\nand\fin Fig. 2(c), where \bvaries between \b = 45\u000eand\n\b =\u000045\u000e. Obviously, one has \b = 0\u000e,\b<0\u000e, and \b>0\u000e\nfor\u000b=\f,\u000b>\f , and\u000b<\f , respectively, which agree with\nthe simulation results for the nanotrack when the edge effect\nis not significant, i.e., when the skyrmion moves in the interior\nof the nanotrack. For example, using Eq. (12), the skyrmion\nhas\b = 14\u000eand\b =\u000014\u000efor\f= 2\u000b= 0:6and\f=\n\u000b=2 = 0:15, respectively, which match the simulation results\natt\u00180ps where the edge effect is negligible [Fig. 1(c)].\nB. Nanotrack with spatially varying \u000bor\f\nWe first demonstrate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially varying \u000band spatially\nuniform\f, i.e.,\u000bis a function of x, as in Eq. (4), and \f=\n0:3. Figure 3(a) shows the trajectories of the current-driven\nskyrmions with different \u000b(x)functions where \u0015\u000b= 2wand\n\f= 0:3. For\u000bmax= 0:75,\u000bmin= 0:12, i.e.,\u000bamp= 0:315,\nthe skyrmion moves in the rightward direction in a sinusoidal\npattern. For \u000bmax= 0:6,\u000bmin= 0:15, i.e.,\u000bamp= 0:225, the\nmaximum transverse shift of skyrmion is reduced in compared\nto that of\u000bamp= 0:315. For\u000bmax= 0:45,\u000bmin= 0:2, i.e.,\n\u000bamp= 0:125, the amplitude of the skyrmion trajectory further4\nFig. 5. Trajectories of current-driven skyrmions with '= 0\u00182\u0019.\u000bamp=\n\famp= 0:225 and\u0015\u000b=\u0015\f= 2w.\ndecreases. \bas a function of xcorresponding to Fig. 3(a) for\ndifferent\u000b(x)functions are given in Fig. 3(b). Figure 3(c)\nshows the trajectories of the current-driven skyrmions with\ndifferent\u0015\u000bwhere\u000bamp= 0:225and\f= 0:3.\bas a function\nofxcorresponding to Fig. 3(c) for different \u0015\u000bare given in\nFig. 3(d).\nWe then investigate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially uniform \u000band spatially\nvarying\f, i.e.,\fis a function of x, as in Eq. (5), and \u000b=\n0:3. Figure 4(a) shows the trajectories of the current-driven\nskyrmions with different \f(x)functions where \u0015\f= 2w,'=\n0and\u000b= 0:3. The results are similar to the case with spatially\nvarying\u000b. For\fmax= 0:75,\fmin= 0:12, i.e.,\famp= 0:315,\nthe skyrmion moves in the rightward direction in a sinusoidal\npattern. For \fmax= 0:6,\fmin= 0:15, i.e.,\famp= 0:225, the\nmaximum transverse shift of skyrmion is reduced in compared\nto that of\famp= 0:315. For\fmax= 0:45,\fmin= 0:2, i.e.,\n\famp= 0:125, the amplitude of the skyrmion trajectory further\ndecreases. \bas a function of xcorresponding to Fig. 4(a) for\ndifferent\f(x)functions are given in Fig. 4(b). Figure 4(c)\nshows the trajectories of the current-driven skyrmions with\ndifferent\u0015\fwhere\famp= 0:225and\u000b= 0:3.\bas a function\nofxcorresponding to Fig. 4(c) for different \u0015\fare given in\nFig. 4(d).\nFrom the skyrmion motion with spatially varying \u000bor\nspatially varying \f, it can be seen that the amplitude of\ntrajectory is proportional to \u000bampor\famp. The wavelength of\ntrajectory is equal to \u0015\u000b;\f, while the amplitude of trajectory is\nproportional to \u0015\u000b;\f.\balso varies with xin a quasi-sinusoidal\nmanner, where the peak value of \b(x)is proportional to \u000bamp,\n\famp, and\u0015\u000b;\f. As shown in Fig. 2(c), when \fis fixed at a\nvalue between \u000bmaxand\u000bmin, larger\u000bampwill lead to larger\npeak value of \b(x). On the other hand, a larger \u0015\u000b;\fallows\na longer time for the skyrmion transverse motion toward a\ncertain direction, which will result in a larger amplitude of\ntrajectory as well as a larger peak value of \b(x).\nFig. 6. \bas a function of xfor skyrmion motion with '= 0\u00182\u0019.\n\u000bamp=\famp= 0:225 and\u0015\u000b=\u0015\f= 2w.\nC. Nanotrack with spatially varying \u000band\f\nWe also demonstrate the in-plane current-driven skyrmion\nmotion in a nanotrack with both spatially varying \u000band\f,\ni.e., both\u000band\fare functions of x, as given in Eq. (4) and\nEq. (5), respectively.\nFigure 5 shows the trajectories of the current-driven\nskyrmions with spatially varying \u000band\fwhere\u000bamp=\n\famp= 0:225and\u0015\u000b=\u0015\f= 2w. Here, we focus on the effect\nof the phase difference between the \u000b(x)and\f(x)functions.\nFor'= 0 and'= 2\u0019, as the\u000b(x)function is identical to\nthe\f(x)function, the skyrmion moves along the central line\nof the nanotrack. For 0<'< 2\u0019, as\u000b(x)could be different\nfrom\f(x)at a certainx, it is shown that the skyrmion moves\ntoward the right direction in a sinusoidal pattern, where the\nphase of trajectory is subject to '. Figure 6 shows \bas a\nfunction of xcorresponding to Fig. 5 for '= 0\u00182\u0019where\n\u000bamp=\famp= 0:225 and\u0015\u000b=\u0015\f= 2w. It shows that\n\b = 0\u000ewhen'= 0 and'= 2\u0019, while it varies with xin\na quasi-sinusoidal manner when 0<'< 2\u0019. The amplitude\nof trajectory as well as the peak value of \b(x)reach their\nmaximum values when '=\u0019.\nIV. C ONCLUSION\nIn conclusion, we have shown the in-plane current-driven\nmotion of a skyrmion in a nanotrack with spatially uniform\n\u000band\f, where \bis determined by \u000band\f, which can vary\nbetween \b = 45\u000eand\b =\u000045\u000ein principle. Then, we\nhave investigated the in-plane current-driven skyrmion motion\nin a nanotrack with spatially sinusoidally varying \u000bor\f.\nThe skyrmion moves on a sinusoidal trajectory, where the\namplitude and wavelength of trajectory can be controlled by\nthe spatial profiles of \u000band\f. The peak value of \b(x)is\nproportional to the amplitudes and wavelengths of \u000b(x)and\n\f(x). In addition, we have demonstrated the in-plane current-\ndriven skyrmion motion in a nanotrack having both spatially\nsinusoidally varying \u000band\fwith the same amplitude and\nwavelength. The skyrmion moves straight along the central5\nline of the nanotrack when \u000b(x)and\f(x)have no phase\ndifference, i.e., '= 0. 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The procedure requires information on the densit y of the structures. Often, it ignores the damping of the osci llations. We\ncomputed marginal posteriors for parameters such as the wav eguide density; the density contrast; the transverse inhom ogeneity length-\nscale; and the magnetic field strength, under the assumption that the oscillations can be modelled as standing magnetohy drodynamic\n(MHD) kink modes damped by resonant absorption. Our results show that the magnetic field strength can be properly inferre d, even\nif the densities inside and outside the structure are largel y unknown. Incorporating observational estimates of plasm a density further\nconstrains the obtained posteriors. The amount of informat ion one is willing to include (a priori) for the density and th e density\ncontrast influences their corresponding posteriors, but ve ry little the inferred magnetic field strength. The decision to include or leave\nout the information on the damping and the damping time-scal es have a minimal impact on the obtained magnetic field streng th.\nIn contrast to the classic method which provides with numeri cal estimates with error bars or possible ranges of variatio n for the\nmagnetic field strength, Bayesian methods o ffer the full distribution of plausibility over the considere d range of possible values. The\nmethods are applied to available datasets of observed trans verse loop oscillations, can be extended to prominence fine s tructures or\nchromospheric spicules and implemented to propagating wav es in addition to standing oscillations.\nKey words. magnetohydrodynamics (MHD) – waves – methods: statistical – Sun: corona – Sun: oscillations\n1. Introduction\nCoronal seismology uses observed and theoretically predic ted\nproperties of magnetohydrodynamic (MHD) waves and oscil-\nlations to infer plasma and field properties. The method was\nsuggested decades ago by Uchida (1970), Rosenberg (1970),\nand Roberts et al. (1984). The method was first applied to the\ninference of the magnetic field strength in coronal loops by\nNakariakov & Ofman (2001). In observations made by the Tran-\nsition Region and Coronal Explorer (TRACE) and reported by\nAschwanden et al. (1999) and Nakariakov et al. (1999), the ob -\nserved lateral displacements of coronal loops were interpr eted\nin terms of the fundamental MHD kink mode of a magnetic\nflux tube. By estimating the phase speed of the waves and as-\nsociating this observable to the theoretical kink speed in t he\nthin tube approximation, the magnetic field strength could b e\ndetermined, upon making a number of assumptions on the val-\nues of the plasma density inside and outside the coronal loop s.\nThe main shortcoming of the method is that values for phys-\nical parameters that cannot be directly measured are assume d.\nSince then, the same method has been widely used to obtain\ninformation on the local magnetic field strength in structur es\nsuch as prominence threads (Lin et al. 2009), chromospheric\nspicules (Zaqarashvili & Erdélyi 2009), or coronal streame rs\n(Chen et al. 2011); in applications of global seismology usi ng\nEIT waves (Ballai 2007; West et al. 2011; Long et al. 2013,\n2017); or in the first application of radio seismology in the\nouter corona by Zaqarashvili et al. (2013). Extended review son coronal seismology can be found in De Moortel (2005);\nBanerjee et al. (2007); Nakariakov (2008); Arregui et al. (2 012);\nDe Moortel & Nakariakov (2012).\nBesides the use of the kinematics of the oscillations, the\navailability of stereoscopic or spectroscopic informatio n has led\nto improvements when constraining the magnetic field streng th.\nVerwichte et al. (2009) presented the first seismological an al-\nysis of a transverse loop oscillation observed by both So-\nlar TErrestrial RElations Observatories (STEREO) spacecr aft.\nVan Doorsselaere et al. (2008) used Hinode /EIS to obtain infor-\nmation on the mass density to infer the local magnetic field\nstrength in a coronal loop with unprecedented accuracy. The cal-\nculations from simplified theoretical models have been foun d\nto be consistent with those deduced from magnetic extrapola -\ntion and spectral methods by Verwichte et al. (2013). They ha ve\nalso been compared to results obtained from improved mod-\nels and realistic numerical simulations in order to assess t heir\nreliability. The non-planarity of coronal loops and the den sity\nvariation along the loop only weakly a ffect the estimates of the\nmagnetic field magnitude (Scott & Ruderman 2012). The pres-\nence of field aligned flows can cause the underestimation of th e\nmagnetic field strength in coronal loops when using the tradi -\ntional seismological methods (Terradas et al. 2010). The st ud-\nies by De Moortel & Pascoe (2009) and Pascoe & De Moortel\n(2014) have shown that the combined e ffect of the loop curva-\nture, the density ratio, and aspect ratio can lead to ambiguo us\nestimates of the magnetic field strength, because of the appe ar-\nance of a secondary mode. Chen & Peter (2015) have used re-\nArticle number, page 1 of 16A&A proofs: manuscript no. ms\nsults from a three-dimensional coronal simulation in which loop\noscillations are present to test the inversions based on cor onal\nseismology. The field derived by coronal seismology is found to\nbe about 15% to 20% smaller that the average field strength in\ntheir simulation.\nCoronal loop oscillations display time damping and this\ninformation has been widely used to infer the charac-\nteristic spatial scales for the variation of the mass den-\nsity across the magnetic field in these structures (see\ne.g., Arregui et al. 2007; Goossens et al. 2008; Pascoe et al.\n2013; Arregui & Asensio Ramos 2014; Arregui et al. 2015;\nPascoe et al. 2016, for a number of examples). The observed\ndamping has been considered in previous inversions, see e.g .,\nPascoe et al. (2016), but whether or not the consideration of\ndamping time and spatial scales influence the seismological in-\nversion of the magnetic field strength remains unknown.\nThe purpose of the present study is twofold. First, we applie d\nBayesian methods to the solution of inverse problems to infe r the\nmagnetic field strength (see Arregui 2018, for a recent revie w on\nBayesian coronal seismology). The reason to use Bayesian an al-\nysis is the lack of direct access to the physical conditions o f in-\nterest which forces us to use indirect observational inform ation\nwhich is always incomplete and uncertain. Extracting infor ma-\ntion on physical parameters by comparison of theoretical pr e-\ndictions with observed data has therefore to be carried out i n a\nprobabilistic framework. This means that our conclusions w ill at\nbest be probabilities, in the form of posterior probability density\nfunctions. These posteriors arise from a principled way of c om-\nbining prior information, model predictions, and observat ions,\nproviding inferences that are conditional on the data. Seco nd, we\ncompared results on the inference of the magnetic field stren gth\nwith and without considering the damping of transverse osci lla-\ntions to see to what extent the inclusion of damping time scal es\nhas an impact on the inference results. The analysis is perfo rmed\nfor standing transverse waves under coronal conditions, bu t can\neasily be generalised to propagating waves and to plasmas wi th\nchromospheric or prominence properties.\nThe layout of the paper is as follows. In Section 2 our infer-\nence method is described. Our results are shown in Section 3\nwhere different inference problems are solved under di fferent\nknowledge circumstances. We present our conclusions in Sec -\ntion 4.\n2. Inference method\nWe adopt the methods of probabilistic inference in the Bayes ian\nframework which consider any inversion problem as the task o f\nestimating the degree of belief on statements about paramet er\nvalues, conditional on observed data. The methods rely on th e\nuse of Bayes theorem,\np(θ|D,M)=p(D|θ,M)p(θ)\np(D), (1)\nwhich says that our state of knowledge on a given parameter se t,\nθ, conditional on the observed data, D, and the assumed theoret-\nical model, M, is a combination of what we know independently\nof the data, the so-called prior p(θ|M), and the likelihood of ob-\ntaining the observed data as a function of the parameter vect or,\np(D|θ,M). Their combination leads to the posterior, p(θ|D,M),\nwhich contains all the available information about the unkn own\nparameters of interest. The denominator is the so-called ev i-\ndence, a factor that accounts for the full probability of the data.As this quantity is independent of the parameter vector, it j ust\nserves as a normalising constant and does not a ffect the shape of\nthe posteriors. In this study, unless otherwise stated, all probabil-\nity densities will be normalised so that the full integral is unity.\nOnce the full posterior is computed, information on a partic -\nular parameter can be obtained by performing an integral of t he\nposterior with respect to the remaining parameters to obtai n the\nso-called marginal posterior\np(θi|D,M)=/integraldisplay\np(θ|D,M)dθ1...dθi−1dθi+1...dθN. (2)\nTo perform the inference and compute either the full posteri or or\nthe marginal posterior distribution of a given parameter, d ifferent\nalternatives are available. In low-dimensional parameter spaces,\none can still compute the full posterior for di fferent combina-\ntions of parameters and then perform a direct numerical inte gra-\ntion to obtain the marginal posterior for a given parameter. This\napproach ceases to be feasible as we increase the complexity\nand dimensionality of our problem. Then, alternative numer ical\nmethods need to be employed to evaluate the relevant distrib u-\ntions by e.g., performing a Markov Chain Monte Carlo (MCMC)\nsampling of the posterior (see e.g., Sharma 2017, for a recen t re-\nview). The simplicity of the models considered in this work s till\nmake possible the use of direct integration over a grid of num er-\nical points, although MCMC methods are also used to further\nconfirm our results and in the prior dependency analysis show n\nin Appendix A .\nThe posterior probability density in Eq. (1) is a derived qua n-\ntity, while both the likelihood and the prior have to be assig ned\nwhen constructing the statistical model. The prior probabi lity\ndistribution contains our state of belief on the values the u n-\nknown parameters can take before considering the observed d ata.\nThis information usually comes from past knowledge and expe -\nrience from which a guess is usually made. The physical model\ncan also impose limitations to the particular value or range of\nvalues a parameter can take on. In this study, di fferent prior dis-\ntributions are employed. When our information on a given pa-\nrameter is limited to a plausible range of variation, we use a uni-\nform prior probability distribution over the considered ra nge of\nthe form\np(θi)=H(θi,θmin\ni,θmax\ni), (3)\nwhere H(x,a,b) is the top-hat function\nH(x,a,b)=1\nb−aa≤x≤b,\n0 otherwise .(4)\nWhen more specific information on a given parameter value\nis available from observations, this knowledge will be used to\nconstruct a more informative prior using the measured value of\nthe parameter and the error reported from observations to de fine\nthe mean,µθi, and standard deviation, σθi, of a Gaussian prior of\nthe form\np(θi)=(2πσ2\nθi)−1/2exp−(θi−µθi)2\n2σ2\nθi. (5)\nFinally, it is important to assess the dependence of the inve rsion\nresult on the prior information. In Appendix A, another prio r dis-\ntributions are employed in an analysis of the e ffect of prior infor-\nmation on the obtained results. They include e.g., a Je ffreys type\nArticle number, page 2 of 16Arregui et al.: Inference of magnetic field strength and dens ity from damped transverse coronal waves\nFig. 1. Posterior probability distributions for (a) the internal A lfvén speed, (b) the density contrast, (c) the internal dens ity, and (d) the magnetic\nfield strength for a loop oscillation event with observed pha se speed vph=1030±410 km s−1under model M1, given by Eqs. (7) and (9). The inferred\nmedian values for the Alfvén speed and the magnetic field stre ngth are vAi=813+330\n−317km s−1andB0=21+12\n−9G respectively, with uncertainties\ngiven at the 68% credible interval. Uniform priors were cons idered in the intervals vAi∈[1,2000] km s−1,ζ∈[1.1,10],ρi∈[10−13,10−11] kg m−3,\nandB0∈[0.1,100] G. We note that despite the visual impression, the margi nal probabilities for vAiandB0at their zero values are zero, because\nof the considered ranges.\nprior to assign a decreasing probability distribution for i ncreas-\ning values of the parameter or Cauchy functions based priors .\nRegarding the likelihood function, we will consider that ob -\nservations are corrupted with Gaussian noise and that they a re\nstatistically independent. Then, a given observed variabl eDand\nits theoretical prediction Dmodelcan be compared by adopting a\nGaussian likelihood of the form\np(D|θ)=1√\n2πσexp−/bracketleftBig\nD−Dmodel(θ)/bracketrightBig2\n2σ2, (6)\nwithσthe uncertainty associated to the measured D.\n3. Results\nThe methods described above are next applied to a number of\nproblems in which the forward and inverse problems, as well a s\nthe base-knowledge information is di fferent. In this way, a step-\nby-step knowledge is acquired on the amount of information t hat\nwe can gather on the unknown parameters, conditional on the\nassumptions of each physical model and the available data wi th\ntheir uncertainty.3.1. Internal Alfvén speed and magnetic field strength\nLet us first consider the simplest inversion problem consist ing of\ninferring the internal Alfvén speed in a coronal loop underg oing\nundamped transverse oscillations interpreted as the funda mental\nMHD kink mode. Assuming that coronal loops can be modelled\nas one-dimensional density enhancements in cylindrical ge om-\netry and under the thin tube approximation, theory relates t he\nobservable phase speed, vph, to the internal Alfvén speed, vAi,\nand the contrast between the internal ρiand external density ρe,\nζ=ρi/ρe, in the following manner\nvph∼vAi/parenleftBigg2ζ\n1+ζ/parenrightBigg1/2\n. (7)\nWe will refer to this as model M1.\nWe first applied our Bayesian scheme to the inference of the\ninternal Alfvén speed of coronal loops, considering a parti cu-\nlar event, first described by Nakariakov et al. (1999), and su bse-\nquently analysed by Nakariakov & Ofman (2001). In the event\nof the 4th of July, 1999, a period of P=360 s and a loop length\nofL=1.9×1010cm were measured. The authors report an esti-\nmated phase speed of vph=1030±410 km s−1after considering\nArticle number, page 3 of 16A&A proofs: manuscript no. ms\nFig. 2. Marginal posterior distributions for the magnetic field str ength for the same loop oscillation event and model as in Fig. 1, using different\nranges for the prior distribution on (a) density contrast an d (b) internal density.\nthe uncertainty on the measured variables. Bayes theorem ap -\nplied to this particular problem tells us that the posterior for the\ntwo unknowns,θ={vAi,ζ}, conditional on the measured phase\nspeed, D=vph, and the assumed model, M1, is a combination of\nthe likelihood of the data as a function of the unknowns and th e\nprior distributions. Explicitly,\np({vAi,ζ}|vph,M1)=p(vph|{vAi,ζ},M1)p({vAi,ζ}|M1)\nZ1, (8)\nwith Z1=/integraltext\np(vph|{vAi,ζ},M1)p({vAi,ζ}|M1)dvAidζthe evi-\ndence. Considering a Gaussian likelihood function and unif orm\nprior distributions for the unknowns over plausible ranges leads\nto the marginal posterior distributions shown in Figs. 1a an d\n1b, which indicate that the internal Alfvén speed can be prop -\nerly inferred (Fig. 1a). Notice that the factor with the squa re\nroot in Eq. (7) is allowed to vary in between 1 and√\n2, when\nζis allowed to vary in between just a little more than 1 and ∞.\nThe classic result is therefore that one could expect vAito be\nconstrained to a narrow range, as pointed out by Arregui et al .\n(2007). However all those values are not equally probable. W hat\nour Bayesian result o ffers is the probability distribution of those\npossible values within the range. On the other hand, the dens ity\ncontrast cannot be inferred with the information on the phas e\nspeed alone (Fig. 1b).\nEquation (7) for the wave phase speed can be expanded to\nincorporate the magnetic field strength, B0, to the inversion. The\nforward problem for model M1can now be formulated as\nvph(ζ,ρ i,B0)=B0√µ0ρi/parenleftBigg2ζ\n1+ζ/parenrightBigg1/2\n, (9)\nwithµ0the magnetic permeability. In this case, Bayes theorem\nprovides us with the posterior for three unknowns, θ={ρi,ζ,B0}\n- internal density, density contrast, and magnetic field str ength.\nThe explicit expression for the Bayes theorem now reads\np({ρi,ζ,B0}|vph,M1)∼p(vph|{ρi,ζ,B0},M1)p({ρi,ζ,B0}|M1).\n(10)\nwhere we have omitted the explicit expression for the eviden ce\nfor brevity.Figures 1c and 1d show the marginal posterior distributions\nfor the loop density and for the magnetic field strength, re-\nspectively. The marginal posterior for the density contras t is\nthe same as the one shown in Fig. 1b. The results now show a\nwell-constrained distribution for the magnetic field stren gth, that\ncan therefore be properly inferred. The same is not true for t he\nloop density, for which no information can be gathered. Let u s\ncompare our result with that by Nakariakov & Ofman (2001).\nNakariakov & Ofman (2001) obtained a numerical estimate of\nB0=13±9 G, upon employing a density of 109.3±0.3cm−3.\nWhen allowing the density to vary from 1 ×109to 6×109cm−3,\na range of variation for B0was obtained. The main advantage of\nour Bayesian result is that, when considering a range of poss ible\nvalues for the density, the marginal posterior in Fig. 1d tel ls us\nhow the plausibility of the corresponding possible values o f the\nmagnetic field strength is distributed. If we wish to use the e s-\ntimated density, the Bayesian method enables to mimic this b y\nemploying a Gaussian prior, as will be discussed in Sect. 3.2 .\nIn summary, by adopting the simplest possible model for\ntransverse loop oscillations in the long wavelength approx ima-\ntion, measuring the phase speed of the waves, and considerin g\nuniform priors for the unknown parameters, the internal Alf vén\nspeed and the magnetic field strength can be inferred, even if the\ndensities inside and outside the waveguide are largely unkn own.\n3.2. Information on plasma density\nOur knowledge on the plasma density inside the waveguide tur ns\nout to be an important matter when inferring the magnetic fiel d\nstrength. In the results above, a large range of possible val ues for\nthe waveguide density was considered, with typical coronal loop\ndensities in the range ρi∈[10−13−10−11] kg m−3, corresponding\nto particle densities in the range n∼[108−1010] cm−3(Priest\n1982). Also, a density contrast in the range ζ∈[1.1−10] was\nfixed. Uniform prior distributions for ρiandζwere considered\nover those ranges.\nWe repeated the inversion of magnetic field strength by first\nconsidering uniform priors over di fferent ranges for both param-\neters,ρiandζ. First, the maximum value of the density con-\ntrast was modified by considering several values from ζ=3 up\ntoζ=50. Figure 2a shows the obtained results. We can see\nthat the extent on density contrast over which a uniform prio r is\nconsidered does not influence much the magnetic field strengt h\nArticle number, page 4 of 16Arregui et al.: Inference of magnetic field strength and dens ity from damped transverse coronal waves\nFig. 3. (a) Prior and posterior distributions for the waveguide den sity in the inversion of Eq. (9) with vph=1030±410 km s−1, under model M1.\n(b) Comparison between the marginal posteriors for magneti c field strength in the same inversion for the cases of uniform and Gaussian prior on\nthe waveguide density. (c) and (d) Comparison between the jo int two-dimensional posterior distributions for the inter nal density of the waveguide\nand the magnetic field strength obtained for the inference wi thvph=1030±410 km s−1, under model M1, for the cases of uniform priors (left) and\na Gaussian prior for the internal density with µρi=1.9×10−12kg m−3andσρi=0.5µρi(right). The inference with the more informative prior on\ndensity leads to B0=13+7\n−6G andρi=(2.2+0.9\n−0.9)×10−12kg m−3. In the bottom panels, the outer boundaries of the light grey and dark grey shaded\nregions indicate the 95% and 68% credible regions.\ninference. A different result is obtained when the inversion is\nperformed considering uniform prior distributions over di fferent\nranges for the internal density. Figure 2b shows marginal po s-\nterior distributions for the magnetic field strength comput ed by\nconsidering a uniform prior for the fixed range of ζ∈[1.1−10]\nand uniform priors over three di fferent ranges for the waveguide\ndensity. We can see that considering the lower density half- range\nwithρi∈[10−13−10−12] kg m−3or the higher density half-range\nwithρi∈[10−12−10−11] kg m−3produce rather different results.\nWhen the full range ρi∈[10−13−10−11] kg m−3is taken, re-\nsults similar to those obtained with the higher density half are\nfound. This range is in close correspondence to the one consi d-\nered by Nakariakov & Ofman (2001) and was chosen for this\nreason in the results above. A note of warning is in order here .\nWhen a parameter is left to vary over several decades, using a\nuniform prior is not the best choice. As the result above show s,\nthe contribution to the marginal posterior from the integra l over\nthe range [10−13– 10−12] is insignificant in comparison to the\ncontribution from the integral over the range [10−12– 10−11]. In\nthis case, considering the density a scale parameter and usi ng\na Jeffreys prior, which gives equal probabilities over decades ina logarithmic scale might be more appropriate, as discussed in\nAppendix A.\nBesides changing the range of variation over which uni-\nform prior distributions are defined, the Bayesian formalis m en-\nables us to incorporate more informative priors. Spectrosc opy\nor the analysis of the Di fferential Emission Measure (DEM)\nenables us to obtain some properties of the emitting coronal\nplasma, such as the density (see e.g., Van Doorsselaere et al .\n2008; Su et al. 2018). When additional knowledge like this be -\ncomes available, the classic approach has been to use it to ob -\ntain a numerical estimate for the magnetic field strength. Fo r in-\nstance, Nakariakov & Ofman (2001) used a density of 109.3±0.3\ncm−3in their calculation. The Bayesian approach enables to self -\nconsistently incorporate this information to update the po steri-\nors, i.e., our state of belief. The proper way to proceed is to re-\ncalculate the posterior using this more informative prior.\nAs an example, let us use the same estimate as\nNakariakov & Ofman (2001), which considering the proton\nmass and a mean molecular mass of around 1.16 in the corona\ntranslates into∼1.9×1012kg m−3, with its corresponding uncer-\ntainty. We can use this information to construct a Gaussian p rior\nfor the density, by considering Eq. (5) for the unknown param e-\nArticle number, page 5 of 16A&A proofs: manuscript no. ms\nFig. 4. Comparison between marginal posterior distributions for t he\nmagnetic field strength computed by solving the same inversi on prob-\nlem as in Fig. 1, under model M1and using a uniform prior over the\nfull range of plasma density values, ρi∈[10−13,10−11] kg m−1and then\nmarginalising (solid-line); taking a cut of the two-dimens ional joint pos-\nterior at the value µρi=1.9×10−12kg m−1(dashed-line) and solving\nthe problem using a Gaussian prior with µρi=1.9×10−12kg m−1and\nσρi=0.5µρi(dotted-line).\nterθi=ρi, centred on the numerical estimate µθi=µρi, and with\nuncertaintyσθi=σρi.\nFigure 3 shows the result of such an inversion for the mag-\nnetic field strength and the plasma density inside the wavegu ide.\nFigure 3a shows the Gaussian prior for the density and its cor re-\nsponding posterior. The resulting posterior closely resem bles the\nassumed prior, although the information on the data has slig htly\naltered the posterior. Figure 3b shows a comparison between the\nmarginal posterior distributions for the magnetic field str ength\nusing the uniform prior and the Gaussian prior in density. Us -\ning the information obtained from measuring the plasma den-\nsity produces a shift in the marginal posterior for the magne tic\nfield towards smaller values and a more constrained distribu -\ntion. The summary of this posterior using the median and er-\nrors at the 68% credible interval leads to B0=13+7\n−6G, in good\nagreement with the numerical estimate by Nakariakov & Ofman\n(2001). Table B.1 in Appendix B shows a comparison between\nprevious estimates of magnetic field strength in a number of r e-\nported events and our Bayesian posterior summaries. In all c ases,\na good agreement is found. In addition, our results provide u s\nwith the full probability distributions. The last two panel s show a\ncomparison between the joint posteriors for internal densi ty and\nmagnetic field strength obtained by employing the uniform pr ior\n(Fig. 3c) and the Gaussian prior (Fig. 3d) for the internal de nsity.\nAs can be seen, the addition of information enables us to furt her\nconstrain our estimates for both unknowns, waveguide densi ty\nand magnetic field strength. We note the lack of symmetry of\nthe joint posterior in Fig 3d, which explains why assumption s\nin density might have an impact on the inferred magnetic field\nstrength and why considering a range in density from 10−12to\n10−11kg m−3leads to a very similar result to considering the full\nrange (see Fig 2b), since that range already covers most of th e\nregion where the joint posterior is large.\nBy looking at Fig. 3c one may wonder if, instead of using a\nGaussian prior on density, it is not possible to solve the inv er-\nsion problem by considering a uniform prior over the full ran ge\nof internal density values and then performing a cut of the jo int\ntwo-dimensional posterior for density and magnetic field at theplasma density values that has been measured (dashed line in\nFig. 3c). Another option could be to simply insert the value o f\nµρiinρiin Eq. (9) and solve that oversimplified inversion prob-\nlem for two unknowns, θ={ζ,B0}. Figure 4 shows the result of\na comparison of marginal posteriors for the options being di s-\ncussed. The solid line is the result of the integration of the full\nposterior over the full range of values for ρi(andζ) and gives the\nmost uncertain result with the possible values for the magne tic\nfield strength extending up to ∼60 G. The dashed-line shows\nthe result of taking a cut of the joint posterior for B0andρiat\nthe measured value for the plasma density. This is the most co n-\nstrained distribution, but it does not take into account the uncer-\ntainty on the measured plasma density. It just considers tha t this\nuncertainty is zero. In our view, simply inserting a measure ment\nofρiwithout its associated uncertainty, even if it provides the\nmore constrained result, is not the best option. The dotted c urve\nis the posterior corresponding to the use of the Gaussian pri or\nfor the density. This exercise shows that what one is willing to\nassume about the plasma density inside the waveguide may infl u-\nence the inference of the magnetic field strength. The advant age\nof the Bayesian approach is that one is forced to explicitly s pec-\nify this “what one is willing to assume” in the definition of th e\npriors when constructing the statistical model. In Appendi x A\na prior dependence analysis of the obtained results is prese nted.\nThe results indicate that changes on what one is willing to ac cept\na priori for the density and density contrast have an e ffect on their\ncorresponding posteriors, but not on the inferred magnetic field\nstrength.\n3.3. Information on wave damping\nTime damping is a commonly observed property in transverse\nloop oscillations, with characteristic damping times of a f ew os-\ncillatory periods. Although inferences of magnetic field st rength\nusing the damping of the oscillations have been presented (s ee\ne.g., Pascoe et al. 2016), the influence of this observable on the\nestimates of the magnetic field strength inferred by seismol ogy is\nunknown. For this reason, we performed the inference includ ing\nthe simplest available model for damping by resonant absorp -\ntion, a plausible mechanism to explain the observed damping\ntime scales (Goossens et al. 2002; Ruderman & Roberts 2002).\nWe therefore consider model M2in which the previous model\nM1is modified by including a non-uniform density layer of\nlength lat the boundary of the waveguide and centred around\nthe radius Rof the tube. Under the long wavelength and thin\nboundary ( l≪R) approximations analytical relationships can\nbe obtained for the observable phase speed and damping time a s\na function of four unknown parameters - internal density, de nsity\ncontrast, magnetic field strength, and transverse inhomoge neity\nlength scale - such that\nvph(ρi,ζ,B0)=B0√µ0ρi/parenleftBigg2ζ\n1+ζ/parenrightBigg1/2\n, (11)\nτd(ρi,ζ,B0,l/R)=2\nπ/parenleftBiggζ+1\nζ−1/parenrightBigg/parenleftBigg1\nl/R/parenrightBigg/parenleftBigg2L\nvph/parenrightBigg\n. (12)\nHere, Lis the length of the loop, a magnitude that is assumed\nto be measurable. The expression for the damping time contai ns\na factor 2/πas a result of the particular assumption of a sinu-\nsoidal density profile for the density at the non-uniform lay er.\nConsidering different alternatives for the density profile leads to\ndifferences in the forward predictions and also in the inverse so -\nArticle number, page 6 of 16Arregui et al.: Inference of magnetic field strength and dens ity from damped transverse coronal waves\nFig. 5. Top row: marginal posterior distributions for (a) magnetic field strength; (b) density contrast; and (c) transverse inh omogeneity length scale\nfor the inversion of problem with forward model M2, given by Eqs. (11) and (12), and a transverse oscillation wi thvph=1030±410 km s−1and\na damping timeτd=500±50 s. Bottom row: joint two-dimensional posterior distribu tions for (d) magnetic field strength and transverse density\ninhomogeneity length-scale; (e) density contrast and tran sverse density inhomogeneity length-scale; and (f) magnet ic field strength and density\ncontrast. The outer boundaries of the light grey and dark gre y shaded regions indicate the 95% and 68% credible regions. T he inferred median\nvalues are B0=20+11\n−8G and l/R=1.2+0.5\n−0.4, with uncertainties given at the 68% credible interval. A fix ed value for the loop length, L=1.9×1010\ncm, was considered in this computation.\nlutions. The latter may be important in strong damping regim es,\nas shown by Arregui et al. (2015).\nEquations (11) and (12) show that even using the simplest\nmodel for resonantly damped oscillations the expressions f or\nthe phase speed and the damping time are coupled and, hence,\nsome degree of influence of the information on the damping\ntime can be expected when inferring the magnetic field streng th.\nWe note that the phase speed is independent of the transverse\ninhomogeneity length scale, a consequence of the adoption o f\nthe thin boundary approximation. Van Doorsselaere et al. (2 004)\nand Arregui et al. (2005) have shown that, outside this appro x-\nimation, the period of the fundamental kink mode depends on\nthe transverse inhomogeneity length-scale, producing sig nificant\nvariations especially for values above l/R=1.\nAs the event of 4th of July used in our previous calcula-\ntions does not come with the corresponding damping time, we\nfirst repeated the inference presented in Sect. 3.1, by inclu d-\ning some reasonable value for the damping time and using the\nforward model given by (11) and (12). Bayes theorem now in-\ncludes additional parameters, θ={ρi,ζ,B0}, and observables,\nD={vph,τd,L}, and can be written as\np({ρi,ζ,B0,l/R}|{vph,τd,L},M2)∼\np({vph,τd,L}|{ρi,ζ,B0,l/R},M2)p({ρi,ζ,B0}|M2). (13)\nAn example inversion result is shown in Fig. 5, which dis-\nplays marginal posterior distributions and joint probabil ity distri-\nbutions for the magnetic field strength, the density contras t, and\nthe transverse density inhomogeneity length scale. As in Se c-\ntion 3.1, we used uniform priors over given ranges for all par am-\neters. Again, the magnetic field strength can be properly con -\nstrained. The inclusion of a damping time leads to a constrai nton the lower limit of density contrast and transverse densit y in-\nhomogeneity length scale, but no restriction can be found on the\nupper limit beyond the assumed prior.\nThe posterior for the magnetic field strength shows a very\nsimilar probability distribution to the inversion result w ithout the\nuse of the damping time (Sect. 3.1). Both the median of the pro b-\nability density and the dispersion are similar to the case wi th-\nout damping. There is a slight decrease in the median for B0for\nlarger damping times. This is more clearly seen in Fig. 6a, we re\nposteriors for the magnetic field strength for di fferent damping\nregimes are displayed. The posteriors between the case with no\ndamping and the three cases with damping show minimal dif-\nferences. Also, once a value for the damping time is consid-\nered, it does not seem to matter that much the precise value\nof the damping time. These results point to a negligible impo r-\ntance of considering the information on the damping time sca le\nof coronal loop oscillations when inferring the magnetic fie ld\nstrength, when the thin tube and thin boundary approximatio ns\nare used. Table B.2 in Appendix B shows that this conclusion\nholds when analysing a large sample of events with damping pr e-\nsented by Goddard et al. (2016). Outside these approximatio ns,\nthe period becomes dependent on the transverse inhomogenei ty\nlength-scale (Van Doorsselaere et al. 2004; Arregui et al. 2 005;\nSoler et al. 2014) and a stronger coupling between period and\ndamping time is expected, which could lead to a more e ffective\ninfluence of the damping on the inferred magnetic field streng th.\nThe remaining panels in Fig. 6 show that the two parameters th at\ndefine the cross-field variation of the density, ζandl/Rcan be\nconstrained better or worse, depending on the actual value o f the\ndamping time. The constraint on the lower limit of l/Ris much\nmore sensitive to the choice of damping time than the lower\nlimit onζ. Constrained posterior distributions for ζandl/Rcan\nalso be obtained using two resonant damping regimes, as show n\nArticle number, page 7 of 16A&A proofs: manuscript no. ms\nFig. 6. Marginal posterior distributions for (a) magnetic field str ength;\n(b) density contrast; and (c) transverse inhomogeneity len gth scale for\nthe inversion of problem with forward model given by Eqs. (11 ) and\n(12) and a transverse oscillation with vph=1030±410 km s−1and\nthree values for the damping time: no damping (solid-line); τd=500 s\n(dotted-line);τd=800 s (dashed-line); and τd=1200 s (dash-dotted-\nline) with an associated uncertainty of 50 s in all cases. The inferred\nmedians with errors at the 68% credible interval are: B0=21+12\n−9G for\nthe undamped case; B0=20+11\n−8G forτd=500s; B0=19+11\n−8G for\nτd=800s; and B0=18+11\n−9G forτd=500s. A fixed value for the loop\nlength, L=1.9×1010cm, was considered in all computations.\nby Arregui et al. (2013) for synthetic data and by Pascoe et al .\n(2017) for observational data.4. Summary and conclusions\nWe applied Bayesian inversion methods to the inference of\nthe magnetic field strength in transversely oscillating cor onal\nwaveguides. The classic approach to the problem has been to\nuse measured values for the period or phase speed together\nwith an analytical approximation to the kink speed in the lon g\nwavelength limit to extract information on the magnetic fiel d\nstrength, upon inserting numerical estimates, either assu med or\nas a result of an indirect measurement, for parameters such a s\nthe density contrast of the waveguide and the internal densi ty\n(Nakariakov & Ofman 2001) . The result is a numerical esti-\nmate with its corresponding error bar or a range of variation\nfor the possible values the magnetic field strength can take o n.\nIn some cases, information on the density gathered from spec -\ntroscopic measurements has enabled to further constrain th e es-\ntimates (see e.g., Van Doorsselaere et al. 2008). Our approa ch\nconsist of adopting Bayesian methods to obtain the global pr ob-\nability density distribution, or posterior, for the unknow ns. This\nposterior can then be marginalised to obtain information on a\nparticular parameter of interest, e.g., the magnetic field s trength,\nwhich assigns a level of plausibility to each considered par am-\neter value. The process self-consistently propagates unce rtainty\nfrom data and between the unknowns of the problem.\nThese methods have been applied to compute probability\ndensity distributions for the magnetic field strength to stu dy\nin detail the possible impact on the variability of density a nd\ndensity contrast and of the observed damping time. This was\ndone by solving different inversion problems in which our base-\nknowledge is different to analyse how the uncertainty on the den-\nsity and the density contrast a ffects the magnetic field strength\ninversion and whether or not the consideration of the dampin g\nmodifies the obtained posteriors significantly.\nWe found that the magnetic field strength can be in-\nferred, even if the densities inside and outside and their ra -\ntio are largely unknown. The obtained marginal posteriors\nshow well constrained distributions. In comparison to e.g. ,\nNakariakov & Ofman (2001), who obtained numerical estimate s\nor a range of possible values as a function of the considered d en-\nsity, our results offer how the relative plausibility between those\npossible values is distributed. When spectroscopic inform ation\non plasma density is available, the method enables to incorp orate\nthis knowledge in a self-consistent manner, further constr aining\nthe inference. In the inference process, both the density in the\nwaveguide and the density contrast with respect to the coron al\nplasma density need to be considered. By considering unifor m\nprior distributions over di fferent ranges in these two parame-\nters the results indicate that the posterior for the magneti c field\nstrength inference is very little dependent on the density c ontrast\nrange that is assumed, but the range of variation for the dens ity\nof the waveguide has an impact on the obtained magnetic field\nstrength distribution. A sensitivity analysis considerin g other al-\nternative priors for density contrast and loop density show s that\nthese priors influence the corresponding posteriors in thes e two\nparameters, but very little the posterior for the magnetic fi eld\nstrength.\nThe observed oscillation damping is practically irrelevan t to\nthe inversion of the magnetic field strength, at least when th e\nthin tube and thin boundary approximations are considered i n\nthe forward problem. However, its inclusion enables to obta in\ninformation on the transverse inhomogeneity length scale o f the\ndensity at the boundary of the waveguide, a parameter direct ly\nrelated to wave heating processes.\nArticle number, page 8 of 16Arregui et al.: Inference of magnetic field strength and dens ity from damped transverse coronal waves\nFig. 7. Summary of values for the magnetic field strength in Table B.2 and their corresponding uncertainties as a function of the l oop length (left)\nand the oscillation period (right). We note that the loop len gths are given without errors. The errors in the magnetic fiel d strength are given at the\n68% credible interval.\nWe applied the methods here presented to a set of observed\ntransverse loop oscillations. The results are displayed in Ta-\nbles B.1 and B.2 in Appendix B. In Table B.1 we compare mag-\nnetic field strength estimates provided by Nakariakov & Ofma n\n(2001), Aschwanden et al. (2002), Goossens et al. (2002),\nVan Doorsselaere et al. (2008), White & Verwichte (2012), an d\nPascoe et al. (2016) with the corresponding results obtaine d us-\ning the Bayesian method. The field strengths are first com-\nputed using a uniform prior on density over the extended rang e\nρi∈[10−13−10−11] kg m−3, which lead to the values Buin Ta-\nble B.1. Then, Gaussian priors on density are employed aroun d\nthe density values estimated or assumed by those authors, wh ich\nlead to the values BGin Table B.1. The summaries of our distri-\nbutions are in agreement with the previous numerical estima tes.\nIn order to confirm our result regarding the minimal influence\nof the damping on the inferred magnetic field strength, when t he\nthin tube and thin boundary expressions for the forward prob -\nlem are used, we compared the inference results, with and wit h-\nout damping, for 52 events compiled by Goddard et al. (2016)\nfrom the catalogue by Zimovets & Nakariakov (2015). The re-\nsults presented in Table B.2 in Appendix B show that practi-\ncally the same posterior summaries are obtained with or with out\ndamping, with differences of only±1 G.\nFigure 7 shows a summary of the magnetic field strengths\nin Table B.2 for the inferences of Bno damping as a function of the\nloop length and of the oscillation period. No clear signatur e of\na trend is found in the case of the magnetic field strength as a\nfunction of the loop length. On the contrary, the inverse rel ation-\nship between magnetic field strength and period, present in t he\nmodel, can be clearly seen from the inferred field strengths.\nThe methods here presented can in principle be ap-\nplied to another magnetic and plasma structures in the so-\nlar atmosphere, such as prominence fine structures (see e.g.\nMontes-Solís & Arregui 2019) or chromospheric spicules.\nAcknowledgments\nWe acknowledge financial support by the Spanish Ministry\nof Economy and Competitiveness (MINECO) through projects\nAYA2014-55456-P (Bayesian Analysis of the Solar Corona) an d\nAYA2014-60476-P (Solar Magnetometry in the Era of Large So-\nlar Telescopes) and FEDER funds. 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V ., Melnik, V . N., Brazhenko, A. I., et al. 2 013, A&A, 555, A55\nZimovets, I. V . & Nakariakov, V . M. 2015, A&A, 577, A4\nAppendix A: Influence of prior information\nOur analysis has shown that the inference of the magnetic fiel d\nstrength can be performed regardless of the fact that the den -\nsity and the density contrast cannot be properly inferred an d are\ntherefore largely unknown. Using di fferent uniform priors for the\ndensity contrast does not a ffect the marginal posterior for the\nmagnetic field strength, as shown in Fig. 2a. The same is not\ntrue for the density of the waveguide, see Fig. 2b. Issues wit h\npriors are evident already from the results shown in Fig. 1. F ig-\nure 1b shows that the marginal posterior for density contras t has\na long tail and is not going to zero at the extremes. In Fig. 1c,\nthe range of variation for the internal density spans over tw o or-\nders of magnitude. Using a uniform prior and integrating ove r\nthe range [10−13– 10−12] or over the range [10−13– 10−11] will\nlead to probabilities that are not of the same order. This mak es\nthe density a scale parameter for which a Je ffreys prior might be\nmore appropriate.\nIn this section, a more detailed prior analysis is presented .\nIn addition to the use of a uniform prior for density contrast ,\nthe inversion of Eq. (9) was performed using di fferent priors for\ndensity contrast and waveguide density.\nFor density contrast, the following alternative priors wer e\nconsidered:\n–A Normal distribution, p(ζ)=N(µζ,σ2), withµζ=5.5 and\nσ=7.75.\n–A Cauchy distribution,\np(ζ)=f(ζ;µζ,γ)=1\nπγ/bracketleftbigg\n1+/parenleftBigζ−µζ\nγ/parenrightBig2/bracketrightbigg, (A.1)\nwithµζ=5.5 andγ=1,2,3,4,5.\n–An exponential, p(ζ)=Cexp(−ζ/4), withζ∈[1.1−10] and\nCsuch that the integral is unity.\nIn the same manner, the following priors for the density of\nthe waveguide were used, as alternatives to the uniform prio r:–A Jeffreys prior, p(ρi)=/bracketleftbigg\nρilog/parenleftbigg\nρmax\ni\nρmin\ni/parenrightbigg/bracketrightbigg−1\n, withρmin\ni=10−13\nkg m−3andρmax\ni=10−11kg m−3, corresponding to the full\nrange considered in Fig. 1.\n–A Normal distribution, p(ρi)=N(µρi,σ2), withµρi=10−12\nkg m−3andσ2=5×10−12kg m−3.\nThe results of the prior analysis for di fferent priors in den-\nsity contrast are shown in Fig. A.1. Each inversion with a dif -\nferent prior corresponds to a row of panels. The leftmost pan -\nels shows the used prior for density contrast, then we show\nthe corresponding marginal posteriors for density contras t, den-\nsity, and magnetic field strength. The results have been com-\nputed using both direct numerical integration and also Mark ov\nChain Monte Carlo (MCMC) sampling of the posterior mak-\ning use of emcee (Foreman-Mackey et al. 2013), as explained in\nMontes-Solís & Arregui (2017). As can be seen, the use of uni-\nform, normal, Cauchy, or exponential priors for the density con-\ntrast influence the marginal posterior obtained for this par ameter.\nBasically, what one gets as posterior is very similar to what we\nhad as input in the prior. In all four cases, the marginal post e-\nrior for the density is una ffected. More importantly, the marginal\nposterior for the magnetic field strength is una ffected. The in-\nference of magnetic field strength is therefore robust in fro nt of\nchanges in the employed prior distribution for density cont rast.\nThe results of the prior analysis for di fferent priors in density\ncontrast are shown in Fig. A.2. Again, each row shows results\nfrom the inversion using a di fferent prior on density, indicated on\nthe leftmost panel. Similarly to the previous case, changes in the\nprior information for the density do a ffect the marginal posterior\nobtained for this parameter, but do not influence significant ly the\nmarginal posteriors for the other two parameters of the prob lem,\ndensity contrast and magnetic field strength. The posterior for\ndensity contrast seems to be completely independent on the i n-\nformation on the density of the waveguide. Only in the case of a\nJeffreys prior for the density, the posterior for the magnetic fie ld\nstrength slightly affected (see rightmost panel in Fig. A.2).\nIn summary, what one is willing to accept a priori about\nthe density and the density contrast a ffects their correspond-\ning posteriors, but very little the inference of the magneti c field\nstrength. The three parameters seem to be rather independen t,\nsince changes in the prior information do not seem to a ffect the\nregions of parameter space where the likelihood is high.\nAppendix B: Application to loop oscillation data\nWe applied our Bayesian methods to existing data of trans-\nverse loop oscillations. Table B.1 shows a comparison be-\ntween magnetic field strength estimates from previous works\nand summaries of posteriors using Bayesian analysis. For ea ch\nevent, the length of the loop, oscillation period, and mea-\nsured phase speed are shown. First, the summary of the pos-\nterior for the internal Alfvén speed is computed using the\nforward model (7) and the full posterior (8), which is then\nmarginalised. Then, the magnetic field strength posterior i s com-\nputed using the forward model (9) and the Bayesian poste-\nrior (10) for a uniform prior on density over the range ρi∈\n[10−13−10−11] kg m−3. The last column shows the more\nconstrained results obtained using a Gaussian prior on den-\nsity centered at the values estimated or assumed by the pre-\nvious studies. Nakariakov & Ofman (2001) estimate a loop\ndensity of 109.3±0.3cm−3, Aschwanden et al. (2002) list their\nestimated densities on their Table III, Van Doorsselaere et al.\n(2008) use an internal electron density of 9 .8±0.3×1014\nArticle number, page 10 of 16Arregui et al.: Inference of magnetic field strength and dens ity from damped transverse coronal waves\nFig. A.1. Solutions to the inversion of Eq. (9) using di fferent prior distributions for the density contrast. The di fferent priors, shown in the left\ncolumn are the uniform prior, normal distribution, five di fferent Cauchy functions and an exponential function, respec tively. The three rightmost\ncolumns show the corresponding posteriors for the density c ontrast, density of the waveguide, and magnetic field streng th, respectively. Solid lines\ncorrespond to results obtained by direct integration of the posterior. Histograms are samples from the MCMC computatio ns.\nFig. A.2. Solutions to the inversion of Eq. (9) using di fferent prior distributions for the waveguide density. The di fferent priors, shown in the left\ncolumn are the Jeffreys prior, and the normal distribution, respectively. The three rightmost columns show the corresponding posteriors for the\ndensity contrast, density of the waveguide, and magnetic fie ld strength, respectively. Solid lines correspond to resul ts obtained by direct integration\nof the posterior. Histograms are samples from the MCMC compu tations.\nArticle number, page 11 of 16A&A proofs: manuscript no. ms\nm−3, White & Verwichte (2012) and Pascoe et al. (2016) as-\nsume the loops have an electron number density of 1015m−3.\nWhen the posteriors computed with the Gaussian priors on den -\nsity are summarised, the median and credible region at the\n68% credible interval agree well with the previous numeri-\ncal estimates for the events presented by Nakariakov & Ofman\n(2001), Aschwanden et al. (2002), Goossens et al. (2002), an d\nVan Doorsselaere et al. (2008).\nWe found a discrepancy between our posterior summaries\nand previous magnetic field strength estimates in the events\nanalysed by White & Verwichte (2012) and Pascoe et al. (2016) .\nWhite & Verwichte (2012) obtained possible ranges of variat ion\nfor the magnetic field strength according to the inequalitie s\nvph√\n2/radicalbig\nµ0˜µmpne≤B≤vph/radicalbig\nµ0˜µmpne. (B.1)\nBy inserting the values µ0=4π×10−7H m−1, ˜µ=1.27,mp=\n1.6726×10−27kg and ne=1015m−3and using the estimated\nphase speeds in Table 1 by White & Verwichte (2012), we obtain\nthe magnetic field strength ranges shown in italics in Table B .1,\nwhich differ from the ranges in Table 3 by White & Verwichte\n(2012) but agree well with our Bayesian posterior summaries\nusing a Gaussian prior on density. Pascoe et al. (2016) find th eir\nmagnetic field strength estimates and associated errors fro m the\nexpression\nB0=CA0/radicalbig\nµ0˜µmpne. (B.2)\nBy inserting the values µ0=4π×10−7H m−1, ˜µ=1.27,\nmp=1.6726×10−27kg and ne=1015m−3and using the es-\ntimated values for CA0in their Table 3, we obtain the magnetic\nfield strength values shown in italics in Table B.1, which di ffer\nfrom the values quoted in Table 3 by Pascoe et al. (2016) but\nagree well with our Bayesian posterior summaries using a Gau s-\nsian prior on density. The source of the discrepancy between our\nresults and those by White & Verwichte (2012) and Pascoe et al .\n(2016) could be a typo in one of the exponents in the square roo t\nterm in Eqs. (B.1) and (B.2) or, equivalenty, the reported re sults\nin White & Verwichte (2012) and Pascoe et al. (2016) would ap-\nply for values of ne=1014m−3rather than the stated 1015m−3.\nA second application to loop oscillation data is shown in Ta-\nble B.2. The list compiled by Goddard et al. (2016) from a cat-\nalogue by Zimovets & Nakariakov (2015) contains 120 events.\nTwo of these events are not analysed, since no loop length mea -\nsurement is available. In 52 cases, estimates of damping tim es\nare given. For these cases, the Bayesian inference of the mag -\nnetic field strength is performed using uniform priors and fo r\ntwo cases. One without considering the damping, using Eqs. ( 9)\nand (10), leading to the posterior summaries Bno damping . Another\nconsidering the damping, using Eqs. (11), (12) and (13), lea ding\nto the posterior summaries Bdamping . As can be appreciated, the\ndifferences in the posterior summaries are minimal, up to ±1 G at\nmost in the median and /or the upper/lower error bars at the 68%\ncredible interval. Hence we confirm with our application to r eal\ndata that considering the damping of the oscillations in the infer-\nence of the magnetic field strength has very little impact, at least\nwhen the thin tube and thin boundary approximations are used .\nThe analysis is completed by performing the inference witho ut\ndamping for 66 cases in which damping was not measured, using\nEqs. (9) and (10).\nArticle number, page 12 of 16Arregui et al.: Inference of magnetic field strength and dens ity from damped transverse coronal waves\nTable B.1. Transverse loop oscillation data, previous inference resu lts (B0) and\nBayesian posterior summaries using uniform ( Bu) and Gaussian ( BG) priors on\ndensity.\nEvent Loop L P vph vAi B0 Bu BG\n(Mm) (s) (km s−1) (km s−1) (G) (G) (G)\nNakariakov & Ofman (2001)\n1 1 130 256 1020 ±132 800+122\n−101− 22+6\n−713+3\n−3\n2 1 190 360 1030 ±410 813+330\n−31713±9 21+12\n−913+7\n−6\nAschwanden et al. (2002)\nGoossens et al. (2002)\na 1 168 261 1287 1011+127\n−11313 28+7\n−912+3\n−3\nb 1 72 265 543 426+55\n−476 12+3\n−45+1\n−1\nd 1 174 316 1101 863+110\n−9511 24+5\n−811+3\n−3\nf 1 204 277 1473 1156+145\n−12516 32+7\n−1015+4\n−4\ng 1 162 272 1191 936+118\n−10310 26+6\n−910+2\n−2\na 3 390 522 1494 1173+148\n−13011 32+8\n−1011+2\n−3\na 4 258 435 1186 931+118\n−10313 25+7\n−812+3\n−3\nc 5 166 143 2322 1823+230\n−20027 50+12\n−1625+6\n−6\na 10 406 423 1920 1508+188\n−16520 42+10\n−1319+5\n−5\na 16 192 185 2076 1631+205\n−18015 45+11\n−1414+3\n−3\na 17 146 396 737 579+73\n−636 16+4\n−55+1\n−1\nVan Doorsselaere et al. (2008)\n1 1 390 296 ±24 2600±500 2045+425\n−40239±8 55+18\n−1932+6\n−6\nWhite & Verwichte (2012)\n1 1 121 ±2 225±40 1080±220 851+185\n−1783.9−5.6 23+8\n−814+4\n−4\n(12.5−17.6)\n1 2 111 ±6 215±5 1030±110 808+108\n−933.8−5.3 22+6\n−713+3\n−3\n(11.9−16.8)\n1 3 132 213 ±9 1240±140 973+133\n−1184.5−6.4 27+7\n−916+4\n−4\n(14.3−20.3)\n1 4 113 ±4 216±30 1050±170 826+148\n−1383.8−5.4 22+7\n−713+4\n−4\n(12.1−17.2)\n2 1 396 520 ±5 1520±150 1193+148\n−1305.6−7.9 33+8\n−1020+5\n−5\n(17.6−24.8)\n2 2 374 596 ±50 1260±160 991+145\n−1354.6−6.5 27+7\n−816+4\n−4\n(14.6−20.6)\n3 1 279 ±3 212±20 2630±360 2068+322\n−2979.6−13.6 56+15\n−1834+9\n−8\n(30.4−42.9)\n3 2 240 ±4 256±20 1880±250 1478+225\n−2086.8−9.7 40+11\n−1324+6\n−6\n(21.7−30.7)\n3 3 241 135 ±9 3570±430 2806+400\n−36013.0−18.4 77+20\n−2546+12\n−11\n(41.2−58.3)\n3 4 159 ±6 115±2 2770±280 2176+275\n−24310.1−14.3 60+14\n−1936+9\n−9\n(32.0−45.3)\n3 5 132 103 ±8 2560±330 2013+300\n−2759.4−13.2 55+15\n−1733+8\n−8\n(29.6−41.8)\nArticle number, page 13 of 16A&A proofs: manuscript no. ms\nTable B.1. continued.\nEvent Loop L P vph vAi B0 Bu BG\n(Mm) (s) (km s−1) (km s−1) (G) (G) (G)\nPascoe et al. (2016)\n43 4 (Loop #1) 222 ±31 284±1 1564±222 1231+198\n−1839.38±2.56 34+9\n−1127+7\n−7\n(30±8)\n31 1 (Loop #2) 162 ±31 459±1 706±136 556+115\n−1104.37±0.88 15+5\n−513+4\n−4\n(14±3)\n32 1 (Loop #3) 234 ±31 250±1 1871±252 1456+228\n−20817.06±2.63 40+11\n−1358+15\n−14\n(54±8)\nTable B.2. Transverse loop oscillation data from Goddard et al. (2016) and\nBayesian posterior summaries with and without damping.\nEvent Loop L P τd vph vAi Bno damping Bdamping\n(Mm) (s) (s) (km s−1) (km s−1) (G) (G)\n1 1 232 205 ±4 320±67 2261±40 1748+165\n−6350+9\n−1650+10\n−16\n1 2 78 247 ±3 646±167 633±8 489+45\n−1514+3\n−514+3\n−4\n2 1 156 398 ±4 - 783 ±7 605+58\n−1817+3\n−5-\n3 1 213 148 ±2 528±108 2886±35 2228+210\n−7364+12\n−2065+13\n−20\n3 2 262 217 ±5 247±28 2413±53 1868+173\n−7354+10\n−1753+10\n−17\n3 3 311 242 ±6 - 2566 ±64 1988+185\n−8357+11\n−18-\n4 1 183 137 ±2 431±90 2664±35 2056+195\n−6559+11\n−1960+11\n−19\n4 2 181 208 ±2 446±60 1739±15 1341+128\n−4038+7\n−1239+7\n−12\n5 1 438 422 ±4 - 2077 ±18 1601+153\n−4857+11\n−18-\n6 1 430 483 ±16 - 1781 ±58 1383+128\n−6540+7\n−13-\n7 1 162 101 ±1 434±78 3195±38 2466+233\n−7871+13\n−2372+14\n−23\n8 1 207 224 ±4 600±60 1845±35 1426+135\n−5041+8\n−1341+8\n−13\n9 1 264 308 ±10 305±59 1712±57 1331+123\n−6538+7\n−1137+7\n−11\n9 2 326 537 ±8 710±286 1214±19 938+88\n−3327+5\n−926+5\n−8\n10 1 397 688 ±10 481±65 1155±17 891+85\n−3025+5\n−825+5\n−8\n10 2 279 509 ±10 - 1097 ±21 848+80\n−3024+5\n−7-\n11 1 78 238 ±4 - 657 ±12 509+48\n−2015+3\n−5-\n11 2 95 231 ±7 - 823 ±24 639+60\n−2818+3\n−5-\n11 3 118 156 ±3 530±90 1513±29 1171+108\n−4334+6\n−1134+7\n−11\n11 4 125 229 ±2 - 1094 ±11 843+80\n−2524+5\n−7-\n11 5 135 305 ±4 - 884 ±10 681+65\n−2019+4\n−6-\n11 6 160 368 ±13 - 870 ±30 676+63\n−3319+3\n−6-\n12 1 148 334 ±4 - 887 ±11 684+65\n−2019+4\n−6-\n15 1 174 458 ±22 - 759 ±37 591+58\n−3517+3\n−5-\n16 1 242 157 ±2 - 3079 ±47 2378+223\n−8068+13\n−21-\n16 2 146 141 ±4 161±38 2071±62 1608+148\n−7546+9\n−1445+9\n−14\n16 3 318 314 ±11 - 2027 ±74 1576+148\n−8045+9\n−14-\n17 1 153 124 ±2 599±275 2464±48 1906+178\n−7054+11\n−1755+11\n−17\n18 1 289 431 ±19 - 1342 ±60 1046+98\n−6329+6\n−9-\n18 2 284 571 ±7 732±208 994±11 766+73\n−2322+4\n−722+4\n−7\n18 3 393 781 ±10 - 1006 ±13 776+75\n−2522+5\n−7-\n19 1 123 584 ±12 - 421 ±9 326+30\n−139+2\n−3-\n19 2 348 676 ±7 993±86 1029±11 793+75\n−2523+4\n−723+4\n−7\nArticle number, page 14 of 16Arregui et al.: Inference of magnetic field strength and dens ity from damped transverse coronal waves\nTable B.2. continued.\nEvent Loop L P τd vph vAi Bno damping Bdamping\n(Mm) (s) (s) (km s−1) (km s−1) (G) (G)\n20 1 253 322 ±14 971±460 1573±68 1226+115\n−7035+7\n−1135+7\n−11\n21 1 499 429 ±121 - 2326 ±654 1833+532\n−51748+20\n−19-\n22 1 288 162 ±7 - 3556 ±145 2768+257\n−15378+15\n−25-\n23 1 365 922 ±24 1151±93 792±21 614+58\n−2517+3\n−517+3\n−6\n24 1 432 1072 ±18 1646±256 806±14 625+58\n−2318+3\n−618+3\n−6\n24 2 427 987 ±17 - 865 ±15 669+63\n−2319+3\n−6-\n24 3 538 1228 ±35 2101±386 877±25 681+63\n−3019+4\n−619+4\n−6\n25 1 156 308 ±7 480±300 1014±22 786+73\n−3322+5\n−722+4\n−7\n25 2 264 438 ±10 - 1205 ±26 933+85\n−3827+5\n−9-\n26 1 473 717 ±8 1123±270 1319±14 1018+95\n−3329+5\n−929+6\n−9\n26 2 185 751 ±11 - 493 ±7 381+35\n−1311+2\n−3-\n27 1 244 917 ±24 - 532 ±14 414+38\n−1812+2\n−4-\n29 1 154 223 ±3 470±37 1384±19 1068+100\n−3531+5\n−1031+6\n−9\n31 1 162 460 ±2 1453±121 704±4 544+50\n−1815+3\n−516+3\n−5\n31 2 138 575 ±5 1054±141 480±5 371+35\n−1311+2\n−311+2\n−3\n31 3 532 694 ±7 - 1534 ±16 1183+113\n−3834+6\n−10-\n32 1 234 257 ±1 933±73 1822±9 1403+135\n−4040+8\n−1341+8\n−13\n32 2 233 203 ±1 1147±291 2298±14 1771+170\n−5351+9\n−1652+10\n−17\n33 1 314 281 ±5 - 2232 ±38 1726+160\n−6050+9\n−16-\n33 2 407 391 ±6 - 2081 ±32 1608+150\n−5546+9\n−15-\n34 1 333 597 ±16 1002±62 1116±30 866+80\n−3825+5\n−825+5\n−8\n35 1 327 527 ±8 - 1241 ±18 958+90\n−3327+5\n−9-\n35 2 312 346 ±6 - 1802 ±31 1393+130\n−5040+8\n−12-\n36 1 282 401 ±6 - 1407 ±21 1086+103\n−3531+6\n−9-\n37 1 358 496 ±13 - 1443 ±38 1118+103\n−4832+6\n−10-\n38 1 224 182 ±2 - 2456 ±24 1893+180\n−5854+10\n−17-\n38 2 270 312 ±5 914±330 1731±27 1338+125\n−4838+7\n−1239+7\n−12\n38 3 424 785 ±13 - 1081 ±17 836+78\n−3024+4\n−8-\n38 4 478 644 ±11 - 1484 ±25 1146+108\n−4033+6\n−10-\n39 1 402 647 ±6 - 1242 ±12 958+90\n−3027+5\n−9-\n39 2 334 641 ±7 - 1042 ±12 803+78\n−2523+5\n−7-\n39 3 376 754 ±22 - 997 ±29 773+73\n−3522+4\n−7-\n39 4 454 858 ±10 - 1058 ±13 816+78\n−2523+5\n−7-\n40 1 171 341 ±4 - 1004 ±11 773+75\n−2322+4\n−7-\n40 2 347 337 ±2 1490±205 2062±11 1588+153\n−4846+9\n−1447+9\n−14\n40 3 325 355 ±42 - 1830 ±216 1438+203\n−18340+10\n−13-\n40 4 258 332 ±2 439±65 1555±11 1198+115\n−3534+7\n−1134+6\n−11\n40 5 297 325 ±1 - 1827 ±7 1408+135\n−4340+8\n−13-\n40 6 425 416 ±2 - 2044 ±12 1576+150\n−4846+8\n−15-\n40 7 353 343 ±4 850±164 2057±22 1586+153\n−4846+9\n−1446+9\n−14\n40 8 238 260 ±5 541±130 1832±34 1416+133\n−5040+8\n−1341+8\n−13\n40 9 473 371 ±3 789±160 2551±21 1966+188\n−5856+11\n−1757+11\n−17\n40 10 238 376 ±2 - 1265 ±6 976+93\n−3028+5\n−9-\n40 11 220 286 ±2 - 1541 ±13 1188+113\n−3534+7\n−11-\n43 1 363 428 ±4 452±87 1695±17 1308+123\n−4338+7\n−1237+7\n−12\nArticle number, page 15 of 16A&A proofs: manuscript no. ms\nTable B.2. continued.\nEvent Loop L P τd vph vAi Bno damping Bdamping\n(Mm) (s) (s) (km s−1) (km s−1) (G) (G)\n43 2 241 216 ±2 566±55 2231±19 1721+163\n−5350+9\n−1650+10\n−15\n43 3 368 501 ±5 902±109 1469±14 1133+108\n−3533+6\n−1133+6\n−10\n43 4 222 310 ±2 - 1434 ±8 1106+105\n−3332+6\n−10-\n43 5 260 270 ±1 840±120 1926±9 1483+143\n−4343+8\n−1444+8\n−13\n44 1 295 434 ±4 945±185 1360±11 1048+100\n−3330+5\n−930+6\n−9\n44 2 512 587 ±11 877±298 1745±34 1351+125\n−5039+7\n−1338+7\n−12\n44 3 352 417 ±8 540±180 1688±34 1306+123\n−5038+7\n−1237+7\n−12\n44 4 202 145 ±3 - 2794 ±58 2163+200\n−8362+12\n−19-\n45 1 92 149 ±2 469±100 1237±20 956+90\n−3327+5\n−928+5\n−9\n46 1 430 724 ±14 - 1188 ±23 918+85\n−3326+5\n−8-\n46 2 498 659 ±7 - 1510 ±15 1166+110\n−3834+6\n−11-\n46 3 384 594 ±6 - 1293 ±13 998+93\n−3329+5\n−9-\n47 1 225 316 ±8 - 1423 ±38 1103+103\n−4832+6\n−10-\n47 2 222 301 ±7 - 1474 ±35 1141+108\n−4533+6\n−11-\n48 1 540 917 ±10 1319±936 1178±12 908+88\n−2826+5\n−926+5\n−8\n48 2 588 946 ±7 1598±130 1244±9 958+93\n−2827+5\n−927+5\n−8\n48 3 597 965 ±13 946±185 1238±16 956+90\n−3327+5\n−927+5\n−8\n48 4 426 554 ±14 - 1538 ±38 1191+110\n−4834+7\n−11-\n48 5 471 950 ±13 - 992 ±13 766+73\n−2522+4\n−7-\n49 1 484 907 ±28 - 1067 ±33 828+78\n−3823+5\n−7-\n49 2 197 464 ±8 - 850 ±15 656+63\n−2319+3\n−6-\n49 4 386 627 ±10 923±155 1231±20 951+90\n−3327+5\n−927+5\n−8\n49 5 191 482 ±11 562±73 793±18 614+58\n−2317+3\n−517+3\n−6\n52 1 183 356 ±7 - 1029 ±21 796+75\n−3023+4\n−7-\n53 1 420 569 ±13 - 1477 ±34 1143+108\n−4533+6\n−11-\n54 1 408 500 ±4 - 1633 ±14 1258+120\n−3836+7\n−11-\n54 2 400 448 ±6 - 1787 ±24 1378+133\n−4540+7\n−13-\n54 3 238 139 ±3 - 3420 ±74 2648+245\n−10376+15\n−24-\n54 4 355 226 ±8 - 3139 ±108 2441+225\n−12369+14\n−21-\n54 5 257 288 ±6 1183±194 1785±37 1381+130\n−5340+7\n−1341+8\n−13\n55 1 405 518 ±14 - 1564 ±44 1213+113\n−5335+7\n−11-\n55 2 477 392 ±10 - 2431 ±63 1883+175\n−7854+11\n−17-\n56 1 403 544 ±8 1243±283 1481±23 1143+108\n−3833+6\n−1033+6\n−10\n56 2 314 713 ±8 1177±178 881±10 681+63\n−2319+4\n−619+4\n−6\n56 3 205 193 ±10 - 2122 ±105 1656+158\n−10547+9\n−15-\n56 4 501 863 ±20 - 1161 ±27 898+85\n−3525+5\n−8-\n56 5 431 810 ±10 1450±308 1064±13 821+78\n−2523+5\n−723+4\n−8\n56 6 392 455 ±12 - 1722 ±45 1336+123\n−5838+7\n−12-\n56 7 457 850 ±33 818±236 1117±45 868+83\n−4825+5\n−824+5\n−8\n56 8 379 638 ±9 - 1187 ±17 916+88\n−3026+5\n−8-\nArticle number, page 16 of 16"    },    {        "title": "1908.03194v5.Annihilation_of_topological_solitons_in_magnetism_with_spin_wave_burst_finale__The_role_of_nonequilibrium_electrons_causing_nonlocal_damping_and_spin_pumping_over_ultrabroadband_frequency_range.pdf",        "content": "Annihilation of topological solitons in magnetism with spin wave burst \fnale: The\nrole of nonequilibrium electrons causing nonlocal damping and spin pumping over\nultrabroadband frequency range\nMarko D. Petrovi\u0013 c,1Utkarsh Bajpai,1Petr Plech\u0013 a\u0014 c,2and Branislav K. Nikoli\u0013 c1,\u0003\n1Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n2Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA\nWe not only reproduce burst of short-wavelength spin waves (SWs) observed in recent experiment\n[S. Woo et al. , Nat. 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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Mickiewicza 30, 30-059 Kraków, Poland\n2National Institute of Advanced Industrial Science and Tech nology,\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan\nVoltage-induced ferromagnetic resonance (V-FMR) in magne tic tunnel junctions (MTJs) with a\nW buffer is investigated. Perpendicular magnetic anisotrop y (PMA) energy is controlled by both\nthickness of a CoFeB free layer deposited directly on the W bu ffer and a post-annealing process at\ndifferent temperatures. The PMA energy as well as the magneti zation damping are determined by\nanalysing field-dependent FMR signals in different field geom etries. An optimized MTJ structure\nenabled excitation of V-FMR at frequencies exceeding 30 GHz . The macrospin modelling is used\nto analyse the field- and angular-dependence of the V-FMR sig nal and to support experimental\nmagnetization damping extraction.\nI. INTRODUCTION\nMagnetic multilayer structures are commonly utilized\nas key elements of magnetic field sensors1and magne-\ntic random access memories2. A stack structure similar\nto the one used in existing applications and related phy-\nsical mechanisms, such as magnetoresistance effect and\nspin-transfer-torque (STT), can be used for future mi-\ncrowave solutions, for example spin torque oscillators3\nor microwave detectors4. One of the main drawbacks of\nthe existing technologies is a relatively high current used\nin the STT-based devices. In order to reduce the power\nconsumption of such devices, a few alternative solutions\nhave been proposed, including spin-orbit torque5,6and\nelectric-field controlled magnetism7,8. The latter typical-\nly requires structures with relatively thick insulators fo r\napplication of a significant electric-field at the insula-\ntor/ferromagnet interface.\nRecently, a few different mechanisms have been pro-\nposed to maximize the effect of an electric field on ma-\ngnetic properties of materials, including diluted magne-\ntic semiconductors9, nitride semiconductors10, charge mi-\ngration in multilayers11and voltage controlled magnetic\nanisotropy (VCMA) in metallic thin films12. It has been\nalready presented that, by utilizing the VCMA effect, dri-\nving the magnetization into a precession at several GHz\nis possible13, which is promising for high-frequency devi-\nces.\nIn this work we present studies on CoFeB/MgO/\nCoFeB-based MTJs deposited on a W buffer14,15. Using a\nrelatively thin CoFeB bottom free layer and different an-\nnealing temperatures, high perpendicular magnetic ani-\nsotropy (PMA) energies of up to 1.5 MJ/m3are achieved,\nwhich, in turn, enables voltage-induced ferromagnetic re-\nsonance (V-FMR) excitation at frequencies exceeding 30\nGHz. In addition, the V-FMR measurements in combina-\ntion with vector network analyser ferromagnetic resonan-\nce (VNA-FMR) investigation16,17were used to determine\nthe magnetization damping in the discussed multilayers.Tabela I. PMA energies of samples with different tCoFeB after\nannealing at increasing temperatures. Values of KPMAin bold\nindicate samples with perpendicular effective anisotropy.\ntCoFeB = 0.9 nm tCoFeB = 1.2 nm\nTA(◦C)KPMA(MJ/m3)KPMA(MJ/m3)\nas dep. 0.5 0.4\n200 0.7 0.6\n250 1.04 0.8\n300 1.19 0.95\n350 1.51 1.12\nII. EXPERIMENT\nMultilayers of the following structure: W(5)/\nCoFeB(tCoFeB)/ MgO(1.8)/ CoFeB(5)/ Ta(5)/ Ru(5)/\nPt(2) (thickness in nm) were deposited using magnetron\nsputtering in a similar conditions to Ref.18. Two thick-\nnesses of the free layer, i.e., tCoFeB = 0.9 and 1.2 nm,\nwere investigated. After the deposition process, magnetic\nproperties were determined with VNA-FMR in 10 ×10\nmm samples by analysing the frequency vs. perpendicu-\nlar magnetic field dependence using the Kittel relation f\n=γ/(2π)√\nBH, whereHis the external magnetic field\nandBis the magnetic field induction. Measurements\nwere repeated after subsequent sample annealing in\nhigh-vacuum furnace at TA= 200, 250, 300 and 350◦C.\nIndependently, the same multilayers were fabricated into\npillars of 2×4µm2using electron-beam lithography\nand ion-beam milling for transport measurements. The\ntransport properties were measured in a probe station\nthat enabled sample rotation at azimuth ( θ) and polar\n(φ) angle with respect to the sample plane, in a magnetic\nfield of up to 1 T with a broadband electrical contact.\nBoth quasi-static (resistance vs. magnetic field) and\ndynamic measurements (DC mixing voltage vs. magnetic\nfield) were performed using a two-point method.2\n/s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53/s48/s49/s48/s50/s48/s48/s49/s48/s50/s48/s51/s48\n/s40/s98/s41/s32/s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s49/s46/s50/s32/s110/s109/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s107/s79/s101/s41/s40/s97/s41/s32/s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s48/s46/s57/s32/s110/s109/s84\n/s65/s32/s40 /s67/s41\n/s32/s51/s53/s48\n/s32/s51/s48/s48\n/s32/s50/s53/s48\n/s32/s50/s48/s48\n/s32/s65/s68/s102/s32/s40/s71/s72/s122/s41\nRysunek 1.fvs.Hdependence of samples with tCoFeB = 0.9\n(a) andtCoFeB = 1.2 nm (b) measured after annealing at diffe-\nrent temperatures using VNA-FMR - points. Lines represents\nbest fits to the model based on the Kittel formula, resulting\ninKPMAsummarized in Table I.\nIII. RESULTS AND DISCUSSION\nFirst, we focus on the wafer-level investigation using\nVNA-FMR. Resonance frequency ( f) vs. perpendicular\nmagnetic field curves of samples with different thicknes-\nses of the free layer ( tCoFeB = 0.9 and 1.2 nm) are presen-\nted in Fig. 1. Experimental points were modelled using\nthe Kittel relation with a CoFeB saturation magnetiza-\ntion value ofµ0MS= 1.6 T14. The PMA energies ( KPMA)\nresulting from fits to the model are gathered in Table\nI. Assuming an infinite-plane sample configuration, the\ndemagnetization energy density KD=µ0M2\nS/2 = 1.02\nMJ/m3, resulting in an effective perpendicular anisotro-\npy for the sample with tCoFeB = 0.9 nm forTA/xgreaterequal250◦C\nand for sample with tCoFeB = 1.2 nm forTA>300◦C.\nThe PMA energy of the sample with tCoFeB = 0.9 nm\nannealed atTA= 350◦C evaluated from the TMR de-\npendence on the magnetic field applied in the sample\nplane18is K = 1.47 MJ/m3, which is in good agreement\nwith the values obtained from VNA-FMR.\nFrom the same type of measurements, the free layer\nmagnetization damping ( α) was calculated from the line-\nwidth (∆H) vs.fslope according to the following Eq.;\n∆H=4πα\nγf+∆H0 (1)\nwhereγ= 1.76×1011Hz/T is the gyromagnetic ratio and\n∆H0is the inhomogeneous broadening19. An example of\nthe∆Hvs.fdependence for the sample with tCoFeB =\n1.2 nm annealed at TA= 300◦C is presented in Fig. 2(a).\nThe calculatedαfor all samples is presented in Fig. 2(c).\nThe values obtained for the sample with tCoFeB = 1.2 nm\nagree well with the W/CoFeB sample of similar thickness\nreported in Ref.20. One can also note a tendency of αto\nincrease with decreasing thickness of CoFeB21.\nNext, we move on to the transport measurement on/s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48\n/s65/s68/s32/s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s49/s46/s50/s32/s110/s109/s44/s32/s86/s78/s65/s45/s70/s77/s82\n/s32/s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s48/s46/s57/s32/s110/s109/s44/s32/s86/s78/s65/s45/s70/s77/s82\n/s32/s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s48/s46/s57/s32/s110/s109/s44/s32/s86/s45/s70/s77/s82/s97 /s42/s49/s48/s45/s51\n/s32\n/s84\n/s65/s32/s40/s176/s67/s41/s40/s99/s41/s102/s32/s40/s71/s72/s122/s41/s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s49/s46/s50/s32/s110/s109/s44/s32/s84\n/s65/s32/s61/s32/s50/s53/s48/s32 /s176/s67\n/s32/s115/s108/s111/s112/s101/s32/s61/s32/s49/s42/s49/s48/s45/s49/s50\n/s32/s84/s47/s72/s122\n/s97 /s32/s61/s32/s48/s46/s48/s49/s52\n/s32/s100/s72/s68 /s72 /s32/s40/s79/s101/s41\n/s40/s97/s41/s40/s98/s41\n/s40/s100/s41/s102/s32/s40/s71/s72/s122/s41/s68 /s72/s32/s40/s79/s101/s41/s75\n/s80/s77/s65/s32/s40/s107/s74/s47/s109/s51/s41/s44/s32/s84\n/s65/s32/s40 /s176/s67/s41\n/s32/s49/s48/s52/s48/s44/s32/s50/s53/s48\n/s32/s49/s49/s57/s48/s44/s32/s51/s48/s48\n/s32/s49/s53/s49/s48/s44/s32/s51/s53/s48/s98/s32 /s40/s176/s41\n/s102/s32/s40/s71/s72/s122/s41\nRysunek 2. (a) Example of ∆Hvs.fdependence for the sam-\nple withtCoFeB = 1.2 nm annealed at 250◦C measured using\nVNA-FMR. Fitting a line resulted in slope of 10−12T/Hz and\nα= 0.014. The linewidth vs. fdependence, presented in (b),\nindicates a non-linear behaviour for small f(and thus small\nH). (c) presents summarized αfor all samples obtained using\nboth VNA-FMR and V-FMR. The origin of a non-linear ∆H\nvs.fis explained in (d), where βis the angle between the\napplied magnetic field vector and the magnetization vector -\na strong non-collinearity is found in simulations for small f\nand highKPMA.\nthe fabricated sample. The crystallization process of Co-\nFeB (initiated from the MgO tunnel barrier) starts after\nannealing at 250◦C22. Therefore, we begin the analysis\nof the V-FMR signal for the samples annealed at this\ntemperatures. The tunnelling magnetoresistance ratio of\nthe fabricated MTJ reaches 40% (measured between an\northogonal and parallel CoFeB magnetization orienta-\ntions) and the resistance-area product is around RA=\n40 kOhm*µm2. Figure 3(a-i) presents DC voltage (a re-\nsult of a small AC current mixing with resistance change\noriginating from the VCMA) Vmixas a function of the\nmagnetic field applied at an angle θ= 60◦from the sam-\nple plane. The resonance peak at higher (smaller) field\noriginates from the free (reference) layer. This depen-\ndence was modelled using a sum of two symmetric and\nasymmetric Lorentz curves:\nVmix=VS1∆H2\n1\n(H−Hr1)2+∆H2\n1\n+VA1(H−Hr1)∆H1\n(H−Hr1)2+∆H2\n1\n+VS2∆H2\n2\n(H−Hr2)2+∆H2\n2\n+VA2(H−Hr2)∆H2\n(H−Hr2)2+∆H2\n2(2)\nwhereHr1(Hr2) is the resonance field of the free (re-\nference) layer at a given f,VS1andVA1(VS2andVA2)\nare the amplitudes of the symmetric and antisymmetric3\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s49 /s50 /s51 /s52 /s53 /s54\n/s53/s48/s49/s48/s48/s49/s53/s48\n/s52 /s54 /s56 /s49/s48 /s49/s50/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s82/s76/s70/s76\n/s49/s50/s49/s49/s49/s48/s57/s56/s55/s54/s53/s32\n/s32\n/s52/s102/s32/s40/s71/s72/s122/s41\n/s32 /s32 /s32 /s32/s86\n/s109/s105/s120/s32/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41\n/s32 /s32 /s32 /s32\n/s72/s32/s40/s107/s79/s101/s41/s40/s105/s41/s40/s104/s41/s40/s103/s41/s40/s102/s41/s40/s101/s41/s40/s100/s41/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s32\n/s40/s107/s41/s40/s106/s41\n/s97 /s32/s61/s32/s48/s46/s48/s48/s53/s57/s32\n/s32/s82/s76/s44/s32 /s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s48/s46/s57/s32/s110/s109/s44/s32 /s84\n/s65/s32/s61/s32/s50/s53/s48/s32 /s176/s67/s32\n/s115/s108/s111/s112/s101/s32/s61/s32/s52/s46/s50/s42/s49/s48/s45/s49/s51\n/s32/s84/s47/s72/s122/s32/s68 /s72/s32/s40/s79/s101/s41/s32\n/s32\n/s102/s32/s40/s71/s72/s122/s41/s70/s76/s44/s32/s115/s108/s111/s112/s101/s32/s61/s32/s56/s46/s53/s42/s49/s48/s45/s49/s51\n/s32/s84/s47/s72/s122/s32\n/s97 /s32/s61/s32/s48/s46/s48/s49/s50\nRysunek 3. (a-i)Vmixvs.Happlied atθ= 60◦andφ= 90◦\ndependence measured with an excitation frequency fbetween\n4 and 12 GHz, for sample with tCoFeB = 0.9 nm annealed\nat 250◦C. Lines represent best fits to Eq. 2. (j) ∆H1vs.f\ndependence for the reference layer (RL), fitting a line resul ted\nin slope of 4.2×10−13T/Hz andα= 0.006, (k)∆H2vs.f\ndependence of the free layer (FL), fitting a line resulted in\nslope of 8.5×10−13T/Hz andα= 0.012, (l) presents a sketch\nof the experimental geometry\nLorentz functions of the free (reference) layer and ∆H1\n(∆H2) is the linewidth of the free (reference) layer. Fit-\nting the∆H1and∆H2vs.fdependence enabled us to\ncalculate the damping of the free and the reference layers\nindependently - Fig. 3(j-k). The results agree well with α\nobtained using the VNA-FMR method and are included\nin Fig. 2(c).\nWithin the limit of the available magnetic field of up\nto 1 T, for the MTJ annealed at 250◦C the maximum\nresonance signal was measured for around f= 20 GHz,\ndepending on the magnetic field configuration23. In order\nto increase the resonance frequency, the fabricated sam-\nple was further annealed at 300◦C, which enhanced the\nPMA energy. The V-FMR signal measured at a nearly\nperpendicular field ( θ= 85◦) is presented in Fig. 4(a-i).\nA much better signal-to-noise ratio was obtained for an\nazimuth angleφ= 0◦, resulting in a symmetric linesha-\npe, contrary to the signal presented in Fig. 3(a-i), where\nφ= 90◦resulted in an asymmetric signal. In the general\ncase, the lineshape depends on the angle of the free layer\nmagnetization vector projection on the sample plane and\nthe reference layer magnetization, which, in our case is\ndefined along theyaxis (Fig. 3(l))24. We also note that,/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s49 /s50 /s51 /s52 /s53 /s54\n/s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48 /s49 /s50 /s51/s48/s49/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53\n/s48/s46/s53/s49/s46/s48/s49/s46/s53/s32\n/s32\n/s51/s49/s50/s57/s50/s55/s50/s53/s50/s51/s50/s49/s49/s57/s49/s55/s32\n/s49/s53\n/s48/s46/s53\n/s48/s46/s50/s53\n/s48/s46/s48/s49\n/s45/s48/s46/s50/s53\n/s45/s48/s46/s53/s102/s32/s40/s71/s72/s122/s41\n/s40/s105/s41/s40/s104/s41/s40/s103/s41/s40/s102/s41/s40/s101/s41/s40/s100/s41/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s32\n/s32 /s32 /s32\n/s32/s86\n/s109/s105/s120/s32/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41\n/s32 /s32 /s32 /s32\n/s72/s32/s40/s107/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s68 /s72/s32/s40/s107/s79/s101/s41/s32\n/s32/s116\n/s67/s111/s70/s101/s66/s32/s61/s32/s48/s46/s57/s32/s110/s109/s44/s32 /s84\n/s65/s32/s61/s32/s51/s48/s48/s32 /s176/s67/s32\n/s115/s108/s111/s112/s101/s32/s61/s32/s49/s46/s54/s42/s49/s48/s45/s49/s50\n/s32/s84/s47/s72/s122/s32\n/s97 /s32/s61/s32/s48/s46/s48/s50/s50\n/s32\n/s72/s32/s40/s107/s79/s101/s41\n/s32 /s32\n/s40/s110/s41/s40/s109/s41/s40/s108/s41/s40/s107/s41/s40/s106/s41\n/s40/s112/s41/s40/s111/s41\n/s32 /s32/s86\n/s109/s105/s120/s32/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41\n/s86\n/s66/s32/s40/s86/s41/s72\n/s48/s32/s40/s107/s79/s101/s41/s86\n/s66/s32/s40/s86/s41\n/s32\nRysunek 4. (a-i)Vmixvs.Happlied atθ= 85◦andφ= 0◦\nwith differentfof the excitation signal of sample with tCoFeB\n= 0.9 nm annealed at 300◦C. Lines represent best fits to Eq.\n2, (j-n) showsVmixvs.Hdependence forf= 17 GHz under\ndifferent bias voltage condition, (o) shows the dependence o f\nthe resonance frequency on VB, (p)∆H1vs.fdependence,\nfitting a line resulted in slope of 1.6 ×10−12T/Hz andα=\n0.024\ncontrary to the previous studies of V-FMR in MTJs25,26,\nthe lineshape is only little affected by the bias voltage\n(VB), which is due to strong PMA in our devices - Fig.\n4(j-n). Therefore, magnetic anisotropy changes induced\nby static bias voltage are weak comparing to high PMA\nenergy.\nTo understand the angular dependence of the V-FMR\nand to confirm the magnetization damping analysis, the\nmacrospin simulations were performed. We used Landau-\nLifshitz-Gilbert equation, where voltage excitation was\nmodelled as sinusoidal changes of KPMA. This, in turn,\ncontributes an alternating term to the effective field,\nwhich determines magnetization dynamics. Afterwards,\nMTJ resistance is calculated and multiplied by the assu-\nmed leakage current associated with the applied voltage,\nresulting inVmix, in the same way as in the experimen-\ntal procedure. We used a similar approach previously in\nV-FMR spin diode effect modelled using micromagnetic\nsimulations27.\nThe simulated dependence of the linewidth vs. ffor\ndifferentKPMAis depicted in Fig. 2(b). A strong de-\nviation from the linear dependence is observed for high\nKPMA, which is a result of a significant non-collinear di-\nrection of the magnetic field and magnetization in this4\ncase presented in Fig. 2(d). At the same time, magnetic\nfields sufficient to saturate the sample would increase the\neffective field to the level where the VCMA excitation wo-\nuld no longer be strong enough to induce significant ma-\ngnetization dynamics27. Therefore, we limit the V-FMR\ninvestigation to samples annealed at 250 and 300◦C.\nIV. SUMMARY\nIn conclusion, ferromagnetic resonance in\nW/CoFeB/MgO/CoFeB was investigated by means\nof wafer-level vector network analyser FMR and voltage-\ninduced FMR in patterned devices. Both the CoFeB\nthickness and the thermal treatment influence magnetic\nanisotropy, which reaches a value of 1.5 MJ/m3, which\nis well above demagnetizing energy. Resonance signals\nfrom both the reference and the free layer are analysed,\nallowing for magnetization damping determination. For\nthin CoFeB free layers the damping between 0.01 and0.02 was measured, independent on annealing conditions,\nwhich is approximately double of the thicker reference\nlayer damping. V-FMR in the MTJ annealed at 300◦C\nis measured up to a high value of f= 31 GHz.\nACKNOWLEDGMENTS\nWe would like to thank prof. Tomasz Stobiecki, prof.\nSławomir Gruszczyński and dr. Sławomir Ziętek for a\nfruitful discussions and technical assistance in the me-\nasurements. This work was supported by the Natio-\nnal Science Centre, Poland, grant No. LIDER/467/L-\n6/14/NCBR/2015 by the Polish National Centre for Re-\nsearch and Development. M.F. acknowledges grant Prelu-\ndium UMO-2015/17/N/ST3/02276 from National Scien-\nce Center, Poland. Microfabrication was performed at\nAcademic Centre for Materials and Nanotechnology of\nAGH University. Numerical calculations were supported\nby PL-GRID infrastructure.\n∗skowron@agh.edu.pl\n1S. Tumański. Thin film magnetoresistive sensors . IOP\nPublishing, Bristol, (2001).\n2S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and\nS. Piramanayagam, Materials Today 20, 530 – 548 (2017).\n3T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli, A. Ho-\nushang, A. A. Awad, P. Dürrenfeld, B. G. Malm, A. Rusu,\nand J. Åkerman, Proceedings of the IEEE 104, 1919–1945\n(2016).\n4W. Skowroński, S. Ziętek, M. Cecot, T. Stobiecki, J. 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Kubota,\nA. Fukushima, Y. Suzuki, and S. Yuasa, Applied Physics\nLetters 105, 192408 (2014).5\n26S. Kanai, M. Gajek, D. C. Worledge, F. Matsukura, and\nH. Ohno, Applied Physics Letters 105, 242409 (2014).\n27M. Frankowski, J. Ch¸ eciński, W. Skowroński, and T. Sto-\nbiecki, Journal of Magnetism and Magnetic Materials 429,11 (2017)."    },    {        "title": "1804.03553v1.Motion_of_a_superconducting_loop_in_an_inhomogeneous_magnetic_field__a_didactic_experiment.pdf",        "content": "arXiv:1804.03553v1  [physics.ed-ph]  10 Apr 2018Motion of a superconducting loop in an inhomogeneous magnet ic field:\na didactic experiment\nMarco Giliberti, Luca Perottia, and Lucio Rossi\nDipartimento di Fisica dell’Universit di Milano, via Celor ia 16, 20133 Milano Italy. and\naDepartment of Physics, Texas Southern University, Houston ,Texas 77004 USA.\n(Dated: November 9, 2018)\nWe present an experiment conductive to an understanding of b oth Faraday’s law and the prop-\nerties of the superconducting state. It consists in the anal ysis of the motion of a superconducting\nloop moving under the influence of gravity in an inhomogeneou s horizontal magnetic field. Gravity,\nconservation of magnetic flux, and friction combine to give d amped harmonic oscillations. The mea-\nsured frequency of oscillation and the damping constant as a function of the magnetic field strength\n(the only free parameter) are in good agreement with the theo retical model.\nPACS numbers: 01.50.Pa, 01.40.Gm\nI. INTRODUCTION\nHigh temperature superconductors have not only brought new ap plications and an increase of the general interest\nin superconductivity, but also the possibility to introduce with relativ e ease university students to actual experiments.\nA number of didactic experiments have been described in the literatu re [1] and laboratory kits are available for\nmany of them [2].\nThe following experiment can help students get acquainted with supe rconductivity by describing and visualizing\nin a fascinating way one of the most well known aspects of supercon ductivity: magnetic levitation induced by super-\ncurrents. While conceptually simple, the complexity of the apparatu s –even in its simpler high temperature version–\nmakes it suitable for a graduate lab; to make it more accessible, video recordings can be made that could be used for\nin class presentations.\nDue to Faraday’s law, a time dependent magnetic field induces curren ts in a conducting circuit. The same result\nis obtained when the circuit moves in an inhomogeneous magnetic field. In the case of an ohmic conductor, these\ncurrents are heavily damped by the circuit resistance R, as the damping time constant L/R – Lbeing the circuit\ninductance– is usually very small. However in the case of a supercond ucting loop the situation is different. In fact in\nthis case –as the resistance is vanishingly small– the induced current s are persistent and the magnetic flux through\nthe loop is therefore constant.\nAllowing a conducting loop to fall under the influence of gravity in a dec reasing inhomogeneous magnetic field, the\ninducedcurrentcausesaforceontheloopopposingitsdownwards acceleration. Inthecasetheloopissuperconducting,\nthe only damping is mechanical and for suitably chosen parameters s uch that the magnetic force is strong enough to\nreversethe acceleration of the loop, the loop itself hangs unsuppo rted while oscillating around its equilibrium point for\nsufficiently long times to allow detection of the oscillations. Note that t his is a very different kind of “levitation” from\nthe one seen in the didactic levitation experiments usually found in the literature [1], which is due to the Meissner\neffect [3] (expulsion of the magnetic field from the superconductor ).\nStarting from an idea of D.McAllister [4] and R. Romer [5] we performed a real experiment where a superconducting\nloop falls in a magnetic field with a sharp discontinuity.\nII. THEORETICAL MODEL\nIn our experiment the loop moved in liquid helium (LHe) and not in vacuum , as discussed by Romer [5], and thus\nhis model must be modified to take into account the dissipation due to the viscosity of the medium and the friction\nbetween the loop and parts of the apparatus. Using a high temperature conductor permits to avoid the use o f LHe; in\nthe following sections we shall indicate in italics changes in the apparatus and procedure for this simpler experiment.\nLet’s consider a rectangular loop of height hand width wthat moves in a given medium under the influence of\ngravity (directed downward) in a magnetic field ¯B(directed out of the page) as in Figure 1.\nLet’s moreover assume that:\n1. the magnetic field ¯Bis horizontal, uniform and of constant magnitude B0forz≤0 andB1< B0forz >0 (note\nthat the zaxis is pointing downward);\n2. The loop is kept vertical and its motion is along the vertical axis so t hat the magnetic force is also vertical and\nthe problem can be considered to be one dimensional;2 \nB=B1zxB=B0z\n \n☎ x \ny \nFIG. 1: The superconducting loop of height hand width l, and the reference frame of the model.\n3. for simplicity, all damping effects, even those due to friction with t he apparatus, are described as a single viscous\nforceFd=−η˙zlinear in the loop vertical velocity ˙ zwhere the dot denotes, as usual, differentiation with respect to\ntime.\nSuppose the loop is released from rest with its lower edge in z= 0 with no initial current. As it falls down, it\nprogressively leaves the region with magnetic field B0and enters the zone with field B1and a current Ibegins to\ncirculate, in order to maintain constant the initial magnetic flux. Whe n the lower edge of the loop is in z∈[0,h], the\nconcatenated outward flux of Bis given by:\nφ=B0lh+(B1−B0)lz+LI, (1)\nwhereIis the current (taken positive if flowing counterclockwise) and Lis the self-inductance of the loop. Therefore,\ndenoting by mthe mass and by Rthe resistance of the loop, from Faraday’s law we have:\nRI=−˙φ= (B0−B1)l˙z−L˙I. (2)\nNote that when the loop leaves the region z∈[0,h], the current does not depend on zany longer but becomes\nconstant: I= 0 forz <0 andI= (B0−B1)lh/Lforz > h. Care must therefore be taken to design the experiment\nto avoid these discontinuities and keep the motion of the loop within z= 0 and z=h. We now write Newton’s law\nof motion for the loop, subjected to the three forces gravitation al, magnetic, and viscous:\nm¨z=mg−η˙z−(B0−B1)lI, (3)\nwheregis the usual gravitational acceleration on the earth surface. Diffe rentiating equation (2) and using equations\n(2) and (3) to eliminate ˙ z, we obtain for Ithe well known equation of damped harmonic oscillations:\n¨I+2γ˙I+ω2I=F. (4)3\nwith:\nγ≡1\n2/parenleftbiggR\nL+η\nm/parenrightbigg\n;\nω≡/bracketleftbiggηR\nmL+(B0−B1)2l2\nmL/bracketrightbigg1/2\n;\nF≡gl(B0−B1)\nL.\nThe equation for zis in general more complicated than above, depending on a time-depe ndent driving force.\nLuckily superconductivity simplifies things: when R= 0, equation 2 reduces to the proportionality relation ˙ z=\n˙IL/[l(B0−B1)]. Since we have chosen both initial conditions z0andI0to be zero, the same proportionality relation\nholds between zandI, and the resulting equation of motion is again the equation for dampe d harmonic motion:\n¨z+2γ˙z+ω2z=g, (5)\nthe only difference with eq. 4 being in the driving term that is now just gitself. If we now suppose that the damping\nis less than critical, i.e. if γ2< ω2, equation 5 has the following solution:\nz(t) =g\nω2/bracketleftBig\n1−e−γt/parenleftBig\ncosΩt+γ\nΩsinΩt/parenrightBig/bracketrightBig\n=g\nω2/bracketleftbigg\n1−e−γt\ncosϕcos(Ωt−ϕ)/bracketrightbigg\n, (6)\nwhere\nΩ = (ω2−γ2)1/2, (7)\nϕ= arctanγ\nΩ. (8)\nIf this is this case, as stated in the introduction, the loop performs oscillations with a frequency Ω and damping time\nconstant τ=γ−1which is almost entirely due to mechanical friction (some tiny electrica l dissipation is present when\nusing a type II superconductor, such as Niobium).\nNote that the initial amplitude of the oscillations is\ng\nω2=gmL\n(B0−B1)2l2(9)\nwhich means that to keep the loop in the region of validity of eq. 6, we n eed\n(B0−B1)>1\nl/radicalbigg\ngmL\nh. (10)\nIII. EXPERIMENTAL SET UP\nThe experiment has been performed with a superconducting loop cu t out of a niobium (Nb) sheet weakly alloyed\nwith 1% titanium, immersed in a LHe bath at 4 .2K, well below the Nb critical temperature (about 9 K). In order\nto minimize the cryogenic losses of LHe, we used a cryostat where th e helium vessel was surrounded by a vessel filed\nwith liquid nitrogen (LN); both vessels were moreover double wall one s, with a vacuum chamber as insulation. A\nschematic view is shown in Figure 2.\nA HTS (High Temperature Superconductor) material could be u sed for the superconducting loop. The most suitable\ncandidate is the YBCO (Yttrium based copper oxide) coated co nductor, since it is produced in thin sheets so it is\nrelatively easier to obtain a small loop without resistive j oints. YBCO has a critical temperature of 85K, therefore\nthe experiment could be done directly in liquid nitrogen, hi ghly simplifying the cryostat and also reducing the cost of\nthe experiment. Using an YBCO loop may enable even to design t he experiment without cryogenic fluid: the loop\ncould be placed in an inner chamber without cryogen and coole d by the LN surrounding the chamber via radiation and\nconduction (the loop could be in a N or He atmosphere). As a fur ther advantage, in this configuration the friction\nwould be greatly reduced (no liquid in the experiment chambe r, only gas). The cooling of the chamber via a cryocooler,\nbeside eliminating the need of cryogenic fluid, greatly redu ce the complexity and operational risk. Use of a cryocooler\nwould facilitate the carrying out of the experiment at varia ble temperature. In this case it would also be easy to show\nwhat happens when the temperature is slowly increased above the critical temperature.4\nFIG. 2: Schematics of the bottom part of the apparatus. The do uble walls between the LN chamber and the LHe one are shown\nwhile the outer double walls are not. The horizontal red line marks the transition from the B0(above) to the B1magnetic field\n(below).\nIn the inner zone of the cryostat we placed an electromagnet, whic h generated the needed magnetic field. The\npoles of the magnet were 6 mmapart and the cross-section of the gap was 30 ×40mm2. To measure the magnetic\nfield in operation condition, a cryogenic Hall probe was placed in a samp le holder that could be moved –acting from\noutside the cryostat– along the axis of the magnet to map the field it self. The superconducting loop, cut from a Nb\nsheet, had the following dimensions: w= 25mm,h= 39mm. Its mass was m= 0.93gand its self-inductance Lwas\n6.4±0.5×10−8H; the minimum value of ( B0−B1) required to keep the loop in the operation region is therefore abou t\n5mT. Two guides kept the loop away from the edges of the magnet poles a nd limited unwanted lateral movements.\nThey were made ofG10 fiberglassand, to reduce friction, they wer ecoveredwith Molykote(bi-solphureofmolybdenus\nwith some graphite powder).\nTo reduce the boiling of LHe in the zone of the moving loop (that could d isturb the oscillations and their visibility),\nthe twocoilsofthe magnethavebeen woundusingverylowresistivity wiressoasto givethe minimum heat production\nonce the magnetic field, suitable to keep frequency of oscillations in a “reasonable” range, has been turned on. For\nB0∼15mT, the calculated evaporation, due to the heat production of the co ils, was about 3 ×10−6m3/h.\nCare has also been taken of always being below the critical magnetic fi eld (which for Nb is about 180 mTwith a\ndensity of current of 1500 A/mm2). The “reasonable” frequency of oscillation has been chosen to be about 10 Hz, so\nthat the motion could be video-recorded, both for didactic purpos es(e.g. in–class presentations) and as a set up\nand measuring tool, as we shall explain in the following. For this purpos e, a common 75 frames/s video recorder was\nused.\nThe cryostat was designed with a double optical window, through th e LN vessel and the LHe chamber, to allow a\ndirect observation of the oscillating loop in the magnet gap between p oles. The video camera could be placed in front\nof it, as a visual aid when setting up the experiment and to record th e oscillations of the loop.\nThe motion of the loop was detected measuring the voltage induced in two pick-up coils by the magnetic flux that\nwas generated by the supercurrents in the moving loop. These coils consisted of 1000 windings of 0 .2mmdiameter\ncopper wire surrounding an area of 25 .8×11.1mm2, for a total thickness of 3 .5mm. They were placed just below\nthe magnet poles, where the motion of the loop took place. The volta ge in the coils was registered by an oscilloscope\n(National VP-5730A ID4N0073B12). As the current in the pick-up coils was neglible, so were the heating around\nthem and the braking effect due to the magnetic field they generate d.5\nIV. EXPERIMENTAL PROCEDURE\nThe experiment can be divided into four time–ordered steps.\n1. The loop was placed inside the magnet gap and kept in position by a th in nylon thread fastened to the top of\nthe cryostat.\n2. The electromagnet was switched on at a given magnetic field.\n3. Vacuum was made in the vacuum chamber; the external section o f the cryostat was then filled with liquid\nnitrogen; finally, LHe was poured into the inner zone so that the tra nsition of the loop to the superconducting state\ncould take place.\nLet us notice that steps 1 and 2 had to precede step 3 because we h ad to place the loop in position, and in the\ndesired magnetic field, while it was still warm. The reason why the loop h ad to become superconducting when the\nmagnetic field was already linked to the loop itself is that inserting an alr eady superconducting loop in a field region\nwould have induced a super-current tending to expel the loop itself .\n4. The thread, used to hang the loop in the proper position, was sud denly disconnected from the top ofthe cryostat,\nso that the loop was let free to fall down. As predicted the loop star ted to oscillate before stopping at the equilibrium\nposition.\nThe use of the video camera was fundamental since it helped to cont rol the filling of the cryostatand the positioning\nof the loop during the setup, and to afterwards check the motion o f the loop itself during the experiment. Moreover,\nlooking at the motion frame after frame, we could also get some roug h measurements of the frequencies of oscillation\nof the loop. Of course, more refined measurements were made by m eans of the pick-up coils, connected to the digital\noscilloscope, as described in Section III. The plots generated by th e oscilloscope (see Figure 3) showed the typical\nbehavior of damped harmonic motions: the time between two maxima w as the period of the oscillation while the\ndamping constant could be evaluated from the decreasing amplitude of the peaks.\nUnfortunately the video camera and the pick-up coils could not be us ed at the same time, since the light source,\nessentialto illuminate the inner cryostat, causedtoomuch noisein t he signalcomingfrom the pick-upcoils. Therefore,\nfor each value of the magnetic field, the experiment had to be divided into two parts: the first one used the video\ncamera for a first analysis of the motion; the second one, without t he camera but with the pick-up coils switched on,\nmeasured the frequencies from the pick-up coils signal.\nV. DATA ANALYSIS AND RESULTS\nWenowdiscusstheresultsoftheexperiment, comparingthemeasu redoscillationfrequencieswiththeonespredicted\nby our model. To determine these latter values, the quantity B0−B1is crucial. For our purposes the magnetic field\nB0could be considered uniform inside the magnet: for a typical operat ing value B0= 30mT,B0is uniform up to 4%\nall through the magnet gap.\nThe situation was different outside the magnet poles, where the mag netic field varied with the position up to 40%.\nSince our model assumes a magnetic field of uniform strength B1outside the region between the magnet poles, a\nsuitable average had to be considered. Our choice was to perform a spatial average limited to the region accessed by\nthe loop. The vertical range of motion outside the magnet gap has b een visually assessed with the help of the video\ncamera to be 30 mm. For this and similar measurements, a graduated scale, calibrated t o match a scale placed in the\ncryostat before filling it and then removed, was attached to the ca mera monitor.\nThe oscillation frequencieswere measured for 11 values of B0−B1, varying from 14 .8mTto 44.2mT. For each\nvalue of the field, 10 to 20 tests were conducted. Measurements t aken in the range 5 .0mT < B 0−B1<14.0mT\nwere dropped for two reasons; the first one being that B0−B1was too close to the 5 .0mTlower bound to keep the\nloop in the operating region. The second reason is that the records taken by the video-camera show that at such low\nvalues of B0−B1the loop often went partially off the guides and oscillated on a non-ver tical plane, contrary to the\nassumptions of the theoretical model we used.\nTwo analysis methods were used to get the oscillation frequency, bo th of them from the pick-up coils voltage\nmeasurements: spectral analysis, using the Microsoft Excel c/circlecopyrtFourier analysis instrument (Figure 4 shows a typical\nexample of the spectra we obtained) and the direct measurement o n the plot of the oscillation period (“plot method”).\nThe “plot method” has been mainly used as control of the numerical procedure.\nThe calculated spectra showed various peaks of different amplitude but only the first two in each plot were clearly\nstronger than the others: see Figure 4. We consistently chose th e frequency of the lowest frequency peak as the\noscillation frequency of the loop, even in the two cases ( B0−B1= 19.7mTandB0−B1= 26.0mT) when the two\nstrongest peaks had nearly the same amplitude. The second peak d oes not usually appear to be harmonic of the first\none; we therefore surmise it might be due to the non-uniformity of t heB1field.6\n\u0001\u0002\u0003\u0001\u0004\u0003\u0001\u0005\u0003\u0001\u0006\u0003\u0003\u0006\u0003\u0005\u0003\u0004\u0003\u0002\u0003\n\u0003 \u0003\u0007\b \u0003\u0007\u0006 \u0003\u0007\t \u0003\u0007\u0005 \u0003\u0007\n\u0001\u0002\u0003\u0004\u0005\u0006\u0005\u0007\u0005\b\t\n\u000b\u0004\f\u000b\u0002\r\u000e\u0005\u0006\u0003\u000f\b\nFIG. 3: A plot of oscillation as detected by the pick up coils a tB0−B1= 35.84±0.75mT.\nThe experimental damping time–constant γwas again obtained in two ways: through the “plot method”, i.e.\nmeasuring, from the plots of the pick-up coils voltage, the time it tak es for the oscillations to decay to half their initial\namplitude, and –when the shape of the main spectral peak was regu lar enough– through a measure of its half width.\nAs in the previous case, the second method was used for control o f the procedure.\nOur results are summarized in Table I and Figure 5, where we compare our experimental frequencies to the\n“theoretical”frequencies, evaluatedfromequation7, with B1calculatedasexplainedaboveandusingtheexperimental\nvalues of γ.The listed error for the “theoretical” frequencies is due to the unc ertainties in these two quantities. In\nparticular, for low magnetic fields it is mainly due to the error in the mea surements of γ,while for large magnetic\nfields it is mainly due to the uncertainty in the value of B1.\nThe agreement between the two sets of data in Figure 5 is almost eve rywhere good; the least square fitting for the\nexperimental data gives a slope of 0 .485±0.041Hz/mT (R2= 0.934) in good agreement with that obtained from the\nfitting of the “theoretical” data: 0 .00499±0.0075Hz/mT (R2= 0.998).\nVI. CONCLUSIONS\nDue to easy to perform experiments and the availability of cheap labo ratory kits, superconductivity has recently\nbecome part of undergraduate lab courses and courses for futu re physics teachers. Here, we have described a didactic\nexperiment that has the advantage of combining superconductivit y with another common lab topic –damped har-7\n\u0001 \u0002\u0001 \u0003\u0001\u0001 \u0003\u0002\u0001 \u0004\u0001\u0001 \u0004\u0002\u0001\n\u0001\u0002\u0003\u0004\u0005\u0003\u0006\u0007\b\t\n\u000b\f\r\u000e\u000f\u0010\u0011\u0012\u0013\u0005\u0014\u0003\nFIG. 4: Oscillation spectrum at B0−B1= 35.84±0.75mT.\nmonic oscillations– thus giving students the opportunity to deepen t heir understanding of both. Besides giving the\nstudents the chance to tackle many different physics topics such a s magnetism, superconductivity, electric currents\nand oscillations, the experiment we propose involves different detec ting devices and data analysis techniques that\ncould easily be presented to students in an almost peer to peer instr uction mode.\nThe experimental data we collected show that the theoretical mod el we adopted is sufficiently good and that it is\ntherefore suitable for a discussion with college students on superc onductivity, Faraday’s law and oscillatory motion\nbased on it and on our experimental set-up. Clearly we could try to im prove our model and the fitting of the data\nbut, for that purpose, more refined mathematics should be used a nd it is our opinion that it could obscure more than\nlighten the physics involved.\nFollowing the present technical analysis, we are planning to soon rep ort on the in-class experimentation that we\nare organizing and that will help us decide whether our experiment ca n help students to get closer to the fascinating\nphenomena of superconductivity through an “unusual” application .\nVII. ACKNOWLEDGMENTS\nWe are very grateful to Mr. Castellazzi and Mr. Ormenese of Video time s.p.a. for their precious help and also to\nthe technicians of L.A.S.A. laboratory for their work and kind sugges tions. We also thank Mr. Giuseppe Baccaglioni\nof L.A.S.A.for having built the pick-up coils and measured the self induct ance of the loop.\n[1] Several didactic experiments are listed in section VIII -B of N.P. Butch, M.C. de Andrade and M.B. Maple, “Resource\nLetter Scy-3: Superconductivity”; Am. J. Phys. 76, 106 (200 8); http://dx.doi.org/10.1119/1.2802574. More recent pa pers8\nMagnet B0−B1Theoretical Spectrum analysis\ncurrent [ A] [±0.75mT] model First peak\nν[Hz]σν[Hz]ν[Hz]σν[Hz]\n0.9 14.84 7.4 0.7 7.0 0.9\n1.0 16.24 8.1 0.8 7.8 2.0\n1.25 19.74 9.8 0.9 12.5 0.9\n1.5 23.24 12 1.0 10.9 0.9\n1.7 26.04 13 1.0 13.7 1.0\n2.0 30.24 15 1.1 15.6 1.0\n2.2 33.04 16 1.2 17.6 1.0\n2.4 35.84 18 1.3 17.6 1.0\n2.5 37.24 19 1.3 18.6 1.0\n2.7 40.04 20 1.4 18.6 1.0\n3.0 44.24 22 1.5 23.4 1.0\nTABLE I: Summary of the experimental results.\n/s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53/s53/s49/s48/s49/s53/s50/s48/s50/s53/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s91/s72/s122/s93\n/s40/s66\n/s48/s32/s45/s32/s66\n/s49/s41/s32/s91/s109/s84/s93\nFIG. 5: Plot of the loop oscillation frequency vs. magnetic fi eld difference ( B0−B1). The black triangles represent direct\nexperimental data, while the red dots represent the “theore tical results”. The two lines represent the least square fit o f each\ndata set.9\nare e.g.: C.P. Strehlow and M.C. Sullivan, “A classroom demo nstration of levitation and suspension of a superconductor\nover a magnetic track”; Am. J. Phys. 77, 847 (2009); http://d x.doi.org/10.1119/1.3095809, M.R. Osorio, D.E. Lahera\nand H. Suderow, “Magnetic levitation on a type-I supercondu ctor as a practical demonstration experiment for students” ;\nEuropean Journal of Physics, Volume 33, Number 5, p 1383 (201 2), A. Bonanno, G. Bozzo, M. Camarca and P. Sapia,\n“An innovative experiment on superconductivity, based on v ideo analysis and non-expensive data acquisition”; Europe an\nJournal of Physics, Volume 36, Number 4, 045010 (2015).\n[2] Several links to providers of kits are available at the “p lay” page of superconductors.org:\nhttp://superconductors.org/play.htm.\n[3] W.Meissner and R. Ochsenfeld, “Ein neuer Effekt bei Eintr itt der Supraleitfhigkeit”. Naturwissenschaften 21 (44) 7 87788\n(1933). doi:10.1007/BF01504252.\n[4] D. McAllister, “Magnetic levitation”; Eur. J. Phys., 9, 232-233, 1988.\n[5] R. Romer, “The motion of a superconducting loop in an inho mogeneous magnetic field: the harmonic oscillator equation\nin an unfamiliar setting”; Eur. J. Phys., 11, 103-106, 1990."    },    {        "title": "1005.2573v1.The_effect_of_spin_magnetization_in_the_damping_of_electron_plasma_oscillations.pdf",        "content": "arXiv:1005.2573v1  [physics.plasm-ph]  14 May 2010The effect of spin magnetization in the damping of electron pl asma oscillations\nPablo S. Moya1,∗and Felipe A. Asenjo1,2,†\n1Departamento de F´ ısica, Facultad de Ciencias, Universida d de Chile, Casilla 653, Santiago, Chile.\n2Departamento de Ciencias, Facultad de Artes Liberales,\nUniversidad Adolfo Ib´ a˜ nez, Diagonal Las Torres 2640, Pe˜ nalol´ en, Santiago, Chile.\n(Dated: November 9, 2018)\nThe effect of spin of particles in the propagation of plasma wa ves is studied using a semi-classical\nkinetic theory for a magnetized plasma. We focus in the simpl e damping effects for the electrostatic\nwave modes besides Landau damping. Without taking into acco unt more quantum effects than spin\ncontribution to Vlasov’s equation, we show that spin produc es a new damping or instability which\nis proportional to the zeroth order magnetization of the sys tem. This correction depends on the\nelectromagnetic part of the wave which is coupled with the sp in vector.\nPACS numbers: 03.65.Sq, 52.25.Dg, 52.25.Xz, 52.27.Gr, 52. 35.Fp\nKeywords: Spin plasma dynamics, Landau damping, spin magne tization\nOneofthe mostimportantphysicalresultsin theprop-\nagation of plasma waves which cannot be deduced by a\nfluid description is Landau damping [1]. This effect pre-\ndicts that an electron plasma wave, in an collisionless\nplasma, suffers damping owingto the wave-particleinter-\naction. Damping mechanism will depend on the velocity\ndistribution of the particles which are often Maxwellian.\nThe mathematicalproceduresto obtainLandaudamping\nare very standard in kinetic theory. Thus, the formalism\nhas been extended for other kind of electron interactions,\nfor example thermal ion Landau damping effects [2], and\nLandau damped electron waves by photons [3] or neutri-\nnos [4].\nOn the other hand, recently there have been a huge\ninterest in the field of plasma physics for the dynamics of\nthe spin of electrons, and how its quantum nature affects\nthe different known modes of propagations [5–7] or other\nproperties [8, 9]. These treatments are useful in, for ex-\nample, astrophysical systems [10] or high-energy lasers\n[11]. Often these effects are important in high density,\nlowtemperatureorstrongmagneticfieldsconditions, but\nit has been shown that for some systems at high temper-\naure the spin dynamics play a crucial role [12].\nIn a previous work [13], we present a first approach to\ncalculate a correctionto Landau damping due to spin. In\nthis letter, we complete the analysis done in the previous\nwork finding a full solution for the damping produced by\nthe spin to electrostatic modes. We will show that this\nnewdamping isproportionalto the spinmagnetizationof\nthe plasma, and it depends on the electromagnetic part\nof the wave.\nTo obtain the correction to the Landau damping pro-\nduced by the spin of the plasma constituents, we use\nthe semi-classical kinetic theory constructed in Ref. [14]\nwhich start from the Pauli Hamiltonian. Here, the dy-\nnamics of the spin of particles is included in a Vlasov\n∗Electronic address: pmoya@levlan.ciencias.uchile.cl\n†Electronic address: fasenjo@levlan.ciencias.uchile.clequation for a generalized distribution function. Other\nquantum corrections, as Bohm potential, spin-spin inter-\naction force and high order force terms in the spin evolu-\ntion equation are neglected. Thus, Vlasov equation, for\nparticles with velocity v, spin vector sand with distri-\nbution function f=f(r,v,s,t) is\n∂f\n∂t+v·∇f+/bracketleftbiggq\nm(E+v×B)+2µe\nm¯h∇(s·B)/bracketrightbigg\n·∇vf\n+2µe\n¯h(s×B)·∇sf= 0,(1)\nwhereq=−eandmare the electron charge and mass\nrespectively, µe=−ge¯h/(4m) is the electron magnetic\nmomentand g≈2.002319istheelectronspinfactor. The\nFermi-Dirac equilibrium distribution function f0(v,s) is\n[14]\nf0(v,s) =B0µe˜f0(v)\n4πkBTsinh(B0µe/kBT)exp/parenleftbigg2µes·B0\n¯hkBT/parenrightbigg\n,\n(2)\nwhereTis the temperature, kBis the Boltzmann con-\nstant,B0is a backgroundmagnetic field ( B0=|B0|) and\n˜f0is the classical Maxwellian distribution function\n˜f0(v) =/parenleftbiggm\n2πkBT/parenrightbigg3/2\nexp/parenleftbigg\n−mv2\n2kBT/parenrightbigg\n,(3)\nwithv=|v|. The distribution function (2) is normalized\nas/integraltext\nf0dvds= 1, where the integration is made over the\nthree degree of freedom in velocity space and the two\ndegree of freedom in spin space.\nNow, we use the kinetic formalism (1) and a simi-\nlar analysis of Ref. [13] to derive the dispersion relation\nfor electron plasma oscillations in a magnetized plasma\nwhich interacts with the spin of the particles. The elec-\ntric and magnetic fields will be perturbed in the form\nE=E1andB=B0+B1respectively. The terms\nwith subscript 0 are the zeroth order equilibrium quan-\ntities, and the terms with subscript 1 are the first order\nperturbed quantitites. The distribution function is per-\nturbed as f(r,v,s,t) =f0(v,s) +ˆf1(r,v,s,t) and we2\nchooseB0=B0ˆz. The perturbed distribution ˆf1will\nhave the form ˆf1(r,v,s,t) =f1(v,s)exp(ik·r−iωt),\nwith similar assumption for other perturbed quantities.\nHere,kandωare the wavenumber and the frequency of\nthe wave.\nLinearizing Eq. (1), with the velocity vand spin sas\nindependet variables and following Ref. [13], we can find\nthe perturbed distribution function as\nf1=−i\nω−k·v/parenleftBigq\nmE1·∇vf0\n+2µe\nm¯h∇(s·B1)·∇vf0+2µe\n¯h(s×B1)·∇sf0/parenrightbigg\n,\n(4)\nbecausev×B1·∇vf0= 0.\nFrom here, we are going to focus in the study of\nspin correction to the Landau damping for electrostatic\nmodes. Let us concentrate the charge density ρof theplasma which is given by ρ=qn0/integraltext\nf1dvds, where n0\nis the equilibrium density. Using the Maxwell equa-\ntionk×E1=ωB1, the perturbed distribution func-\ntion (4), and defining the quantities E⊥=k×E1/kand\nE/bardbl=k·E1/kwithk=|k|, the charge density becomes\nin\nρ=−iqn0/integraldisplay\nds/integraldisplay∞\n−∞dv\nω−k·v/parenleftBigq\nmkE/bardbl(k·∇vf0)\n+i2µek\nm¯hω(s·E⊥)k·∇vf0+2µek\n¯hω(s×E⊥)·∇sf0/parenrightbigg\n.\n(5)\nThischargedensitymustbeusedinthePoisson’sequa-\ntionikE/bardbl= 4πρto obtain the dispersion relation for\nelectrostatic modes. In this way, the dispersion relation\nis\n1 =ω2\np\nk2/integraldisplay\nds/integraldisplay∞\n−∞dv\nk·v−ω/parenleftbigg\n1+2iµek2\nq¯hω/parenleftbiggs·E⊥\nE/bardbl/parenrightbigg/parenrightbigg\nk·∇vf0+ω2\np\nk2/integraldisplay\nds/integraldisplay∞\n−∞dv\nk·v−ω/parenleftbigg2µek2m\nq¯hω/parenrightbigg/parenleftbiggs×E⊥\nE/bardbl/parenrightbigg\n·∇sf0,\n(6)\nwhereω2\np= 4πe2n0/mis the square of the plasma fre-\nquency. When the spin contribution is neglected ( µe=\n0), we reobtain the classical dispersion relation for elec-\ntrostatic modes.\nTo solve the dispersion relation (6), we need to evalu-\nate the two integral involving the spin contribution. The\nintegration in the two degree of freedom in spin space\nis done in spherical coordinates such that ds≡dΩs=\nd(cosθs)dφswhere the subindex sis for spin coordi-\nnates. In the same sense, the spin vector will be s=\n−¯h/2ˆs=−¯h/2(sinθscosφsˆx+sinθssinφsˆy+cosθsˆz),and∇sf0= 2µeB0sinθsf0/(¯hkBT)ˆθ. The choice on the\nspin orientation is to minimize the magnetic moment\nenergy, which is consistent with paramagnetism [15].\nOn the other hand, the above integrations in velocity\nand spin space can be simplified introducing the one-\ndimensional distribution\nF0(u,s)≡/integraldisplay\nf0δ/parenleftbigg\nu−k·v\nk/parenrightbigg\ndv. (7)\nWe can use (7) to rewrite the dispersion relation (6) as\n1 =ω2\np\nk2/integraldisplay\ndΩs/integraldisplay∞\n−∞du\nu−ω/k/parenleftbigg\n1−iµek2\nqω/parenleftbiggˆs·E⊥\nE/bardbl/parenrightbigg/parenrightbigg∂F0\n∂u−2ω2\npµ2\nemB0\nq¯hωkBT/integraldisplay\ndΩs/integraldisplay∞\n−∞dusinθsF0\nu−ω/k(ˆs×E⊥)·ˆθ\nE/bardbl.(8)\nThe integralsin Eq. (8) must be evaluated as a contour\nintegral considering the singularity at uφ≡ω/k. This is\nthe origin of classical Landau damping. We consider the\ncase of large phase velocity uφand weak damping, where\nthe pole lies near the real uaxis. In this case, F0and\n∂F0/∂uare both small near uφ. Neglecting the thermal\ncorrection to the real part of the frequency, the first twointegrals are given by [16]\n/integraldisplay\ndΩs/integraldisplay∞\n−∞du\nu−ω/k∂F0\n∂u≃k2\nω2+iπ∂˜F0\n∂u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nu=uφ,(9)3\n/integraldisplay\ndΩs/integraldisplay∞\n−∞du\nu−ω/kˆs·E⊥\nE/bardbl∂F0\n∂u≃\n−χη(α)\nk2\nω2+iπ∂˜F0\n∂u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nu=uφ\n,(10)\nwhereχ= ˆz·E⊥/E/bardbl,α=µeB0/kBTandη(x) =\ncoth(x)−1/xis the Langevin function. ˜F0comes from\n(7) and it is defined as\n˜F0(u)≡/integraldisplay\n˜f0δ/parenleftbigg\nu−k·v\nk/parenrightbigg\ndv. (11)\nThe third integral in dispersion relation (8) vanish due\nthe only relevant spin contribution is anti parallel to the\nbackground magnetic field. Then, the dispersion relation\n(8) becomes\n1 =\nω2\np\nω2+iπω2\np\nk2∂˜F0\n∂u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nuφ\n/parenleftbigg\n1+iµek2\nqωχη(α)/parenrightbigg\n.(12)\nWe seek afrequency which hasa real and an imaginary\npart given by ω=ωr+iωisuch that ωi≪ωr. Using this\nin Eq.(12), and solving for the real and imaginary parts,\nwe can obtain the frequency for the electrostatic modes\nω=ωr\n1+iπω2\nr\n2k2∂˜F0\n∂u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nuφ\n+ik2M0χ\n2n0q,(13)\nwhere, neglecting terms of order ( ∂˜F/∂u)|2\nuφ, the real\npart of the frequency is given by\nωr=ωp\n1−M0χπωp\n2qn0∂˜F0\n∂u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nuφ\n,(14)\nandM0=n0µeη(α) is value of the spin magnetization\nof the system. This is because the distribution f0of\nEq. (2), gives the zeroth order magnetization of the sys-\ntemM0= (2µen0/¯h)/integraltext\nsf0dvds=M0ˆz[14, 17]. Thus,\nusing Eqs. (13) and (14) the imaginary part of the fre-\nquency is\nωi=πω3\np\n2k2∂˜F0\n∂u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nuφ+k2M0χ\n2n0q, (15)\nFrom (15) we note that there is a correction in the\nimaginary part of the frequency which appears due to\nthe magnetization of the plasma due to spin of their con-\nstituents. This correction depends on the electromag-\nnetic part of the waves, which is coupled with spin vector\nof each particle. As we discuss in our previous work [13],\nthe interaction of spin with the perturbed magnetic field\nin a magnetized plasma is the responsible of this contri-\nbution to the damping of an electrostatic mode. More-\nover, in the present analysis we show that the energytrasferbetween wavesand particlesdepends onthe shape\nof the equilibrium function (classical Landau damping)\nand also on the spin magnetization of the plasma.\nThe spin damping correction of Eq. (15) depends on\nthe ratio of the magnetization and the charge of each\nparticle, i.e, M0/n0q= ¯hgη(α)/4m. It is expected that\nspin effects will be important at low temperatures, high\ndensities and huge magnetic fields, and due to the de-\npendences of ηon theαparameter we can see that, in\nfact, the spin correction to the Landau damping is higher\nfor high values of density and background magnetic field,\nand for low temperatures. However, this spin damping\nis proportional to ¯ hand the main effect is the classical\nLandau damping.\nBesides, we can see that the correction to the damping\nis proportional to χ, which depends on the transversal\npart of the electromagnetic wave. If χ >0 in the case\nof an electron plasma, η(α)<0 and the correction is a\ndamping. Also, forthesameelectronplasma,if χ <0the\ncorrection is an instability. When the wave has no elec-\ntromagnetic transverse component |E⊥|= 0, then χ= 0\nand there are no spin correction to the damping. The\nexact value of this coefficient should be obtained solving\nthe dynamical Maxwell equations with the current den-\nsityj=qn0/integraltext\nvf1dvds. The complete implications of the\nvalue ofχis being studied.\nOn the other hand, from Eq. (14) we note that there is\na correction in the frequency of plasma oscillations. This\ncorrections depends on the magnetization of the plasma\nand also on the shape of the distribution function. As\nspin corrections are proportional to ¯ hand the derivative\nof the distribution function is small, as in the case of the\nimaginary part, the correction is small and the frequency\nof electron waves is near ωpas expected.\nIn conclusion, we have shown that the incorporationof\nspin to kinetic theory in a magnetized plasma produces\ncorrections to the classical Landau damping for electro-\nstatic waves that depends on the magnetization of the\nsystem, and is due to the coupling between the spin vec-\ntor and the electromagnetic part of the wave. In other\nwords, in addition to the Landau mechanism to trans-\nfer energy from waves to particles, the inclusion of spin\nallows the energy transfer through the quantum inter-\naction between spin and magnetic fields. However, the\ncorrections are of ¯ horder and, in the case of electron\nplasma and mawellian distribution functions, the electro-\nstatic wave will show an evolution similiar to its classical\ndynamics when the spin is not included. In addition we\nshown that spin contribution also introduces a correction\nto the frequency of plasma oscillations which is also pro-\nportional to ¯ hand it is due to the magnetization of the\nplasma.\nAll of these results show the importance of the ki-\nnetic theory of plasmas. The same formalism can be\nused to introduce other quantum contributions to classi-\ncal plasma physics and derive new correctionsand effects\nfor strongly coupled plasmas, as well as plasmas in pres-\nence of large magnetic fields when quantum effects are4\nrelevant.\nPablo S. Moya is grateful to CONICyT D-21070397Doctoral Fellowship.\n[1] L. D. Landau, J. Phys. U.S.S.R. 10, 25 (1946).\n[2] R. J. Goldston, and P. H. Rutherford, Introduction to\nPlasma Physics , Institute of Physics Publishing, Bristol\nand Philadelphia (1995).\n[3] R. Bingham, J. T. Mendon¸ ca, and J. M. Dawson, Phys.\nRev. Lett. 78, 247 (1997).\n[4] L. O. Silva, R. Bingham, J. M. Dawson, J. T. Mendon¸ ca,\nand P. K. Shukla, Phys. Lett. A 270, 265 (2000).\n[5] P. K. Shukla, and L. Stenflo, Phys. Lett. A 357, 229\n(2006).\n[6] A. P. Misra, Phys. Plasmas 14, 064501 (2007).\n[7] P. K. Shukla, Phys. Lett. A 369, 312 (2007).\n[8] A. P. Misra, and N. K. Gosh, Phys. Lett. A 372, 6412\n(2008).\n[9] M. Marklund, B. Eliasson, and P. K. Shukla, Phys. Rev.\nE76, 067401 (2007).[10] M. G. Baring, and A. K. Harding, Astrophys. J. 547, 929\n(2001).\n[11] D. Kremp, Th. Bornath, M. Bonitz, and M. Schlanges,\nPhys. Rev. E 60, 4725 (1999).\n[12] G. Brodin, M. Marklund, and G. Manfredi, Phys. Rev.\nLett.100, 175001 (2008).\n[13] F. A. Asenjo, Phys. Lett. A 373, 48 (2009).\n[14] G. Brodin, M. Marklund, J. Zamanian, ˚A. Ericsson, and\nP. L. Mana, Phys. Rev. Lett. 101, 245002 (2008).\n[15] G. Brodin and M. Marklund, New J. Phys. 9, 277 (2007).\n[16] F. F. Chen, Introduction to Plasma Physics and Con-\ntrolled Fusion , Plenum Press, New York (1984).\n[17] M. Marklund, and G. Brodin, Phys. Rev. Lett. 98,\n025001 (2007)."    },    {        "title": "1003.5344v1.Giant_magnetic_broadening_of_ferromagnetic_resonance_in_a_GMR_Co_Ag_Co_Gd_quadlayer.pdf",        "content": "arXiv:1003.5344v1  [cond-mat.mtrl-sci]  28 Mar 2010Giant magnetic broadening of ferromagnetic resonance in a G MR Co/Ag/Co/Gd\nquadlayer.\nS. Demirtas and M. B. Salamon\nUniversity of Texas at Dallas\nA. R. Koymen\nUniversity of Texas at Arlington\n(Dated: December 14, 2018)\nBoth magnetic-resonance damping and the giant magnetoresi stance effect have been predicted\nto be strongly affected by the local density of states in thin f erromagnetic films. We employ the\nantiferromagnetic coupling between Co and Gd to provide a sp ontaneous change from parallel to an-\ntiparallel alignment of two Co films. A sharp increase in magn etic damping accompanies the change\nfrom parallel to antiparallel alignment, analogous to resi stivity changes in giant magnetoresistance.\nThe discovery of giant magnetoresistance (GMR) by\nBaibichetal.[1]hasledtoimportantapplicationsinmag-\nnetic recording and data storage. Nonetheless, a fun-\ndamental understanding of the microscopic mechanism\nremains a subject of continuing research.[2, 3] Early\nwork[4, 5] considered spin-dependent scattering to be the\nprimary mechanism for GMR effects, and indeed such\nscattering can considerably enhance them.[6] However,\nSchep, et al.[7] were the first to demonstrate that sig-\nnificant GMR (for currents perpendicular to the mag-\nnetic layers (CPP) at least) is possible in a perfect mag-\nnetic superlattice, a consequence ofs-d hybridization and\nresultant differential localization of electronic states be-\ntweeenparallel(P) and antiparallel(AP) alignment. The\nsame quantum-well states strongly modify the effective-\nness of scatterersat the interface[3], thereby contributing\nto GMR for in-plane currents (CIP) as well. The aim\nof this paper is to provide independent evidence for sub-\nstantial changes in the local density of states accompa-\nnying a transition from P to AP alignment. Exploiting\nthe strong antiferromagnetic coupling between Co and\nGd, we fabricated a GMR structure that spontaneously\nreverses the relative orientation of two Co layers as the\ntemperature is reduced. Upon reversal from P to AP\nalignment, the width of the ferromagnetic resonance line\nof the free Co layer sharply changes its temperature de-\npendence. We interpret these results in the context of\nthe so-called torque-correlation model of ferromagnetic\ndamping, [8–10] applicable to Co, in which the linewidth\nis directly related to the local density of states; by anal-\nogy, we term the increased broadening Giant Magneto-\nBroadening (GMB).\nWe have prepared a trilayer structure of Co/Ag/Co\nwith an underlying Gd layer; the Ag layer is suffi-\nciently thick that there is no exchange coupling of the\ntwo Co layers. Co and Gd are strongly coupled\nantiferromagnetically.[11] Above, and somewhat below,\nthe Curie temperature of Gd, the two Co layers are fer-\nromagnetically aligned in a modest magnetic field. As\nthe temperature is reduced, the magnetic moment of Gd/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s48/s48/s48/s48/s46/s48/s48/s48/s48/s50/s48/s46/s48/s48/s48/s48/s52/s48/s46/s48/s48/s48/s48/s54/s48/s46/s48/s48/s48/s48/s56/s48/s46/s48/s48/s48/s49/s48/s77/s97/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s32/s91/s101/s109/s117/s93\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s91/s75/s93/s72/s61/s49/s48/s48/s32/s79/s101\nFIG. 1: Magnetic moment as a function of temperature for\nthe [Co 40 ˚A/Gd 100 ˚A] bilayer. Minimum corresponds to\nTcomp.\nincreases. Below the compensation temperature Tcomp,\nthe Gd moment exceeds that of its adjacent Co layer,\ncausing it to align with the magnetic field, producing AP\nalignment of the two Co layers. In Fig. 1, we show the\nlow-field magnetization of a Ag(10 nm)/Co(4nm) bilayer\non Gd(10 nm). The minimum in net magnetization at\nTcomp= 170 K reflects the point at which the magneti-\nzation of the underlying Gd and its adjacent Co layer are\nequaland opposite, orientedperpendicularto the applied\nfield. The small paramagnetic moment at Tcompresults\nfrom the canting of the opposing moments toward the\napplied field direction.\nMultilayer samples were created at room temperature\nusing a dc magnetron sputtering system. The base pres-\nsure of the deposition chamber was 10−9Torr. Ultra\nhigh purity argon gas was used and the deposition pres-2\nsure was 3 mTorr. An in situquartz thickness monitor,\ncalibrated by a stylus profilometer, measures the deposi-\ntion thicknesses. Samples were sputtered from pure Gd,\nCo and Ag targets on Si (100) substrates. Ag layers 200\nAngstrom ( ˚A) thick were used as buffer layers in all sam-\nples. The Co(1)/Ag/Co(2)/Gd multilayer was created\nwith a 100 ˚A nonmagnetic Ag spacer between the two 4\nnm-Co layers, thick enough to suppress any long range\nexchange interactions. A 100- ˚A Ag cap layer completed\nthe deposition. The Curie temperature TCof the Gd\nthin film is 240 K, somewhat below the bulk value.\nThe absorption spectrum as a function of applied\nmagnetic field for the Co(1)/Ag/Co(2)/Gd multilayer is\nshown in Fig. 2 at room temperature. The microwave\nfrequency is 10 GHz and the applied field is in the plane\nof the sample. Two Lorentzian derivative fits are also\nshown in Fig. 2 to identify two different resonances. Sep-\narationofthe adjacent absorptionpeaks can be made be-\ncause, as shown previously,[12] a proximate layer of Gd\nreduces the field for resonance and significantly increases\nthe linewidth of Co thin films. This leads to the conclu-\nsion that the broader reasonance is associated with the\nCo(2) layer. Fig 3 shows the temperature dependence\nof the linewidth associated with Co(1) and Co(2) reso-\nnances. Above the Curie temperature of Gd ( TC= 240\nK), both resonancelines broadenslightly with decreasing\ntemperature. Below TCthe Co(2) resonanceis no longer\nseen while the Co(1) resonance first broadens abruptly\nand then continues to increase with decreasing tempera-\nture to the compensation point, Tcomp= 170 K. Below\nTcomp, the linewidth increases much more strongly with\ndecreasing temperature, exceeding the resonant field be-\nlow 100 K.\nFerromagnetic resonance is generally treated phe-\nnomenologically via the Landau-Lifshitz-Gilbert (LLG)\nequation of motion,[13]\nd− →m\ndt=−γ− →m×− →H+α− →m×d− →m\ndt. (1)\nwhere− →mis the reduced magnetization vector, γ,the gy-\nromagnetic ratio and α,the Gilbert damping parameter.\nRelaxation in metallic ferromagnet films has convention-\nally been attributed to the transfer of angular momen-\ntum from the precessing magnetization to the spin of\nthe conduction electrons via s-dexchange and the subse-\nquent relaxation of the conduction electron polarization\nvia spin-dependent scattering.[14] More recently, atten-\ntion has been focused on the so-called torque-correlation\nmodel first introduced by Kambersky.[15] In this pro-\ncess, the time-dependent magnetization induces charge-\ncurrents in the conduction electrons via the spin-orbit\ninteraction. These, in turn, exert torque on the mag-\nnetization, transferring angular momentum to the lattice\nvia the relaxation of charge currents. The longer the re-\nlaxation time τof these currents, the greater the torque\nand the broader the line. For intraband transitions,/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48/s45/s49/s50/s48/s48/s48/s45/s54/s48/s48/s48/s48/s54/s48/s48/s48/s49/s50/s48/s48/s48\n/s32/s32/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s91/s97/s46/s117/s46/s93\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s91/s71/s93/s51/s48/s48/s32/s75\nFIG. 2: FMR absorption spectra for the [Co 40 ˚A/Ag 100 ˚A\n/Co 40˚A/Gd 100 ˚A] film at room temperature. Linewidths\nwere found making two Lorentzian fits to the overall absorp-\ntion spectra\nGilmore, et al.[10] have shown that\nα(T) =γτ(T)\n2µ0m/summationdisplay\nnk|Γn(k)|2/parenleftbigg\n−∂f\n∂ε/parenrightbigg\n,(2)\nwhereτis the orbital relaxation time of the conduc-\ntion electron, Γ n(k) is the torque matrix element from\nthe spin-orbit interaction, and ( −∂f/∂ε) is the negative\nderivative of the Fermi function. The sum is over band\nindices. The interplay between the two mechanisms has\nbeen discussed by several authors.[9, 16] By artificially\nchanging the Fermi energy in their band calculations,\nGilmore et al. demonstrate specfically that the summa-\ntion in Eq. (2) follows the density of states for Co and\nother ferromagnetic metals. Note that the linewidth is\nrelated to the Gilbert parameter by ∆ H= 1.16ωα/γ,\nwhereω/2π= 10 GHz is the applied microwave fre-\nquency.\nAs seen in Fig. 3, the Co(1) linewidth gradually in-\ncreases with decreasing temperature from TCtoTcomp\nand then increases more rapidly below; this is the GMB\neffect. A linewidth that increases with decreasing tem-\nperature is indicative [9] that the torque-correlation pro-\ncess dominates over spin damping, evidently becoming\neven more dominant below Tcomp. In the absence of\ntorque-correlation processes, spin-damping, which varies\nτ(T)−1, would require a mechanism that, upon rever-\nsal of the Co(2) magnetization, increases with decreas-\ning temperature at a rate that overcomes the increase\ninτ(T). The band structure of the Co(1) layer, on the\nother hand, will change dramatically upon the transition\nfrom P to AP alignment.[7], thereby changing the den-3\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48/s52/s53/s48\n/s67/s111/s40/s49/s41/s67/s111/s40/s50/s41/s47/s71/s100\n/s84\n/s99/s111/s109/s112/s67/s111/s40/s49/s41/s47/s65/s103/s47/s67/s111/s40/s50/s41/s47/s71/s100\n/s67/s111/s40/s49/s41/s32/s105/s115/s32/s97/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108/s32/s116/s111/s32/s67/s111/s40/s50/s41\n/s67/s111/s40/s49/s41/s32/s105/s115/s32/s112/s97/s114/s97/s108/s108/s101/s108/s32/s116/s111/s32/s67/s111/s40/s50/s41\n/s32/s32/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s91/s79/s101/s93\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s91/s75/s93\nFIG. 3: FMR linewidth as a function of temperature for par-\nallel and antiparallel alignment of Co layers in [Co 40 ˚A/Ag\n100˚A /Co 40 ˚A/Gd 100 ˚A] film.\nsity of states in the Co(1) layer. Further, Binder et al.[3]\nshowedthat impuritieslocated within a Colayerin GMR\nstructures exhibit dramatically larger relaxation rates in\nAP vs P alignment, again reflecting an increase in the lo-\ncal density of states. Impurities located at the interface\nbetween Co and Cu, in their calculation, are seven times\nmore effective as scatterers in the AP configuration; the\neffect is even larger for impurities in the center of the Co\nlayer. Similarly, the torque matrix element Γ n(k), which\ntracks with the density of states, [10] should reflect the\nsame increasein localdensity ofstates in the AP configu-\nration. We attribute the seven-fold increase in the slope\nof ∆H(T) shown in Fig. 3, therefore, to an increase in\nthe summation in Eq.(2) and consequently, to a stronger\ndependence on τ(T). Further, Steiauf and F¨ ahnle[17]\nhave shown, in the context of the torque-correlation ap-\nproach, that band-structure effects in lower-dimensional\nstructures dramatically increase the Gilbert parameter\nof Co relative to the bulk metal. We suggest that, in\nthe single layer considered by Steiauf and F¨ ahnle, both\nspin sub-bands are localized, much as in the case of AP\nalignment, while only one sub-band is localized in the P\nconfiguration. We conclude that the largeenchancement\nof the temperature dependence of the linewidth in our\nGMR structure–the GMB effect–confirms both the dom-\ninance of the torque-correlation process in spin damping\nand the importance of electron localization in the GMR\neffect.\nThere have been, of course, many studies of magnetic\nrelaxation in thin metallic films and multilayers. For ex-\nample, an experiment by Urban, et al.[18] found that\nthe relaxation rate for a thin Fe layer was larger when asecond Fe layer, separated by an Au spacer, was added.\nBecause the increase depends on the thickness dof the\nresonating layer, they ascribed it to torques that occur\nat single ferromagnetic-normal metal interfaces, with no\nrole proposed for the thicker Fe layer beyond acting as a\nsinkforspincurrents. Wesuggestthatlocalizationeffects\nmay play a role, even though the conduction electrons in\nFe are less polarized than in Co. A very similar exper-\niment [19] showed that when the resonance of the two\nlayers in an Fe/Au/Fe are made to coincide by judicious\nchoice of in-plane field angle, the linewidths are equal\nand narrowest. This was interpreted in terms of spin\npumping between the two layers. In that picture, the\noff-resonance ferromagnetic layer acts as a perfect spin\nsink, except when the two layers have a common reso-\nnant field. Then spin currents generated in each layer\ncompensate the spin-sink effect of the other.However,the\nresonances coincide when the effective field is the same\nin each layer, which may also maximize ferromagnetic\nalignment and minimize localization. As seen in Fig. 2\nin the present experiment, the Co(2) and Co(1) reso-\nnances overlap at room temperature, and therefore each\nmay be narrowed by spin pumping. Below Tc,however,\nthe Co(2) resonance is no longer detected, with the anti-\nferromagnetic coupling to the ferromagnetic moment of\nGd shifting the resonance out of the observed field range.\nAs a consequence, we expect dynamical coupling due to\nspin-pumping to disappear below the Gd transition, giv-\ningrisetotheobservedjumpinthelinewidthoftheCo(1)\nresonance.\nTo summarize, we argue that the change in the tem-\nperature dependence of the ferromagnetic linewidth that\noccurs at the transition between P and AP alignment,\nprovides independent confirmation of the role of quan-\ntum confinement in GMR structures. At the same time,\nit provides further evidence that the torque-correlation\nmodel playsasubstantialrolein spin relaxationin metal-\nlic ferromagnets, especially in Co, which is nearly a half-\nmetal. Clearly,similarexperimentsusingFeandpermal-\nloy, where the torque-correlationmodel may be less dom-\ninant, are clearly in order.\nOne of us (ARK) wishes to acknowlege the support of\nthe Welch Foundation through Grant No. Y-1215.\n[1] M. N. Baibich, J. M. Broto, A. Fert, Nguyen Van Dau,\nF. Petroff, P. Etienne, G. Creuzet, A. Friederich and J.\nChazelas, Phys. Rev. Lett. 61, 2472 (1988).\n[2] P. Zahn, J. Binder, I. Mertig, R. Zeller and P.H. Ded-\nerichs, Phys. Rev. Lett. 80, 4309 (1998).\n[3] J. Binder, P. Zahn, and I. Mertig, J. Appl. Phys. 89,\n7107 (2001).\n[4] R. E. Camley and J. Barna´ s, Phys. Rev. Lett. 63, 664\n(1989).\n[5] P. Levy, S. Zhang, and A. Fert, Phys. Rev. Lett. 65, 16434\n(1990).\n[6] P. Zahn, I.Mertig, M. Richter andH. Eschrig, Phys. Rev.\nLett.75, 2996 (1995).\n[7] Kees M. Schep, Paul J. Kelly and Gerrit E. W. Bauer,\nPhys. Rev. Lett. 74, 586 (1995).\n[8] J. Kune˘ s and V. Kambersk´ y, Phys. Rev. B65, 212411\n(2002).\n[9] K. Gilmore, Y. U. Idzerda and M. D. Stiles, Phys. Rev.\nLett.99,027204((2007)..\n[10] K. Gilmore, Y. U. Idzerda and M. D. Stiles, J. Appl.\nPhys.103, 07D303 (2008).\n[11] S. Demirtas, M. R. Hossu, R. E. Camley, H. C. Mireles,\nand A. R. Koymen, Phys. Rev. B 72, 184433 (2005).\n[12] S. Demirtas, R. E. Camley, Z. Celinsky, M. R. Hossu, A.\nR. Koymen, C. Yuand M. J. Pechan ( arXiv:1002.4889v1[cond-mat.mtrl-sci])\n[13] T.L. Gilbert, Phys. Rev. 100, 1243 (1955).\n[14] Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[15] V. Kambersky, Can. J. Phys. 48, 2906 (1970).\n[16] Y. Tserkovnyak, G. A. Fiete and B. I. Halperin, Appl.\nPhys. Lett. 54, 5234 (2004).\n[17] D. Staiauf and M. F¨ ahnle, Phys. Rev. B 73, 064450\n(2005).\n[18] K. Urban, G. Woltersdorf and B. Heinrich, Phys. Rev.\nLett.87, 217204 (2001).\n[19] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Bratas ,\nR. Urban and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003)."    },    {        "title": "1511.04802v1.Determination_of_intrinsic_damping_of_perpendicularly_magnetized_ultrathin_films_from_time_resolved_precessional_magnetization_measurements.pdf",        "content": "1 \n Determination of intrinsic damping of perpendicularly magnetized \nultrathin films from time resolved precessional magnetization \nmeasurements  \n \nAmir Capua1,*, See -hun Yang1, Timothy Phung1, Stuart S. P. Parkin1,2 \n \n1 IBM Research Division, Almaden Research Center, 650 Harry Rd., San Jose,  California \n95120, USA  \n2 Max Planck  Institute for Microstructure Physics, Halle (Saale), D -06120, Germany  \n \n*e-mail: acapua@us.ibm.com  \nPACS number(s) : 75.78. -n \n \n \nAbstract:  \nMagnetization dynamics are strongly influenced by damping, namely the loss of spin \nangular momentum from the magnetic system to the lattice. An “effective” damping \nconstant  αeff is often determined experimentally from the spectral linewidth of the free \ninduction decay of the magnetization after the system  is excited to its non -equilibrium state . \nSuch an  αeff, however, reflects both intrinsic damping as well as inhomogeneous \nbroadening that arises , for example,  from spatial variations of the anisotropy field.  In this \npaper we compare measurements of the m agnetization dynamics in ultrathin non -epitaxial \nfilms having perpendicular magnetic anisotropy using  two different techniques,  time-\nresolved magneto optical Kerr effect (TRMOKE ) and hybrid  optical -electrical  \nferromagnetic resonance  (OFMR) . By using a n external  magnetic field that is applied at \nvery small angles  to the film plane  in the TRMOKE studies , we develop  an explicit closed -\nform analytical expression  for the TRMOKE spectral linewidth and show how this can be \nused to reliably extract the intrinsic Gilbert damping constant. The damping constant \ndetermined in this way is in exc ellent agreement with that determined from the  OFMR \nmethod  on the same samples. Our studies indicate  that the asymptotic high -field approach \nthat is often used in the TRMOKE method to distinguish the  intrinsic damping  from the 2 \n effective damping may result in significant error , because such high external  magnetic \nfields are required  to make this approach valid that they  are out of reach . The error becomes \nlarger the lower is the intrinsic damping  constant, and thus may account for the \nanomalously high damping constants that are  often  reported  in TRMOKE studies . In \nconventional ferromagnetic resonance ( FMR ) studies , inhomogeneous contributions can \nbe readily distinguished from intrinsic damping contributions from the magnetic field \ndependence of the FMR linewidth. Using the analogous approach, w e show how reliable \nvalues of the intrinsic damping can be extracted from  TRMOKE in two distinct magnetic  \nsystems with significant perpendicular magnetic anisotropy: ultrathin CoFeB layers and \nCo/Ni/Co trilayers.  \n \n  3 \n I. Introduction  \nSpintronic nano -devices have been identified in recent years as one of the most \npromisin g emerging technologies for future low power microelectronic circuits1, 2. In the \nheart of the dynamical spin -state transi tion stands the energy loss  parameter of the Gilbert \ndamping . Its accurate dete rmination is of paramount importance  as it determines the \nperformance of key building blocks required for  spin manipulation  such as t he switching  \ncurrent threshold of the spin transfer torq ue magnetic tunnel junction  (MTJ)  used in \nmagnetic random access memory (MRAM) as w ell as the skyrmion velocities  and the \ndomain wall motion current threshold . Up-scaling for high logic and data capacities while \nobtaining stability with high retention energies  require in addition that large  magnetic  \nanisotropies be ind uced. T hese cannot be achieved simply by engineering the geometrical \nasymmetries in the nanometer -scale range , but rather  require harnessing the induced spin-\norbit interaction a t the interface of the  ferromagnet ic film to obtain perpendicular  magnetic  \nanisotropy  (PMA)2. Hence an increasing  effort is invested in the quest for perpendicular \nmagnetized materials  having large anisotropies with low Gilbert damping3-11. \nTwo distinct families of experimental methods  are typically used for measurement \nof Gilbert damping , namely,  time-resolved pump -probe and continuous microwave \nstimulated ferromagnetic resonance ( FMR ), either of which can be implemented using \noptical and/or electrical methods . While in some cases good agreement between these \ndistinct techniques have been reported12, 13, there is often significant disagreement between \nthe methods14, 15.  4 \n When the time resolved pump -probe  method  is implemented using the magneto \noptical Kerr effect ( TRMOKE), a clear advantage over the FMR  method  is gained  in the \nability to  operate at  very high fields and frequencies16, 17. On the other hand , the FMR \nmethod a llows operation over a wide r range of geometrical configurations . The \nfundamental geometrical restriction of the TRMOKE comes from the fact that  the \nmagnetization  precession s are initiated from  the perturbation of the  effective  anisotropy \nfield by the pump  pulse , by momentarily  increasing the lattice temperature18, 19. In cases \nwhere  the torque exerted by the  effective  anisotropy field is n egligible , the pump pulse \ncannot sufficiently perturb the magnetization . Such a case occurs  for example  whenever \nthe magnetization lays in the plane of the sample in uniaxial  thin films having \nperpendicular magneti c anisotropy . Similar limitations exist if the magnetic field i s applied  \nperpendicular to the film . Hence in TRMOKE experiments , the external field is usually \napplied at  angles  typically not smaller than about \n10\n  from  either the film plane or its \nnormal.  This fact has however the consequence that the steady state magnetization \norientation , determined by the bal ancing condition for  the torques , cannot be described  \nusing a n explicit -form algebraic expression , but rather a numerical approach  should be \ntaken5. Alternatively , the dynamics  can be  described using an effective damping  from \nwhich the intrinsic damping , or at least an upper bound o f its value , is estimated at the high \nmagnetic field  limit with the limit being  undetermined . These approaches are hence less \nintuitive while the latter does not indicate directly on the energy losses but rather on the  \ncombination of the energy loss rate , coherence  time of the spin ensemble  and geometry of \nthe measurement .  5 \n In this paper , we present an approach where the TRMOKE system is operated while \napplying the  magnetic field at  very sm all angles with respect to the sample plane. This \nenables  us to use explicit closed -form analytical expressions derived for a perfectly in -\nplane external magnetic field as an approximate  solution. Hence , extraction of the intrinsic \nGilbert damping using an analytical model becomes possible without the need to drive the \nsystem to the high magnetic field limit providing at the same time an intuitive \nunderstanding of the measured responses. The validity of t he method  is verified using a \nhighly sensitive hybrid optical -electrical FMR system (OFMR) capable of operating with \na perfectly in -plane magnetic field where the analytical expressions hold. In particular, we \nbring to test the high -field asymptotic approa ch used for evaluation of the intrinsic damping \nfrom the effective damping and show that in order for it to truly indicate the intrinsic \ndamping, extremely high fields need to be applied. Our analysis reveals the  resonance \nfrequency dispersion relation as well as the inhomogeneous broadening to be the source of \nthis requirement which becomes more difficult  to fulfill the smaller the intrinsic damping \nis. The presented method is applied  on two distinct families of technologically relevant \nperpendicularly mag netized systems; CoFeB4, 6 and Co/Ni/Co20-23. Interestingly, the results \nindicate  that the Ta seed layer  thickness used in  CoFeB  films  strongly affects the intrinsic \ndamping , while t he static characteristics of the  films remain intact . In the Co/Ni/Co trilayer \nsystem which has in contrast a large effective anisotropy field, unexpected ly large spectral \nlinewidth s are measured when  the external  magnetic field is comparable to the  effective  \nanisotropy field, which cannot be explained by the  conventional model  of no n-interacting \nspins  describing the inhomogeneous broadening . This  suggest s that under the low stiffness  6 \n conditions associated with such bias fi elds, cooperative  exchange  interactions, as two \nmagnon scattering, become relevant8, 24.  \nII. EXPERIMENT  \nThe experiments present ed were carried out on three PMA samples: two samples \nconsist ed of Co36Fe44B20 which differed by the thickness of the underlayer  and a third \nsample consisting of Co/Ni/Co trilayer . The CoFeB samples were characterized by low \neffective anisotropy  (Hkeff) values as well as by small distribution of its value  in contrast to \nthe Co /Ni/Co trilayer system . We define here Hkeff as 2Ku/Ms-4πMs where Ku is the \nanisotropy energy constant and Ms being the saturation magnetization.   \nThe structure s of the two CoFeB samples were 50Ta|11CoFeB |11MgO |30Ta, and \n100Ta|11CoFeB |11MgO |30Ta (units are in Å)  and had  similar Ms value s of 1200 emu/cc  \nand Hkeff of 1400 Oe and 1350 Oe respectively.  The t hird system studied was  \n100AlO x|20TaN |15Pt|8Pt 75Bi25|3Co|7Ni|1.5Co |50TaN  with Ms of 600 emu/cc  and Hkeff \nvalue  of about 4200 Oe . All samples were grown on oxidized Si substrates  using DC \nmagnetron sputtering  and exhibited sharp perpendicular switching characteristics . The \nsamples consisting of CoFeB were annealed for  30 min  at \n275\n  C in contrast to the \nCo/Ni/Co which was measured as deposited.  Since the resultant film has a polycrystalline  \ntexture , the in -plane anisotropy is averaged out and the films are regarded as uniaxial \ncrystals with the symmetry axis being perpendicular to the film plane.  7 \n The t wo configurations of the experimental setup were driven by a Ti:Sapphire  laser \nemitting 70 fs pulses at 800 nm  having energy of 6 nJ. In the first configuration a standard  \npolar  pump -probe TRMOKE was implemented with the probe  pulse being a ttenuated by \n15 dB compared to the  pump pulse. Both  beams were focused on the sample to an estimated \nspot size of 10.5 m defined by the full width at half maximum (FWHM) . In the hybrid \noptical -electrical  OFMR system , the Ti:Sapphire laser  served to pro be the magnetization \nstate via  the magneto -optical  Kerr effect after being attenuated  to pulse energies of about \n200 pJ  and was phase -locked with a microwave oscillator in a similar configuration to the \none reported in Ref. [ 25]. For this measurement , the film was patterned into a  20 m x 20 \nm square island with a Au wire  deposited in proximity  to it, which  was driven by the \nmicrowave signal. Prior to reaching the sample, the probing laser beam traversed the \noptical delay line that enabled mapping of the time axis and in particular the  out of plane -\nmz component of the magnetization  as in the  polar TRMOKE  experiment . With this \nconfiguration the OFMR realizes  a conventional FMR system where the magnetization \nstate is read in the time-domain using the magneto optical Kerr effect  and hence its high \nsensitivity . The OFMR  system therefore  enables operation  even when the external field is \napplied fully in the sample plane.  \nIII. RESULTS AND DISCUSSION  \nA. TRMOKE measurements on 50 Å-Ta CoFeB  film  \nThe first experiments we present  were performed  on the 50  Å-Ta CoFeB  system  \nwhich is similar to the one studied in Ref. [4]. The TRMOKE measurement was carried 8 \n out at two angles  of applied magnetic field,  \nH, of \n4\n and \n1\n  measured from the surface \nplane as indicated in Fig. 1. We de fine here in addition the comple mentary angle measured \nfrom the surface normal, \n2HH   . Having i ts origin in the effective anisotropy, the \ntorque  generated by the optical pump  is proportional to \n cos( )sins keffMH   with θ being \nthe angle of the magnetization relative to the normal of the sample plane.  Hence, f or \n1H\n, the angle θ becomes close to \n/2 , and the resultant torque generated by the optical pump \nis not strong enough to  initiate reasonable precessions . For the same reason, the maximum \nfield measureable for the \n1H\n  case is significantly lower than for the \n4H\n  case. This \nis clearly demonstrated in the m easured MOKE signals for the two \nH  angles in Fig. 2 (a). \nWhile for \n4H\n  the precessional motion  is clearly seen even at a bias field of 12 kOe, \nwith \n1H\n  the precessions are hardly observable already at a bias field of 5.5 kOe.  \nAdditionally, it is also possible that the  lower signal to noise ratio observed for  \n1H\n may \nbe due to a breakdown into domains with the almost in -plane applied magnetic field26. \nAfter reduction of the background signal, the measured data can be fitted to a decaying \nsinusoidal response from which t he frequency and decay time can be extracted in the usual \nmanner 6 (Fig. 2(b)) . The measured precession frequency as a function of the applied \nexternal field , H0, is plotted in Fig. 3(a). Significant differences near Hkeff are observed for \nmerely a change of three degrees in the angle of the applied magnetic field . In particular, \nthe trace for \n1H\n  exhibits a minimum point at approximately  Hkeff in contrast to the \nmonotonic behavior of the \n4H\n  case. The theoretical dependence of the resonance 9 \n frequency  on the magnetic  bias field expressed in normalized units, \n/keffH , with \n  \nbeing the resonance angular frequency and \n  the gyromagnetic ratio, is presented in Fig. \n3(b) for several  representative  angles of the applied  field. The resonance frequency at the \nvicinity of Hkeff is very sensitive to slight changes in the angle of the applied field  as \nobserved also in the experiment . Actually the derivative of the resonance frequency with \nrespect to the applied field  at the vicinity of Hkeff is even more sensitive where it diverges \nfor \n90\n  but reaches a value of zero for the slightest angle divergence.  A discrepancy \nbetween the measurement and the theoretical solution exists however.  At field values much \nhigher than Hkeff the precession frequenc y should be identical for all angles (Fig. 3(b)) but \nin practice the resonance frequency measured for \nH  of \n4\n  is consistently higher by nearly \n2 GHz  than at \n1\n . The t heory  also predicts that for the case of \n4\n , the resonance frequencies \nshould  exhibit a minimum point as well which is not observed  in the measurement . The \norigin of the difference is not clear and may be related to the inhomogeneities in the local \nfields  or to the higher orders of the  interface induced anisotropy  which were neglected in \nthe theoretical calculation .  \nIn Fig. 3(c), we plot the effective Lorentzian resonance linewidth in the frequency \ndomain , \neff , defined by \n2/eff eff  with \neff being the measured decay time extracted \nfrom the measured responses. Decompos ing the measured linewidth t o an  intrinsic \ncontribution that represent s the energy loss es upon precession  and an extrinsic contribution \nwhich represent s the inhomogeneities in the local fields and is not related  to energy loss of 10 \n the spin system , we express  the linewidth as : \nint eff IH     . \nint  is given by the \nSmit -Suhl formula27, 28 and equals \n2/  with \n  denoting  the intrinsic spin  precession  decay \ntime where as \nIH  represents the dispersion in the resonance frequencies due to the \ninhomogeneities. If the variations  in the resonance frequency  are assumed to be primarily \ncaused by variations in the local effective anisotrop y field \nkeffH , \nIH  may be given by : \n/IH keff keff d dH H  \n. For the case of \n/2H  or \n0H , \neff  has a closed  \nmathematical  form.  In PMA  films  with bias field applied in the sample plane , the \nexpression  for \neff  becomes : \n      \n0\n002\n00\n0\n0022\n002         for        H\n2\n2     for         Heff keff keff keff\nkeff\nkeff keff\neff keff keff\nkeffkeffHH H H H\nH H H\nHH HH H HHH HH \n      \n\n\n              ,    (1) \nwith \n denot ing the Gilbert damping . The first term s in Eq. (1) stem from  the intrinsic \ndamping , while the second term s stem  from the inhomogeneous broadening . Eq. (1)  shows \nthat while the contribution of the intrinsic part to the total spectral linewidth  is finite, as the \nexternal field approaches  Hkeff either from higher or lower field values, the inhomogeneous \ncontribution diverges. Equation  (1) further shows that for H0 >> Hkeff , the slope of \neff  \nbecomes \n2  with a constant offset given by \n/2keffH . Although Eq. (1) is valid only \nfor\n/2H , it is still  instructi ve to apply it on the measured linewidth for the  \n4H\n  case.  11 \n The theoretical intrinsic linewidth for \n/2H , inhomogeneous contribution and the sum \nof the two a fter fitting \n  and \nkeffH  in the range H 0 > 5000 Oe  are plotted in Fig. 3(c). The \nresul tant fitting values were 0.023 ±0.002 for the Gilbert damping and 175 Oe for \nkeffH . At \nexternal fields comparable to Hkeff the theoretical expression derived for the  \ninhomogeneous broadening for a perfectly in -plane field does not describe properly the \nexperiment . In the theoretical analysis , at fields comparable to Hkeff, the derivative \n0/d dH\ndiverges  and therefore also the derivative \n/keff d dH  as understood from Fig. 3(b).  In the \nexperiment however , \n/2H  and the actual derivative \n/keff d dH  approache s zero. \nHence any variation in Hkeff result s in minor  variation of the frequency . This mean s that the \ncontribution of the inhomogeneous broadening to the total linewidth is suppressed near \nHkeff in the experiment as opposed to being expanded in  the theoretical calculation which \nwas carried out for  \n/2H . The result  is an overestimate d theoretical linewidth  near \nHkeff. After reduction of the inhomogeneous broadening , the extracted  intrinsic measured \nlinewidth is  presented in Fig. 3(c) as well  showing the deviation from  the theor etical \nintrinsic contribution as the field approaches  Hkeff.  \nTo further investigate the e ffect of tilt ing the magnetic field , we study the TRMOKE \nresponses for the \n1H\n  case.  The measured linewidth for this  case is presented in Fig. \n3(d). In contrast to the \n4H\n  case, the measured linewidth now increases at fields near \nHkeff as expected theoretically . Furthermore,  the measured linewidth  for the \n1H\n  case is 12 \n well describe d by Eq. (1) even in the vicinity of Hkeff as well as for bias  fields smaller than \nHkeff. The fitting result s in the same damping value of 0.023 ±0.0015 as with the \n4H\n  \ncase,  and a variation in \nkeffH  of 155 Oe, which is 20 Oe smaller  than the value fitted for \nthe \n4H\n  case.  \nWe next turn to examine  the G ilbert damping. In the absence of the demagnetization \nand crystalline anisotrop y fields, the expression  for the intrinsic Gilbert damping is given \nby:  \n                                                     \n1 .                  (2) \nOnce the anisotropy and the demagnetization field s are included , the expression for the \nintrinsic Gilbert damping becomes :  \n          \n 0\n0\n0\n0\n001                                         for           \n21       for           \n2keff\nkeff\nkeff keffdHHHd\ndHHHd H H H H \n   \n   \n  ,          (3) \nand is valid  only for \n2H  and for  crystals having uniaxial symmetry. At oth er angles \na numerical method5 should be used  to relate the precession decay time to the Gilbert \ndamping. Eq. (3) is merely the intrinsic contribution in Eq. (1)  written in the form \nresembling Eq. (2) . At high fields both Eq s. (2) and (3) converge  to the same result since  13 \n \n1 0dH\nd. As seen in Fig. 3(b), at bias field s comparable to Hkeff the additional derivative \nterm of Eq. (3) becomes very significant . When substituting the measured  decay time,\neff\n, for \n , Eq. (2) gives what is often interpreted as  the “effective ” damping , αeff, from which \nthe intrinsic damping is measured by evaluating it at high fields when the damping becomes  \nasymptotically field independent. Additionally, t he asymptotic limit should be reached \nwith respect to  the inhomogeneous contribution of  Eq. (1). In Fig. 3(e), we plot the effective \ndamping  using \neff  and Eq. (2) . We further show the intrinsic damping value after  \nextracting the intrinsic linewidth  and using Eq. (3). Examining first the effective damping \nvalues, we see that for the two angles , the values are distinctively  different at low fields  \nbut converge at approximately 41 00 Oe  (Beyond  5500 Oe the data  for the \n1H\n  case \ncould not be measured).  In fact , the behavior of the effective damping seems to be related \nto the dependence of the resonance frequency on H0 (Fig. 3(a)) in which  for the \n1H\n  \ncase reaches an extremum while  the \n4H\n  case exhibits a monotonic behavior . Since Eq. \n(2) lacks  the derivative term \n0/dH d , near Hkeff the effective damping is related to the \nGilbert damping by the relation: \n01\neffd\ndH  for H0 > Hkeff. Furthermore, since  \n does not \ndepend on the magnetic field to the first order, the dependence  of the effective damping,\neff , on \nthe bias field stems from the derivative  term \n0 d dH which becomes larger and eventually \ndiverges to infinity  when the magnetic field reaches Hkeff as can be inferred from Fig. 3(b) \nfor the case of \n0H\n  for which Eq. (3) was derived . Hence the increase in \neff at bias 14 \n fields near Hkeff. The same considerations apply also for H0 < Hkeff. As the angle \nH  increases , \nthis analysis becomes  valid only for bias fields which are large enough or small enough \nrelative to Hkeff. When examined  separately, each effective damping trace may give the \nimpression that  at the higher fields  it has become bias field independent and reached its \nasymptotic value from which two very distinct  Gilbert  damping  values of ~0.027 and \n~0.039 are extracted at field values of 12 kOe and 5.5 kOe  for the \n4H\n and \n1H\nmeasurements , respectively . These values are also rather different from the intrinsic \ndamping value of 0.023  extracted using  the analytical model . In contrast  to the effective \ndamping , the intrinsic damping obtained from the analytical model  reveal s a constant and \ncontinuous behavior  which is field and angle independent.  The presumably negative values \nmeasure d for the \n4H\n case stem of course  from the fac t that the expressions in Eqs. (1) \nand (3)  are derived for the  \n2H  case. The error in using the effective damping in \nconjunction with the asymptotic approximation compared to using  the analytical model is \ntherefore 17% and 70% for the \n4H\n and \n1H\n  measurements respectively.  \nIt is important in addition to understand th e conse quence of using Eq. (2) rathe r than \nEq. (3) . In Fig. 3(f) we present the error  in the damping value  after accounting for the \ninhomogeneous broadening using Eq. (2) instead of the complete  expression of  Eq. (3) . As \nexpected , the error increases as the applied field approaches Hkeff. For the measurement \ntaken with \n4H\n  the error is significantly smaller due to the smaller value of the \nderivative\n0/d dH .  15 \n As mentioned previously, i n order to evaluate the intrinsic damping from the total \nmeasured linewidth , the asymptotic limit should be reached with respect to the \ninhomogeneous broadening as well  (Eq. (1) ). In Fig s. 3(c) and 3(d) we see that this is not \nthe case  where the contribution  of the inhomogeneous linewidth is still large compared to \nthe intrinsic l inewidth . Examining Figs. 3(d) and 3(f) for the case of \n1H\n , we see that \nthe overall error of 70% resulting in the asymptotic evaluation stems from both the  \ncontribution of  inhomogeneous broadening as well as from the use of Eq. ( 2) rather than \nEq. (3) while for \n4H\n  (Figs. 3(c) and 3(f)) the error of 17% is solely due to contribution \nof the inhomogeneous broadening which was not as negligible as conceived when applying \nthe asymptotic approximation .  \nB. Comparison of TRMOKE and OFMR measurements in 100  Å-Ta CoFeB  \nfilm \nWe next turn to  study the magnetization dynamics using the OFMR system where \nthe precession s are driven with the microwave signal . Hence, the external magnetic field \ncan be applied perfectly in the sample plane. The 100 Å-Ta CoFeB sample  was used for \nthis experiment. Before patterning the film for the OFMR measurement, a TRMOKE \nmeasurement was carried out at \n4H\n  which  exhibited a similar  behavior to that observed \nwith the sample having  50 Å Ta  as a seeding layer . The  dependence of the  resonance \nfrequency on the magnetic  field as well  as the measured linewidth and its  different \ncontributions  are presented in Figs. 4(a) and 4(b). Before reduction of the inhomogeneous 16 \n broadening the asymptotic effective damping was measured to be ~0.0168 while after \nextraction  of the intrinsic damping a value of 0.0109 ±0.0015 was measured marking a \ndifference of 54%  (Fig. 4(c)). The fitted \nkeffH  was 205 Oe. Fig. 4(b) shows that the origin \nof the error stems from significan t contribution of the inhomogeneous broadening \ncompared to the intrinsi c contribution which plays a mor e significant role when the \ndamping is low.  By us ing the criteria  for the  minimum  field that results in  \n10IH eff   \nto estimat e the point where the asymptotic approximation would be valid , we arrive to a \nvalue of at least 4.6 T which is rather impractic al. The threshold of this minimal f ield is \nhighly dependent on the damping so that for a lower damping an even higher field would \nbe required.  \nAn example of a measured trace using the OFMR system  at a low microwave \nfrequency of 2.5 GHz  is presented in Fig. 4(d). The square root of the magn etization \namplitude  (out of plane mz component)  while preserving its sign  is plotted to show detail . \nThe high sensitivity of the OFMR system enable s operation  at very low frequencies and \nbias fields. For every frequency  and DC  magnetic  field value , several  cycles of the \nmagnetization precession  were recorded  by scanning the optical delay  line. The magnetic \nfield was then swept  to fully capture the resonance . The trace should be examined  \nseparately in two sections, be low Hkeff and above Hkeff (marked in the figure by black dashed \nline). For frequencies of up to \nkeffH  two resonances are crossed as indicated by the guiding \nred dashed line which represents the out -of-phase component of the magnetization, namely \nthe imaginary part of the magnetic susceptibility . Hence the cross section along this line 17 \n gives  the field dependent absorption spectrum  from which the resonance frequency and \nlinewi dth can be identified. This spectrum is show n in Fig. 4( e) together with the fitted \nlorentzian lineshapes  for bias fields below and above Hkeff. The resultant resonance \nfrequencies  of all measurements are plotted in addition in Fig. 4(a).  \nThe resonance linewidth s extracted  for bias fields larger than Hkeff, are presented in \nFig. 4(f). Here the effective magnetic field linewidth , ΔHeff, that includes the contribution \nof the inhomogeneous broadening derived  from the same principles that led to Eq. (1) with \n/2H\n is given by: \n              \n02\n2\n0\n00\n2 2\n0211                     for       2\n4\n2\n                   \n        with       keff\neff keff keff\nkeff\nkeff keff\neff keff\nkeff\nkeffHH H H H\nH\nHH HHHH H H\nHH\n\n\n\n\n\n\n      \n      \n \n       \n 0      for       keff HH   (5).  \nThe second terms in Eq. (5) denote  the contribution of the inhomogeneous broadening , \nIHH\n, and are frequency dependent as opposed to the case where the field is applied out \nof the sample plane9. The dispersion in the effective anisotropy , \nkeffH , and the intrinsic \nGilbert damping  were found by fitting the linewidth in the  seemingly  linear range at \nfrequencies larger th an 7.5 GHz . The contribution s of the intrinsic  and inhomogeneous \nparts  and the ir sum are presented as well in Fig. 4(f).  18 \n It is apparent that the measured linewidth at the lower frequencies is much broader \nthan the theoretical one.  The reason for that lies in the fact that in practice the bias field is \nnot applied perfectly in the sample plane as well as in the fact that there migh t be locally  \ndifferent orientation s of the polycrystalline  grains  due to the natural imperfections of the \ninterfaces that further result  in angle distribution of \nH . Since  the measured  field linewidth \nis a projection of the spectral  linewidth into the  magnetic  field domain, the relation between \nthe frequency and the field  intrinsic  linewidth s is given by: \n1\nint int\n0dHdH   \n . The \nintrinsic linewidth , \nint , in the frequency domain near Hkeff is finite , as easily seen from \nEq. (1)  while the derivative term near Hkeff is zero for even the slightest angle misalignment  \nas already seen. H ence the field-domain  linewidth  diverg es to infinity as observed \nexperimentally. The inhomogeneous broadening component does not diverge in that \nmanner  but is rather suppressed . To show that the excessive linewidth at low field s is \nindeed related to the derivative of \n0/d dH  we empirically multiply the total theoretical \nlinewidth by the factor \n0 / ( )d d H  which  turns out to fit  the data surprisingly well (Fig. \n4(f)). This  is merely a phenomenological qualitative description, and a rigorous description \nshould still be derived.   \nThe fitted linewidth of Fig. 4(f) results in  the intrinsic damping value of \n0.011 ±0.0005  and is identical to the value obtained by the TRMOKE  method . Often \nconcerns regarding the differences between the TRMOKE and FMR measurements such \nas spin wave emission away from the pump laser spot in the TRMOKE29, increase of 19 \n damping due to thermal heating  by the pump pulse  as well as differences in the  nature of \nthe inhomogeneous broadening  are raised. Such effects do not seem to be significant here . \nAdditionally , it is worth noting that s ince the linewidth seems to reach a  linear  dependence \nwith respect to  the field  at high fields , it may be naively fitted using a constant frequency -\nindependent inhomogeneous broadening factor . In that case an underestimated value of \n~0.0096 would have been obtained . The origin of this misinterpretation is seen clearly by \nexamining the  inhomogeneous broadening contribu tion in Fig. 4(f) that show s it as well to \nexhibit a seemingly linear dependence at the high fields. Regarding the  inhomogeneous \nbroadening , the anisotropy field dispersion,  \neffKH , obtained with the TRMOKE was 205 \nOe while the value  obtained from the OFMR system was 169 Oe . Although these values  \nare of the same order of magnitude , the difference is rather significant. It is possible that \nthe discrepancy  is related to the differences in the measurement  techniques. For instance, \nthe fact that both  the pump and probe beams have the same spot size may cause an uneven \nexcitation across the probed region in the case of the TRMOKE measurement  while in the \ncase of the OFMR measurement the amplitude of the microwave field decays at increasin g \ndistances away from the microwire. These effects may  be reflect ed in the measurements as \ninhomogeneous broadening. Nevertheless , the measured intrinsic damping values are \nsimilar.  \nFinally , we compare the effective damping of the OFMR and the TRMOKE \nmeasu rements without correcting for the inhomogeneous broadening in Fig. 4(g). The 20 \n figure shows a deviation in the low field values which  is by now understood to  be unrelated \nto the energy losses  of the system .  \nFurthermore, we observe that the thickness of the Ta underlayer affects the \ndamping. The comparison  of the 50 Å -Ta CoFeB and the 100 Å -Ta CoFeB samples  shows \nthat the increase by merely 50 Å of Ta, reduced significantly the damping while leaving \nthe anisot ropy field unaffected.  \nC. TRMOKE and OFMR measurements in Co /Ni/Co film  \nIn the last set of measurements we study the Co /Ni/Co film which has distinctively \ndifferent static properties compared to the CoFeB samples . The sample was studied  using \nthe TRMOKE setup at two \nH  angles of \n1\n  and \n4\n  and using the OFMR system at\n0H\n. The resultant  resonance frequency traces are depicted in Fig. 5(a). The spectral linewidth \nmeasure d for \n4H\n using  the TRMOKE  setup  is presented in Fig . 5(b). A linear fit at the \nquasi linear high field range results in a large damping value of 0.081 ±0.015 and in a very \nlarge \neffKH  of 630 Oe . The large damping is attributed to the efficient spin pumping into \nPtBi30 layer  having large spin -orbit coupling . When the angle of the  applied  magnetic f ield \nis reduced to  \n1H\n  a clearer picture of the  contribution  of the inhomogeneous broadening \nto the total linewidth is obtained (Fig. 5(c)) revealing  that it cannot explain solely the \nmeasured  spectral  linewidth s. While the theoretical model predicts that  the increase in \nbandwidth spans  a relatively  narrow  field range around Hkeff, the measurement shows an \nincrease over a much larger range around Hkeff. The linewidth broadening originating from 21 \n the anisotropy dispersion was theoretically calculated under the assumption of a small \nperturbation of the resonance frequency. A large \nkeffH  value was measured however  from \nthe TRMOKE measurement taken at \n4H\n . Calculating numerically the exact variation \nof the resonance frequency improved slightly the fit but definitely  did not resolve the \ndiscrepancy  (not presented) . From this fact we understand that  there should be an  additional  \nsource  contributing  to the line broadening  at least near Hkeff. A possible explanation may \nbe related to the low stiffness27 associated with the \n0 keff HH  conditions . Under such \nconditions weaker torques which are usually neglected may become relevant24, 31. These  \ntorques could possibly originate from  dipolar or exchange coupling resulting in two \nmagnon scattering processes  or even in a breakdown into magnetic domains as described \nby Grolier et al.26. From the limited data range at this angle, the damping could not be \nmeasured.  \nThe OFMR system  enabl ed a wider range of fields and frequencies than the  ones \nmeasured with the TRMOKE  for \n1H\n  (Fig. 5(a)). Fig. 5(d) presents t he measured  \nOFMR linewidth . The quasi -linear regime of the linewidth seems to be reached  at \nfrequencies of 12 GHz corresponding to bias field values which are larger than 7500 Oe . \nThe resultant intrinsic damping after fitting to this range was 0.09±0.005  with a \nkeffH  of \n660 Oe  which differ  by approximately 10% from the  values obtained from the  TRMOKE \nmeasurement . The effective measured damping is plotted in Fig. 5(e). The asymptotic \ndamping value , though not fully reached  for this high damping sample , would be about 0.1.  22 \n This represents an error of about 10% which is smaller compared to the errors of 17% and \n54% encountered in the CoFeB samples because of the larger damping of the Co /Ni/Co \nsample.   \nD. Considerations of two -magnon scattering  \nIn general, two -magnon spin wave scattering by impurities may exist in our \nmeasurements  at all field ranges32, 33, not only near Hkeff as suggested in the discussion of \nthe previous section32, 33. The resultant additional linewidth broadening would then be \nregarded as an extr insic contribution to the damping34-36. While in isotropic films which \nexhibit low crystalline anisotropy or in films having in -plane  crystalline anisotropy, two -\nmagnon scattering is maximized when the external field is applied in the film plane, in \nPMA films this is not necessarily the case and the highest efficiency of two -magnon \nscattering may be obtained at some oblique angle35.  \nIn films where two -magnon scatt ering is significant , the measured linewidth should \nexhibit  an additional  nonlinear dependence on the external field which  cannot be accounted \nfor by the present model . In such case, a s trong dependence on the external  field would be  \nobserved  for fields below Hkeff  due to the variation in the  orientation of the magnet ization \nwith the external magnetic field. At higher fields the dependence on the external field is \nexpected to be moderate35.  \nWhile at bias field values below Hkeff our data is relatively limited, at external \nmagnetic fields that are larger than Hkeff, the observed linewidth seems to be described well 23 \n by our model  resulting in a field independent Gilbert damping coefficient . This seems to \nsupport our model that the scattering of spin waves does not have a prominent effect. It is \npossible however that a moderate dependence on the bias field, especially at high field \nvalues, may have been “linearized” and classified as intrinsic damping.  \n \nIV. CONCLUSION  \nIn conclusion, in this paper we studied the time domain magnetization dynamics  in \nnon-epitaxial  thin films  having perpendicular magnetic  anisotropy  using the TRMOKE  and \nOFMR  system s. The analytical model used to interpret the magnetization dynamics from \nthe TRMOKE  responses indicated that the asymptotic high -field approach often used to \ndistinguish the intrinsic damping from the effective damping may result in significant error  \nthat increases the lower the damping is . Two sources for the error  were  identified  while \nvalidity of th e asymptotic  approach was shown to require very high magnetic fields. \nAdditionally, the effective damping was shown to be highly affected by the derivative of \nthe resonance frequency with respect to  the magnetic field  \n0/d dH . The analytical \napproach  developed here was verified by use of the OFMR measurement showing excellent \nagreement whenever the intrinsic damping was compared and ruled out the possibility of \nthermal heating by the laser or  emission of spin waves away from  the probed  area.  \n  24 \n As to the systems studied, a  large impact of the seed layer  on the intrinsic damping \nwith minor effect on the static characteristics  of the CoFeB system was observed  and may \ngreatly aid in engineering the proper materials for the MTJ. Interestingly, the use of the \nanalytical model enabled identification of  an additional exchange torque when low stiffness \nconditions prevailed. 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Baberschke, D. Spoddig, R. Meckenstock, J. Pelzl, Z. Frait, \nand D. L. Mills, Physical Review B 68, 060102 (2003).  \n34 H. Suhl, Magnetics, IEEE Transactions on 34, 1834 (1998).  \n35 M. J. Hurben and C. E. Patton, Journal of Applied Physics 83, 4344 (1998).  \n36 D. L. Mills and S. M. Rezende, in Spin Dynamics in Confined Magnetic Structures II , edited by \nB. Hillebrands and K. Ounadjela (Springer Berlin Heidelberg, 2003).  \n \n \n  27 \n Figure 1  \n \nFIG 1. Illustration of the angles \nH , \nH and \n . M and H0 vectors denote the magnetization \nand external magnetic field , respectively.  \n  \n28 \n Figure 2 \n \nFIG. 2. Measured TRMOKE  responses  at \nH  angles of \n4\n and \n1\n . (a) TRMOKE signal at \nlow and high external magnetic field values. Traces are shifted for clarity. (b) Measured \nmagnetization responses after reduction of background signal (open circles) \n29 \n superimposed  with the fitted decaying sine  wave  (solid lines). Traces are shifted and \nnormalized to have the same peak  amplitude. Data presented for  low and high external \nmagnetic field values.  \n  30 \n Figure 3 \n  \nFIG. 3. TRMOKE measurements at \n4H\n  and \n1H\n . (a) Measured resonance \nfrequency versus magnetic field. (b) Theoretical dependence of resonance frequency on \nmagnetic field  presented in normalized units . (c) & (d)  Measured linewidth (blue) , fitted \ntheoretical con tributions to l inewidth (green, cyan, magenta) and extracted intrinsic \nlinewidth from measurement (red) for \n4H\n  and \n1H\n , respectively. (e) Intrinsic and \neffective damping. (f) Error in damping value when using Eq. (2) instead  of Eq. (3 ).  \n31 \n Figure 4 \n \nFIG. 4. TRMOKE and OFMR measurements at \n4H\n  and \n0H\n , respectively. (a) \nMeasured resonance frequency versus magnetic field. (b)  Measured linewidth (blue), fitted \ntheoretical contributions to linewidth (green, cyan, magenta) and extracted intrinsic \n32 \n linewidth from measurement (red) using the TRMOKE with  \n4H\n . (c) Intrinsic and \neffective damping  using TRMOKE . (d) Representative OFMR trace at 2.5 GHz.  The \nfunction sign( mz)(mz)1/2 is plotted.  (e) Field dependent absorption spectrum  (blue)  \nextracted from the cross section along the red dashed lined of (d) together with fitted \nlorentzian lineshapes (red).  (f) Measured linewidth (blue), fitted theoretical contributions \nto linewidth (green, cyan, black) and empirical fit that describes the angle misalignment \n(magenta) using the OFMR with \n0H\n . (g) Effective damping using the OFMR and \nTRMOKE .    33 \n Figure 5 \n \nFIG. 5. TRMOKE at \n4H\n  and \n1H\n  and OFMR measurement at \n0H\n  for Co/Ni/Co \nsample . (a) Measured resonance frequency versus magnetic field. (b ) Measured linewidth \n(blue), fitted theoretical contributions to linewidth (green, cyan, magenta) and extracted \nintrinsic linewidth from measurement (red) using the TRMOKE with \n4H\n . (c) \nMeasured linewidth (blue), fitted theoretical c ontributions to linewidth (green, cyan, \nmagenta) using the TRMOKE with \n1H\n . (d) Measured linewidth (blue), fitted \ntheoretical contributions to linewidth (green, cyan, black) using the OFMR with \n0H\n . \n34 \n (e) Effecti ve (blue)  and intrinsic  (black ) damping using the TRMOKE at \n4H\n  and \neffective damping measured with the OFMR at \n0H\n  (red).  "    },    {        "title": "1608.00984v2.Ferromagnetic_Damping_Anti_damping_in_a_Periodic_2D_Helical_surface__A_Non_Equilibrium_Keldysh_Green_Function_Approach.pdf",        "content": "arXiv:1608.00984v2  [cond-mat.mes-hall]  13 Aug 2016Ferromagnetic Damping/Anti-damping in a Periodic 2D Helic al surface; A\nNon-Equilibrium Keldysh Green Function Approach\nFarzad Mahfouzi1,∗and Nicholas Kioussis1\n1Department of Physics, California State University, North ridge, California 91330-8268, USA\nIn this paper, we investigate theoretically the spin-orbit torque as well as the Gilbert damping for\na two band model of a 2D helical surface state with a Ferromagn etic (FM) exchange coupling. We\ndecompose the density matrix into the Fermi sea and Fermi sur face components and obtain their\ncontributions to the electronic transport as well as the spi n-orbit torque (SOT). Furthermore, we\nobtain the expression for the Gilbert damping due to the surf ace state of a 3D Topological Insulator\n(TI) and predicted its dependence on the direction of the mag netization precession axis.\nPACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nThe spin-transfer torque (STT) is a phenomenon in\nwhich spin current of large enough density injected into\na ferromagnetic layer switches its magnetization from\none static configuration to another [1]. The origin of\nSTT is absorption of itinerant flow of angular momen-\ntum components normal to the magnetization direc-\ntion. It represents one of the central phenomena of the\nsecond-generation spintronics, focused on manipulation\nof coherent spin states, since reduction of current den-\nsities (currently of the order 106-108A/cm2) required\nfor STT-based magnetization switching is expected to\nbring commercially viable magnetic random access mem-\nory (MRAM) [2]. The rich nonequilibrium physics [3]\narising in the interplay of spin currents carried by fast\nconduction electrons and collective magnetization dy-\nnamics, viewed as the slow classical degree of freedom,\nis of great fundamental interest.\nVery recent experiments [4, 5] and theoretical stud-\nies[6] havesoughtSTT innontraditionalsetupswhich do\nnot involvetheusual two(spin-polarizingandfree) F lay-\ners with noncollinear magnetizations [3], but rely instead\non the spin-orbit coupling (SOC) effects in structures\nlacking inversion symmetry. Such “SO torques” [7] have\nbeen detected [4] in Pt/Co/AlO xlateral devices where\ncurrent flows in the plane of Co layer. Concurrently, the\nrecent discovery [8] of three-dimensional (3D) topologi-\ncal insulators (TIs), which possess a usual band gap in\nthe bulk while hosting metallic surfaces whose massless\nDirac electrons have spins locked with their momenta\ndue to the strong Rashba-type SOC, has led to theoreti-\ncal proposals to employ these exotic states of matter for\nspintronics [9] and STT in particular [10]. For example,\nmagnetizationofa ferromagneticfilm with perpendicular\nanisotropy deposited on the TI surface could be switched\nby interfacial quantum Hall current [10].\nIn this paper, we investigate the dynamical properties\nof a FM/3DTI heterostructure, where the F overlayer\n∗farzad.mahfouzi@gmail.comcovers a TI surface and the device is periodic along in-\nplanex−ydirections. The effect of the F overlayer is a\nproximityinduced exchangefield −∆surf/vector m·/vectorσ/2superim-\nposed on the Dirac cone dispersion. For a partially cov-\nered FM/TI heterostructure, the spin-momentum-locked\nDirac electrons flip their spin upon entering into the in-\nterface region, thereby inducing a large antidamping-like\nSOT on the FM [15–17]. The antidamping-like SOT\ndriven by this mechanism which is unique to the sur-\nface of TIs has been predicted in Ref. [17], where a time-\ndependent nonequilibrium Green function [18] (NEGF)-\nbased framework was developed. The formalism made it\npossible to separate different torque components in the\npresenceofarbitraryspin-flipprocesseswithinthedevice.\nSimilaranti-dampingtorqueshasalsobeen predicted[19]\nto exist due to the Berry phase in periodic structures\nwhere the device is considered infinite in in-plane direc-\ntions and a Kubo formula was used to describe the SOT\nas a linear response to homoginiuos electric field at the\ninterface. However, the connection between the two ap-\nproaches is not clear and one of the goals of the current\npaper is to address the similarities and the differences\nbetween the two. In the following we present the theo-\nretical formalism of the SOT and damping in the regime\nofslowlyvaryingparametersofaperiodicsysteminspace\nand time.\nGenerally, in a quantum system with slowly varying\nparameters in space and/or time, the system stays close\ntoits equilibrium state ( i.e.adiabaticregime)and the ef-\nfects of the nonadiabaticity is taken into account pertur-\nbatively using adiabatic expansion. Conventionally, this\nexpansion is performed using Wigner representation [20]\nafter the separation of the fast and slow variations in\nspace and/or time. [21] The slow variation implies that\nthe NEGFs vary slowly with the central space ( time ),\n/vector xc= (/vector x+/vector x′)/2(tc= (t+t′)/2 ), while they changefast\nwith the relative space (time), /vector xr=/vector x−/vector x′(tr=t−t′).\nHere we use an alternative approach, where we consider\n(x,t) and (/vector xr,tr) as the natural variables to describe the\nclose to adiabatic apace-time evolution of NEGFs and\nthen perform the following Fourier transform\nˇG(/vector xt;/vector x′t′) =/integraldisplaydE\n2πd/vectork\nΩkeiE(t−t′)+i/vectork·(/vector x−/vector x′)ˇG/vectorkE(/vector xt).(1)2\nwhere, Ω kis the volume of the phase space that the\n/vectork-integration is being performed. The standard Dyson\nequation of motion for ˇG(/vector xt;/vector x′t′) is cumbersome to ma-\nnipulate[22,23]orsolvenumerically,[24]sotheyareusu-\nally transformed to some other representation.[11] Gen-\neralizing the equation to take into account slowly varying\ntime and spatial dependence of the Hamiltonian we ob-\ntain,\nˇG=/parenleftbigg\nGrG<\n0Ga/parenrightbigg\n, (2)\n=/parenleftbigg\nGr,−1\nad−iDxtΣ<\n0 Ga,−1\nad−iDxt/parenrightbigg−1\n,\nwhere,\nGr,−1\nad= (E−iη)1−H(/vectork,t)−µ(/vector x),(3a)\nΣ<=−2iηf(E−i∂\n∂t−µ(/vector x)), (3b)\nDxt=∂\n∂t+∂H\n∂/vectork·/vector∇, (3c)\nand,η=/planckover2pi1/2τis the phenomenological broadening pa-\nrameter, where τis the relaxation time. It is worth\nmentioning that for a finite ηthe number of particles\nis not conserved, and a more accurate interpretation of\nthe introduced broadening might be to consider it as an\nenergy-independent scape rate of electrons to fictitious\nreservoirs attached to the positions /vector x. Consequently, a\nfinite broadening could be interpreted as the existence\nof an interface in the model between each atom in the\nsystem and the reservoir that is spread homogeneously\nalong the infinite periodic system.\nEq. (2) shows that the effect of the space/time varia-\ntion is to replace E→E−i∂/∂tand/vectork→/vectork−i/vector∇in the\nequation of motion for the GFs in stationary state. To\nthe lowest order with respect to the derivatives we can\nwrite,\nˇG=ˇGad−i∂ˇGad\n∂E∂ˇG−1\nad\n∂tˇGad−i∂ˇGad\n∂/vectork·/vector∇ˇG−1\nadˇGad,\n(4)\nwhere,\nˇG−1\nad=/parenleftbigg\nGr,−1\nad−2iηf(E−µ(/vector x))\n0 Ga,−1\nad/parenrightbigg\n.(5)\nFor the density matrix of the system, ρ(t) =1\niG<(t,t),\nwe obtain,\nρneq\n/vectork,t≈ −/integraldisplaydE\n2πℜ/parenleftbigg\n[D(Gr\nad),Gr\nad]f+2iηD(Gr\nad)Ga\nad∂f\n∂E/parenrightbigg\n(6)\nwhereD=∂\n∂t−/vector∇µ·∂\n∂/vectorkis the differential operator act-\ning on the slowly varying parameters in space and time.The details of the derivation is presented in Appendix.A.\nThe density matrix in Eq. (6) is the central formula of\nthe paper and consists of two terms; the first term con-\ntains the equilibrium Fermi distribution function from\nthe electrons bellow the Fermi surface occupying a slowly\n(linearly) varying single particle states that has only in-\nterband contributions and can as well be formulated in\nterms of the Berry phase as we will show the following\nsections, and; the second term corresponds to the elec-\ntrons with Fermi energy (at zero temperature we have,\n∂f/∂E=δ(E−EF)) which are the only electrons al-\nlowed to get excited in the presence of the slowly varying\nperturbations. The fact that the first term originates\nfrom the assumption that the electric field is constant\ninside the metallic FM suggests that this term might dis-\nappearoncethescreeningeffect isincluded. Onthe other\nhand, duetothe factthat thesecondtermcorrespondsto\nthe nonequilibrium electrons injected from the fictitious\nreservoirs attached to the device through the scape rate\nη, it might capture the possible physical processes that\noccur at the contact region and makes it more suitable\nfor the calculation of the relevant physical observables in\nsuch systems.\nUsing the expression for the nonequilibrium density\nmatrix the local spin density can be obtained from,\n/vectorSneq(t) =/angb∇acketleft/vector σ/angb∇acket∇ightneq=1\n4π2/integraldisplay\nd2/vectorkTr[ρneq\n/vectork,t/vector σ],(7)\nwhere/angb∇acketleft.../angb∇acket∇ightneqrefers to the ensemble average over many-\nbody states out of equilibrium demonstrated by the\nnonequilibrium density matrix of the electrons and, Tr\nrefers to the trace. In this case the time derivative in the\ndifferential operator Dleads to the damping of the dy-\nnamics of the ferromagnet while the momentum deriva-\ntive leads to either damping or anti-damping of the FM\ndynamics depending on the direction of the applied elec-\ntric field. In the followingsection we apply the formalism\nto a two band helical surface state model attached to a\nFM.\nII. SOT AND DAMPING OF A HELICAL 2D\nSURFACE\nA two band Hamiltonian model for the system can be\ngenerally written as,\nH(/vectork,t) =ε0(/vectork)1+/vectorh(/vectork,t)·/vectorσ (8)\nwhere,/vectorh=/vectorhso(/vectork)+∆xc(/vectork)\n2/vector m(t), with/vectorhso(/vectork) =−/vectorhso(−/vectork)\nand ∆ xc(/vectork) = ∆ xc(−/vectork) being spin-orbit and magnetic\nexchange coupling terms respectively. In particular in\nthe case of Rashba type helical states we have /vectorhso=\nαsoˆez×/vectork. In this case for the adiabatic single particle\nGF we have,\nGr\nad(E,t) =(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2(9)3\nFrom Eq. (7) for the local spin density, we obtain (See\nAppendix B for details),\n/vectorSneq(t) =/integraldisplayd2/vectork\n4π2/parenleftBigg/vectorh×D/vectorh\n2|/vectorh|3(f1−f2)−(/vector∇µ·/vector v0)/vectorh\n2η|/vectorh|(f′\n1−f′\n2)\n+(/vectorh×D/vectorh\n2|/vectorh|2+ηD/vectorh−1\nη(/vectorh·D/vectorh)/vectorh\n2|/vectorh|2)(f′\n1+f′\n2)/parenrightBigg\n(10)\nwhere,f1,2=f(ε0± |/vectorh|) and/vector v0=∂ε0/∂/vectorkis the group\nvelocity of electrons in the absence of the SOI. Here, we\nassumeη≪ |/vectorh|which corresponds to a system close to\nthe ballistic regime. In this expression we kept the ηD/vectorh\nbecause of its unique vector orientation characteristics.\nAs it becomes clear in the following, the first term in\nEq. (10) is a topological quantity which in the presence\nof an electric field becomes dissipative and leads to an\nanti-damping torque. The second term in this expression\nleads to the Rashba-Edelstein field-like torque which is a\nnondissipative observable. The third term has the exact\nformasthe firstterm with the difference that it is strictly\na Fermi surface quantity. The fourth term, also leads to\na field like torque that as we will see in the following has\nsimilar features as the Rashba-Edelstein effect. It is im-\nportant to pay attention that unlike the first term, the\nrest of the terms in Eq. (10) are solely due to the flow\nof the non-equilibrium electrons on the Fermi surface.\nFurthermore, we notice that the terms that lead to dissi-\npation in the presence of an electric field ( D ≡/vector∇µ·∂\n∂/vectork)\nbecome nondissipative when we consider D ≡∂/∂tand\nvice versa.\nA. Surface State of a 3D-TI\nIn the case of the surface state of a 3D-TI, as an ap-\nproximation we can ignore ε0(/vectork) and consider the helical\nterm as the only kinetic term of the Hamiltonian. In this\ncasethelocalchargecurrentandthenonequilibriumlocal\nspin density share a similar expression, /vectorI=/angb∇acketleft∂(/vectorh·/vector σ)/∂/vectork/angb∇acket∇ight.\nFor the conductivity, analogous to Eq. (10), we obtain,\nσij=e/integraldisplayd2/vectork\n4π2\n/vectorh·∂/vectorh\n∂ki×∂/vectorh\n∂kj\n2|/vectorh|2/parenleftBigg\nf1−f2\n|/vectorh|+f′\n1+f′\n2/parenrightBigg\nδi/negationslash=j\n+−η|∂/vectorh\n∂ki|2+1\nη(∂|/vectorh|2\n∂ki)2\n2|/vectorh|2(f′\n1+f′\n2)δij\n(11)\nThis shows that the Fermi sea component of the density\nmatrixcontributesonlytotheanomalousHallconductiv-\nity which is in terms of a winding number. On the other\nhand, the second term is finite only for the longitudinal\ncomponents of the conductivity and can be rewritten in\nterms of the group velocity of the electrons in the system\nwhich leads to the Drude-like formula.Should the linear dispersion approximation for the ki-\nnetic term in the Hamiltonian be valid in the range of\nthe energy scale corresponding to the magnetic exchange\ncoupling ∆ xc(i.e. when vF≫∆xc), the effect of the in-\nplane component of the magnetic exchange coupling is to\nshift the Dirac point (i.e. center of the k-space integra-\ntion) which does not affect the result ofthe k-integration.\nIn this case after performing the partial time-momentum\nderivatives, ( D(/vectorh) =∆xc\n2∂/vector m\n∂t−vFˆez×/vector∇µ), we use\n/vectorh(/vectork,t) =vFˆez×/vectork+∆xc\n2mz(t)ˆez, to obtain,\n/vectorSneq(t) =/integraldisplaykdk\n4π|/vectorh|2/parenleftBigg\n/vectorS1f1−f2\n|/vectorh|+(/vectorS1+/vectorS2)(f′\n1+f′\n2)/parenrightBigg\n,\n(12)\nwhere,\n/vectorS1(/vectork,t) =∆2\nxc\n4mz(t)ˆez×∂/vector m\n∂t+∆xcvF\n2mz(t)/vector∇µ(13)\n/vectorS2(/vectork,t) =∆xc\n4η(2η2−v2\nF|k|2)(∂mx\n∂tˆex+∂my\n∂tˆey)\n+∆xc\n4η(2η2−∆2\nxcm2\nz\n2)∂mz\n∂tˆez\n−vF\nη(η2−v2\nF|k|2\n2)ˆez×/vector∇µ (14)\nThe dynamics of the FM obeys the LLG equation where\nthe conductions electrons insert torque on the FM mo-\nments through the magnetic exchange coupling,\n∂/vector m\n∂t=/vector m×\nγ/vectorBext+∆xc\n2/vectorSneq(t)−/summationdisplay\nijαij\n0∂mi\n∂tˆej\n\n(15)\nwhere,αij\n0=αji\n0, withi,j=x,y,z, is the intrinsic\nGilbert damping tensor of the FM in the absence of\nthe TI surface state and /vectorBextis the total magnetic field\napplied on the FM aside from the contribution of the\nnonequilibrium electrons.\nWhile the terms that consist of /vector∇µare called SOT,\nthe ones that contain∂/vector m\n∂tare generally responsible for\nthe damping of the FM dynamics. However, we no-\ntice that ˆ ez×∂/vector m\n∂tterm in Eq. (13) which arises from\nthe Berry curvature, becomes mz∂/vector m\n∂tin the LLG equa-\ntion that does not contribute to the damping and only\nrenormalizes the coefficient of the left hand side of the\nEq. (15). The second term in the Eq. (13), is the\nanti-damping SOT pointing along ( ez×/vector∇µ)-axis. The\ncone angle dependence of the anti-damping term can\nbe checked by assuming an electric field along the x-\naxis when the FM precesses around the y-axis, (i.e.\n/vector m(t) = cos(θ)ˆey+sin(θ)cos(ωt)ˆex+sin(θ)sin(ωt)ˆez). In\nthis case the average of the SOT along the y-axis in one\nperiod of the precession leads to the average of the an-\ntidamping SOT that shows a sin2(θ) dependence, which\nis typical for the damping-like torques. Keeping in mind4\nthat in this section we consider vF≫∆xc, the first and\nsecond terms in Eq. (14) show that the Gilbert damp-\ning increases as the precession axis goes from in-plane ( x\nory) to out of plane ( z) direction. Furthermore, when\nthe precession axis is in-plane (e.g. along y-axis), the\ndamping rate due to the oscillation of the out of plane\ncomponent of the magnetization ( ∂mz/∂t) has a sin4(θ)\ndependence that can be ignored for low power measure-\nment of the Gilbert damping θ≪1. This leaves us with\nthe contribution from the in-plane magnetization oscilla-\ntion (∂mx∂t) only. Therefore, the Gilbert damping for\nin-plane magnetization becomes half of the case when\nmagnetization is out-of-plane. The anisotropic depen-\ndence of the Gilbert damping can be used to verify the\nexistence of the surface state of the 3DTI as well as the\nproximity induce magnetization at the interface between\na FM and a 3DTI. Finally, the third term in Eq. (14)\ndemonstrates a field like SOT with the same vector field\ncharacteristics as the Rashba-Edelstein effect.\nIII. CONCLUSION\nIn conclusion, we have developed a linear response\nNEGF framework which provides unified treatment of\nboth spin torque and damping due to SOC at interfaces.\nWe obtained the expressions for both damping and anti-\ndamping torques in the presence of a linear gradiance of\nthe electric field and adiabatic time dependence of the\nmagnetization dynamics for a helical state correspond-\ning to the surface state of a 3D topological insulator.\nWe present the exact expressions for the damping/anti-\ndamping SOT as well as the field like torques and showed\nthat, (i); Both Fermi surface and Fermi sea contribute\nsimilarly to the anti-damping SOT as well as the Hall\nconductivity and, ( ii); The Gilbert damping due to the\nsurface state of a 3D TI when the magnetization is in-\nplane is less than the Gilbert damping when it is in the\nout-of-plane direction. This dependence can be used as\na unique signature of the helicity of the surface states\nof the 3DTIs and the presence of the proximity induced\nmagnetic exchange from the FM overlayer.\nACKNOWLEDGMENTS\nWe thank Branislav K. Nikoli´ c for the fruitful discus-\nsions. F. M. and N. K. were supported by NSF PREM\nGrant No. 1205734.Appendix A: Derivation of the Density Matrix\nUsing Eqs. (2) and . (4) it is straightforwardto obtain,\nG<=(Gr\nad−Ga\nad)f−2ηf′∇µ·∂Gr\nad\n∂kGa\nad\n+i∂G<\nad\n∂E∂H\n∂tGa\nad+i∂Gr\nad\n∂E∂H\n∂tG<\nad\n+i∂G<\nad\n∂k·∇HGa\nad+i∂Gr\nad\n∂k·∇HG<\nad(A1)\nWe plug in the expression for the adiabatic lesser GF in\nequilibrium, G<\nad= 2iηfGr\nadGa\nad= (Gr\nad−Ga\nad)f, and\nobtain,\nG<= (Gr\nad−Ga\nad)f−2ηf′∇µ·∂Gr\nad\n∂kGa\nad\n+if∂(Gr\nad−Ga\nad)\n∂E∂H\n∂tGa\nad+if∂Gr\nad\n∂E∂H\n∂t(Gr\nad−Ga\nad)\n+if′(Gr\nad−Ga\nad)∂H\n∂tGa\nad+if∂(Gr\nad−Ga\nad)\n∂k·∇HGa\nad\n+if∂Gr\nad\n∂k·∇H(Gr\nad−Ga\nad). (A2)\nExpanding the terms, leads to,\nG<=/parenleftbigg\nGr\nad−Ga\nad+iGa\nad∂H(t)\n∂t∂Ga\nad\n∂E\n−iGr\nad∂H\n∂k·∇µ(x)∂Gr\nad\n∂E−i∂Ga\nad\n∂E∂H(t)\n∂tGa\nad\n+i∂Gr\nad\n∂E∂H\n∂k·∇µ(x)Gr\nad/parenrightbigg\nf\n+if′Gr\nad∇µ·∂H\n∂k(Gr\nad−Ga\nad)\n+if′(Gr\nad−Ga\nad)∂H\n∂tGa\nad, (A3)\nwhere,forthefirstandthirdlineswehaveusedtheiranti-\nHermitian forms instead. Since to calculate the density\nmatrix we integrate G<over energy, we can use integra-\ntion by parts and obtain,\nG<=/parenleftbigg\nGr\nad−Ga\nad+2iGa\nad∂H(t)\n∂t∂Ga\nad\n∂E\n−2iGr\nad∂H\n∂k·∇µ(x)∂Gr\nad\n∂E/parenrightbigg\nf\n−if′/parenleftbigg\nGr\nad∇µ·∂H\n∂kGa\nad−Gr\nad∂H\n∂tGa\nad/parenrightbigg\n= (Gr\nad−Ga\nad)f\n+i/parenleftbigg\n2Gr\nadD(H)∂Gr\nad\n∂Ef+Gr\nadD(H)Ga\nadf′/parenrightbigg\n.(A4)\nwhere we define, D=∂\n∂t− ∇µ·∂\n∂k. Finally, using the\nidentity,\n2Gr\nadD(H)∂Gr\nad\n∂E=Gr\nadD(H)∂Gr\nad\n∂E−∂Gr\nad\n∂ED(H)Gr\nad\n+∂(Gr\nadD(H)Gr\nad)\n∂E, (A5)5\nand performing the differential over the energy, we arrive\nat Eq. (6).\nAppendix B: Derivation of the Local Spin Density\nFrom Eqs. (6) and (7), the local spin density can be written as, /vectorSneq=/vectorS(1)\nneq+/vectorS(2)\nneq, where /vectorS(1)\nneqis due to the\nnonequilibrium electrons at the Fermi surface and for a two band mo del can be calculated as the following,\n/vectorS(1)\nneq≈−2ηℑ/integraldisplaydE\n2πTr\n/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig/parenleftBig\n(E−ε0+iη)1+/vectorh·/vectorσ/parenrightBig\n((E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)2−|/vectorh|2)/vectorσ\n+D(1\n(E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)1+/vectorh·/vectorσ)/vectorσ((E−ε0−iη)1+/vectorh·/vectorσ)\n(E−ε0+iη)2−|/vectorh|2/bracketrightBigg\nf′(E) (B1)\nPreforming the trace over the Pauli matrix, we obtain,\n/vectorS(1)\nneq≈ −4η/integraldisplaydE\n2π/parenleftBigg\nηD/vectorh+D(/vectorh)×/vectorh\n((E−ε0−iη)2−|/vectorh|2)(E−ε0+iη)2−|/vectorh|2)\n+4ℑ(/parenleftBig\n/vectorh·D(/vectorh)+(E−ε0−iη)(/vector∇µ·/vector v0)/parenrightBig\n(E−ε0)/vectorh\n((E−ε0−iη)2−|/vectorh|2)2((E−ε0+iη)2−|/vectorh|2))\nf′(E) (B2)\nIn the limit of small broadening, η≪ |/vectorh|, we obtain,\n/vectorS(1)\nneq≈−ηD/vectorh−/vectorh×D/vectorh+1\nη/vectorh·D(/vectorh)/vectorh\n2|/vectorh|2/parenleftBig\nf′(ε0+|/vectorh|)+f′(ε0−|/vectorh|)/parenrightBig\n−1\nη(/vector∇µ·/vector v0)/vectorh\n2|/vectorh|/parenleftBig\nf′(ε0+|/vectorh|)−f′(ε0−|/vectorh|)/parenrightBig\n(B3)\nFor the Fermi sea contribution to the nonequilibrium local spin densit y we have,\n/vectorS(2)\nneq≈ℜ/integraldisplaydE\n2πTr/bracketleftBigg\n∂\n∂E/parenleftBigg\n(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenrightBigg/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/vectorσ\n−(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig∂\n∂E/parenleftBigg\n(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenrightBigg\n/vectorσ/bracketrightBigg\nf(E). (B4)\nSimilarly, we obtain,\n/vectorS(2)\nneq≈ℜ/integraldisplaydE\n2πTr\n/bracketleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01,(E−ε0−iη)1+/vectorh·/vectorσ/bracketrightBig\n((E−ε0−iη)2−|/vectorh|2)2/vectorσ\nf(E)\n≈2ℑ/integraldisplaydE\nπD(/vectorh)×/vectorh\n((E−ε0−iη)2−|/vectorh|2)2f(E)\n≈−1\n2D(/vectorh)×/vectorh\n|/vectorh|3/parenleftBig\nf(ε0+|/vectorh|)−f(ε0−|/vectorh|)/parenrightBig\n(B5)\n[1] D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320, 1190 (2008).[2] J. Katine and E. E. Fullerton, Journal of Magnetism and\nMagnetic Materials 320, 1217 (2008).6\n[3] C. Wang et al., Nature Physics 7, 496 (2011).\n[4] I. M. Miron et al., Nature Materials 9, 230 (2010).\n[5] L. Liu et al., Phys. Rev. Lett. 109, 096602 (2012).\n[6] A.Manchon andS. Zhang, Physical ReviewB 78, 212405\n(2008).\n[7] P. Gambardella and I. M. Miron, Philos. Transact. A\nMath. Phys. Eng. Sci. 369, 3175 (2011).\n[8] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[9] D. Pesin and A. H. MacDonald, Nature Mater. 11, 409\n(2012).\n[10] I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802\n(2010).\n[11] F. Mahfouzi, J. Fabian, N. Nagaosa, and B. K. Nikoli´ c,\nPhys. Rev. B 85, 054406 (2012).\n[12] A. Manchon, Phys. Rev. B 83, 172403 (2011).[13] E. Zhao, C. Zhang, and M. Lababidi, Phys. Rev. B 82,\n205331 (2010).\n[14] N. P. Butch et al., Phys. Rev. B 81, 241301 (2010).\n[15] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Nat Phys 4,\n273 (2008).\n[16] F. Mahfouzi, B. K.Nikoli´ c, S.-H.Chen, andC.-R.Chang ,\nPhys. Rev. B 82, 195440 (2010).\n[17] F. Mahfouzi, B. K. Nikoli´ c, and N. Kioussis, Phys. Rev.\nB93, 115419 (2016).\n[18] L. Keldysh, Sov. Phys. JETP 20, 1018 (1965).\n[19] H. K. et al, Nature Nanotechnology 9, 211 (2014).\n[20] H. Haug and A.-P. Jauho, Quantum kinetics in transport\nand optics of semiconductors (Springer-Verlag, Berlin,\nADDRESS, 2008).\n[21] N. Bode et al., Phys. Rev. B 85, 115440 (2012).\n[22] L. Arrachea, Phys. Rev. B 75, 035319 (2007).\n[23] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B\n50, 5528 (1994).\n[24] B. Gaury et al., Physics Reports 534, 1 (2014)."    },    {        "title": "1308.0450v2.Spin_pumping_damping_and_magnetic_proximity_effect_in_Pd_and_Pt_spin_sink_layers.pdf",        "content": "arXiv:1308.0450v2  [cond-mat.mes-hall]  5 Apr 2016Spin pumping damping and magnetic proximity effect in Pd and P t spin-sink layers\nM. Caminale,1,2,∗A. Ghosh,2,†S. Auffret,2U. Ebels,2K. Ollefs,3F. Wilhelm,4A. Rogalev,4and W.E. Bailey1,5,‡\n1Fondation Nanosciences, F-38000 Grenoble, France\n2SPINTEC, Univ. Grenoble Alpes / CEA / CNRS, F-38000 Grenoble , France\n3Fakult¨ at f¨ ur Physik and Center for Nanointegration (CENI DE),\nUniversit¨ at Duisburg-Essen, 47057 Duisburg, Germany\n4European Synchrotron Radiation Facility (ESRF), 38054 Gre noble Cedex, France\n5Dept. of Applied Physics & Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: April 6, 2016)\nWe investigated the spin pumping damping contributed by par amagnetic layers (Pd, Pt) in both\ndirect and indirect contact with ferromagnetic Ni 81Fe19films. We find a nearly linear dependence\nof the interface-related Gilbert damping enhancement ∆ αon the heavy-metal spin-sink layer thick-\nnesses t Nin direct-contact Ni 81Fe19/(Pd, Pt) junctions, whereas an exponential dependence is o b-\nservedwhenNi 81Fe19and(Pd, Pt)areseparated by3nmCu. Weattributethequasi-l inear thickness\ndependence to the presence of induced moments in Pt, Pd near t he interface with Ni 81Fe19, quan-\ntified using X-ray magnetic circular dichroism (XMCD) measu rements. Our results show that the\nscattering of pure spin current is configuration-dependent in these systems and cannot be described\nby a single characteristic length.\nI. INTRODUCTION\nAs a novel means of conversion between charge- and\nspin-currents,spinHallphenomenahaverecentlyopened\nup new possibilities in magneto-electronics, with poten-\ntial applications in mesocale spin torques and electrical\nmanipulation of domain walls1–9. However, several as-\npects of the scattering mechanisms involved in spin cur-\nrent flow across thin films and interfaces are not entirely\nunderstood. Fundamental studies of spin current flow in\nferromagnet/non-magnetic-meal (F/N) heterostructures\nin the form of continuous films have attempted to iso-\nlate the contributions of interface roughness, microstruc-\nture and impurities10–12. magnet/non-magnetic-meal\n(F/N) heterostructures in the form of continuous films\nhave attempted to isolate the contributions of interface\nroughness, microstructure and impurities10–12. Proto-\ntypical systems in this class of studies are Ni 81Fe19/Pt\n(Py/Pt)3,5–7,13–18andNi 81Fe19/Pd(Py/Pd)8,11,14,16,19,20\nbilayers. Inthesesystems, PtandPdareemployedeither\nas efficient spin-sinks or spin/charge current transform-\ners, in spin pumping and spin Hall experiments, respec-\ntively. Pd and Pt are metals with high paramagnetic\nsusceptibility and when placed in contact with a ferro-\nmagnetic layer (eg. Py, Ni, Co or Fe) a finite magnetic\nmoment is induced at the interface by direct exchange\ncoupling21–24.\nThe role of the magnetic proximity effect on interface\nspin transport properties is still under debate. Zhang et\nal.25havereportedthat induced magnetic momentsin Pt\nand Pd films in direct contact with Py correlate strongly\nreduced spin Hall conductivities. This is ascribed to a\nspin splitting of the chemical potential and on the energy\ndependence of the intrinsic spin Hall effect. In standard\nspin pumping theory26, possible induced moments in N\nare supposed to be a priori included in calculations of\nthe spin-mixing conductance g↑↓of a F|N interface27,28,which tends to be insensitive to their presence.\nRecent theoretical works, on the other hand, propose\nthe need of a generalized spin pumping formalisms in-\ncluding spin flip and spin orbit interaction at the F |N\ninterface, in order to justify discrepancies between exper-\nimental and calculated values of mixing conductance29,30.\nAt present, it is still an open issue whether and how\nproximity-induced magnetic moments in F/N junctions\narelinked to the varietyofthe spin-transportphenomena\nreported in literature10,17,31.\nHere, we present an experimental study of the pro-\ntotypical systems: Py/Pd, Pt and Py/Cu/Pd, Pt het-\nerostructures. The objective of our study is to address\nthe role of proximity induced magnetic moments in spin\npumpingdamping. Tothisend, weemployedtwocomple-\nmentary experimental techniques. X-ray magnetic circu-\nlar dichroism (XMCD) is an element sensitive technique\nwhich allows us to quantify any static proximity-induced\nmagnetic moments in Pt and Pd. Ferromagnetic reso-\nnance(FMR) measurementsprovideindirectinformation\non the spin currents pumped out the Py layer by the pre-\ncessing magnetization, through the characterization of\nthe Pd, Pt thickness dependence of the interface-related\nGilbert damping α. In Fig. 3(Sec.IIIB), comparative\nmeasurements in Py/Cu/N and Py/N structures show a\nchange of the N thickness dependence of ∆ α(tN) from\nan exponential to a linear-like behavior. A change in\n∆α(tN) indicates a transformation in the spin scattering\nmechanism occurring at the interface, ascribed here to\nthe presence of induced moments in directly exchange\ncoupled F/N systems. Theoretical works predicted a\ndeviation from a conventional N-thickness dependence\nwhen interface spin-flip scattering is considered in the\npumping model29,30, howeverno functional form waspro-\nvided. For Py/N systems, we find that the experimental\nthickness dependence cannot be described by standard\nmodels16,26,32, but rather a linear function reproduces2\nthe data to a better degree of accuracy, by introducing\na different characteristic length. We speculate that the\nspatial extent of spin current absorption in F/N systems\nshows an inverse proportionality to interfacial exchange\ncoupling energy, obtained from XMCD, as proposed be-\nfore for spin polarized, decoupled interfaces in F 1/Cu/F 2\nheterostructures14.\nII. EXPERIMENT\nThe heterostructures were fabricated by DC mag-\nnetron sputtering on ion-cleaned Si/SiO 2substrates in\nthe form of substrate/seed/multilayer/cap stacks, where\nTa(5nm)/Cu(5nm) bilayerwasemployedas seed. Ta/Cu\nis employed to promote <111>growth in Py and subse-\nquent fcc layers (Pd, Pt), and Ta is known to not affect\nthe dampingstrongly17,32,33. Different stacksweregrown\nasmultilayer for each measurement.\nFor FMR measurements, we have multilayer =\nPy(tF)/N(tN), Py(tF)/Cu(3nm)/N( tN) with N = Pd,\nPt; an Al(3nm) film, oxidized in air, was used as\ncap. The smallest N layer thickness tNdeposited is\n0.4nm, the maximum interdiffusion length observed\nfor similar multilayers34. Samples with multilayer =\nPy(tN)/Cu(3nm) and no sink layer were also fabricated\nas reference for evaluation of the Gilbert damping en-\nhancement due tothe Pd orPtlayer. The tN-dependence\nmeasurementsofFMRweretakenforPythicknesses tF=\n5and 10nm. Results from the tF= 10nm data set are\nshown in Appendix A. Measurements of the FMR were\ncarried out at fixed frequency ωin the 4-24 Ghz range,\nby means of an in-house apparatus featuring an external\nmagnetic field up to 0.9T parallel to a coplanar waveg-\nuide with a broad center conductor width of 350 µm.\nFor XMCD measurements, given the low X-ray ab-\nsorption cross-sectionpresented by Pt and Pd absorption\nedges, a special set of samples was prepared, consisting\nof 20 repeats per structure in order to obtain sufficiently\nhighsignal-to-noiseratio. Inthiscase,wehave multilayer\n= [Py(5nm)/N] 20, with N = Pd(2.5nm) and Pt(1nm);\nCu(5nm)/Py(5nm)/Al(3nm) was deposited as cap. The\nPt and Pd thicknesses were chosen to yield a damping\nenhancement equal about to half of the respective satu-\nration value (as it will be shown later), i.e. a thicknesses\nfor which the F/N interface is formed but the damp-\ning enhancement is still increasing. XMCD experiments\nwere carried out at the Circular Polarization Beamline\nID-12 of the European Synchrotron Radiation Facility\n(ESRF)35. Measurementsweretakenintotalfluorescence\nyield detection mode, at grazing incidence of 10◦, with\neither left or right circular helicity of the photon beam,\nswitching a 0.9T static magnetic field at each photon en-\nergy value (further details on the method are in Ref.22).\nNo correction for self-absorption effects is needed; how-\never XMCD spectra measured at the L 2,3edges of Pd\nhave to be corrected for incomplete circular polarization\nrate of monochromatic X-rays which is 12% and 22% at/s49/s49/s46/s53/s52 /s49/s49/s46/s53/s54 /s49/s49/s46/s53/s56 /s49/s49/s46/s54/s48 /s49/s49/s46/s54/s50 /s49/s51/s46/s50/s54 /s49/s51/s46/s50/s56 /s49/s51/s46/s51/s48 /s49/s51/s46/s51/s50 /s49/s51/s46/s51/s52/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s116/s32/s88/s65/s83\n/s32/s65/s117/s32/s88/s65/s83\n/s80/s100/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s80/s116\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54\n/s32\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\n/s51/s46/s49/s52 /s51/s46/s49/s54 /s51/s46/s49/s56 /s51/s46/s50/s48 /s51/s46/s50/s50 /s51/s46/s51/s48 /s51/s46/s51/s50 /s51/s46/s51/s52 /s51/s46/s51/s54 /s51/s46/s51/s56/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53\n/s32/s80/s100/s32/s88/s65/s83\n/s32/s65/s103/s32/s88/s65/s83/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s45/s48/s46/s48/s57/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\nFIG. 1. (Color online) X-ray absorption (XAS, left axis) and\nmagnetic circular dichroism (XMCD, right axis) spectra at\nthe L-edges of Pt (top panel) and Pd (bottom panel) for\n[Py(5nm)/Pt(1nm)] 20and [Py(5nm)/Pd(2.5nm)] 20multilay-\ners. The dashed traces represent XAS spectra at L-edge of\nAg and Au used as background of Pd and Pt, respectively, to\nextract the values of induced magnetic moment reported in\nTab.I.\nL3and L2, respectively. The circular polarization rate is\nin excess of 95 % at the L 2,3edges of Pt.\nIII. RESULTS AND ANALYSIS\nIn order to study how the proximity-induced mag-\nnetic moments may affect the absorption of spin-currents\nthrough interfaces, the static moment induced in Pt, Pd\nlayers in direct contact with ferromagnetic Py in char-\nacterized first, by means of XMCD. The value of the\ninduced moment extracted for the two Py/N systems is\nusedtoestimatetheinterfacialexchangeenergyactingon\nthe two paramagnets. Afterwards, the dynamic response\nofthe magnetization is addressedby FMR measurements\nin Py/N(direct contact) and Py/Cu/N(indirect contact)\nheterostructures. From FMR measurements carried out\non both configurations as a function of N thickness, the\ndamping enhancement due to the presence of the spin-\nsink layers Pt and Pd is obtained from the frequency-\ndependence of the FMR linewidth. The relation between\nthe static induced moment and the spin pumping damp-\ning is discussed by comparing the results of the direct\nwith indirect contact systems.3\n-0.06-0.04-0.020.00XMCD\n11.5911.5811.5711.5611.55\nPhoton Energy (keV)1.0\n0.5\n0.0\n3210tCu (nm) Area\n 1 nm Cu\n 0.5 nm Cu\n 0 nm Cu\nFIG. 2. (Color online) XMCD spectra at the L 3edge of Pt\nfor [Py(5nm)/Cu(t Cu)/Pt(1nm)] 15, with t Cu= 0, 0.5 and\n1nm. As inset the ares of the peak is plot as a function of Cu\nthickness.\nA. XMCD: Probing the induced magnetic moment\nIn Fig.1we report X-ray absorption (XAS) and mag-\nnetic circular dichroism (XMCD) spectra at the L2,3\nedges of Pt (top panel) and Pd (bottom panel) taken on\nPy(5nm)/Pt(1nm) |20and Py(5nm)/Pd(2.5nm) |20, re-\nspectively. Rather intense XMCD signals have been de-\ntected at both Pt and Pd L 2,3edges, showing unambigu-\nously that a strong magnetic moment is induced by di-\nrect exchange coupling at the Py |N interface. The static\ninduced moment is expected to be ferromagnetically cou-\npled with the magnetization in Py21. From the integrals\nof XMCD spectra, the induced magnetic moment on the\nPt, Pd sitesis determined byapplying the sum rulesasin\nRef.22(and references therein). In Py/Pd(2.5nm) |20, Pd\natoms bear a moment of 0.12 µB/at, averaged over the\nwhole volume of the volume, with an orbital-to-spinratio\nmL/mS= 0.05. In Py/Pt(1nm) |20, a magnetic moment\n0.27µB/at is found on Pt, comparable to that reported\nfor Ni/Pt epitaxial multilayers23, with a relatively high\norbitalcharacter mL/mS= 0.18, ascomparedwithPdin-\nduced moment. The large difference in volume-averaged\ninduced moment per atom comes from the different film\nthickness, hence volume, for Pt and Pd. Assuming that\nthe induced magnetic moment is confined to the first\natomic layers at the interface with Py23,24, one could\nestimate 0.32 µB/at for Pd and 0.30 µB/at for Pt36.\nWhen a3nm thick Cu interlayeris introducedbetween\nPy and N, a two ordersofmagnitude smaller induced mo-\nment (0.0036 µB/at) was found for 2.5nm Pd22, while Pt\nshowed an XMCD signal of the order of the experimen-\ntal sensitivity, ∼0.5·10−3µB/at. In Fig. 2XMCD\nspectra at the L 3edge of Pt are shown for Cu interlayer\nthicknesses 0, 0.5 and 1nm. For 0.5nm Cu the integral\nof XMCD signal at the L 3edge shrinks to 30%, while\nfor 1nm it is reduced to zero within experimental error.\nThis result could be explained either by a 3d growth of\nthe Cu layer, allowing a fraction of the Pt layer to be in\ndirect contact with Py for Cu coverages of 0.5nm, or byNχmol37S37N0abulktN/angbracketleftM/angbracketrightMiJex\n[cm3/mol][1/eV·at][nm][nm][µB/at][µB/at][meV]\n10−4\nPd5.5±0.29.30.83±0.030.3892.50.1160.3242\nPt1.96±0.13.70.74±0.040.3921.00.270.30109\nTABLE I. Spin-sink layer N properties in Py/N heterostruc-\ntures: experimental molar susceptibility χmolat 20◦C; den-\nsity of states N0calculated from tabulated χmol; Stoner pa-\nrameter S from Ref.37; bulk lattice parameter a; layer thick-\nnessestN; volume averaged induced magnetic moment /angbracketleftM/angbracketright\nfrom XMCD measurement in Fig. 1; interface magnetic mo-\nmentMi36; Py|N interfacial exchange energy per interface\natomJex(Eq.1).\ndiffusion of magnetic Ni atoms in Cu on a scale shorter\nthan 1nm. The film then becomes continuous, and at\n1nm coverage, no direct exchange coupling takes place\nbetween Py and Pt layers. For FMR measurements pre-\nsented in the followingsection, a 3nm thick Cu interlayer\nis employed, reducing also any possible indirect exchange\ncoupling.\nFrom the values of induced moments in Pd and Pt,\nwe can make a step forward and estimate the interfacial\nexchange coupling energies for the two cases. Equating\ninteratomic exchange energy Jexand Zeeman energy for\nan interface paramagnetic atom, we have (see Appendix\nB1for the derivation)\nJex=1\n2/angbracketleftM/angbracketright\nµBN0StN\nti(1)\nwhere/angbracketleftM/angbracketrightis the thickness-averaged paramagnetic\nmoment, N0is the single-spin density of states (in\neV−1at−1), S is the Stoner factor and ti= 2∗a/√\n3\nis the polarized interface-layer thickness36. The 1/2fac-\ntor accounts for the fact that in XMCD measurements\nthe N layer has both interfaces in contact with F. Un-\nder the simplifying assumption that all the magnetic mo-\nment is confined to the interface N layer and assuming\nexperimental bulk susceptibility parameters for χv, we\nobtainJPd\nex= 42 meV for Pd and JPt\nex= 109 meV for\nPt (results and properties are summarized in Tab. I).\nHere the difference in estimated Jex, despite roughly\nequalMi, comes from the larger Stoner factor S for\nPd. A stronger interfacial exchange energy in Pt de-\nnotes a stronger orbital hybridization, yielding possibly\na higher orbital character of the interfacial magnetic mo-\nment in the ferromagnetic Py counterpart21. For com-\nparison, we consider the interatomic exchange param-\netersJexin ferromagnetic Py and Co, investigated in\nRef.14.Jexis estimated from the respective Curie tem-\nperatures TC, through Jex≃6kBTC/(m/µB)2, wherem\nis the atomic moment in µB/at (see Appendix B2). Ex-\nperimental Curie temperatures of 870K and 1388K give\nJCo\nex=293meV for Co and JPy\nex=393meV for Py, which\nare of the same order of the value calculated for Pt (de-\ntails about calculation in Appendix B2).4\n1.0\n0.5\n0.0\n14121086420\ntPt (nm) Py/Pt\n Py/Cu/Pt12\n10\n8\n6\n4\n2\n0\n468\n12468\n10\ntPt (nm) Py/Pt\n tc = 2.4 nm\n Py/Cu/Pt\n λα = 1.8 nm\n1.0\n0.5\n0.0∆α / ∆α0\n14121086420\ntPd (nm) Py/Pd\n Py/Cu/Pd6\n5\n4\n3\n2\n1\n0∆α (x103)\n468\n12468\n10\ntPd (nm) Py/Pd\n tc = 5.0 nm\n Py/Cu/Pd\n λα = 5.8 nma) b)\nc) d)\nFIG. 3. (Color online) Damping enhancement ∆ α, due to\npumped spin current absorption, as a function of thickness tN\nfor Py(5nm)/N and Py(5nm)/Cu(3nm)/N heterostructures,\nwith N = Pd( tN) (panels a,c), Pt( tN) (panels b,d). Solid\nlines result from a fit with exponential function (Eq. 2) with\ndecay length λα. Dashed lines represents instead a linear-\ncutoff behavior (Eq. 3) fortN< tc. Please notice in panels a,\nc the x-axis is in logarithmic scale. In panels b, c the dampin g\nenhancement is normalized to the respective saturation val ue\n∆α0.\nIn the following, the effect of these static induced mo-\nments on the spin pumping damping of the heterostruc-\ntures characterized will be discussed.\nB. FMR: damping enhancement\nThe main result of our work is now shown in Figure\n3. In Fig. 3the damping enhancement ∆ αis plotted as\na function of the spin-sink layer thickness tN, for Py/Pd,\nPy/Cu/Pd (panels a, c) and Py/Pt, Py/Cu/Pt (panels\nb, d). The enhancement ∆ αis compared with the damp-\ningαof a reference structure Py(5nm)/Cu, excluding\nthe sink layer N. Each value of αresults from established\nanalysis of the linewidth of 11 FMR traces13,14, employ-\ning ag-factor equal to 2.09 as a constant fit parameter\nfor all samples.\nIn Py/Cu/N systems (Fig. 3, green square markers),\n∆αrises with increasing tNthickness to similar satura-\ntion values ∆ α0= 0.0027, 0.0031 for Pd and Pt, but\nreachedondifferentlengthscales,giventhedifferentchar-Ng↑↓\neff(Py|Cu/N) λαg↑↓\neff(Py|N)tc\n[nm−2][nm] [nm−2][nm]\nPd 7.2 5.8±0.2145.0±0.3\nPt 8.3 2.4±0.1321.8±0.2\nTABLE II. Mixing conductance values extracted from the\ndamping enhancement ∆ αat saturation in Fig. 3, and re-\nspective length scales (see text for details).\nacteristic spin relaxation lengths of the two materials.\nFrom the saturation value, an effective mixing conduc-\ntanceg↑↓\neff(Py|Cu/N) = 7 .2−8.3nm−2is deduced in the\nframeworkof standardspin pumping picture13,17,19, with\nPy saturation magnetization µ0Ms= 1.04T. The fact\nthat the spin-mixing conductance is not material depen-\ndent indicates that similar Cu |N interfaces are formed.\nThe thickness dependence is well described by the expo-\nnentialfunction14,20\n∆α(tN) = ∆α0(1−exp(−2tN/λN\nα)) (2)\nas shown by the fit in Fig. 3a-b (continuous line). As\na result, exponential decay lengths λPt\nα= 1.8nm and\nλPd\nα= 5.8nm are obtained for Pt and Pd, respectively.\nWhen the Pt, Pd spin-sink layers come into direct con-\ntact with the ferromagnetic Py, the damping enhance-\nment ∆α(tN) changes dramatically. In Py/N systems\n(Fig.3a-b, trianglemarkers),the damping saturationval-\nues become ∆ αPt\n0= 0.0119 and ∆ αPd\n0= 0.0054 for Pt\nand Pd, respectively a factor ∼2 and∼4 larger as com-\npared to Py/Cu/N. Within the spin-pumping descrip-\ntion, a largerdamping enhancement implies a largerspin-\ncurrentdensitypumped outoftheferromagnetacrossthe\ninterface and depolarized in the sink.\nIn Py/N heterostructures, because of the magnetic\nproximity effect, few atomic layers in N are ferromagnet-\nically polarized, with a magnetic moment decaying with\ndistance from the Py |N interface. The higher value of\ndamping at saturation might therefore be interpreted as\nthe result of a magnetic bi-layer structure, with a thin\nferromagnetic N characterized by high damping αN\nhigh\ncoupled to a low damping αF\nlowferromagnetic Py38. To\ninvestigate whether damping is of bi-layer type, or truly\ninterfacial, in Fig 4we show the tFthickness dependence\nof the damping enhancement ∆ α, for a Py( tF)/Pt(4 nm)\nseries of samples. The power law thickness dependence\nadheres very closely to t−1\nF, as shown in the logarithmic\nplot. The assumption of composite damping for syn-\nchronous precession, as ∆ α(t1) = (α1t1+α2t2)/(t1+t2),\nshown here for t2= 0.25nm and 1.0nm, cannot follow\nan inverse thickness dependence over the decade of ∆ α\nobserved. Damping is therefore observed to have a pure\ninterfacial character.\nIn this case, the mixing conductances calculated from\nthe saturation values are g↑↓\neff(Py|Pd) = 14nm−2and\ng↑↓\neff(Py|Pt) = 32nm−2. From ab initio calculations\nwithin a standard spin-pumping formalism in diffusive\nfilms10,29, it is found g↑↓\neff(Py|Pd) = 23nm−2for Pd and5\n10 3 4 56789 20 30 40\ntF (nm)110\n2345678920∆α(10−3)Py(tF)/Pt(4)\ninterfacial, K=0.055\nbilayer, t2=0.25 nm\n             t2=1.0 nm\n26101520 30\ntF (nm)051015202530α(10−3)Py(t)/Pt(4nm)\nPy(t)\nFIG. 4. Logarithmic plot of the damping enhancement ∆ α\n(triangle markers) as a function of the Py layer thickness tF,\nin Py(t F)/Pt(4nm). Solid and dashed lines represents, re-\nspectively, fits according to the spin pumping ( interfacial )\nmodel ∆ α=Kt−1\nFand to a αlow(tF)/αhigh(t2)bilayermodel,\nwitht2= 0.25,1.0nm.Inset: Gilbert damping αfor Py(t F)\n(square markers) and Py(t F)/Pt(4nm) (round markers).\ng↑↓\neff(Py|Pt) = 22nm−2for Pt. Theoretical spin mixing\nconductance from a standard picture does reproduce the\nexperimental order of magnitude, but it misses the 2.3\nfactor between the Py |Pt and Py |Pd interfaces. Beyond\na standard pumping picture, Liu and coworkers29intro-\nduce spin-flipping scattering at the interface and calcu-\nlate from first principles, for ideal interfaces in finite dif-\nfusive films: g↑↓\neff(Py|Pd) = 15nm−2, in excellent agree-\nment with the experimental value here reported for Pd\n(Tab.II), andg↑↓\neff(Py|Pt) = 25nm−2. Zhang et al.10\nsuggest an increase up to 25% of the mixing conductance\ncan be obtained by introducing magnetic layers on the\nPt side. The results here reported support the emerging\nidea that a generalized model of spin pumping including\nspin-orbit coupling and induced magnetic moments at\nF|N interfaces may be required to describe the response\nof heterostructures involving heavy elements.\nThe saturation value of damping enhancement at ∆ α0\nas a function of the Cu interlayer thickness is shown in\nFig.5to follow the same trend of the XMCD signal\n(dashed line), reported from Fig. 2. Indeed, it is found\nthat the augmented ∆ α0in Py/N junctions is drasti-\ncally reduced by the insertion of 0.5nm Cu at the Py |N\ninterface17, and the saturation of the Py/Cu/N configu-\nration is already reached for 1nm of Cu interlayer. As\nsoonasacontinuousinterlayerisformedandnomagnetic\nmoment is induced in N, ∆ α0is substantially constant\nwith increasing Cu thickness.\nThe N-thickness dependence of ∆ α(tN) in Py/N sys-\ntems before saturation is addressed in the following. At\nvariance with the Py/Cu/N case, the thickness depen-\ndence of ∆ αis not anymore well described by an expo-1.0\n0.5\n0.0\n3210\ntCu (nm)1.0\n0.5\n0.0L3 XMCD area (norm.)Pt(3nm)\n XMCD Pt\n3210\ntCu (nm)1.0\n0.5\n0.0∆α0 (norm.)Pd(7nm)\n XMCD Pt\nFIG. 5. (Color online) Normalized damping enhancement ∆ α\n(left axis), due to spin pumping, as a function of interlayer\nthickness tCufor Py(5nm)/Cu( tCu)/N heterostructures, with\nN = Pd(7nm), N = Pt(3nm). The dashed line represents the\nXMCD signal (right axis) reported from inset in Fig. 2.\nnential behavior, as an exponential fit (with exponential\ndecay length as only free fit parameter) fails to repro-\nduce the increase of ∆ αtowards saturation (solid lines\nin Fig.3a-b). More rigorous fitting functions employed\nin spin pumping experiments, within standard spin trans-\nport theory16,26,32, cannot as well reproduce the experi-\nmental data (see Appendix. A). It is worth mentioning\nthat the same change of trend between the two configu-\nrations was observed for the same stacks with a 10nm\nthick Py layer (data shown in Appendix A, Fig.7). A\nchange of the functional dependence of ∆ αontNre-\nflects a change in the spin-depolarization processes the\npumped spin current undergoes, as for instance shown in\nRef.30when interfacial spin-orbit coupling is introduced\nin the spin-pumping formalism. Experimentally, a linear\nthickness dependence with sharp cutoff has been shown\nto characterize spin-current absorption in spin-sink lay-\ners exhibiting ferromagnetic order at the interface, as re-\nported for F 1/Cu/F 2(tF2) junctions with F = Py, Co,\nCoFeB14. Given the presence of ferromagnetic order in\nN at the interface of F/N structures, the data are tenta-\ntively fit with a linear function\n∆α= ∆α0tN/tN\nc (3)\nThis linear function better reproduce the sharp rise of\n∆α(dashed lines in Fig. 3a-b) and gives cutoff thick-\nnessestPt\nc= 2.4±0.2nm and tPd\nc= 5.0±0.3nm for Pt\nand Pd, respectively. The linearization is ascribed to the\npresence of ferromagnetic order in the paramagnetic Pd,\nPt spin-sink layers at the interface with the ferromag-\nnetic Py. The linear trend extends beyond the thickness\nfor which a continuous layer is already formed (less than\n1nm), and, especially for Pd, far beyond the distance\nwithin the non-uniform, induced moment is confined (up\nto0.9nm). InRef.14, thecutoff tcinF/Cu/Fheterostruc-\nturesis proposedto be on the orderofthe transversespin\ncoherence length λJin ferromagnetically ordered layers.\nλJcan be expressed in terms of the exchange splitting6\n6\n5\n4\n3\n2\n1\n0tc(nm)\n2520151050\n1/Jex(eV-1)PtPd\nPy\nCo\nFIG. 6. Effect of direct exchange strength on length scale of\nspin current absorption. Cutoff thickness tcextracted from\nthe ∆α(tN) data in Fig. 3as a function of reciprocal interfa-\ncial exchange energy 1 /Jexextracted from XMCD in Fig. 1.\nLabels are given in terms of Jex. The Co and Py points are\nfrom Ref.14.\nenergyJex,\nλJ=hvg\n2Jex(4)\nwherevgis the electronic group velocity at the Fermi\nlevel. This form, found from hot-electron Mott\npolarimetry1, is expressed equivalently for free electrons\nasπ/|k↑−k↓|, which is a scaling length for geometrical\ndephasing in spin momentum transfer2. Electrons which\nenter the spin-sink at E Fdo so at a distribution of angles\nwith respect to the interface normal, traverse a distribu-\ntionofpathlengths, andprecessbydifferentangles(from\nminority to majority or vice versa ) before being reflected\nback into the pumping ferromagnet. For a constant vg,\nit is therefore predicted that tcis inversely proportional\nto the exchange energy Jex.\nIn Figure 6we plot the dependence of the cutoff thick-\nnesstN\ncupontheinverseoftheestimatedexchangeenergy\nJex(Tab.I),asextractedfromtheXMCDmeasurements.\nA proportionality is roughly verified, as proposed for the\ntransverse spin coherence length across spin polarized in-\nterfaces. Under the simplistic assumption that tc=λJ,\nfrom the slope of the line we extract a Fermi velocity\nof∼0.1·106m/s (Eq.4), of the order of magnitude ex-\npectedforthematerialsconsidered39,40. Thesedatashow\nthat, up to a certain extent, length scale for spin-current\nscattering shares common physical origin in ferromag-\nnetic layers and paramagnetic heavy-metals, such as Pd\nand Pt, under the influence of magnetic proximity effect.\nThis unexpected results is observed in spite of the fact\nthat F 1/Cu/F 2and F/N systems present fundamental\ndifferences. In F/N structure, the induced moment in N\nis expected to be directly exchange coupled with the fer-\nromagneticcounterpart. Whereasin F 1/Cu/F 2, themag-\nnetic moment in F 2(off-resonance) are only weakly cou-\npled with the precession occurring in F 1(in-resonance),\nthrough spin-orbit torque and possible RKKY interac-tion. Magnetization dynamics in N might therefore be\nexpected with its own pumped spin current, albeit, to\nthe best of our knowledge, no experimental evidence\nof a dynamic response of proximity induced moments\nwas reported so far. From these considerations and the\nexperimental findings, counter-intuitively the proximity-\ninduced magnetic moments appear not to be involved in\nthe production of spin current, but rather to contribute\nexclusively with an additional spin-depolarization mech-\nanism at the interface.\nIV. CONCLUSIONS\nWe have investigated the effect of induced magnetic\nmoments in heavy metals at Py/Pt and Py/Pd inter-\nfaces on the absorption of pumped spin currents, by\nanalyzing ferromagnetic resonance spectra with varying\nPt, Pd thicknesses. Static, proximity-induced magnetic\nmoments amount to 0.32 and 0.3 µB/atom in Pd and\nPt, respectively, at the interface with Py, as deduced\nfrom XMCD measurements taken at the L 2,3edges. We\nhave shown that when the proximity induced moment\nin Pt and Pd is present, an onset of a linear-like thick-\nness dependence of the damping is observed, in con-\ntrast with an exponential trend shown by Py/Cu/Pd\nand Py/Cu/Pt systems, for which no induced moment\nis measured. These results point to the presence of an\nadditional spin-flip process occurringat the interface and\nto a change of the character of spin current absorption\nin the ultrathin Pd and Pt paramagnets because of the\ninterfacial spin polarization. The range of linear increase\nis proposed to be inversely proportional to the interfa-\ncial exchange energy in Py/Pt and Py/Pd, inferred from\nXMCD data.\nWEB acknowledges the Universit´ e Joseph Fourier and\nFondation Nanosciences for his research stay at SPIN-\nTEC. This work was supported in part by the U.S. NSF-\nECCS-0925829 and the EU EP7 NMP3-SL-2012-280879\nCRONOS. MC is financed by Fondation Nanosciences.\nAppendix A: N-thickness dependence\nIn order to confirm the results presented in the\nmanuscript, additional sample series with thicker Py\nlayer were fabricated and measured. The experimental\nresults for 10nm thick Py layer are shown in Fig.s 7and\n8forPdandPt, respectively. Wehavepresentedthedata\nhere, rather than including them with the other plots in\nFigure3, to keep the figures from being overcrowded. As\nexpected when doubling the ferromagnet thickness, the\nsaturation values ∆ α0are about half of those measured\nfor 5nm Py (Fig. 3). Confirming the data presented in\nthe manuscript, it is observedagaina changeof thickness\ndependence of ∆ α(tN), fromexponential for Py/Cu/N\n(solid lines; Eq. 2,λα= 4.8nm and 1.4nm for Pd and Pt7\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPd (nm)Py/Pd\nPy/Cu/Pd\n Eq. 3\n,  Eq. 2\n [16, 32] - ρPd=1.4E-7\n [16, 32] - ρPd->ρ(tPd)\nFIG. 7. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPd\nfor Py(10nm)/Pd and Py(10nm)/Cu(3nm)/Pd heterostruc-\ntures. Solid lines result from a fit with exponential functio n\n(Eq.2) with decay λα. Dashed lines represents instead a\nlinear-cutoff behavior (Eq. 3) fortPd< tc. Short-dash and\npoint-dash traces are fit to the data, employing equations\nfrom standard spin transport theory (see text for details)16,32.\nIn bottom panel, ∆ αis normalized to the respective satura-\ntion value.\nrespectively) to linear-like for Py/N (dashed lines; Eq. 3,\ntc= 5.3nm and 2nm for Pd and Pt respectively).\nThe experimental data are also fitted with a set of\nequations derived from standard theory of diffusive spin\ntransport16,26,32, describing the the dependence of ∆ α\non the thickness of adjacent metallic layers (either N or\nCu/N in our case) as follow\n∆α=γ¯h\n4πMstFMg↑↓\n1+g↑↓/gx\next(A1)\nwith (Eq. 7 in Ref.16, and Eq. 6 in Ref.32)\ngN\next=gNtanhtN/λN\nsd\ngCu/N\next=gCugCucothtN/λN\nsd+gNcothtCu/λCu\nsd\ngCucothtN/λN\nsdcothtCu/λCu\nsd+gN(A2)\nwheregx=σx/λx\nsd,σxandλx\nsdare the electrical conduc-\ntivity and spin diffusion length of the non magnetic layer\nx. For the thin Cu layer, we used a resistivity ρCu=\n1×107Ωm and a spin diffusion length λCu\nsd= 170nm32.\nFor the Pt and Pd layers, two fitting models in which the\nconductivity of the films is either constant or thickness\ndependent areconsidered, as recently proposed by Boone\nand coworkers16. The values of conductivity, as taken di-\nrectly from Ref.16, will influence the spin diffusion length\nλN\nsdand spin mixing conductance g↑↓resulting from the\nfit, but will not affect the conclusions drawn about the\noverall trend. When a constant resistivity is used (short-\ndash, blue lines), the model basically corresponds to the6\n5\n4\n3\n2\n1\n0∆α (x10-3)\n345678\n12345678\n10\ntPt (nm) Py/Pt\n Py/Cu/Pt\n Eq. 3\n, Eq. 2\n [16, 32] - ρPt=1.7E-7\n [16, 32] - ρPt->ρ(tPt)\nFIG. 8. Damping enhancement ∆ α, due to pumped\nspin current absorption, as a function of thickness tPtfor\nPy(10nm)/Pt and Py(10nm)/Cu(3nm)/Pt heterostructures.\nSolid lines result from a fit with exponential function (Eq. 2)\nwith decay λα. Dashed lines represents instead a linear-cutoff\nbehavior (Eq. 3) fortPt< tc. Short-dash and point-dash\ntraces are fit to the data, employing equations from standard\nspin transport theory (see text for details)16,32. In bottom\npanel, ∆ αis normalized to the respective saturation value.\nsimple exponential function in Eq. 2. It nicely repro-\nduces the data in the indirect contact case (Py/Cu/N)\nfor both Pd (Fig. 7) and Pt (Fig. 8), but it fails to fit\nthe direct contact (Py/N) configuration. When a thick-\nness dependent resistivity of the form ρN=ρb\nN+ρs\nN/tNis\nused (dash-point, cyan lines)16, in Py/Cu/N systems, no\nsignificant difference with the other functions is observed\nfor Pt, while for Pd a deviation from experimental trend\nis observed below 1.5nm. In Py/Nsystems, the fit better\ndescribes the rise at thicknesses shorter than the charac-\nteristic relaxation length, while deviates from the data\naround the saturation range.\nModels from standard spin transport theory cannot\nsatisfactorily describe the experimental data for the di-\nrect contact Py/N systems. For this reason a different\nmechanismforthe spin depolarizationprocesseshasbeen\nproposed, considering the presence of induced magnetic\nmoments in N in contact with the ferromagnetic layer.\nAppendix B: Interfacial interatomic exchange\n1. Paramagnets\nWe will show estimates for exchange energy based\non XMCD-measured moments in [Py/(Pt, Pd)] repeatsu-\nperlattices. Calculations of susceptibility are validated\nagainst experimental data for Pd and Pt. Bulk suscepti-\nbilities will be used to infer interfacial exchange parame-8\ntersJi\nex.\na. Pauli susceptibility For an itinerant electron sys-\ntem characterized by a density of states at the Fermi\nenergyN0, if an energy ∆ Esplits the spin-up and spin-\ndown electrons, the magnetization resulting from the\n(single-spin) exchange energy ∆ Eis\nM=µB/parenleftbig\nN↑−N↓/parenrightbig\n= 2µBN0S∆E(B1)\nwhereN0is the density of states in # /eV/at,Sis the\nStoner parameter, and 2∆ Eis the exchange splitting in\neV. Moments are then given in µB/at. Solving for ∆ E,\n∆E=M\n2µBN0S(B2)\nIf the exchange splitting is generated through the ap-\nplication of a magnetic field, ∆ E=µBH,\nµBH=M\n2µBN0S(B3)\nand the dimensionless volume magnetic susceptibility\ncan be expressed\nχv≡M\nH= 2µ2\nBN0S (B4)\nIn this expression, the prefactor can be evaluated\nthrough\nµ2\nB= 59.218 eV ˚A3(B5)\nso with N0[=]/eV/at, χvtakes units of volume per\natom, and is then also called an atomic susceptibility, in\ncm3/at, as printed in Ref37.\nb. Molar susceptiblity Experimentalvaluesaretabu-\nlated as molar susceptibilities. The atomic susceptibility\nχvcanbe contrastedwith the masssusceptibility χmand\nmolar susceptibility χmol\nχmass=χv\nρχmol=ATWT\nρχv(B6)\nwhere ATWT is the atomic weight (g/mol) and ρis\nthe density (g/cm3). These have units of χmass[=]cm3/g\nandχmol[=]cm3/mol. The molar susceptibility χmolis\nthen\nχmol= 2µ2\nBN0NAS (B7)\nin cm3/mol, where µBis the Bohr magneton, and\n2N0S=χmol\nNAµ2\nB(B8)Eq.B8providesaconvenentmethodtoestimateexper-\nimental unknowns, the density of states N0and Stoner\nparameter S, from measurements of χmol.\nExample: for Pd, the low-temperature measurement\n(differentfromtheroom-temperaturemeasurementinTa-\nbleI) isχmol∼7.0×10−4cm3/mol. In the denomina-\ntor, (NAµ2\nB) = 2.622 ×10−6Ry·cm3/mol, The value\n2N0Sconsistent with the experiment is 266/(Ry-at) or\n19.6/(eV-at). For the tabulated measurement of S= 9.3,\nthe inferred density of states is then N0= 1.05/eV/at.\nc. Interfacial exchange We canassumethat the Zee-\nman energy per interface atom is equal to its exchange\nenergy, through the Heisenberg form\nM2\np\nχvVat= 2Ji\nexsfsp (B9)\nwhereMpis the magnetization of the paramagnet,\nwith the atomic moment of the paramagnet mpin terms\nof its per-atom spin sp,\nMp=mp\nVatmp= 2µBsp (B10)\nVatis the volume of the paramagnetic site, sf,pare the\nper-atom spin numbers for the ferromagnetic and para-\nmagnetic sites, and Ji\nexis the (interatomic) exchange en-\nergy acting on the paramagnetic site from the ferromag-\nneticlayersonthe othersideoftheinterface. Interatomic\nexchangeenergyhasbeen distinguished fromintraatomic\n(Stoner) exchange involved in flipping the spin of a single\nelectron. Rewriting Eq B9,\nM2\np\nχvVat= 2Ji\nexsfMp\n2µBVat (B11)\nifsf= 1/2, appropriate for 4 πMs∼10 kG,\nJi\nex= 2µBMp\nχv(B12)\nand substituting for χvthrough Eq B4,\nJi\nex=Mp\nµBN0S(B13)\nIn the XMCD experiment, we measure the thickness-\naveraged magnetization as < M > in a [F/N]nsuper-\nlattice. We make a simplifying assumption that the ex-\nchange acts only on nearest-neighbors and so only the\nnear-interface atomic layer has a substantial magnetiza-\ntion. We can then estimate Mpfrom< M >through\n< M > t p= 2Mpti (B14)9\nwheretiis the polarized interface-layer thickness of\nN36. Since the interface exists on both sides of the N\nlayer, 2tiis the thickness in contact with F. Finally,\nJex=1\n2< M >\nµBN0Stp\nti(B15)\nThe exchange energy acting on each interface atom,\nfrom all neighbors, is JPt\nex= 109 meV for Pt and\nJPd\nex= 42 meV for Pd. Per nearest neighbor for an\nideal F/N(111) interface, it is JPy|Pt= 36 meV and\nJPy|Pd= 14 meV. Per nearest neighbor for an inter-\nmixed interface (6 nn), the values drop to 18 meV and 7\nmeV, respectively.\nSince explicit calculations for these systems are not\nin the literature, we can compare indirectly with theo-\nretical values. Dennler41showed that at a (3 d)F/(4d)N\ninterface (e.g. Co/Rh), there is a geometrical enhance-\nment in the moment induced in Nper nearest-neighbor\nofF. The 4d Natoms near the Finterface have larger\ninduced magnetic moments per nn of Fby a factor of\nfour. Specific calculations exist of JF|N(per neighbor)\nfor dilute Co impurities in Pt and dilute Fe impurties in\nPd42.JFe−Pd∼3 meV is calculated, roughly indepen-\ndent of composition up to 20% Fe. If this value is scaled\nup by a factor of four, to be consistent with the inter-\nface geometry in the XMCD experiment, it is ∼12 meV,\ncomparable with the value for Pd, assuming intermixing.\nTherefore the values calculated have the correct order of\nmagnitude.\n2. Ferromagnets\nThe Weiss molecular field,\nHW=βMs (B16)\nwhereβis a constant of order 103, can be used to give\nan estimate of the Curie temperature, as\nTC=µBgJJ(J+1)\n3kBHW (B17)\nDensity functional theory calculations have been used\ntoestimatethemolecularfieldrecently42,43; forspintype,\ntheJ(J+1) term is substutited with < s >2, giving an\nestimate of\nTC=2< s >2J0\n3kB(B18)where< s >is the number of spins on the atom as in\nEqB10; see the text by St¨ ohr and Siegmann44.< s >\ncan be estimated from m=1.07µBfor Py and 1.7 µBfor\nCo, respectively. Then\nJ0≃6kBTC\n(m/µB)2(B19)\nwith experimental Curie temperatures of 870 and 1388\nK, respectively, gives estimates of J0= 293 meV for Co\nandJ0= 393 meV for Py.\nNote that there is also a much older, simpler method.\nKikuchi45has related the exchange energies to the Curie\ntemperature for FCC lattices through\nJ= 0.247kBTC (B20)\nTaking 12 NN, 12 Jgives a total energy of 222 meV for\nPy (870 K) and 358 meV for FCC Co (1400K), not too\nfar off from the DFT estimates.\nd. Other estimates TheJ0exchange parameter is\ninteratomic, describing the interaction between spin-\nclusters located on atoms. Reversing the spin of one of\nthese clusters would change the energy J0. The Stoner\nexchange ∆ is different, since it is the energy involved in\nreversing the spin of a single electron in the electron sea.\nGenerally ∆ is understood to be greater than J0because\nit involvesmore coloumbrepulsion; interatomic exchange\ncan be screened more easily by spelectrons.\nThis exchange energy is that which is measured by\nphotoemissionandinversephotoemission. Measurements\nare quite different for Py and Co. Himpsel40finds an\nexchange splitting of ∆ = 270 meV for Py, which is not\ntoo far away from the Weiss J0value. For Co, however,\nthe value is between 0.9 and 1.2 eV, different by a factor\nof four. For Co the splitting needs to be estimated by a\ncombination of photoemission and inverse photoemission\nbecause the splitting straddles EF.46.\nFor comparison with the paramagnetic values of Ji\nex,\nwe use the J0estimates, since they both involve a bal-\nance between Zeeman energy (here in the Weiss field)\nand Heisenberg interatomic exchange. 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Lett. 64, 1059\n(1990)."    },    {        "title": "0808.0119v1.Field_Driven_Domain_Wall_Dynamics_in_GaMnAs_Films_with_Perpendicular_Anisotropy.pdf",        "content": "arXiv:0808.0119v1  [cond-mat.mtrl-sci]  1 Aug 2008APS/123-QED\nField-Driven Domain-Wall Dynamics in GaMnAs Films with\nPerpendicular Anisotropy\nA. Dourlata,b, V. Jeudya,c, A. Lemaˆ ıtred, C. Gourdona,b\naCNRS, UMR7588, Institut des Nanosciences de Paris,\n140 rue de Lourmel, Paris, F-75015 France\nbUniversit ´ePierre et Marie Curie-Paris 6, UMR 7588, INSP, Paris, France\ncUniversit ´ede Cergy-Pontoise, 95000 Cergy-Pontoise, France\ndLaboratoire de Photonique et Nanostructures, CNRS, UPR 20\nRoute de Nozay, Marcoussis, F-91460 France\n(Dated: October 24, 2018)\nAbstract\nWe combine magneto-optical imaging and a magnetic field puls e technique to study domain wall\ndynamics in a ferromagnetic (Ga,Mn)As layer with perpendic ular easy axis. Contrary to ultrathin\nmetallic layers, the depinningfield is foundto besmaller th an the Walker field, thereby allowing for\nthe observation of the steady and precessional flow regimes. The domain wall width and damping\nparameters are determined self-consistently. The damping , 30 times larger than the one deduced\nfrom ferromagnetic resonance, is shown to essentially orig inate from the non-conservation of the\nmagnetization modulus. An unpredicted damping resonance a nd a dissipation regime associated\nwith the existence of horizontal Bloch lines are also reveal ed.\nPACS numbers: 75.50.Pp, 75.60.Ch, 75.70.Ak\n1In ferromagnetic systems, domain wall (DW) motion driven by a magn etic field [1, 2, 3,\n4, 5] or a spin-polarized current [6, 7, 8] presents a variety of dyn amical regimes. Depending\non the field strength, several regimes characterized by the dyna mics of the magnetization\nvector inside the DW and the DW mobility (field derivative of the velocity ) are predicted to\noccur. Theoretically, the dissipation-limited regimes were mostly inve stigated in a system\nconsisting of an ideal ferromagnetic film with uniaxial perpendicular a nisotropy, subject to\na magnetic field Hparallel to the easy axis [1, 2, 3, 4]. In the Walker steady regime, the DW\nstructure is stationary and a linear velocity v=µHis expected up to the Walker field [2, 4].\nAbove this field the precession of the DW magnetization around Hleads to a DW back and\nforth motion, reducing its average velocity [3]. This regime may becom e unstable, leading\nto the nucleation and propagation of Bloch lines inside the DW [1, 9]. In t he high-field\nrange of the precessional regime the damping torque becomes larg e enough to move the DW\nwith linear velocity but reduced mobility with respect to the Walker reg ime. These various\ndynamical regimes could be observed in the past in micrometer-thick garnet films [4, 5]\nand very recently in nanowires with in-plane magnetization [10, 11]. Ho wever, in ultrathin\nmetallic ferromagnetic films with perpendicular magnetization dissipat ion-limited regimes\nare masked by thermally-activated creep and depinning regimes [12, 13]. For instance, in\nPt/Co/Pt layers only the high field linear precessional regime could be observed owing to\nthe strong pinning of DWs [12].\nIn ferromagnetic semiconductors, DW dynamics has been explored only recently. The\ncreep regime has been observed in GaMnAs thin films and wires [8, 14]. H owever, the\nstrength of the applied field was too small to reach the dissipation-lim ited regimes. The\nobservation of these regimes is of prime importance to determine th e relevant dissipation\nprocesses involved in DW motion. In particular, the nature of the fe rromagnetism (hole-\nmediated interaction between diluted Mn ions [15, 16]) could lead to spe cific features in the\nDW dynamics. Moreover, the understanding of dissipation process es in GaMnAs should\nhave important implications for the study of current-driven DW mot ion. It should help\nto discriminate between the damping contribution and the spin trans fer one, which is not\nyet well understood [7]. GaMnAs is a good candidate for experimenta l investigation of this\nquestion since the current density required to move a DW was found to be two orders of\nmagnitude smaller than in metallic nanowires [17].\nIn this letter, we report on DW dynamics in GaMnAs over a wide range o f magnetic\n2field and temperature values. A quantitative analysis of the DW dyna mics leads to the\nidentification of the steady and precessional regimes and to a self- consistent determination\nof the DW width and damping parameters. This damping is shown to invo lve the variation\nof the magnetization magnitude. We also identify a dynamical regime c onsistent with the\nexistence of horizontal Bloch lines. Finally, DW dynamics reveals an un predicted velocity\npeak.\nThe sample consists of an annealed Ga 0.93Mn0.07As epilayer of thickness d=50 nm grown\non a relaxed Ga 0.902In0.098As buffer deposited on a GaAs substrate. After annealing, the\nCurie temperature TCis 130 K and the magnetic easy axis is perpendicular to the sample\nplane. Kerr microscopy is used for the direct observation of DW mot ion (see Ref. [18, 19] for\nmore details). The sample is placed in a helium-flow cryostat. The DW ve locity is measured\nusing a magnetic pulse field technique [12, 20]. For low velocity ( v <10−3m s−1), pulses are\ngenerated by a conventional external coil (rise time ≈100 ms, maximum amplitude 60 mT).\nThe DW velocity is determined from a set of snapshots tracking the D W position during\na field plateau (see image (a) in Fig. 1). For higher velocity ( v >10−3m s−1), pulses are\ngenerated by a small coil (rise time ≈200 ns, maximum amplitude 250 mT) of diameter\n≈1 mm placed inside the cryostat, onto the sample surface. DW motion is driven by a\nseries of pulses of constant amplitude Hand increasing duration τ. The snapshots recorded\nbefore and after each pulse are used to determine the DW displacem ent as a function of τ\nas shown in images (b) and (c) of Fig. 1. The DW velocity reported in Fig . 1 corresponds\nto a linear fit of this curve (not shown). This procedure eliminates th e effects of the pulse\nrise and decay times.\nA typical velocity curve for field-driven DW dynamics is shown in Fig. 1 ( T= 80 K).\nSeveraldynamical regimescanbeidentified. For1mT < µ0H <2.5mT- creep regime - DWs\nare rough and present spatially inhomogeneous displacements (imag e (a) in Fig. 1). The\nfree DW motion is impeded by defects, some of them appearing as poin tlike or long linear\ndefects [18, 19]. To determine theaverage velocity, DWdisplacemen ts were onlymeasured in\nareaswithnostrongpinningdefects. Theresultsarereportedint heinsetofFig.1. visfound\nto increase over 6 orders of magnitude. The good agreement with a fitv=v0exp(−aH−1/4)\nsuggests that DWs follow a creep regime described by the motion of a n elastic string in the\npresence of a random pinning potential [13]. However, the systema tic investigation of this\nregime is out of the scope of this paper. For 2.5 mT < µ0H <8 mT- depinning regime - the\n3velocity varies linearly with the field. The DW roughness decreases (im age (b) in Fig. 1) and\nthedomainsexpandalmost isotropically. Onestill noticespointlike def ects aroundwhich the\nDWs fold up, creating lamellar domains [18, 19]. This regime, separating the creep and flow\nregimes, corresponds to the depinning regime. For µ0H >8 mT- flow regimes - the DWs\nare smooth and the displacements homogeneous (image (c) in Fig. 1) , thereby indicating\nthat the DW dynamics is no more limited by pinning. The v(H) curve (Fig. 1) presents the\nmain features of the predicted dissipation-limited DW dynamics [1, 2, 3 , 4]. A velocity peak\n(v= 10.5 m s−1forµ0H= 8.2 mT) is followed by a region with negative differential mobility\n(8.2 mT< µ0H <35 mT). The DW velocity decreases down to 7 m s−1forµ0H= 35 mT.\nForµ0H >50 mT, the velocity increases again with the field. vis proportional to H\nas expected for the high field precessional regime, except near µ0H= 92 mT where an\nunpredicted second velocity peak is observed. In order to determ ine whether those features\nof the DW dynamics are systematically observed, measurements we re performed over a wide\ntemperature range (0.03 < T/T C<0.92).\nTheresultsarereportedinFig.2. Qualitatively, the v(H)curves showaweakdependence\non temperature for the depinning and flow regimes. The main differen ce concerns the upper\nboundary Hupof the investigated field range. At low temperature ( T= 4-50 K), Hup\ncorresponds to the maximum available field amplitude (250 mT). At high er temperature\n(T= 65-120 K) Hupis limited by nucleation. The distance between domains nucleated\nduring a pulse is too small for an accurate measurement of the DW dis placement. The\nanomalous velocity peak in the high field precessional regime is system atically observed, as\nshown by the arrows in Fig. 2.\nIn the following the main features of the DW dynamics are analyzed qu antitatively.\nIn uniaxial ferromagnetic films, the anisotropy is characterized by the quality factor\nQ= 2Ku/µ0M2\ns, whereKuistheuniaxialanisotropy constant and Msthesaturationmagne-\ntization [4]. Q is found in the range 8.6-14, which denotes strong uniax ial anisotropy [21]. In\nthatcase, thevelocityintheWalkersteadyregimeisgivenby vst(H) =µstµ0H=γ∆µ0H/α,\nwhereγis the gyromagnetic factor (1.76 1011Hz T−1), ∆ the DW width parameter, and α\nthe damping parameter [4]. In the high field range of the precessiona l regime the velocity is\ngiven by vprec(H) =µprecµ0H=γ∆µ0Hα/(1+α2) [4].\nThosepredictions arecompared totheobserved dynamical regime s. Since theflow regime\nis reached close to the velocity peak the mobility µstis obtained by adjusting a straight line\n4tangent to the experimental curve (Fig. 3 (a) and (b)). µprecis obtained from a linear fit of\nthe high-field regime (Fig. 3(a)). The ratio µprec/µst=α2/(1+α2) yieldsα. ∆ is obtained\nas ∆ =µprec(1 +α2)/αγ. As shown in Fig. 4(a) ∆ varies weakly with the temperature.\nMoreover, the ∆( T) values lie between the two boundaries deduced independently from\ndomain theory for the same sample [21]. This very good quantitative a greement confirms\nthat theprocedureused forthe determinationof ∆and αisrelevant. This also demonstrates\nthattheprecessional andsteady regimesareindeedobserved ex perimentally, thelatterbeing\nreached only near the Walker velocity peak as shown in Fig. 3(b).\nThe damping coefficient α≈0.3 is also weakly dependent on the temperature (Fig. 4(b)).\nA similar value ( α≈0.15) was obtained from the decay of the magnetization precession in -\nduced by optical excitation [22]. Surprisingly, the DW damping is more t han one order of\nmagnitudelargerthanthedampingdeduced fromthefrequency de pendence oftheferromag-\nnetic resonance (FMR) linewidth: α≈0.01 for this sample [23], a value in agreement with\ntheoretical predictions for GaMnAs ( α≈0.02−0.03) [24]. A similar discrepancy was re-\nportedforgarnet films with small α-values[9, 25]. Theoretically, theDWequation ofmotion\nis derived from the Landau-Lifshitz-Gilbert (LLG) equation for the magnetization. The dis-\nsipation is classically described by the phenomenological Gilbert dampin g coefficient αthat\naccounts for the relaxation of the direction of the magnetization v ector [1, 2, 3, 4]. However,\ndissipation processes are expected to differ for uniform magnetiza tion (FMR) and for a mov-\ning DW. A deviation from the linear steady regime is predicted when the interaction of the\nmoving DW with thermal magnons is taken into account [26]. However, this interaction can\nbe safely discarded here since it would give a significant contribution o nly at high velocity,\nabove≈300 m s−1for this sample. In the low velocity range, additional relaxation term s in\nthe LLG equation, taking into account the non-conservation of th e magnetization modulus\nand heat exchange with a thermostat, can lead to a DW dynamical da mping larger than the\nGilbert (FMR) damping [27, 28]. For a magnetization response time τM=χ/bardbl/γMsαFMR\nshorter than the DW transit time τt= ∆/v, the dissipation for DW motion in the steady\nregime is described by a dynamical damping [27] αDW=αFMR/bracketleftbigg\n1+16/parenleftBig\n4πχ/bardblQ/parenrightBig2/3α2\nFMR/bracketrightbigg\n,\nwhereχ/bardblis the longitudinal magnetic susceptibility (CGS units). Using this equa tion with\nαFMR≈0.01 yields χ/bardbl≈10−4, in reasonable agreement with theoretical estimations based\non spin-wave theory [29, 30]. Taking into account the anisotropy ga p, one finds χ/bardblin the\nrange 4 10−6-10−5forT= 12 K and 8 10−5-2 10−4forT= 80 K. For the measured velocities\n5onefinds τM≪τt, thereby justifying theuse ofthis model. Wepoint out thatthe the oretical\nprediction of enhanced DW dynamical damping in the steady regime als o accounts well for\nthe damping in the high-field precessional regime.\nLet us now extend the analysis of the experimental velocity curves beyond the 1D theory\nof DW motion. As calculated and shown in Fig. 3(b), the 1D theory pre dicts a maximum\nvelocity (Walker velocity vW=γ∆µ0Ms/2) at the Walker field HW=αMs/2. ForHjust\naboveHWthe DW back and forth motion due to the DW magnetization precessio n around\nthe applied field should lead to a decrease of the time-averaged veloc ity and hence to a\nregion of negative differential mobility [3]. In contrast, the experime ntalv(H) curve just\nshows a change of slope with still a constant positive mobility, yet str ongly reduced with\nrespect to the steady regime (Fig. 3(b)). In this field range, the D W structure is expected to\nbe unstable. A solution for DW propagation with generation and prop agation of horizontal\nBloch lines through the film has been proposed [1]. For sufficiently larg eαthis model\npredicts a viscous-like drag with decreased mobility µBL=γ∆/[α(1+π2Λ/2α2a)] with\nΛ = ∆√Qthe exchange length and aof the order of the film thickness [1]. The linear fit of\nthe velocity curve just above the Walker field for the set of investig ated temperatures yields\nain the range 86 ±50 nm, consistent with the sample thickness d= 50 nm. Given the fact\nthat the exchange length d/10<Λ< d/5 is not very much smaller than d, contrary to the\nassumption of the model, the agreement is quite satisfactory. It s trongly suggests that DW\nmotion above the Walker field is slowed down by the repetitive generat ion of one Bloch line\nat one surface of the film and subsequent propagation and annihilat ion at the other surface.\nLet us now discuss the intriguing velocity peak observed around 90- 120 mT (Fig. 2). To\nour knowledge, such a peak has been neither predicted nor observ ed in ferromagnetic sys-\ntems. Sinceit occurswithin thelinearprecessional regime, where th evelocity isproportional\nto the damping parameter, it can be ascribed to a damping resonanc e. In order to char-\nacterize its temperature dependence, the velocity curves are fit ted using a field-dependent\ndamping α(H) =α+δαexp/bracketleftBig\n−(H−Hp)2/2σ2/bracketrightBig\n. As shown in Fig. 2 (inset) the resonance\namplitude ( δα=0.04 at 60 K) decreases linearly with the temperature, extrapolat ing to zero\natT≈120 K, close to TC. The resonance field Hpvaries slightly with the temperature. The\ncorrespondingenergy¯ hω= ¯hγµ0/radicalBig\nH2−H2\nW/(1+α2)isoftheorderof10 µeV(ω= 17GHz).\nIt may correspond to transitions between confined states of volu me magnons, as calculated\nfrom the dispersion curve using the spin stiffness constant determ ined from this work and\n6ref. [21]. The damping resonance might also be related to DW excitatio ns (flexural modes),\nwhose energy spectrum lies inside the anisotropy gap of the volume m agnons (43 µeV at\n80 K). These excitations have been calculated for a static or moving DW in the steady\nregime [31, 32] but not in the precessional one.\nDespite the small value of the saturation magnetization and hence o f the Walker field\nin GaMnAs, we could observe the dissipation-limited flow regimes beyon d the creep and\ndepinning regimes. The steady as well as the precessional regime ar e described by a DW\ndynamical damping, which is found to be much larger than the FMR dam ping, in agreement\nwith theoretical predictions considering non-conservation of the magnetization modulus.\nHowever, contrary to the classical assumption, a single, field-inde pendent, damping constant\ncannot account for the whole DW dynamics. The existence of a damp ing resonance inside\nthe precessional regime suggests an additional dissipation mechan ism and calls for further\ntheoretical investigations of dissipation processes in this regime.\nWe gratefully acknowledge fruitful discussions with Jacques Ferr´ e, Alain Mauger and\nAndrejs Cebers. This work was in parts supported by R´ egion Ile de France under contract\nIF07-800/R with C’Nano IdF at CNRS.\n[1] J. C. Slonczewski, J. Appl. Phys. 44, 1759 (1973); ibid.45, 2705 (1974)\n[2] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974).\n[3] A. Hubert, Theorie der Dom¨ anenw¨ ande in Geordneten Medien , Springer, Berlin (1974).\n[4] A.P. Malozemoff and J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials , Aca-\ndemic, New York (1979).\n[5] F.H. de Leeuw, R. van den Doel and U. Enz, Rep. Prog. Phys. 43, 659 (1980).\n[6] L. Berger, J. Appl. Phys. 55, 1954 (1984).\n[7] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Eur. Phys. 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Kita, F. Matsukura, and H. Ohno, Phy s. Rev. Lett 93, 216602 (2004).\n[18] A. Dourlat, V. Jeudy, C. Testelin, F. Bernardot, K. Khaz en, C. Gourdon, L. Thevenard, L.\nLargeau, O. Mauguin, A. Lemaˆ ıtre, J. Appl. Phys. 102, 1 (2007).\n[19] L. Thevenard, L. Largeau, O. Mauguin, G. Patriarche, A L emaˆ ıtre, N. Vernier, and J. Ferr´ e,\nPhys. Rev. B 73, 195331 (2006).\n[20] A. Dourlat, V. Jeudy, L. Thevenard, A. Lemaˆ ıtre, and C. Gourdon, J. Supercond. Nov. Magn.\n20, 453 (2007).\n[21] C. Gourdon, A. Dourlat, V. Jeudy, K. Khazen, H. J. von Bar deleben, L. Thevenard, A.\nLemaˆ ıtre, Phys. Rev. B 76, 241301(R) (2007).\n[22] J. Qi, Y. Xu, N.H. Tolk, X. Liu, J.K. Furdyna and I.E. Pera kis, J. Appl. Phys. 91, 112506\n(2007).\n[23] K. Khazen, H.J. von Bardeleben, M. Cubukcu, J. L. Cantin , V. Novak, K. Olejnik, M. Cukr,\nL. Thevenard, A. Lemaˆ ıtre, submitted to Phys. Rev. B.\n[24] J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. K. Furdyna , W. A. Atkinson, and A. H.\nMacDonald, Phys. Rev. B 69, 085209 (2004).\n[25] G. P. Vella-Coleiro, D. H. Smith, and L. G. van Uitert, Ap pl. Phys. Lett. 21, 36 (1972).\n[26] B. A. Ivanov, Yu. N. Mitsai, and N. V. Shakhova, Sov. Phys . JETP60, 168 (1984).\n[27] H.L. Huang, V. L. Sobolev, and S. C. Chen, J. Appl. Phys. 81, 4066 (1997).\n[28] V. G. Bar’yakhtar, B. A. Ivanov, A.L. Sukstanskii, and E . Yu. Melikhov, Phys. Rev. B 56,\n619 (1997).\n[29] T. Holstein and H. Primakoff, Phys. Rev 58, 1098 (1940)\n[30] B.E Argyle, S. H. Charp, and E. W. Pugh, Phys. Rev. 132, 2051 (1963)\n8[31] A. A. Thiele, Phys. Rev. B 7, 391 (1973).\n[32] E. Schl¨ omann, IEEE Trans. Magn. 10, 11 (1974).\n9FIGURE CAPTIONS\nFIG. 1 Magnetic field dependence of the DW velocity at T= 80 K. The error bars\nrepresent the standard deviation of the velocity distribution. Ins et: low field region in\nsemi-logarithmic scale with a fit (dashed line) according to the creep m odel [13]. Labels\n(a), (b), and (c) refer to the corresponding images in the creep, depinning and flow regimes,\nrespectively. Images (b) and (c) are differential images showing th e DW displacement as\nindicated by black arrows.\nFIG. 2 (Color online) Temperature dependence of the DW field-veloc ity curves. Each\ncurve is up-shifted by 10 m s−1with respect to the previous one. Inset: temperature de-\npendence of the resonance field Hpand damping δαof the velocity peak indicated by arrows.\nFIG. 3 (Color online) Comparison between theoretical predictions a nd experimental\nresults (black circles). Dashed blue lines in (a): linear velocity in the st eady andprecessional\nregimes with mobility µstandµprec, respectively. Solid blue curves in (b): velocity in the\nsteady regime (below HW) and in the unstable precessional regime (above HW) calculated\nusing the experimentally determined values of ∆ and α(this work) and Ms[21]; dotted red\nline: linear fit of the velocity in the Bloch line regime.\nFIG. 4 (Color online) (a) Comparison between the DW parameter ∆ ob tained from\nDW dynamics (full circles) and the boundaries for ∆ obtained from do main theory (open\nsymbols) [21]. (b) Temperature dependence of the DW damping para meterα.\n10FIG. 1:\n110 50 100 150 200 250 300 0 20 40 60 80 100 0.00 0.04 0.08 \n10 \n0120 K 115 K 110 K 100 K 90 K 80 K 65 K 50 K 4 K \n  DW velocity (m s -1 )\nµ0H (mT) \n  δα\nTemperature (K) 050 100 150  \n \nµ0Hp (mT) \nFIG. 2:\n120 5 10 15 0510 15 \n0 50 100 150 200 010 20 30 \nµst \nvW\nHW b\n  DW velocity (m s -1 )\nµ0H (mT) µprec µst \naT=80 K \n  DW velocity (m s -1 )\nµ0H (mT) \nFIG. 3:\n130 20 40 60 80 100 120 01234\n0 20 40 60 80 100 120 0.0 0.2 0.4 0.6 \na\n  DW parameter ∆\nTemperature (K) b\n  DW damping parameter α\nTemperature (K) \nFIG. 4:\n14"    },    {        "title": "1210.3530v1.Reversal_of_magnetization_of_a_single_domain_magnetic_particle_by_the_ac_field_of_time_dependent_frequency.pdf",        "content": "Reversal of magnetization of a single-domain magnetic particle\nby the ac field of time-dependent frequency\nLiufei Cai, D. A. Garanin, and E. M. Chudnovsky\nPhysics Department, Lehman College, City University of New York\n250 Bedford Park Boulevard West, Bronx, New York 10468-1589, USA\n(Dated: October 31, 2018)\nWe report numerical and analytical studies of the reversal of the magnetic moment of a single-\ndomain magnetic particle by a circularly polarized ac field of time-dependent frequency. For the\ntime-linear frequency sweep, the phase diagrams are computed that illustrate the dependence of the\nreversal on the frequency sweep rate v, the amplitude of the ac field h, the magnetic anisotropy field\nd, and the damping parameter \u000b. It is shown that the most efficient magnetization reversal requires\na non-linear time dependence of the frequency, !(t), for which an exact analytical formula is derived\nwith account of damping. The necessary condition of the reversal is h>\u000bd. Implementation of a\nsmall-scale magnetization reversal is proposed in which a nanomagnet is electromagnetically coupled\nto two weak superconducting links controlled by the voltage. Dynamics of such a system is analyzed\nwith account of the back effect of the magnet on the superconducting links.\nPACS numbers: 75.60.Jk, 84.40.-x, 75.50.Tt, 85.25.Cp\nI. INTRODUCTION\nIn recent years a significant effort has been made\nto achieve magnetization reversal in nanostructures, as-\nsistedbythelowamplitudeacfieldintheradiofrequency\nrange. The idea is rather simple. The dc magnetic field\nrequired to reverse the magnetization of a single-domain\nmagnetic particle, the so-called anisotropy field, is typ-\nically in the range 0.01-0.1T. The field of this strength\nat the location of the particle is not easy to develop fast.\nThe ac magnetic field that one can typically develop in\nthe radio-frequency range would be two orders of mag-\nnitude weaker. Applied in a resonant fashion, it could\nincrease the amplitude of the precession of the magnetic\nmoment, sometimes leading to its full reversal, in the\nsame way as weak pushes of a pendulum at the frequency\nof its mechanical oscillation can flip the pendulum over\nthetop. However, thestudyofbothproblemshowsalack\nof robust reversal. In many cases the attempted reversal\nresults in a chaotic behavior that may be interesting on\nits own.\nMagnetization reversal by ultrashot magnetic field\npulses produced by a high-energy electron beam has been\nstudied by Back et al.1in perpendicularly magnetized\nCoPt films. Schumacher et al2studied phase coherent\nprecessional magnetization reversal in spin valves by a\npulse of the transverse field of a few hundred picosecond\nduration produced by the electric current.\nLater, significant number of experiments focused on\nmicrowave-assisted reversal in smaller structures and\nindividual single-domain magnets with a strong static\nfield applied to reduce the barrier. Thirion et al3at-\ntempted magnetization reversal in static fields below\nthe anisotropy field, assisted by a linearly polarized mi-\ncrowave field, in 20-nm-diameter Co particles placed on\nthe bridge of a micro-SQUID. They were able to repro-\nduce the Stoner-Wohlfarth astroid4and study the de-\npendence of the reversal on the frequency and durationof the ac pulse. Enhancement of the magnetization re-\nversal by microwave magnetic fields in nanometer Co\nstrips has been demonstrated by Grollier et al.6Nem-\nbach et al7and Nozaki et al8used magnetic force mi-\ncroscopy to measure microwave assisted magnetization\nreversal in individual submicron Co and permalloy par-\nticles. Microwave-assisted magnetization switching in\npermalloy tunnel junctions has been demonstrated by\nMoriyama et al9. Podbielski et al studied magnetiza-\ntion reversal in microscopic permalloy rings at GHz fre-\nquency. They observed non-linear spin dynamics and\nobtained experimental phase diagram of the reversal\nas function of microwave frequency and power.10Using\ntime-resolved magneto-optic Kerr microscopy, Wolters-\ndorf and Back11detected enhancement of magnetization\nswitching in single-domain permalloy elements subjected\ntotheresonantmicrowavefield. Microwave-assistedmag-\nnetization reversal in single-domain permalloy nanoele-\nments has been studied by Nembach et al.13Wang et\nal14have investigated experimentally the competition\nbetween damping and pumping for microwave-assisted\nmagnetization reversal in FeCo thin films.\nTheoretical research in this area mostly focused on the\nmagnetization reversal assisted by the ac field of con-\nstantfrequency.15Non-linearmagnetizationdynamicsin-\nduced by such a field that results in a chaotic behavior\nhas been studied by Bertotti et al.16,17Denisov et al18\naddressed magnetization of nanoparticles in a rotating\nmagnetic field. Synchronization and chaos induced in\nthe damped dynamics of a single-domain particle by the\nac field of constant frequency has been investigated by\nSun and Wang.5Nonlinear-dynamical-system approach\nto the microwave-assisted magnetization dynamics was\nreviewed by Bertotti et al.15Micromagnetic modelling\nof microwave-assisted magnetic recording was performed\nby Wang et al.12Constant frequency microwave switch-\ning magnetic grains coupled by exchange interaction has\nbeen investigated by Igarashi et al.19Okamoto et al ad-arXiv:1210.3530v1  [cond-mat.mes-hall]  12 Oct 20122\ndressed stability of the magnetization switching by lin-\nearly and circularly polarized waves. Magnetization re-\nversal in a resonant cavity has been studied by Yukalov\nand Yukalova.22\nFewer number of theoretical papers have considered\ndynamics of the magnetization of a nanoparticle gener-\nated by the ac magnetic field of variable frequency. May-\nergoltz et al23developed the inverse problem approach\nto the precessional switching of the magnetization by a\nlinearly polarized pulse of the magnetic field. Rivkin and\nKetterson24obtained the optimal time dependence of the\nmicrowave frequency in the absence of damping, as well\nas the condition of the reversal in the presence of damp-\ning. Magnetization reversal by a linearly polarized ac\nfield of frequency that depends linearly on time has been\nstudied by two of the authors.25Barros et al21developed\nan optimization method in which the energy consump-\ntion needed for reversal is minimized with respect to the\ntime dependence of the amplitude and frequency of mi-\ncrowaves.\nA few general points need to be made before address-\ning the problem of the reversal of the magnetization by\nthe microwaves. Firstly, a robust magnetization reversal\ncan be effectively achieved only with a circularly polar-\nized ac field. Indeed, photons of circular polarization\nhave a definite orientation of their spin projection, while\nphotons with linear polarization are in a superposition\nof spin states. Consequently, photons with the right cir-\ncular polarization, when absorbed by the magnet, drive\nthe magnetization in one direction towards the reversal,\nwhile linearly polarized photons can be both absorbed\nand emitted and, therefore, do not necessarily reverse\nthe magnetization. Secondly, the photons are effectively\nabsorbed only when they are in resonance with the spin\nlevels. The latter are not equidistant on the magnetic\nquantum number, that is, on the projection of the mag-\nneticmomentonthedirectionoftheeffectivefield. Thus,\nas the magnetic moment reverses, the photon frequency\nthat can be resonantly absorbed by the magnet changes\nwith time, so that the frequency of the microwave field\nhas to be adjusted. Damping of the precession adds an-\nother dimension to this problem as the power of the ac\nfield that is pumped into the magnet should exceed the\nrate of energy dissipation. Analysis shows that circularly\npolarized small-amplitude ac field of a time-dependent\nfrequency that follows the condition of the resonance is\nsufficient for achieving magnetization reversal. The case\nofazerostaticfieldisofthehighestpracticalimportance.\nThe typical wavelength of microwaves that are in res-\nonance with the precession of the magnetic moment is\nin the centimeter range. Thus, one of the challenges for\npotential applications of the microwave-assisted magne-\ntization reversal for, e.g., computer technology, consists\nof the generation of a circularly polarized ac field of suf-\nficient amplitude at the position of a nanoscale single-\ndomain particle. In Ref. 25 a suggestion has been made\nto use the ac field generated by a superconducting weak\nlink. If one is not turned off by the necessity to goto lower temperatures (which is probably, inevitable for\nmagnetic memory of ultra-high density), the advantage\nof this method would be the possibility to control the\ntime dependence of the frequency by voltage across the\nlink. Interaction between a nanomagnet and a Josephson\njunction has been subject of intensive research. Micro-\nSQUID setup has been used by Jamet et al to observe\nswitching of the magnetization in a 3nm Co cluster26,27,\nsee also review 28. Ferromagnetic resonance in permalloy\nfilms grown on Nb substrate has been studied by Bell et\nal.29Petcović et al30investigated experimentally the di-\nrectdynamicalcouplingofspinmodesandasupercurrent\nin a ferromagnetic junction, following theoretical study\nof this system by Houzet.31Current-phase relation in a\nJosephsonjunctioncoupledwithamagneticdothasbeen\ninvestigated theoretically by Samokhvalov.32Most of the\nresearch in this area focused on the proximity effect33–35\nrather than on electromagnetic interaction.\nIn this paper we study magnetization dynamics of a\nsingle-domain uniaxial magnetic particle in zero static\nfield, induced by a circularly polarized ac field of con-\nstant amplitude but variable frequency. The model is\nformulated in Section II. General properties of the mag-\nnetization reversal are studied in Section III. Numeri-\ncal results for the time-linear frequency sweep are pre-\nsented in Section IV where the phase diagrams are com-\nputed for the dependence of the magnetization switching\non parameters. They are the frequency sweep rate, the\namplitude of the ac field, the magnetic anisotropy field,\nand the damping parameter. Analytical results for the\ntime-linear sweep, that are generally in good agreement\nwith numerical results, are given in Section VA. In Sec-\ntion VB we obtain the exact analytical solution for non-\nlinear time dependence of the frequency that provides\nthe fastest magnetization reversal. The model in which\ncircularly polarized ac field is generated by two supercon-\nducting weak links is studied in Section VI with account\nof the back effect of the dynamics of the magnetic mo-\nment on the links. Our conclusions and suggestions for\nexperiment are presented in Section VII.\nII. THE MODEL\nThe energy of a single-domain magnetic particle with\nan uniaxial anisotropy in a circularly-polarized ac field\nhas the form\nH=\u0000KVM2\nz\u0000VMxhcos \b(t)\u0000VMyhsin \b(t):(1)\nHereKis the magnetic anisotropy constant, Vis parti-\ncle’s volume, Mis the magnetization, his the amplitude\nof the ac field, and \b(t)is the phase related to the time-\ndependent frequency as\n_\b(t)\u0011!(t): (2)\nOne of the cases we consider is that of the frequency\nlinearly changing with time,\n!(t) =\u0000vt; (3)3\nwhere \b(t) =\u0000vt2=2. Theothercasethatwillbestudied\nhereisanon-lineartimedependenceofthefrequencythat\nprovides the fastest magnetization reversal.\nIt is convenient to recast the problem in terms of a\nclassical spin s=M=Ms,jsj= 1, whereMsis the sat-\nuration magnetization. The Landau-Lifshitz equation of\nmotion for this spin has the form\n_ s=\r[s\u0002He\u000b]\u0000\u000b\r[s\u0002[s\u0002He\u000b]];(4)\nwhere\ris the gyromagnetic ratio, \u000bis dimensionless\ndamping coefficient and\nHe\u000b=\u00001\nV@H\n@M= 2dszez+hexcos \b(t) +heysin \b(t)\n(5)\nis the effective field. Here d\u0011Ha=KMsis the\nanisotropy field. In the initial state the spin points in\nthe negative- zdirection, s(\u00001) =\u0000ez:\nFurther it is convenient to switch to the coordinate\nframe rotating around the zaxis together with the mag-\nnetic field, so that in this frame the magnetic field is\nstatic. As the result, in the new frame the spin acquires a\nrotation opposite to that of the ac field in the initial (lab-\noratory) frame. Thus in the rotating frame the Landau-\nLifshitz has the form\n_ s= [s\u0002(\rHe\u000b+\n(t))]\u0000\u000b\r[s\u0002[s\u0002He\u000b]];(6)\nwhere\nHe\u000b= 2dszez+hex; \n(t) =!(t)ez:(7)\nIII. GENERAL PROPERTIES OF THE\nMAGNETIZATION REVERSAL\nWith the sign choice in Eq. (3), the ac field at negative\ntimes is precessing in the same direction as the magnetic\nmoment, thus it excites magnetic resonance and may\ncause magnetization reversal. In the ideal case, as we\nwill see below, the resonance condition holds during the\nwhole reversal. After the magnetic moment overcomes\nthe barrier, sz>0, it changes its precession direction,\nand so does the ac field.\nIn the rotating frame, the field !(t)=\rsweeping at a\nlinear rate makes the problem resembling that of the\nLandau-Zener(LZ)effectthatcanbeformulatedinterms\nof the evolution of a classical spin described by a Larmor\nequation. There are three modifications, however: (i)\nuniaxial anisotropy and (ii) damping added to the model\nand (iii) sweeping !(t)in the negative direction. Because\nof the latter, the initial state of the spin in the rotating\nframe is the high-energy state with szopposite to !, see\nEq. (3). To the contrast, in the regular LZ effect the\ninitial spin state is the low-energy state. In the absence\nof anisotropy and damping, the initial orientation of the\nspin and the sweep direction do not matter. However,\nin the general case the situation does depend on these\nfactors.In particular, in the absence of damping one can mul-\ntiply\rhe\u000b+\n(t)by\u00001that only makes the spin precess\nin the opposite direction but does not affect its reversal.\nThe resulting model is a model with a positive sweep,\nsuch as the regular LZ problem, while the anisotropy be-\ncomes easy-plane, d < 0. It was shown36that in this\ncase for a sweep slow enough the system adiabatically\nfollows the time-dependent lowest-energy state and the\nspin switching is very efficient. In our original model\n(with\u000b= 0) the magnetization reversal is similar. Only\ninstead of adiabatically following the lowest-energy state,\nthe spin adiabatically follows the highest-energy state, in\nwhich it was at the beginning.\nThis adiabatic solution corresponds to the maximum\nof the energy in the rotating frame\nH=(VMs) =\u0000ds2\nz\u0000sxh\u0000(!=\r)sz(8)\nThe maximal energy corresponds to sy= 0. Usingsx=\n\u0000p\n1\u0000s2z(oppositetothetransversefield)andrequiring\ndH=dsz= 0, one obtains the equation\n2dsz+hsz=p\n1\u0000s2z+!=\r= 0 (9)\nfor the energy maximum. Since in practical conditions\nh\u001cd, an approximate solution of this 4th-order alge-\nbraic equation for the adiabatic spin value reads\nsz=8\n><\n>:\u00001; !> 2\rd\n\u0000!=(2\rd);j!j\u00142\rd\n1; !<\u00002\rd:(10)\nNote that this solution is independent of h. Nonzero\nvalues ofhcause rounding at the borders of the cen-\ntral regionj!j\u00142\rdwhere the reversal occurs. In the\nlaboratory frame, the spin is precessing during adiabatic\nreversal being phase-locked to the ac field.\nFor\u000b= 0the magnetization reversal can be achieved\nfor whatever small ac field h. In the case of a nonzero\ndamping, there is a dissipative torque acting towards the\nenergy minima, and the ac field hhas to exceed a thresh-\nold value to overcome this torque. Below we will see that\nthe magnetization reversal requires\nh>\u000bd (11)\nthat is much easier to fulfill than h > 2din the case of\na static field. As the torque due to the transverse field\nis maximal when the magnetic moment is perpendicular\nto it, in the presence of damping the magnetic moment\ngoes out of the x-zplane during the reversal.\nIV. NUMERICAL - MAGNETIZATION\nREVERSAL BY THE TIME-LINEAR\nFREQUENCY SWEEP\nA. Time dependences of reversing magnetization\nThe results of numerical solution of Eq. (6) in the un-\ndamped case \u000b= 0for a small frequency sweep rate are4\nFigure 1: Almost adiabatic magnetization reversal at zero\ndamping.\nFigure 2: Non-adiabatic magnetization reversal at zero damp-\ning.\nshown in Fig. 1. Magnetization reversal in this case is\nalmost adiabatic and sz(t)is well described by Eq. (10)\nwith rounding at the borders of the reversal interval due\nto a small value of h=d. The reversal is practically con-\nfined to the z-xplane andsyis small. Numerical results\nfor a faster sweep rate are shown in Fig. 2. Here there\nis still magnetization reversal but it is not adiabatic and\nthe final value of szis smaller than one. Because of this,\nthe magnetic moment is precessing around the zaxis, as\nmanifested by sxandsy. During reversal the magneti-\nzation is substantially deviating from the z-xplane. For\nlarger sweep rates the reversal quickly becomes impossi-\nble.\nFig. 3 shows that in the damped case the magnetic\nmoment substantially deviates from the z-xplane. Still,\noverall the reversal in this case is close to adiabatic. In-\ncreasing the sweep rate leads to a non-adiabatic regime\nshown in Fig. 4. Here transverse spin components are os-\ncillating and the dependence of szis jagged. This shows\nthat, in the laboratory frame, the phase locking between\nthe magnetic moment and the ac field is about to break.\nFigure 3: Almost adiabatic magnetization reversal for \u000b=\n0:02.\nFigure 4: Non-adiabatic magnetization reversal for \u000b= 0:02.\nIn spite of all this, there is a complete reversal because\nthe damping finally brings the magnetic moment to the\nbottom of the potential well (c.f. Fig. 2). For a faster\nsweep the reversal disappears and the magnetic moment\nlands in the initial well, sz=\u00001. In the case of a slow\nsweep shown in Fig. 5 an instability can develop that\nleads to the breakdown of the phase locking and to faster\nrelaxation of the magnetic moment towards one of the\ntwo potential wells. The final value of sz(1 or -1) be-\nhaves irregularly vs sweep rate. This regime is not in-\nteresting for applications aimed at achieving as fast as\npossible reversal.\nB. Phase diagram of the magnetization reversal by\nthe time-linear frequency sweep\nDependence of the final value of szon the amplitude\nof the ac field hand frequency sweep rate vdefines the\n“phasediagram” ofthemagnetizationreversal. Intheun-\ndamped case the numerically calculated phase diagram is\nshown in Fig. 6. The final szis color-coded: black corre-5\nFigure 5: Instability in slow magnetization reversal for \u000b=\n0:02.\nFigure 6: Phase diagram of the magnetization reversal in the\nundamped case.\nsponds tosz=\u00001(non-reversal) and red corresponds to\nsz= 1(reversal). One can see that the reversal requires\nhsufficiently large and vsufficiently small. The curva-\nture of the phase boundary at small handvsuggests a\nfractional power. Careful examenation of this region of\nthe phase diagram shows that the reversal condition has\nthe form\nv\n2\r2d2<c\u0012h\nd\u00134=3\n; c'1:6 (12)\nath=d\u001c1and\u000b= 0.\nPhase diagram of the magnetization reversal in the\ndamped case \u000b= 0:02is shown in Fig. 7. It is similar to\nFig. 6 but there is a threshold for the magnetization re-\nversal onhand the phase-boundary line goes linearly at\nsmallv. Computations for different values of \u000bsuggest\nthat the reversal requires h=d>\u000b.\nOne can compute other types of phase diagrams for\nthe magnetization reversal that show a compact rever-\nsal region and the whole boundary line. The most\nFigure 7: Phase diagram of the magnetization reversal for\n\u000b= 0:02. Yellow line: Eq. (26).\nFigure 8: Efficiency-type phase diagram of the magnetization\nreversal for \u000b= 0:1.\nuseful of these phase diagrams uses the parameters\u0000\n\u000bd=h; \u000bv= (\r2h2)\u0001\n. Indeed, the area of the magnetiza-\ntion reversal is the compact region 0< \u000bd=h < 1and\nv=h2is inversely proportional to the energy of the ac\nfield injected during the time of the reversal by the lin-\near frequency sweep\nt(linear)\nrev =2\rd\nv: (13)\nThe maximum of v=h2corresponds to the minimal in-\njected energy and thus to the maximal efficiency of the\nreversal. Figs. 8 and 9 show that the maximal effi-\nciency of the time-linear frequency sweep corresponds to\n\u000bd=h\u00190:5. Also in these figures one can see that there\nis no reversal if the sweep rate is too low, especially for\nlow ac fields on the right side.6\nFigure 9: Efficiency-type phase diagram of the magnetization\nreversal for \u000b= 0:01.\nV. ANALYTICAL\nAnalytical investigation of the magnetization reversal\nis more convenient in spherical coordinates\nsz= cos\u0012; sx= sin\u0012cos'; sy= sin\u0012sin':\n(14)\nAfter neglecting the ac field in the dissipation term, Eq.\n(6) becomes\n_\u0012=\rhsin'\u0000\u000b\rdsin 2\u0012 (15)\n_'=\u00002\rdcos\u0012\u0000!(t) +\rhcos'cot\u0012:(16)\nA. Linear frequency sweep\nIn the case of a linear frequency sweep, Eq. (3), one\ncan rewrite the equation of motion for the spin in terms\nof the dimensionless time variable\n\u001c=vt=(2\rd): (17)\nThe resulting equation of motion has the form\nd\u0012=d\u001c =bsin'\u0000\u000basin 2\u0012 (18)\nd'=d\u001c =\u00002a(cos\u0012\u0000\u001c) +bcos'cot\u0012;(19)\nwhere\na\u00112\r2d2\nv; b\u00112\r2dh\nv(20)\ncharacterise the sweep rate. Another important parame-\nter is\nA=\u000bd=h: (21)\nSinceais a large parameter, phase locking of the mag-\nnetic moment to the ac field and thus efficient reversal\nFigure 10:f(\u001c)of Eq. (24).\nrequires cos\u0012\u0018=\u001cin the reversal region j\u001cj<1. Ifcos\u0012\nonly slightly deviates from this form, this causes strong\noscillations of 'and thus the breakdown of the phase\nlocking. Setting cos\u0012=\u001c, from Eq. (18) one obtains the\nphase-locking condition for 'in the form\nsin'=1\nbd\u0012\nd\u001c+Asin 2\u0012: (22)\nThe term on the left of this formula is proportional to\nthe torque acting on the spin from the ac field. This\ntorque has to ensure temporal change of \u0012(i.e., reversal)\nand compensate for the dissipative torque that is acting\ntoward potential wells. One can see that damping ham-\npers climbing the barrier by the magnetic moment. The\nmaximal damping torque is realized at \u0012= 3\u0019=4, where\nsin 2\u0012=\u00001. Since the reversal implies d\u0012=d\u001c < 0;it is\nclear that for A > 1the ac torque cannot overcome the\ndamping torque. Thus, the necessary condition for the\nmagnetization reversal is\nA<1; (23)\nwhile the more restricting sufficient condition requires\nthat the right-hand side of Eq. (22) does not drop below\n-1 for all\u001c. The latter requires the frequency sweep rate\nto be not too fast. Using cos\u0012=\u001c, one can rewrite this\ncondition in the form\nmaxf(\u001c)<1; f (\u001c)\u0011\u00002A\u001cp\n1\u0000\u001c2+1\nbp\n1\u0000\u001c2:\n(24)\nBecause of the inertial term, f(\u001c)shown in Fig. 10 di-\nverges at the borders of the reversal interval. This is,\nhowever, an artefact of neglecting the rounding of the de-\npendencesz(\u001c)at\u001c=\u00061because of the finite value of\nh. When this effect is taken into account, there are max-\nima around \u001c=\u00061instead of divergences. Thus, spin\nreversal can break down either because of the inertial ef-\nfect near\u001c=\u00001or because of the effect of dissipation\nnear\u0012= 3\u0019=4, i.e.,\u001c=\u00001=p\n2, depending on which one\noccurs at a smaller sweep rate.7\nLet us first consider the dissipative breakdown of the\nmagnetization reversal for Aslightly below 1 that hap-\npens at a small sweep rate. In this case 1=b/vis small\nand the second term in f(\u001c)in Eq. (24) is a perturba-\ntion. Thus the value of this term can be taken at the\nunperturbed dissipative maximum \u001c=\u00001=p\n2. Using\n\u00002\u001cp\n1\u0000\u001c2= 1andp\n1\u0000\u001c2= 1=p\n2, one obtains the\nreversal condition\nA+p\n2\nb<1: (25)\nThis can be rewritten in real units as\nv\n2\r2d2<1p\n2\u0012h\nd\u0000\u000b\u0013\n(26)\nand it is in a reasonable agreement with the numerical\nresults in Fig. 7. Although this expression formally sur-\nvives in the dissipationless limit \u000b!0, it becomes inap-\nplicablein thislimit. Herethebreakdownofspin reversal\nis due to the inertial effect.\nTo investigate the latter, one needs a more accurate\napproximation for f(\u001c)in Eq. (24) near \u001c=\u00001that\ntranforms divergence into a maximum. This can be done\nby solving Eq. (9) although it is difficult to do it ana-\nlytically in general. Instead, since the maximum should\nbe close to \u001c=\u00001, we can solve this equation exactly at\n\u001c=\u00001, which is much easier. A perturbative solution\nforh=d\u001c1yields\nd_s=d\u001c\u0018=2=3; sin\u0012\u0018=(h=d)1=3:(27)\nand then one obtains\n\u0000d\u0012\nd\u001c=1\nsin\u0012dcos\u0012\nd\u001c=2\n3\u0012d\nh\u00131=3\n:(28)\nReplacing in Eq. (24) 1=p\n1\u0000\u001c2by this result and using\nEq. (20), one obtains the dissipationless reversal condi-\ntion\nv\n2\r2d2<3\n2\u0012h\nd\u00134=3\n; (29)\nin a reasonable agreement with the numerical result, Eq.\n(12).\nThe combined reversal condition obtained from Eqs.\n(26) and (12) is thus\nv\n2\r2d2<min(\n1p\n2\u0012h\nd\u0000\u000b\u0013\n;3\n2\u0012h\nd\u00134=3)\n:(30)\nLet us shortly discuss the stability of our spin-reversal\nsolutions that, in the laboratory frame, is the stability of\nphase locking between the spin and the ac field at slow\nfrequency sweep. Linearizing Eqs. (15) and (16) around\nthe static solution ( \u0012;')at a fixed time, one obtains the\ndeviation (\u000e\u0012;\u000e' )/e\u0015t. For the orientations closer to\nthe wells, 3\u0019=4<\u0012\u0014\u0019and0\u0014\u0012<\u0019= 4, one has\u0015<0and phase locking is stable. However, for the orientations\ncloser to the barrier, \u0019=4<\u0012< 3\u0019=4, one has\u0015>0and\nphase locking is unstable. Thus the barrier has to be\ncrossed fast enough during reversal before the instability\ndevelops. Considering the process quasi-statically, one\ncan write\n(\u000e\u0012;\u000e' )\u0018exp2\n4t\u0002\nt0dt0\u0015(t0)3\n5 (31)\nand use the stability criterion\u00010\nt0dt\u0015(t)<1, wheret0is\nthe time of entering the instability region and the top of\nthe barrier is reached at t= 0:After some algebra one\narrives at the stability criterium\n\u000b\n3p\n2<v\n2\r2d2: (32)\nA boundary of this kind is seen in Figs. 8 and 9 close to\nthe bottom.\nB. Optimal frequency sweep\nThe magnetization reversal can be optimized by apply-\ningatime-nonlinearfrequencysweep. Amongallpossible\ncases the so-called “optimal sweep” stands out as a rota-\ntionofthemagneticmomentinoneplane(intherotating\nframe) with '=\u0000\u0019=2, that is with the moment being\nalways perpendicular to the ac field. It is easy to see that\nthis provides the maximal torque on the magnetization\nduring the reversal. With\n!(t) =\u00002\rdcos\u0012 (33)\nEq. (16) self-consistently yields _'= 0. Then Eq. (15)\ntakes the form\n_\u0012=\u0000\rh\u0000\u000b\rdsin 2\u0012: (34)\nIntegratingthisequationwiththeinitialcondition \u0012(0) =\n\u0019, one obtains\ntan\u0012=\u0000sin\u0000~t\u0001\ncos\u0000~t\u0000arcsinA\u0001; (35)\nwhere ~t\u0011p\n1\u0000A2\rhtandAis defined by Eq. (21).\nAfter some algebra the expression for the optimal sweep\ncan be transformed to the most convenient form:\nsz= cos\u0012=\u0000cos\u0000~t\u0000arcsinA\u0001\nq\n1\u0000Acos\u0000\n2~t+ arccosA\u0001(36)\nillustrated in Fig. 11. Together with Eq. (33) it gives\na non-linear time dependence of the frequency of the ac\nfield that provides the fastest reversal of the magnetic\nmoment. This exact result, that generalizes the result of8\nFigure 11: Optimal magnetization reversal for different values\nofA=\u000bd=h.\nRef. 24 obtained in the absence of damping, must have\nimportant practical applications.\nIn the dissipationless case, A!0, the optimal mag-\nnetization reversal is described by a pure cosine function\nthat is a Rabi precession of the magnetic moment around\nthe ac field. In the general case, the reversal is mainly\ndue to the cos term in the numerator, whereas the de-\nnominator only affects the shape of the switching curve,\nmaking it non-sinusoidal in the presence of damping.\nEq. (36) and Fig. 11 describe the optimal reversal\nduring the time\ntrev=\u0019p\n1\u0000A2\rh; A\u0011\u000bd\nh<1(37)\nIt is instructive to compare this time with the time of\nthe magnetization reversal for the linear sweep, defined\nby Eq. (13) (notice that the total time of the process\nmay be longer). In the dissipationless case, the maximal\nsweep speed is given by Eq. (12). For that speed one\nobtains the minimal reversal time\nt(linear)\nrev =4\n3\rh\u0012d\nh\u00131=3\n(38)\nthatislongerthanthetimegivenbyEq. (37)with A= 0.\nIn the dissipative case near A= 1, the maximal sweep\nrate for the linear sweep follows from Eq. (26). This\nyields\nt(linear)\nrev =2p\n2\n\rh1\n1\u0000A: (39)\nForj1\u0000Aj\u001c1this is also is much longer than the time\ngiven by Eq. (37). In the relevant region A\u00181Eqs. (37)\nand (39) are comparable. However, one has to keep in\nmind that the linear frequency sweep has to begin with a\nfrequency beyond the resonance range, so that the actual\nreversal time of the linear sweep is somewhat longer than\nabove.\nThe total energy input of the ac power needed for the\nreversal satisfies\nE/h2trev: (40)For the optimal sweep one has\nE/hp\n1\u0000A2=h2\nq\nh2\u0000(\u000bd)2: (41)\nThe minimum of this function, 2\u000bd, is achieved at\nh=p\n2\u000bd (42)\n(that is atA=\u000bd=h = 1=p\n2).\nFor the linear sweep one has\nE(linear)/h2t(linear)\nrev =h\n1\u0000A=h2\nh\u0000\u000bd:(43)\nThe minimum of this function, 4\u000bd, is achieved at\nh= 2\u000bd (44)\n(that is, at A=\u000bd=h = 0:5).\nWe see that the magnetization reversal by the opti-\nmal sweep requires both a smaller amplitude of the ac\nfield and a smaller total energy input, as compared to\nthe linear sweep. In both cases the injected energy at\nthe maximal efficiency is proportional to \u000band thus the\nefficiency itself is inversely proportional to \u000b. (The lat-\nter was multiplied by \u000bin Figs. 8 and 9 to make them\napproximately scale with \u000b.)\nVI. REVERSAL OF THE MAGNETIZATION\nBY JOSEPHSON CURRENTS\nPointed switching of the magnetization of a nanomag-\nnet by the ac field of varying frequency may be achieved\nbycouplingthemagnettoaweaksuperconductinglink25.\nAdvantage of this method consists of the possibility to\ncontrol the time dependence of the frequency by the volt-\nageacrossthelink, V(t). AshasbeendiscussedintheIn-\ntroduction the most effective switching occurs when the\nac field is circularly polarized. This requires two weak\nlinks shown in Fig. 12. In addition to the previous prob-\nlem one should now take into account the back effect of\nthe magnet on the weak links. As we shall see below, our\nresults for the optimal sweep permit generalization that\nprovides exact time dependences of voltages on the two\nlinks needed to obtain full reversal of the magnetization.\nEach superconducting weak link interacting with the\nmagnet contributes the term\nEJ=\u0000EJcos\u0014\n\u000e\u00002\u0019\n\b0\u00022\n1dr\u0001A(r;t)\u0015\n(45)\nto the total energy. Here EJ=~Ic=(2e)is the Joseph-\nson energy of the link, with Icbeing the critical current.\nThe argument of cosine is the gauge invariant phase that\nconsists of two contributions. The first contribution sat-\nisfying _\u000e= 2eV(t)=~comes from the voltage across the\nlink, while the second contribution is due to the vector9\nFigure 12: Geometry used in the model: Nanomagnet makes\nthe right angle with two parallel superconducting weak links\nof length 2Lat a distance afrom the magnet.\npotential of the magnet integrated between the terminals\nof the link; \b0= 2\u0019~c=(2e)being the flux quantum. Fol-\nlowing the footsteps of Ref. 25, it is easy to show that\ninteraction of the magnet with two Josephson junctions\nleads to a modified expression (5) for the effective field,\nHe\u000b= 2dszez+hJ[sin(\u000ey\u0000ksx)ex+ sin(\u000ex\u0000ksy)ey]\n(46)\nwherehJ=kEJ=(MsV)is the amplitude of the ac mag-\nnetic field created by the junction at the position of the\nnanomagnet,\nd\u000ex;y\ndt=2eVx;y\n~; (47)\nVx;yare the voltages across the junctions, and\nk=4\u0019MsV\na\b0Lp\nL2+a2(48)\nis a dimensionless spin-feedback coupling coefficient.\nFor the time-linear voltage, dependence of the effective\nfield in Eq. (46) on oscillating transverse spin compo-\nnents is detrimental for reversal. This happens because\nthe oscillating additions to the otherwise time-smooth\nphases disturb phase locking between the magnetization\nand the ac field and cause non-adiabaticity. Fig. 13\nshows that non-adiabaticity in the zero-damping case be-\ncomes pronounced already for small values of the feed-\nback coefficient k. In the realistic damped case the nega-\ntive influence of finite kis even stronger. The instability\nof the phase locking for \u0019=4<\u0012< 3\u0019=4discussed at the\nend of Sec. VA exponentially increases the mismatch\nbetween the directions of the ac field and the magneti-\nzation arizing because of the feedback. As a result, the\nmagnetization randomly lands in one of the two wells, as\nshown in Fig. 14.\nFigure 13: Detrimental influence of the magnetization feed-\nback on the Josephson junction in the case of linear frequency\nsweep and zero damping.\nFigure 14: Influence of the magnetization feedback in the\ndamped case with linear sweep.\nIn order to reduce the effect of the Josephson junctions\non the magnet to the effect of a circularly polarized field,\none can require that\n\u000ex\u0000ksy=\u000e(t); \u000ey\u0000ksx=\u000e(t) +\u0019\n2;(49)\nwhere\u000e(t)is a phase. This allows one to use the results\nof the previous section for the optimal sweep requiring\n_\u000e(t) =!(t) =\u00002\rdcos\u0012. For such a sweep, sx;y(t)\nin the laboratory frame are precessing diring reversal as\nfollows\nsx= sin\u0012(~t) cos\u000e(~t) =sin(~t) cos\u000e(~t)p\n1\u0000Acos(2 ~t+ arccosA)\nsy= sin\u0012(~t) sin\u000e(~t) =sin(~t) sin\u000e(~t)p\n1\u0000Acos(2 ~t+ arccosA);(50)\nwhereA=\u000bd=hJ,~t=p\n1\u0000A2\rhJt, and sin\u0012(~t)was10\nFigure 15: The optimal choice of Vx(purple) and Vy(gold)\nacross the weak links at A=k= 0:5. Time dependence of sz\nis shown by blue dots.\nobtained by combining Eqs. (35) and (36). The time-\ndependent phase is given by\n\u000e(t) =\u0002t\n0!(t0)dt0=\u00002d=hJp\n1\u0000A2\u0002~t\n0cos\u0012(~t0)d~t0:(51)\nwith!(t)defined by Eqs. (33) and \u0012(t)given by (36). In\naccordance with Eq. (49), the optimal time dependence\nof the voltages across the junctions become\nVx(t) =~\n2e[!(t)+k_sy]; Vy(t) =~\n2e[!(t)+k_sx]:(52)\nThis dependence is shown in Fig. 15. The time depen-\ndence ofszin the figure is the same as for the optimal\nfreqency sweep in Fig. 11. Oscillations of the voltages are\ndue to the terms in Eq. (52) that depend on sx;y. They\nare weak as long as kis small. Oscillations disappear in\nthe limit of k!0, makingVx;yin that limit to follow the\nsmooth time dependence of the optimal frequency sweep\nobtained in the previous section.\nHaving practical applications in mind, it is interesting\nto test the stability of the reversal described by the above\nequations against high-frequency voltage noise. This can\nbe done by writing\nV0\nx;y(t) =Vx;y(t) +\u000f~\rd\neFx;y(t) (53)\nwithFx;ybeing uniform random functions of time be-\ntween\u00001and+1and\u000frepresenting the relative strength\nof the noise. Quite remarkably, as is illustrated in Fig.\n16, the full magnetization reversal may occur even in the\npresence of a strong noise. At \u000f= 1this happens with\nmore than 0.99 probability. With less than 0.01 prob-\nability the magnetic moment bounces back to sz=\u00001\nbefore it reaches sz= 1. This can be traced to the fact\nthat the high-frequency noise in most cases averages out\nin the phase \u000ebecause the latter is proportional to the\ntime integral of the voltage.\nFigure 16: Optimal magnetization reversal in the presence of\nvoltage noise in the weak links.\nVII. DISCUSSION AND CONCLUSIONS\nWe have studied numerically and analytically a\nmicrowave-assisted reversal of the magnetic moment of a\nsingle-domain magnetic particle. We conclude from our\nstudies that a circularly polarized ac field that has spe-\ncific time dependence of the frequency can be an effective\ntool for switching the magnetization. The corresponding\nphysical mechanism consists of the resonant absorbtion\nof photons of the spin projection that ensures the consis-\ntent change in the projection of the magnetic moment.\nEmission of excitations by the magnetic moment inhibits\nthis process. In the micromagnetic theory it is described\nby the phenomenological dimensionless small parameter\n\u000bthat can be independently measured. In single-domain\nparticles this parameter is usually greater than in bulk\nmaterials and is typically of order 0:01\u00000:1.37The con-\ndition on the power of the ac field needed to overcome\ndamping and reverse the magnetization is h > \u000bd. For,\ne,g., the anisotropy field dof order 0:01T and\u000bof order\n0:01, one obtains hof order 0:0001T for the amplitude of\ntheacfield, whichisareasonablevaluefromthepractical\npoint of view.\nWe have studied linear and nonlinear time dependence\nof the frequency of the ac field. It has been demonstrated\nthat the linear case, !=\u0000vt, resembles the Landau-\nZener problem. Magnetization reversal has been demon-\nstrated numerically and the phase diagrams have been\nobtained that show the range of v,h,d, and\u000bthat\nprovide the reversal. They show that for the reversal\nto occur, the frequency sweep must be sufficiently slow,\nbut not too slow when the damping is finite. The linear\ncase has also been studied analytically. Condition (30)\nhas been obtained for the upper bound on the frequency\nsweep rate. For the values of the parameters used above,\nthat upper bound is in the ballpark of 107GHz/s. The\nminimalreversaltimeforthetime-linearsweepisoforder\n(\rh)\u00001.11\nWehavealsostudiedatime-nonlinearfrequencysweep.\nExact analytical solution for !(t)that provides the\nfastest reversal has been obtained with account of damp-\ning. It is given by equations (33) and (36). This finding\nmay have important practical application. We call this\nsweep the optimal sweep. It has been demonstrated that,\nbesides ensuring the fasted magnetization switch, it also\npumps less energy into the system as compared to the\nlinear sweep. In both cases the injected energy is pro-\nportional to \u000b.\nCircularly polarized ac field can be generated by cou-\npling a single-domain particle electromagnetically to two\nweak superconducting links whose phases are displaced\nby\u0019=2withrespecttoeachother. Oneadvantageofsuch\na system is that the time dependence of the frequency of\nthe ac field generated by the links can be controlled by\nvoltage. This problem has been studied by us with ac-count of the back effect of the magnetic moment on the\nlinks. Magnetization reversal has been demonstrated nu-\nmerically and analytical expressions have been derived\nfor the time dependence of the voltages across the links\nthat provide the fasted magnetization reversal. One re-\nmarkable property of this system is weak dependence of\nthe reversal dynamics on the voltage noise.\nVIII. ACKNOWLEDGEMENTS\nD.G. is thankful to H. Kachkachi, N. 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Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F\u0013 \u0010sica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nMagnetic damping is a key metric for emerging technologies based on magnetic nanoparticles,\nsuch as spin torque memory and high-resolution biomagnetic imaging. Despite its importance,\nunderstanding of magnetic dissipation in nanoscale ferromagnets remains elusive, and the damping\nis often treated as a phenomenological constant. Here we report the discovery of a giant frequency-\ndependent nonlinear damping that strongly alters the response of a nanoscale ferromagnet to spin\ntorque and microwave magnetic \feld. This novel damping mechanism originates from three-magnon\nscattering that is strongly enhanced by geometric con\fnement of magnons in the nanomagnet. We\nshow that the giant nonlinear damping can invert the e\u000bect of spin torque on a nanomagnet leading\nto a surprising current-induced enhancement of damping by an antidamping torque. Our work\nadvances understanding of magnetic dynamics in nanoscale ferromagnets and spin torque devices.\nI. INTRODUCTION\nNanoscale magnetic particles are the core components\nof several emerging technologies such as nonvolatile spin\ntorque memory [1], spin torque oscillators [2{7], targeted\ndrug delivery, and high-resolution biomagnetic imaging\n[8{11]. Control of magnetic damping holds the key to\nimproving the performance of many nanomagnet-based\npractical applications. In biomagnetic characterization\ntechniques such as magnetic resonance imaging [12], re-\nlaxometry [13], and magnetic particle imaging [14, 15],\nmagnetic damping a\u000bects nanoparticles relaxation times\nand image resolution. In spin torque memory and oscil-\nlators, magnetic damping determines the electrical cur-\nrent necessary for magnetic switching [1] and generation\nof auto-oscillations [16] and thereby determines energy-\ne\u000eciency of these technologies. The performance of\nnanomagnet-based microwave detectors is also directly\na\u000bected by the damping [17{19]. Despite its impor-\ntance across multiple disciplines, magnetic damping in\nnanoparticles is poorly understood and is usually mod-\neled as a phenomenological constant [6, 16].\nIn this article, we experimentally demonstrate that a\nferromagnetic nanoparticle can exhibit dynamics quali-\ntatively di\u000berent from those predicted by the constant\ndamping model. We show that nonlinear contributions\nto the damping can be unusually strong and the damp-\ning parameter itself can exhibit resonant frequency de-\npendence. Our work demonstrates that nonlinear damp-\ning in nanomagnets is qualitatively di\u000berent from that in\nbulk ferromagnets and requires a new theoretical frame-\nwork for its description. We show both experimentally\nand theoretically that such resonant nonlinear damping\noriginates from multi-magnon scattering in a magnetic\n\u0003igorb@ucr.edusystem with a discrete spectrum of magnons induced by\ngeometric con\fnement.\nWe also discover that the resonant nonlinear damping\ndramatically alters the response of a nanomagnet to spin\ntorque. Spin torque arising from injection of spin cur-\nrents polarized opposite to the direction of magnetization\nacts as negative damping [2]. We \fnd, however, that the\ne\u000bect of such antidamping spin torque is reversed, lead-\ning to an enhanced dissipation due to the nonlinear res-\nonant scattering. This counterintuitive behavior should\nhave signi\fcant impact on the operation of spin torque\nbased memory [1], oscillators [2{7] and microwave detec-\ntors [17{19].\nII. RESULTS\nA. Spin wave spectroscopy\nWe study nonlinear spin wave dynamics in nanoscale\nelliptical magnetic tunnel junctions (MTJs) that consist\nof a CoFeB free layer (FL), an MgO tunnel barrier, and a\nsynthetic antiferromagnet (SAF) pinned layer [20]. Spec-\ntral properties of the FL spin wave modes are studied in a\nvariety of MTJs with both in-plane and perpendicular-to-\nplane equilibrium orientations of the FL and SAF magne-\ntization. We observe strong resonant nonlinear damping\nin both the in-plane and the perpendicular MTJs, which\npoints to the universality of the e\u000bect.\nWe employ spin torque ferromagnetic resonance (ST-\nFMR) to measure magnetic damping of the FL spin wave\nmodes. In this technique, a microwave drive current\nIacsin(2\u0019ft) applied to the MTJ excites oscillations of\nmagnetization at the drive frequency f. The resulting\nmagnetoresistance oscillations Racsin(2\u0019ft+\u001e) generate\na direct voltage Vmix. Peaks in ST-FMR spectra Vmix(f)\narise from resonant excitation of spin wave eigenmodes\nof the MTJ [21{28]. To improve signal-to-noise ratio,arXiv:1803.10925v1  [cond-mat.mes-hall]  29 Mar 20182\n0 0.5 1 1.5 2H1H2\n36912\nField (kOe)0 0.5 1 1.5 2H2H1\n00.20.4\nField (kOe)Linewidth (GHz)Experiment\nSimulationFrequency (GHz)\n+1\n1a b\n<Vmix>~\nFIG. 1. Spin wave spectra in a nanoscale MTJ. (a) Normalized ST-FMR spectra h~Vmix(f)iof spin wave eigenmodes in a\nperpendicular MTJ device (Sample 1) measured as a function of out-of-plane magnetic \feld. Resonance peaks arising from\nthree low frequency modes of the MTJ free layer j0i,j1i, and j2iare observed. (b) Spectral linewidth of the quasi-uniform\nj0ispin wave mode as a function of out-of-plane magnetic \feld. Strong linewidth enhancement is observed in the resonant\nthree-magnon regime at H1andH2.\nthe magnitude of external magnetic \feld Happlied par-\nallel to the free layer magnetization is modulated, and\na \feld-derivative signal ~Vmix(f) = dVmix(f)=dHis mea-\nsured via lock-in detection technique [20]. Vmix(f) can\nthen be obtained via numerical integration (Supplemen-\ntal Material).\nFigure 1(a) shows ST-FMR spectra ~Vmix(f) measured\nas a function of out-of-plane magnetic \feld Hfor an el-\nliptical 52 nm\u000262 nm perpendicular MTJ device (Sam-\nple 1). Three spin wave eigenmodes with nearly linear\nfrequency-\feld relation fn(H) are clearly visible in the\nspectra. Micromagnetic simulations (Supplemental Ma-\nterial) reveal that these modes are three lowest frequency\nspin wave eigenmodes of the FL (Supplemental Material).\nThe lowest frequency (quasi-uniform) mode j0iis node-\nless and has spatially uniform phase. Each of the two\nhigher-order modes jni(n= 1;2) has a single node at\nthe FL center that is either perpendicular ( n= 1) or\nparallel (n= 2) to the ellipse long axis.\nThe spectral linewidth of the resonances in Fig. 1(a)\ncan be used for evaluation of the mode damping. The\nquasi-uniform mode j0iresonance visibly broadens at\ntwo magnetic \feld values: H1= 0:74 kOe (4 GHz) and\nH2= 1:34 kOe (6 GHz). Near H1, the modej1iresonance\nalso broadens and exhibits splitting, same behavior is ob-\nserved for the mode j2iatH2. At these \felds, the higher-\norder mode frequency is twice that of the quasi-uniform\nmodefn= 2f0. This shows that three-magnon con\ru-\nence [29{33] is the mechanism of the quasi-uniform mode\ndamping increase: two magnons of the quasi-uniform\nmodej0imerge into a single magnon of the higher-ordermodejni.\nThe most striking feature of the quasi-uniform mode\nresonance near H1is its split-peak shape with a local min-\nimum at the resonance frequency. Such a lineshape can-\nnot be \ft by the standard Lorentzian curve with symmet-\nric and antisymmetric components [20]. We therefore use\na double-peak \ftting function (Supplemental Material)\nto quantify the e\u000bective linewidth \u0001 f0of the resonance\npro\fle. For applied \felds su\u000eciently far from H1, the\nST-FMR curve recovers its single-peak shape and \u0001 f0\nis determined as half width of the standard Lorentzian\n\ftting function [20]. Figure 1(b) shows \u0001 f0as a function\nofHand demonstrates a large increase of the linewidth\nnear the \felds of the resonant three-magnon regime H1\nandH2. The stepwise increase of \u0001 f0nearH1is a result\nof the ST-FMR curve transition between the split-peak\nand single-peak shapes. For \felds near H2, the resonance\npro\fle broadens but does not develop a visible split-peak\nlineshape. As a result, \u0001 f0(H) is a smooth function in\nthe vicinity of H2.\nB. E\u000bect of spin torque\nIn MTJs, direct bias current Idcapplied across the\njunction exerts spin torque on the FL magnetization, act-\ning as antidamping for Idc>0 and as positive damping\nforIdc<0 [22, 34]. The antidamping spin torque in-\ncreases the amplitude of the FL spin wave modes [22, 35]\nand decreases their spectral linewidth [36]. We can em-\nploy spin torque from Idcto control the amplitude of spin3\nFIG. 2. E\u000bect of spin torque on spin wave resonance lineshape. (a)-(b) Spin wave resonance lineshapes in the nonresonant\nregime at H > H 1for di\u000berent values of direct bias current Idc. (c)-(d) Spin wave resonance lineshapes in the resonant three-\nmagnon regime at H=H1. (a), (c) Measured ST-FMR spectra (Sample 2). (b), (d) Solutions of Eqs. (3) and (4). Identical\nbias current values Idc(displayed in (a) are used in (a)-(d).\nwave eigenmodes excited in ST-FMR measurements, and\nthereby study the crossover between linear and nonlinear\nregimes of spin wave resonance.\nFigure 2 shows the dependence of ST-FMR resonance\ncurve of thej0imodeVmix(f) onIdcfor a 50 nm\u0002110 nm\nelliptical in-plane MTJ (Sample 2). For in-plane mag-\nnetic \feld values far from the three-magnon resonance\n\feldsHn, the amplitude of ST-FMR resonance curve\nVmix(f) shown in Fig. 2(a) monotonically increases with\nincreasing antidamping spin torque, as expected. At\nH=H1, the antidamping spin torque has a radically\ndi\u000berent and rather surprising e\u000bect on the resonance\ncurve. As illustrated in Fig. 2(c), increasing antidamp-\ning spin torque \frst broadens the resonance at H=H1\nand then transforms a single-peak resonance lineshape\ninto a split-peak lineshape with a local minimum at the\nresonance frequency f0. The data in Fig. 2 demonstrate\nthat the unusual split-peak lineshape of the resonance is\nonly observed when (i) the three-magnon scattering of\nthe quasi-uniform mode is allowed by the conservation of\nenergy and (ii) the amplitude of the mode is su\u000eciently\nhigh, con\frming that the observed e\u000bect is resonant and\nnonlinear in nature.\nFig. 2(c) reveals that antidamping spin torque can in-\ncrease the spectral linewidth and the e\u000bective damping\nof the quasi-uniform spin mode if the mode undergoes\nresonant three-magnon scattering. Figure 3 further illus-\ntrates this counterintuitive e\u000bect. It shows the linewidth\nof the quasi-uniform mode of a 50 nm \u0002110 nm elliptical\nin-plane MTJ (Sample 3) measured as a function of bias\ncurrent. In Fig. 3, blue symbols show the linewidth mea-\nsured at an in-plane magnetic \feld su\u000eciently far fromthe three-magnon resonance \felds Hn. At this \feld, the\nexpected quasi-linear dependence of the linewidth on Idc\nis observed for currents well below the critical current\nfor the excitation of auto-oscillatory magnetic dynamics.\nNear the critical current, the linewidth increases due to\na combination of the fold-over e\u000bect [37{39] and ther-\nmally activated switching between the large- and small-\namplitude oscillatory states of the fold-over regime [22].\nThe red symbols in Fig. 3 show the linewidth measured\nin the resonant three-magnon regime at H=H1. In con-\ntrast to the nonresonant regime, the linewidth increases\nwith increasingjIdcjfor both current polarities. Fur-\nthermore, the maximum linewidth is measured for the\nantidamping current polarity.\nIII. THEORETICAL MODEL\nNonlinear interactions among spin wave eigenmodes\nof a ferromagnet give rise to a number of spectacu-\nlar magneto-dynamic phenomena such as Suhl instabil-\nity of the uniform precession of magnetization [40, 41],\nspin wave self-focusing [42] and magnetic soliton forma-\ntion [43{45]. In bulk ferromagnets, nonlinear interac-\ntions generally couple each spin wave eigenmode to a\ncontinuum of other modes via energy- and momentum-\nconserving multi-magnon scattering [40]. This kinemat-\nically allowed scattering limits the achievable amplitude\nof spin wave modes and leads to broadening of the spin\nwave resonance. These processes lead to a resonance\nbroadening [40, 46{48] and cannot explain the observed\nsplit-peak lineshape of the resonance. In nanoscale ferro-4\nmagnets, geometric con\fnement discretizes the spin wave\nspectrum and thereby generally eliminates the kinemati-\ncally allowed multi-magnon scattering. This suppression\nof nonlinear scattering enables persistent excitation of\nspin waves with very large amplitudes [49] as observed in\nnanomagnet-based spin torque oscillators [2, 50]. Tun-\nability of the spin wave spectrum by external magnetic\n\feld, however, can lead to a resonant restoration of the\nenergy-conserving scattering [31]. The description of\nnonlinear spin wave resonance in the nanoscale ferromag-\nnet geometry therefore requires a new theoretical frame-\nwork. To derive the theory of resonant nonlinear damp-\ning in a nanomagnet, we start with a model Hamilto-\nnian that explicitly takes into account resonant nonlinear\nscattering between the quasi-uniform mode and a higher-\norder spin wave mode (in reduced units with ~\u00111):\nH=!0aya+!nbyb+\t0\n2ayayaa+\tn\n2bybybb (1)\n+( naaby+ \u0003\nnayayb)\n+\u0010\b\nexp(\u0000i!t)ay+ exp(i!t)a\t\nwhereay,aandby,bare the magnon creation and an-\nnihilation operators for the quasi-uniform mode j0iwith\nfrequency!0and for the higher-order spin wave mode\njnimode with frequency !n, respectively. The non-\nlinear mode coupling term proportional to the coupling\nstrength parameter  ndescribes the annihilation of two\nj0imagnons and creation of one jnimagnon, as well as\nthe inverse process. The Hamiltonian is written in the\nresonant approximation, where small nonresonant terms\nsuch asaab,aaayare neglected. The terms proportional\nto \t 0and \t ndescribe the intrinsic nonlinear frequency\nshifts [51] of the modes j0iandjni. The last term de-\nscribes the excitation of the quasi-uniform mode by an\nexternal ac drive with the amplitude \u0010and frequency !.\nWe further de\fne classically a dissipation function Q,\nwhere\u000b0and\u000bnare the intrinsic linear damping param-\neters of the modes j0iandjni[52{54]:\nQ=day\ndtda\ndt(\u000b0+\u00110aya) +dby\ndtdb\ndt(\u000bn+\u0011nbyb) (2)\nFor generality, Eq. (2) includes intrinsic nonlinear\ndamping [16] of the modes j0iandjnidescribed by the\nnonlinearity parameters \u00110and\u0011n. However, our analy-\nsis below shows that the split-peak resonance lineshape\nis predicted by our theory even if \u00110and\u0011nare set equal\nto zero.\nEquations describing the nonlinear dynamics of the\ntwo coupled spin wave modes of the system follow from\nEq. (1) and Eq. (2):\nida\ndt=@H\n@ay+@Q\n@(day=dt)(3)\nidb\ndt=@H\n@by+@Q\n@(dby=dt)(4)\nIt can be shown (Supplemental Material) that these\nequations have a periodic solution a= \u0016aexp (\u0000i!t) and\n100 50 0 50 10000.10.20.30.40.5\nI (A)f0 (GHz)Resonant regime\nNonresonant regimeFIG. 3. E\u000bect of spin torque on linewidth. Linewidth of the\nquasi-uniform spin wave mode as a function of the applied\ndirect bias current (Sample 3): blue symbols { in the non-\nresonant regime H6=H1and red symbols { in the resonant\nthree-magnon regime H=H1. Lines are numerical \fts using\nEqs. (3) and (4).\nb=\u0016bexp (\u0000i2!t), where \u0016a,\u0016bare the complex spin wave\nmode amplitudes. For such periodic solution, Eqs. (3)\nand (4) are reduced to a set of two nonlinear algebraic\nequations for absolute values of the spin wave mode am-\nplitudesj\u0016ajandj\u0016bj, which can be solved numerically.\nSince the ST-FMR signal is proportional to j\u0016aj2(Supple-\nmental Material), the calculated j\u0016aj2(!) function can be\ndirectly compared to the measured ST-FMR resonance\nlineshape.\nWe employ the solution of Eqs. (3) and (4) to \ft the\n\feld dependence of the quasi-uniform mode linewidth in\nFig. 1(b). In this \ftting procedure, the resonance line-\nshapej\u0016aj2(!) is calculated, and its spectral linewidth\n\u0001!0is found numerically. The resonance frequencies !0\nand!nare directly determined from the ST-FMR data\nin Fig. 1(a). The intrinsic damping parameters \u000b0and\n\u000bnnearH1andH2are found from linear interpolations\nof the ST-FMR linewidths \u0001 f0and \u0001fnmeasured at\n\felds far from H1andH2. We \fnd that \u0001 !0weakly\ndepends on the nonlinearity parameters \t and \u0011, and\nthus these parameters are set to zero (Supplemental Ma-\nterial). We also \fnd that the calculated linewidth \u0001 !0\ndepends on the product of the drive amplitude \u0010and\nmode coupling strength  n, but is nearly insensitive to\nthe individual values of \u0010and nas long as\u0010\u0001 n= const\n(Supplemental Material). Therefore, we use \u0010\u0001 nas a\nsingle \ftting parameter in this \ftting procedure. Solid\nline in Fig. 1(b) shows the calculated \feld dependence\nof the quasi-uniform mode linewidth on magnetic \feld.\nThe agreement of this single-parameter \ft with the ex-\nperiment is excellent.\nFigures 2(b) and 2(d) illustrate that Eqs. (3) and\n(4) not only describe the \feld dependence of ST-FMR\nlinewidth but also qualitatively reproduce the spectral5\nlineshapes of the measured ST-FMR resonances as well\nas the e\u000bect of the antidamping spin torque on the line-\nshapes. Fig. 2(b) shows the dependence of the calculated\nlineshapej\u0016aj2(!) on antidamping spin torque for a mag-\nnetic \feldHfar from the three-magnon resonance \felds\nHn. At this nonresonant \feld, increasing antidamping\nspin torque induces the fold-over of the resonance curve\n[37] without resonance peak splitting. The dependence of\nj\u0016aj2(!) on antidamping spin torque for H=H1is shown\nin Fig. 2(d). At this \feld, the resonance peak in j\u0016aj2(!)\n\frst broadens with increasing antidamping spin torque\nand then splits, in qualitative agreement with the ex-\nperimental ST-FMR data in Fig. 2(c). Our calculations\n(Supplemental Material) reveal that while the nonlinear-\nity parameters \t 0,\u00110, \tnand\u0011nhave little e\u000bect on\nthe linewidth \u0001 !0, they modify the lineshape of the res-\nonance. Given that the nonlinearity parameter values\nare not well known for the systems studied here, we do\nnot attempt to quantitatively \ft the measured ST-FMR\nlineshapes.\nEquations (3) and (4) also quantitatively explain\nthe observed dependence of the quasi-uniform mode\nlinewidth \u0001 !0on direct bias current Idc. Assuming an-\ntidamping spin torque linear in bias current [36, 55, 56]:\n\u000b0!\u000b0(1\u0000Idc=Ij0i\nc),\u000bn!\u000bn(1\u0000Idc=Ijni\nc), where\nIjni\nc>Ij0i\ncare the critical currents, we \ft the measured\nbias dependence of ST-FMR linewidth in Fig. 3 by solv-\ning Eqs. (3) and (4). The solid lines in Fig. 3 are the best\nnumerical \fts, where \u0010\u0001 nandIcare used as indepen-\ndent \ftting parameters. The rest of the parameters in\nEqs. (3) and (4) are directly determined from the experi-\nment following the procedure used for \ftting the data in\nFig. 1(b). Theoretical curves in Fig. 3 capture the main\nfeature of the data at the three-magnon resonance \feld\nH1{ increase of the linewidth with increasing antidamp-\ning spin torque.\nIV. DISCUSSION\nFurther insight into the mechanisms of the nonlinear\nspin wave resonance peak splitting and broadening by an-\ntidamping spin torque can be gained by neglecting the in-\ntrinsic nonlinearities \t nand\u0011nof the higher-order mode\njni. Setting \t n= 0 and\u0011n= 0 in Eqs. (3) and (4) allows\nus to reduce the equation of motion for the quasi-uniform\nmode amplitudej\u0016ajto the standard equation for a single-\nmode damped driven oscillator (Supplemental Material)\nwhere a constant damping parameter \u000b0is replaced by\nan e\u000bective frequency-dependent nonlinear damping pa-\nrameter\u000be\u000b\n0:\n\u000be\u000b\n0=\u000b0+\u0014\n\u00110+4\u000bn 2\nn\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(5)and the resonance frequency is replaced by an e\u000bective\nresonance frequency:\n!e\u000b\n0=!0+\u0014\n\t0+2j nj2(2!\u0000!n)\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(6)\nEquation (5) clearly shows that the damping parame-\nter of the quasi-uniform mode itself becomes a resonant\nfunction of the drive frequency with a maximum at half\nthe frequency of the higher order mode ( !=1\n2!n). The\namplitude and the width of this resonance in \u000be\u000b\n0(!) are\ndetermined by the intrinsic damping parameter \u000bnof\nthe higher-order mode jni. If\u000bnis su\u000eciently small,\nthe quasi-uniform mode damping is strongly enhanced\nat!=1\n2!n, which leads to a decrease of the quasi-\nuniform mode amplitude at this drive frequency. If the\ndrive frequency is shifted away from1\n2!nto either higher\nor lower values, the damping decreases, which can re-\nsult in an increase of the quasi-uniform mode amplitude\nj\u0016aj. Therefore, the amplitude of the quasi-uniform mode\nj\u0016aj(!) can exhibit a local minimum at !=1\n2!n. Due to\nits nonlinear origin, the tendency to form a local min-\nimum inj\u0016aj(!) at1\n2!nis enhanced with increasing j\u0016aj.\nSincej\u0016ajis large near the resonance frequency !0, tun-\ning!0to be equal to1\n2!ngreatly ampli\fes the e\u000bect of\nlocal minimum formation in j\u0016aj(!). This qualitative ar-\ngument based on Equation (5) explains the data in Fig. 2\n{ the split-peak nonlinear resonance of the quasi-uniform\nmode is only observed when external magnetic \feld tunes\nthe spin wave eigenmode frequencies to the three-magnon\nresonance condition !0=1\n2!n.\nEquation (6) reveals that the nonlinear frequency shift\nof the quasi-uniform mode is also a resonant function of\nthe drive frequency. In contrast to the nonlinear damping\nresonance described by Equation (5), the frequency shift\nresonance is an antisymmetric function of !\u00001\n2!n. The\nnonlinear shift is negative for ! <1\n2!nand thus causes\na fold-over towards lower frequencies while it is positive\nfor!>1\n2!ncausing fold-over towards higher frequencies.\nAt the center of the resonance pro\fle, the three-magnon\nprocess induces no frequency shift. This double-sided\nfold-over also contributes to the formation of the split-\npeak lineshape of the resonance shown in Figs. 2(c) and\n2(d) and to the linewidth broadening. As with the non-\nlinear damping resonance, the antisymmetric nonlinear\nfrequency shift and the double-sided fold-over become\ngreatly ampli\fed when the spin wave mode frequencies\nare tuned near the three-magnon resonance !0=1\n2!n.\nEquations (5) and (6) also shed light on the origin\nof the quasi-uniform mode line broadening by the an-\ntidamping spin torque. The antidamping spin torque in-\ncreases the quasi-uniform mode amplitude j\u0016ajvia transfer\nof angular momentum from spin current to the mode [57].\nSince the nonlinear damping and the nonlinear frequency\nshift are both proportional to j\u0016aj2and both contribute to\nthe line broadening, the antidamping spin torque can in-\ndeed give rise to the line broadening. Equation (5) reveals\ntwo competing e\u000bects of the antidamping spin torque on\nthe quasi-uniform mode damping parameter \u000be\u000b\n0: spin6\ntorque from Idcdecreases the linear component of the\ndamping parameter \u000b0!\u000b0(1\u0000Idc=Ij0i\nc) and increases\nthe nonlinear component via increased j\u0016aj2. Whether the\nantidamping spin torque decreases or increases the spec-\ntral linewidth of the mode depends on the system param-\neters. Our numerical solution of Eqs. (3) and (4) shown\nin Fig. 3 clearly demonstrates that the antidamping spin\ntorque can strongly increase the linewidth of the quasi-\nuniform mode when the three-magnon resonance condi-\ntion!0=1\n2!nis satis\fed. Furthermore, we \fnd that\nthe three-magnon process exhibits no threshold behav-\nior upon increasing amplitude (Supplemental Material)\nor decreasing intrinsic damping.\nThe key requirement for observation of the resonant\nnonlinear damping is the discreteness of the magnon\nspectrum imposed by geometric con\fnement in the\nnanoscale ferromagnet. The split-peak nonlinear reso-\nnance discovered in this work cannot be realized in bulk\nferromagnets because the three-magnon resonance con-\ndition in bulk is not only valid at the uniform mode\nfrequency!0=1\n2!nbut instead in a broad frequency\nrange. Owing to the magnon spectrum continuity in\nbulk, shifting the excitation frequency away from !0does\nnot suppress the three-magnon scattering of the uniform\nmode { it simply shifts it from one group of magnons to\nanother [29, 40]. Therefore, the amplitude of the uni-\nform mode does not increase when the drive frequency is\nshifted away from !0and the split-peak resonance is not\nrealized.\nWe expect that the resonant nonlinear damping dis-\ncovered in this work will have strong impact on the\nperformance of spin torque devices such as spin torque\nmagnetic memory, spin torque nanooscillators and spin\ntorque microwave detectors. Since all these devices rely\non large-amplitude oscillations of magnetization driven\nby spin torque, the amplitude limiting resulting from the\nresonant nonlinear damping is expected to have detri-\nmental e\u000bect on the device performance.V. CONCLUSIONS\nIn conclusion, our measurements demonstrate that\nmagnetic damping of spin wave modes in a nanoscale\nferromagnet has a strong nonlinear component of reso-\nnant character that appears at a discrete set of magnetic\n\felds corresponding to resonant three-magnon scattering.\nThis strong resonant nonlinearity can give rise to unusual\nspin wave resonance pro\fle with a local minimum at the\nresonance frequency in sharp contrast to the properties\nof the linear and nonlinear spin wave resonances in bulk\nferromagnets. The resonant nonlinearity has a profound\ne\u000bect on the response of the nanomagnet to spin torque.\nAntidamping spin torque, that reduces the quasi-uniform\nspin wave mode damping at magnetic \felds far from the\nresonant three-magnon regime, can strongly enhance the\ndamping in the resonant regime. This inversion of the\ne\u000bect of spin torque on magnetization dynamics by the\nresonant nonlinearity is expected to have signi\fcant im-\npact on the performance of nanoscale spin torque devices\nsuch as magnetic memory and spin torque oscillators.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFoundation through Grants No. DMR-1610146, No.\nEFMA-1641989 and No. ECCS-1708885. We also ac-\nknowledge support by the Army Research O\u000ece through\nGrant No. W911NF-16-1-0472 and Defense Threat Re-\nduction Agency through Grant No. HDTRA1-16-1-0025.\nA. M. G. thanks CAPES Foundation, Ministry of Educa-\ntion of Brazil for \fnancial support. R.E.A acknowledges\nFinanciamiento Basal para Centros Cienti\fcos y Tec-\nnologicos de Excelencia under project FB 0807 (Chile),\nand Grant ICM P10-061-F by Fondo de Innovacion para\nla Competitividad-MINECON. 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Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F´ ısica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nI. METHODS\nA. Linewidth evaluation\nAll measurements presented were carried out with magnetic field applied along the easy axis of the MTJ devices so\nthat the magnetic moments of the free and pinned layers are collinear to each other. In this geometry, the ST-FMR sig-\nnals are dominated by photo-resistance contribution and are proportional to the square of the transverse component of\nthe dynamic magnetization magnetization [1], which allows us to directly compare calculated |a|2(ω) resonance curves\nto measured ST-FMR resonance curves ˜Vmix(f) and toVmix(f) approximated by numerical integration/integraltext˜Vmix(f)df.\nWhenVmix(f) and|a|2(ω) are single-peak curves, they are fit to a sum of symmetric and antisymmetric Lorentzian\ncurves with identical central frequencies and linewidth parameters as described in Ref. [2], and the spectral linewidth\nis determined as half-width at the half-maximum of the symmetric Lorentzian curve.\nIn order to quantify the linewidth of the split-peak resonance profile, we introduce a fitting function that is a sum\nof two Lorentzian curves with different central frequencies separated by δf. The half width of the resonance profile\n∆f0is then defined as the average of the half widths of the two Lorentzians plus δf/2.\nSupplemental Figure 1. Spatial profiles of spin wave eigenmodes. Normalized amplitude and phase of the three lowest frequency\nspin wave eigenmodes of the MTJ free layer, given by micromagnetic simulations.\nB. Micromagnetic simulations\nMicromagnetic simulations were performed using OOMMF software [3, 4]. To account for all magnetic interactions\nin the MTJ, a three dimensional model was employed with three ferromagnetic layers: free, SAF top and SAF bottom.\nWe use material parameters obtained from the measurements and/or their accepted literature values (see Ref. [2] for2\nthe MTJ structure and fabrication details). Magnetization dynamics is excited by a combined pulse of spin torque\nand Oersted field, resulting from a sinc-shaped spatially uniform current pulse. The spatial profile of the Oersted\nfield corresponds to that of a long wire with elliptical cross section. The direction of the spin torque vector acting\non the free layer is determined by the magnetization orientation of the SAF top layer. The spectrum of spin wave\neigenmodes is obtained via fast Fourier transform (FFT) of the time dependent components of the layers’ magnetic\nmoment. Spatial mapping of the resulting Fourier amplitude and phase at a given frequency provides the mode\nprofiles (Supplmental Fig. 1). The observed excitations are confirmed to be spin wave modes localized to the free\nlayer. SAF modes are found at much higher frequencies than the free layer modes, and their frequencies are found to\nbe incommensurable to the free layer quasi-uniform mode frequency [5].\nII. SOLUTION OF THE EQUATIONS OF MOTION\nThe Hamiltonian equations of motion describing the coupled dissipative dynamics of the quasi-uniform ( a) and the\nhigher-order ( b) spin wave modes are:\nida\ndt=∂H\n∂a†+∂Q\n∂(da†/dt)(1)\nidb\ndt=∂H\n∂b†+∂Q\n∂(db†/dt)(2)\nwhereHis the Hamiltonian of the system and Qis the dissipation function, given by:\nH=ω0a†a+ωnb†b+1\n2Ψ0a†a†aa+1\n2Ψnb†b†bb+ (ψ∗\nnaab†+ψna†a†b) +ζ{exp(−iωt)a†+ exp(iωt)a} (3)\nQ=da†\ndtda\ndt(α0+η0a†a) +db†\ndtdb\ndt(αn+ηnb†b) (4)\nBy using Eq. (3) and Eq. (4) in Eq. (1) and Eq. (2), the Hamiltonian equations can be written as:\nida\ndt−(α0+η0a†a)da\ndt=ω0a+ 2ψna†b+ Ψ 0a†aa+ζexp(−iωt) (5)\nidb\ndt−(αn+ηnb†b)db\ndt=ωnb+ψ∗\nnaa+ Ψnb†bb (6)\nUsing a periodic ansatz a= ¯aexp(−iωt) andb=¯bexp(−2iωt) in Eq. (5) and Eq. (6), where ¯ aand¯bare complex\namplitudes, reduces the Hamiltonian equations to a set of two algebraic equation for the complex amplitudes:\n/parenleftbig\nω−ω0−Ψ0|¯a|2+i(α0+η0|¯a|2)ω/parenrightbig\n¯a−2ψn¯a∗¯b=ζ (7)\n/parenleftbig\n2ω−ωn−Ψn|¯b|2+ 2i(αn+ηn|¯b|2)ω/parenrightbig¯b=ψ∗\nn¯a2(8)\nWe solve Eq. (8) for ¯band multiply the numerator and denominator of this expression by the complex conjugate of\nthe denominator:\n¯b=ψ∗\nn¯a2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n−i2(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(9)\nthen we multiply Eq. (9) by2ψn¯a∗\n¯aand evaluate the real and imaginary parts.\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|22/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(10)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|2−4(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(11)\nBy taking the modulus of Eq. (8), we obtain:\n|¯a|2=|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2 (12)3\nUsing Eq. (12) in Eqs. (10-11), we derive:\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(13)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(14)\nTaking the modulus squared of Eq. (7):\n/braceleftBigg/parenleftbigg\nω−ω0−Ψ0|¯a|2−/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2\n+/parenleftbigg\n(α0+η0|¯a|2)ω−/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2/bracerightBigg\n|¯a|2=ζ2(15)\nand using Equations (12)–(14) in Eq. (15) gives us an algebraic equation for the absolute value of the higher order\nmode amplitude|¯b|:\n\n\n\nω−ω0−Ψ0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2−2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n+\n\n/parenleftbigg\nα0+η0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2/parenrightbigg\nω−−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n\n×\n|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2=ζ2\n(16)\nAfter numerically solving Eq. (16) for |¯b|, and using it in Eq. (12), we can calculate the amplitude of the quasi-uniform\nmode|¯a|.\nIII. EFFECTS OF THE DRIVE AMPLITUDE AND INTRINSIC NONLINEARITITES\nTo understand the impact of the intrinsic nonlinearity parameters (Ψ 0, Ψn,η0,ηn) on the quasi-uniform spin wave\nmode resonance, we plot the numerical solution of Eq. (16) in Supplemental Figure 2. Each panel of this figure shows\na reference lineshape of the resonance calculated with all intrinsic nonlinearity parameters set to zero (red curve) and\na lineshape calculated with one of the intrinsic nonlinearity parameter different from zero (blue curve). This figure\nreveals that increasing η0decreases the mode amplitude and slightly increases the linewidth. Increasing ηndecreases\nthe degree of the double-peak lineshape splitting. Increasing Ψ nincreases the lineshape asymmetry. Increasing Ψ 0\nincreases lineshape asymmetry and induces fold-over.\nSupplemental Figure 3 shows the linewidth as a function of the drive amplitude for three scenarios, where the\nintrinsic nonlinearities Ψ 0, Ψn,η0,ηnare set to zero for simplicity. If the coupling parameter is zero, ψn= 0, the\nlinewidth does not depend on the drive amplitude, as expected for a single-mode linear oscillator. The second case\ndemonstrates that the linewidth remains constant when the product ψn·ζis constant. For a constant non-zero\ncoupling parameter, the linewidth shows an increase with the drive amplitude. This observation allows us to employ a\nsingle fitting parameter ( ψn·ζ) to fit the data in Fig. 1b. This conjecture can be confirmed analytically by introducing\na normalized spin wave amplitude ˆ a=ψn¯a, which allows us to rewrite Eq. (16) omitting all intrinsic nonlinearities\ninto the following form:\nω/bracketleftbigg\n1 +iα0+i4αn|ˆa|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\nˆa−ω0ˆa−2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2|ˆa|2ˆa=ψnζ (17)\nThis equation describes an effective single-mode nonlinear oscillator with renormalized excitation amplitude ψnζ.4\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nη0\n2 2.5 3 3.500.010.02\nFrequency (GHz)|a|20\nηn\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨ0\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨn|a|2|a|2|a|2a b\nc d\nSupplemental Figure 2. Effect of intrinsic nonlinearities on the quasi-uniform spin wave resonance lineshape. Spectral lineshape\nof the quasi-uniform spin wave mode resonance |¯a|2(ω) at the three-magnon resonance condition 2 ω0=ωncalculated by\nnumerically solving Eq. (16). The red curve is a reference lineshape calculated with all intrinsic nonlinearity parameters\n(η0,ηn,Ψ0,Ψn) set to zero. The blue lineshape in each panel is calculated with one of the intrinsic nonlinearity parameters set\nto a non-zero value: (a) η0= 1.325·10−24J, (b)ηn= 3.313·10−24J, (c) Ψ 0= 1.325·10−24J, (d) Ψ n= 1.325·10−23J. Other\nparameters employed in the calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662,αn= 0.03042 atIdc= 0;\nψn·ζ=h2·0.006 GHz2, wherehis the Planck constant.\n0 0.05 0.1 0.15 0.20.150.20.25\nζ h-1 (GHz)∆f0 (GHz)\n(i)(ii)(iii)\nSupplemental Figure 3. Effect of the drive amplitude on linewidth in the resonant three-magnon regime. Calculated linewidth\nof the quasi-uniform spin wave mode as a function of the drive amplitude ζfor different values of the mode coupling parameter\nψn. (i) Green: ψn= 0, (ii) red: variable ψnwith a constraint ψn·ζ=h2·0.006 GHz2, and (iii) blue: ψn=h·0.1 GHz. All\nintrinsic nonlinearity parameters: Ψ 0, Ψn,η0andηnare set to zero. his the Planck constant. Other parameters employed in\nthe calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662 andαn= 0.03042 atIdc= 0.5\nIV. EFFECTIVE SINGLE-MODE NONLINEAR OSCILLATOR APPROXIMATION\nIf we neglect intrinsic nonlinearities Ψ nandηnof the higher order spin wave mode, Eq. (16) can be reduced to a\ncubic equation for ¯ aand solved analytically. This approximation allows us to obtain several important qualitative\ninsights into the properties of the resonant nonlinear damping of the quasi-uniform mode. By setting Ψ n= 0 and\nηn= 0 in Eq. (8), we obtain an exact solution for ¯b:\n¯b=ψ∗\nn¯a2\n2ω(1 +iαn)−ωn(18)\nUsing this result, we reduce Eq. (16) to a cubic algebraic equation for ¯ a:\nω/bracketleftbigg\n1 +i(α0+η0|¯a|2) +i4|ψn|2αn|¯a|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n¯a−ω0¯a−/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2¯a=ζ (19)\nThis equation describes the amplitude ¯ aof an effective single-mode nonlinear oscillator.\nIt is evident from Eq. (19) that the frequency of the quasi-uniform mode experiences a nonlinear shift:\nωeff\n0=ω0+/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(20)\nThe nonlinear frequency shift has a well-pronounced antisymmetric resonant character near the resonance frequency\nωn/2, that arises from the resonant three-magnon scattering.\nFurther, it is clear from Eq. (19) that the effective damping of the quasi-uniform mode also acquires a term arising\nfrom the three-magnon interaction:\nαeff\n0=α0+/bracketleftbigg\nη0+4|ψn|2αn\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(21)\nThe last term describes a resonant enhancement of the nonlinear damping by three-magnon scattering near the\nresonance frequency ωn/2. Strikingly, the magnitude of the resonant damping enhancement at ωn/2 increases when\nthe intrinsic damping of the higher order mode αndecreases. In the limit αn→0, the effective damping becomes\nαeff\n0→α0+/bracketleftbigg\nη0+2π|ψn|2\nωδ(2ω−ωn)/bracketrightbigg\n|¯a|2(22)\nwhereδis Dirac’s delta function. Equation (21) suggests that the effective damping of the quasi-uniform mode αeff\n0\ncan increase with increasing antidamping spin torque applied to the nanomagnet. Indeed, the antidamping spin torque\ntends to increase the amplitude [6] of the quasi-uniform mode |¯a|and decrease the intrinsic damping parameter of the\nhigher order mode αn→αn(1−Idc/I|n/angbracketright\nc), both enhancing the nonlinear damping term in Eq. (19). For a sufficiently\nlarge mode coupling parameter ψn, the enhancement of the nonlinear damping term by the antidamping spin torque\ncan exceed the reduction of the linear damping parameter α0→α0(1−Idc/I|0/angbracketright\nc) by the torque, leading to an increase\nofαeff\n0byIdc>0 and broadening of the quasi-uniform mode resonance by the antidamping spin torque. This scenario\nis indeed realized in the MTJ devices studied here as demonstrated by the data and calculations in Fig. 3.\nV. MODE COUPLING PARAMETER\nIn this Supplementary Note, we discuss how the coupling parameter between the spin wave modes, ψnin Eq. (3),\ncan be calculated. We consider a very thin, magnetically soft ferromagnetic disk with elliptical cross section, that\nis magnetized in-plane. Within a classical micromagnetic model, we include Zeeman, dipolar and exchange terms in\nthe free energy. An applied field Halong thexdirection (long axis of the ellipse) magnetizes the sample to a nearly\nuniform state. Through a classical Holstein-Primakoff transformation [7] we introduce variables c(/vector x,t) andc∗(/vector x,t) to\ndescribe the magnetization such that the magnetization magnitude is conserved:\nmx= 1−cc∗, m +=c√2−cc∗, m−=c∗√2−cc∗, (23)\nwhere/vector m=/vectorM/Ms, andm±≡mz±imy. Approximating the exchange energy to the fourth order in candc∗, the\nnormalized free energy of the disk, U≡E/4πM2\ns, is given by\nU/similarequal−hx/integraldisplay\n(1−cc∗) dV+ (lex)2/integraldisplay/bracketleftbigg\n/vector∇c·/vector∇c∗+1\n4c2(/vector∇c∗)2+1\n4c∗2(/vector∇c)2/bracketrightbigg\ndV−1\n2/integraldisplay\ndV/vectorhD(/vector m)·/vector m, (24)6\nwithhx≡H/4πMs,lex≡/radicalbig\nA/2πM2sis the exchange length, and /vectorhD(/vector m) =/vectorHD(/vector m)/4πMsis the normalized\ndemagnetizing field. The Landau-Lifshitz equations of motion in the new variables are: i˙c=δU/δc∗,i˙c∗=−δU/δc\nwitht/prime= 4πMs|γ|t.\nAssuming the normal modes involved in three magnon scattering dominate the magnetization dynamics, the free\nenergy in Eq. (24) can be written in terms of amplitudes of these modes, by expressing cin terms of aandb:\nc(/vector x,t)/similarequala(t)f(/vector x) +a∗(t)g(/vector x) +b(t)p(/vector x) +b∗(t)q(/vector x) (25)\nThe functions f,g,p,q can be determined from calculating the linear modes of oscillation of the sample. The terms of\nthe free energy proportional to aab∗anda∗a∗bdescribe the three-magnon process and the magnitude of these terms\ngives the coupling parameter ψn.\nIf the magnetization state is approximated as exactly uniform, the dipolar energy for a very thin film may be\napproximated as UD=m2\nz/2 = (c+c∗)2(1−cc∗/2), and in this case all three-magnon terms are zero. However, when\nthe effects due to the sample edges (such as spatial inhomogeneity of the demagnetization field and edge roughness)\nare taken into account, the equilibrium magnetization configuration is generally nonuniform. In this case, there are\nnon-zero three-magnon terms in the free energy expression. An explicit calculation of the corresponding overlap\nintegrals is necessary for a quantitative prediction of ψn. Refs. [8, 9] show such extensive calculations for circular disks\nand include explicit expressions for the exchange and dipolar energies.\n[1] Michael Harder, Yongsheng Gui, and Can-Ming Hu, “Electrical detection of magnetization dynamics via spin rectification\neffects,” Phys. Rep. 661, 1–59 (2016).\n[2] A. M. Gon¸ calves, I. Barsukov, Y.-J. Chen, L. Yang, J. A. Katine, and I. N. Krivorotov, “Spin torque ferromagnetic\nresonance with magnetic field modulation,” Appl. Phys. Lett. 103, 172406 (2013).\n[3] M. J. Donahue and D. G. Porter, OOMMF User’s Guide (National Institute of Standards and Technology, Gaithersburg,\nMD, 1999).\n[4] Robert D. McMichael and Mark D. Stiles, “Magnetic normal modes of nanoelements,” J. Appl. Phys. 97, 10J901 (2005).\n[5] P. S. Keatley, V. V. Kruglyak, A. Neudert, R. J. Hicken, V. D. Poimanov, J. R. Childress, and J. A. Katine, “Resonant\nenhancement of damping within the free layer of a microscale magnetic tunnel valve,” J. Appl. Phys. 117, 17B301 (2015).\n[6] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, “Spin-transfer-driven\nferromagnetic resonance of individual nanomagnets,” Phys. Rev. Lett. 96, 227601 (2006).\n[7] T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58,\n1098–1113 (1940).\n[8] D. Mancilla-Almonacid and R. E. Arias, “Instabilities of spin torque driven auto-oscillations of a ferromagnetic disk mag-\nnetized in plane,” Phys. Rev. B 93, 224416 (2016).\n[9] D. Mancilla-Almonacid and R. E. Arias, “Spin-wave modes in ferromagnetic nanodisks, their excitation via alternating\ncurrents and fields, and auto-oscillations,” Phys. Rev. B 95, 214424 (2017)."    },    {        "title": "2012.14733v1.Twist_induced_Near_field_Thermal_Switch_Using_Nonreciprocal_Surface_Magnon_Polaritons.pdf",        "content": "Twist-induced Near-field Thermal Switch Using\nNonreciprocal Surface Magnon-Polaritons\nJiebin Peng,yGaomin Tang,\u0003,zLuqin Wang,yRair Macêdo,{Hong Chen,y\nand Jie Ren\u0003,y\nyCenter for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics,\nShanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School\nof Physics Science and Engineering, Tongji University, 200092 Shanghai, China\nzDepartment of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n{James Watt School of Engineering, Electronics & Nanoscale Engineering Division, University\nof Glasgow, Glasgow G128QQ, United Kingdom\nE-mail: gaomin.tang@unibas.ch; xonics@tongji.edu.cn\nAbstract\nWe explore that two ferromagnetic insulator slabs host a strong twist-induced near-field\nradiative heat transfer in the presence of twisted magnetic fields. Using the formalism of fluc-\ntuational electrodynamics, we find the existence of large twist-induced thermal switch ratio\nin large damping condition and nonmonotonic twist manipulation for heat transfer in small\ndamping condition, associated with the different twist-induced effects of nonreciprocal ellip-\ntic surface magnon-polaritons, hyperbolic surface magnon-polaritons, and twist-non-resonant\nsurface magnon-polaritons. Moreover, the near-field radiative heat transfer can be significantly\nenhanced by the twist-non-resonant surface magnon-polaritons in the ultra-small damping con-\ndition. Such twist-induced effect is applicable for other kinds of anisotropic slabs with time-\nreversal symmetry breaking. Our findings provide a way to twisted and magnetic control in\nnanoscale thermal management and improve it with twistronics concepts.\n1arXiv:2012.14733v1  [cond-mat.mes-hall]  29 Dec 2020A key component for manipulating radiative heat flow at the nanoscale is near-field radiative\nheat transfer, which can exceed Planck’s blackbody radiation limit1by orders of magnitude due to\nthe presence of evanescent modes.2–12Two types of surface modes have been commonly studied\nin near-field heat transfer; one is surface plasmon-polaritons13–23and the other is surface phonon-\npolaritons.24–29In addition, surface magnon-polaritons (SMPs), hybrid collective excitations due\nto the coupling between magnons and electromagnetic fields,30–32also has functional associations\nto thermal management in nanotechnologies. For instance, in magnetic recording devices, a mag-\nnetic read/write head touches above the disk surface with nanometers separation. At such a short\ndistance, SMPs should play a significant role in the near-field thermal manipulation of magnetic\nrecording devices. Moreover, due to the high gyrotropic optical effect,33SMPs in uniaxial ferro-\nmagnetic insulator (FMI) are nonreciprocal. Such nonreciprocal behavior can break Kirchhoff’s\nlaw34and paves the way for the exploitation of radiative thermal transfer at nanoscale.\nRecently, twistronics becomes an emerging research topic since the electronic state can be ma-\nnipulated through the “twist angle\" between two layers, leading to flat-band superconductivity,35,36\nmoiré excitons,37stacking-dependent interlayer magnetism38and other exotic electronic proper-\nties. Similar twist-induced concepts have been demonstrated in photonics, such as moiré photonics\ncrystal,39moiré hyperbolic metasurfaces40and photonic magic angles.41,42Motivated by these ex-\notic discoveries, several works have shown the development of tunable radiative heat flow between\ntwo-dimensional materials and biaxial crystals43–47through twist. With the analogous principle,\nwe explore the effects of radiative thermal twistronics between the uniaxial FMIs with external\nmagnetic fields, where the twist and nonreciprocal phenomena can both arise in the domain of\nthermal management.\nIn this Letter, we consider to manipulate near-field radiative heat transfer through the twist\nbetween two uniaxial FMIs. Nonreciprocal SMPs emerge at the interface between vacuum and gy-\nrotropic FMIs with asymmetric permeability tensor. Based on the nonreciprocity, we demonstrate\na large twist-induced near-field thermal switch effect with a moderate external magnetic field. Un-\nder ultra-small damping condition, we show an unusual twist-induced near-field thermal transfer\n2enhancement due to the presence of twist-non-resonant SMPs.\n(a)\n(c)\n𝑇2,𝜇2\n𝑇1,𝜇1d\nxz\ny\nθθ\nx z\ny\n(b)\n(d)\nω𝑢\nω𝑑y΄x΄θVacuum \nBulk MP \nSMP \nd = 2mm d = 100nmx'z'\ny'\nFigure 1: (a) A schematic setup for radiative heat transfer between two FMIs with vacuum sep-\narationd. The bottom and top slabs have the temperature T1andT2, respectively. The y(y0)\naxis is along the direction of the satuation magnetisation in the bottom (top) FMI. The magnetic\nfields in each slab is applied along the direction of the corresponding satuation magnetization. The\ntwist angle\u0012is defined by the anticlockwise rotation of x0y0z0coordinate system with respect to\nxyzcoordinate system. (b) Dispersion relation of nonreciprocal SMP with a single vacuum-FMI\ninterface. (c) Energy transmission coefficient Z(!;q;\u001e = 0) with gap distance d= 2mm. The\ncyan dashed line and the red dash-dotted line are the same as in (b). The black dotted line shows\nthe nonreciprocal symmetric and asymmetric modes of SMPs. (d) Energy transmission coefficient\nZ(!;q;\u001e = 0:2\u0019)with gap distance d= 100 nm. The damping constant \u000bis0:01in (c) and (d).\nRadiative heat transfer.– We consider near-field radiative heat transfer between two FMIs with\ntemperatures T1(2)=T\u0006\u0001T=2and twist angle \u0012[See Fig. 1(a)]. A Cartesian coordinate system\nxyz (x0y0z0) is defined at the bottom (top) slab and the y(y0) axis is along the direction of the\napplied magnetic field and saturation magnetisation. The twist angle \u0012is defined as the angle\nbetween the y0andyaxis. We define the heat transfer coefficient \u0014as\u0014= lim \u0001T!0J=\u0001TwithJ\n3the heat flux. From fluctuational electrodynamics,3,4the heat transfer coefficient can be expressed\nas\n\u0014(T;\u0012) =Z1\n0d!\n2\u0019\u0016h!@N\n@TZ1\n0dq\n2\u0019qZ2\u0019\n0d\u001e\n2\u0019Z(!;q;\u001e ); (1)\nwhereqis the in-plane wave vector and \u001ethe in-plane azimuthal angle. In the above expression,\n@N=@T is the derivative of the Bose distribution function with respect to the temperature. We\nconsider the relative heat transfer coefficient scaled by the black-body limit \u0014b= 4\u001bbT3with\n\u001bb=\u00192k4\nB=(60\u0016h3c2). The energy transmission coefficient Z(!;q;\u001e )with twist angle \u0012reads\nZ=8\n><\n>:Tr[(I\u0000Ry\n2R2)D(I\u0000R1Ry\n1)Dy]; q<!=c;\nTr[(Ry\n2\u0000R2)D(R1\u0000Ry\n1)Dy]e\u00002j\f0jd; q>!=c;(2)\nwhere\f0=p\n(!=c)2\u0000q2is the out-of-plane wave vector in vacuum and Ithe identity matrix.\nThe Fabry-Perot-like denominator matrix is written as D= (I\u0000R1R2e2i\f0d)\u00001. In our setup, the\nreflection coefficient matrix Rawitha= 1;2is written as\nRa=2\n64ra\nssra\nsp\nra\npsra\npp3\n75 (3)\nwhere superscripts sandpdenote the polarization states. The reflection coefficients can be cal-\nculated by the transfer matrix methods48and the details are given in the Supplemental Material.49\nFor later convenience, we also define the integrated energy transmission coefficient, i.e. Z(!;\u001e),\nwhich is the energy transmission coefficient after an integration over the wave vector q.\nBy applying a magnetic field along the y-direction in the bottom FMI, the permeability tensor\nhas the form31,33\n\u0016=2\n66664\u0016xx\u0016xy\u0016xz\n\u0016yz\u0016yy\u0016yz\n\u0016zx\u0016zy\u0016zz3\n77775=2\n66664\u0016r0\u0000i\u0016i\n0 1 0\ni\u0016i0\u0016r3\n77775; (4)\n4The diagonal and off-diagonal terms are, respectively, expressed as \u0016r= 1 +!m(!0+i\u000b!)\n(!0+i\u000b!)2\u0000!2and\n\u0016i=!m!\n(!0+i\u000b!)2\u0000!2with frequency !and magnetic precession damping constant \u000b. The magnetic\nresonance frequencies !0=\u00160\rhand!m=\u00160\rmsare due to the external magnetic field hand\nthe saturation magnetization mswith the gyromagnetic ratio \r. The relative permittivity of the\nFMI is assumed to be a constant. For the top FMI, the relative permeability tensor is expressed as\n\u00160=R(\u0012)\u0016RT(\u0012)with the rotation matrix R(\u0012)alongzaxis. During the numerical calculation,\nwe adopt the parameters of yttrium iron garnet (YIG) with the relative permittivity \u000f= 14:5,50gy-\nromagnetic ratio \r=2\u0019= 28 GHz/T51and saturation magnetization \u00160ms= 0:28T.52The applied\nmagnetic field \u00160his taken as 0:4T. Such set of parameters results in SMPs at microwave fre-\nquency range so that we consider the radiative heat transfer at the cryogenic environment (around\n4K).\nNonreciprocal surface magnon-polaritons.– At a single vacuum-FMI interface, there exists\nSMPs of which the dispersion is nonreciprocal. The implicit dispersion relation for SMPs is49\n\f0+ (\u0016r\f1\u0000i\u0016iq)=(\u00162\nr\u0000\u00162\ni) = 0: (5)\nwhere\f1=p\n\u000f\u0016eff(!=c)2\u0000q2is the out-of-plane wave vector inside the FMI and \u0016eff= (\u00162\nr\u0000\n\u00162\ni)=\u0016r. Figure. 1(b) indicates the nonreciprocal dispersion of SMPs (gray line) outside the light\ncone (red dash-dotted line), together with the symmetric dispersion of bulk magnon-polations, that\nis,q=p\u000f\u0016eff!=c(cyan dashed line). We highlight that SMPs exist at the band gap region of FMI\nand the high- qSMPs only exist at positive wave vector region, which is useful in manipulating\nnear-field heat transfer.\nFor the case of two FMIs with millimeter separation, SMPs from two interfaces can be coupled.\nFigure. 1(c) shows the energy transmission coefficient between two FMIs at zero azimuthal and\ntwist angle, that is, Z(!;q;\u001e = 0;\u0012= 0) . We can observe that there exists an asymmetric trans-\nmission coefficient both for bulk MPs (the region outside the light cone and inside the dispersion\nrelation of MPs) and SMPs (the near-unity line inside the band gap), with respect to the in-plane\n5wave vector. The two near-unity lines for SMPs are consistent with the implicit dispersion relation\nof SMPs as follows\n\f0+ tanh(j\f0jd=2)(\f1\u0016r\u0000iq\u0016i)=(\u00162\nr\u0000\u00162\ni) = 0; (6)\n\f0+ coth(j\f0jd=2)(\f1\u0016r\u0000iq\u0016i)=(\u00162\nr\u0000\u00162\ni) = 0: (7)\nIn the absence of the contributions from \u0016i, Eqs. (6) and (7) can be reduced to dispersion relations\nsimilar to those of surface phonon-polaritons.\nIn addition, the optical properties of FMI are anisotropic in the x-zplane when there is nonzero\nazimuthal angle. To qualitatively analyze the anisotropic effects, we show the energy trans-\nmission coefficient with a nonzero azimuthal angle in Fig. 1(d), where the near-unity lines be-\ntween frequency !uand!dexpand as a near-unity spot. Here, !uand!dare the\u0016-near-zero\nfrequencies with azimuthal angles \u001e= 0 and\u001e= 0:2\u0019, respectively, and are determined by\n\u0016r(!u=d) cos2\u001e+ sin2\u001e= 0. In the region between !uand!d, the diagonal terms of permeability\ntensor inx-zplane have the opposite sign, that is, \u0016xx>0,\u0016yy>0and\u0016zz<0.49It is similar to\ntype-I hyperbolic metamaterial45,53without considering the off-diagonal term in the permeability\ntensor. Comparing with that of \u001e= 0 condition, i.e., \u0016xx<0,\u0016yy>0and\u0016zz<0, the twist-\ninduced hyperbolic SMPs emerge at x-zplane when \u001e= 0:2\u0019. Fig. 1(d) proves the existence of\nsuch hyperbolic SMPs and also shows that it can provide more channels for radiative heat transfer.\nSo this azimuthal-angle dependent hyperbolic mode can contribute to a enhancement of radiative\nheat transfer. The coexistence of nonreciprocal and anisotropic effects in FMI is helpful for twisted\nand magnetic thermal management.\nTwist-induced Near-field Thermal Switch.– To study the twist-induced thermal switch mediated\nby the nonreciprocal SMPs, the thermal switch ratio R\u0014(\u0012)is defined as\nR\u0014(\u0012) =\u0014(\u0012)=\u0014min (8)\nwhere\u0014minis the minimal heat transfer coefficient by changing the twist angle \u0012.\n6α=0.01 α=0.001α=0.1\nθ=0θ=0.5πθ=1.5π\nθ=π θ=π(a)\n(c)(b)\n(d)\n0\n( , )\n01×1052×105\n(cm−1)\n𝜙-𝜋 𝜋Figure 2: (a) Twist-induced near-field thermal switch ratio as a function of twist angle with differ-\nent damping constants \u000b. (b) The contour for integrated energy transfer coefficient in !-\u001espace\nat single vacuum-FMI interface with different twist angles. (c)-(d) The spectral function of heat\ntransfer coefficient with different damping constants and twist angles.\n7Fig. 2(a) shows the switch ratio with different damping constants \u000b. It can be seen that the\nswitch ratio is maximal at the parallel configuration ( \u0012= 0). At large damping conditions, the\ngreen-dotted line in Fig. 2(a) indicates that the switch ratio reaches about 9. The physical mecha-\nnism of such a large switch ratio can be related to the match or mismatch of the integrated energy\ntransmission coefficient in the !-\u001espace. As shown in Fig. 2(b), the overlap region of the in-\ntegrated energy transmission coefficient reaches maximal value in parallel configuration. With\nincreasing or decreasing the twist angle \u0012, the central region of the integrated energy transmission\ncoefficient at the twisted FMI will shift left or right in !-\u001espace and the overlap between two FMIs\nreaches the minimum value in anti-parallel configuration. These twist-induced mismatch effects\nresult in a large thermal switch ratio.\nUnder a small damping, the switch ratio is nonmonotonic with respect to the twist angle, as\nindicated by the red solid line in Fig. 2(a). Such angle-dependent behavior is similar to the ther-\nmal magnetoresistance between two magneto-optical plasmonic particles at a large applied mag-\nnetic field.54To explore this different angle dependence at small damping condition, we show the\nspectral function \u0014!by varying the twist angle in Figs. 2(c) and 2(d). The twist angle strongly\nmodulates the height and the width of the spectral function peaks at 0< \u0012 < \u0019= 2. However,\nwhen\u0019=2< \u0012 < \u0019 , the high-frequency peak in spectral function almost disappear and the width\nof the low-frequency peak becomes broader with \u0012increasing. Such results qualitatively indicate\nthat there are several nonreciprocal SMPs taking part in the heat transfer with different angle de-\npendence. The isofrequency contour for energy transmission coefficient at qx-qyspace in Fig. 3(a)\nnumerically verify that statement and we show three kinds of SMPs: elliptic SMPs, hyperbolic\nSMPs,53and twist-non-resonant SMPs. The different twist-induced tunneling and competition\nbetween those modes lead to above nonmonotonic twist manipulation for heat transfer.\nFigure 3(a) shows the different twist-induced energy transmission coefficient of the above men-\ntioned SMPs in qx-qyspace. Due to the nonreciprocal properties of SMPs, the tunneling of three\nkinds of SMPs only occur at a positive qxregion, except in antiparallel configuration. On the\none hand, the vertical slice contours in Fig. 3(a) indicate that there is a transition between hy-\n8ω(GHz)1213141516\nθ0π/4π/23π/4π\n00.10.2𝑞𝑥𝑞𝑦\n𝑞𝑥𝑞𝑦 ①\n②\n③16 GHz\n13 GHz④\nθ= 0 θ= π/4 θ= π/2 θ= 3π/4 θ= π(b)⑤\nθ= 0 θ= π/4 θ= π/2 θ= 3π/4 θ= πα=0.01\nα= 0.001θ=0 (a)α= 0.01\n, ( , )xyqq\n( , )\n00.10.200.51.0\n-\n\n0\n-\n\n0\n-\n\n0\n-\n\n0\n-\n\n0\n𝜙 𝜙 𝜙 𝜙 𝜙012  ×105\n(cm−1)\n012  ×105\n(cm−1)Figure 3: (a) Twist-induced energy transmission coefficient with different frequency in qx-qyspace.\nLeft-vertical slice figures are the energy transmission coefficient with zero twist angle with fre-\nquency increasing. Right-transverse slice figures are the energy transmission coefficient at fixed\nfrequency with the twist angle increasing. (b) Integrated energy transmission coefficient in !-\u001e\nspace with different twist angles and damping constants.\n9perbolic SMPs and elliptic SMPs with an increase of frequency. We highlight that the isofre-\nquency contours of the energy transmission coefficient can be almost flat at !\u001915GHz and\nresult in a sharp peak in the spectral function (Figs. 2(c) and 2(d)). In that scenario, such flat-\ntening transition behavior allows the SMPs bands of each individual FMI hybridize and strongly\ncoupled to each other with large wavenumbers and involves a dramatic increase of the local den-\nsity of states for near-field radiative heat transfer. On the other hand, Fig. 3(a) also indicates that\nthe elliptic SMPs and hyperbolic SMPs propagate at the open-angle ( \u0000\u001em< \u001e < \u001e m), where\n\u001em= arctanp\n1=[\u0016i(!)\u0000\u0016r(!)]. But the twist-non-resonant SMPs emerge when ! < 15GHz\nand is not bounded by the open-angle \u001embecause it originates in the twist-induced anisotropic in\nx\u0000zplane. The horizontal slice figures in Fig. 3(a) demonstrate the twist-induced effects of three\nkinds of SMPs: monotonically decreasing for elliptic SMPs and hyperbolic SMPs and nonmono-\ntonic dependence for twist-non-resonant SMPs at 0<\u0012<\u0019 . The competition mechanism among\nthree kinds of modes can be understood from the integrated energy transmission coefficient in\n!-\u001espace with different damping constants (Fig. 3(b)). When \u000b= 0:01, elliptic SMPs, and hyper-\nbolic SMPs play an equal role for radiative heat transfer comparing with twist-non-resonant SMPs,\nwhich leads to an almost monotonically decreased thermal switch ratio. In the small damping con-\ndition, i.e.,\u000b= 0:001, the twist-non-resonant SMPs will play the dominant role for radiative heat\ntransfer, which is induced by the optical gyrotropy and leads to a \u0012anisotropy in the radiative heat\ntransfer.\nBesides, we find an optimal damping constant for maximizing the heat transfer coefficient in\nFig. 4(a): the magnitude of heat current can be enhanced almost one order in ultra-small damp-\ning condition comparing with the isotropic case and the heat flux is monotonically decreased at\nantiparallel configuration ( \u0012=\u0019). Based on fluctuation electrodynamics, heat flux between two\nsemi-infinite systems is proportional to the imaginary part of the permeability and the magnitude\nof heat current could be reduced to zero when the damping constant approach zero or a large value.\nBut the heat transfer coefficient between two FMIs reaches a fixed value in zero damping constant\nlimit. We demonstrate that twist-non-resonant SMPs play the dominant role in ultra-small damp-\n10⑤①\n②③④(a) (b)\n( , , )q\nFigure 4: (a) Heat transfer coefficient as a function of damping constant \u000bwith different twist\nangle. Gray-solid line is the heat transfer coefficient between two isotropic slab i.e. \u0016xx=\u0016yy=\n\u0016zz. (b) Energy transmission coefficient in !-qspace with different azimuthal angle \u001e. (1\r-5\r)\nmeans that the azimuthal angles \u001eare from 0:1\u0019to0:5\u0019with step 0:1\u0019, respectively. The twist\nangle\u0012is zero and the damping constant \u000bis 0.001.\ning conditions and the mechanism is slightly different from near-field radiative heat transfer in\nmultilayer structure due to multiple surface-states coupling.55The intrinsic relation between twist-\nnon-resonant SMPs and the heat transfer coefficient is demonstrated at Fig. 4(b): the near-unity\nregion in energy transmission coefficient contour with different azimuthal angle can fill in the gi-\nant area in!\u0000qspace and the local density of states for twist-non-resonant SMPs can be boosted\nwithout the constraint of ultra-small damping condition. It also demonstrates that the local density\nof states for elliptic SMPs and hyperbolic SMPs (the thin-solid line in Fig. 4(b)) is not be enhanced\nand plays little contribution for heat transfer in a ultra-small damping condition. As a whole, the\ndifferent\u000band\u0012dependence of elliptic SMPs, hyperbolic SMPs, and twist-non-resonant SMPs\nresult in above twist-induced manipulation for near-field radiative heat transfer.\nTo conclude, we have studied twist-induced near-field radiative heat transfer between two FMIs\nthrough nonreciprocal SMPs. We find a large and nonmonotonic twist-induced near-field thermal\nswitch ratio. In addition, the near-field radiative heat transfer can be enhanced by the contribu-\n11tions from the twist-non-resonant SMPs under ultra-small damping condition. Our results provide\ninsights for active near-field heat transfer control by engineered twists.\nAcknowledgement\nJ.-P., L.-W., H.-C., and J.-R. are supported by the National Key Research Program of China (Grant\nNo. 2016YFA0301101), National Natural Science Foundation of China (No. 11935010, No.\n11775159 and No. 61621001), the Shanghai Science and Technology Committee (Grants No.\n18ZR1442800 and No. 18JC1410900), and the Opening Project of Shanghai Key Laboratory of\nSpecial Artificial Microstructure Materials and Technology. G.-T. thanks the financial support from\nthe Swiss National Science Foundation (SNSF) and the NCCR Quantum Science and Technology.\nRair Macedo acknowledges support from the Leverhulme Trust and the University of Glasgow\nthrough LKAS funds.\nSupporting Information Available\nIn the Supplemental Material, we derive the dispersion relation and the reflection coefficients of\nthe surface magnon polariton.\nDispersion relation of surface magnon polariton\nTo find the dispersion relation of surface magnon polaritons (SMP) at a single vacuum-FMI inter-\nface, we employ Maxwell equations,\nr\u0002E=\u0000@tB; (9)\nr\u0002H=@tD; (10)\nwithB=\u00160\u0016HandD=\u000f0\u000fE. The SMP is transverse electric (TE or s) polarized and the\ntransverse magnetic (TM or p-polarized) mode does not exist for the case where only single FMI\n12slab is considered. By applying an in-plane magnetic field along the y-direction, the electric fields\nin vacuum E0and in FMI E1propagate along the x-direction and decay along the z-direction with\nthe expressions\nE0(x;z;t ) = ^yEeiqx\u0000i\f0ze\u0000i!t; Im(\f0)<0; (11)\nE1(x;z;t ) = ^yEeiqx+i\f1ze\u0000i!t; Im(\f1)<0; (12)\nwhereqis the in-plane wave vector along the x-direction. The out-of-plane wave vector in vacuum\nand FMI are denoted as \f0and\f1, respectively. The corresponding magnetic fields are expressed\nas\nB0=1(x;z;t ) =i\n!(^x@z\u0000^z@x)(E0=1\u0001^y): (13)\nFromH= (\u00160\u0016)\u00001B, the magnetic field strengths are\nH0=i\n!\u00160[^x@z\u0000^z@x](E0\u0001^y); (14)\nH1=i\n!\u00160(\u00162\nr\u0000\u00162\ni)[^x(\u0000i\u0016i@x+\u0016r@z) + ^z(\u0000\u0016r@x\u0000i\u0016i@z)](E1\u0001^y): (15)\nUsing Eq. (10) in both vacuum and FMI, one has\n\f2\n0+q2=k2\n0; (16)\n\f2\n1+q2=\u000f\u0016e\u000bk2\n0; (17)\nwithk0=!=c and\u0016e\u000b= (\u00162\nr\u0000\u00162\ni)=\u0016r. Using the interface conditions for the magnetic field\nstrengths, H0\u0001^x=H1\u0001^x, the implicit dispersion relation for the SMPs can be obtained with\n\f0+ (\u0016r\f1\u0000i\u0016iq)=(\u00162\nr\u0000\u00162\ni) = 0: (18)\nFrom Eqs. (16), (17) and (18), the dispersion relation of SMP can be numerically obtained.\n13Reflection coefficients\nIn this section, we obtain the reflection coefficients by taking the anisotropic effect into account.\nFor the case where incidence plane is at an angle \u001ewith respect to the x-axis, the effective perme-\nability tensor is\n\u00160=R\u0016RT=2\n66664\u0016xx\u0016xy\u0016xz\n\u0016yx\u0016yy\u0016yz\n\u0016zx\u0016zy\u0016zz3\n77775; (19)\nwhereRis the rotation matrix with\nR=2\n66664cos\u001esin\u001e0\n\u0000sin\u001ecos\u001e0\n0 0 13\n77775: (20)\nAlthough the SMP is s-polarized by considering a single FMI slab, the p-polarized mode exists\nbetween two FMI slabs as well due to the anisotropic permeability tensor. We focus on the interface\nbetween the lower FMI slab and vacuum. The genernal form of the electric and magnetic fields\ninside the FMI slab can be written as\nE= (Ex;Ey;Ez)e\u0000i!t+iqx; (21)\nH= (Hx;Hy;Hz)e\u0000i!t+iqx; (22)\nwhere the superscript0in the space variables x0,y0andz0is dropped for simplicity. From the\nMaxwell equations, Eqs. (9) and (10), we can get the differential equation\nd\ndz2\n66666664Ex\nEy\n\u000bHx\n\u000bHy3\n77777775=iK2\n66666664Ex\nEy\n\u000bHx\n\u000bHy3\n77777775(23)\n14with\u000b=p\n\u00160=\u000f0and\nK=2\n666666640q\u0016yz=\u0016zzk0(\u0016yx\u0000\u0016yz\u0016zx=\u0016zz)k0(\u0016yy\u0000\u0016yz\u0016zy=\u0016zz)\u0000q2=(k0\u000f)\n0\u0000q\u0016xz=\u0016zzk0(\u0000\u0016xx+\u0016xz\u0016zx=\u0016zz)k0(\u0000\u0016xy+\u0016xz\u0016zy=\u0016zz)\n0\u0000k0\u000f+q2=(k0\u0016zz)\u0000q\u0016zx=\u0016zz \u0000q\u0016zy=\u0016zz\nk0\u000f 0 0 03\n77777775:\n(24)\nBy solving this differential equation, we get\n[Ex(z);Ey(z);\u000bHy(z);\u000bHy(z)] =X2\nm=1cm[u1;m;u2;m;u3;m;u4;m]eikmz; (25)\nwherekmandui;mare, respectively, the eigenvalue and eigenvector of matrix K. SinceKis a\nfour-by-four matrix, we have four eigenvalues: two of them satisfy Im(km)<0and the other two\nIm(km)>0. We takekmwith Im(km)<0, of which the subscripts are denoted as m= 1;2, to\nensure that the electromagnetic fields vanish at z!\u00001 .\nIn the vacuum, the incoming electric and magnetic fields can be, respectively, writen as\nEin= [es\nin^y+ep\nin(\f0^x\u0000q^z)=k0]ei!t\u0000iqx\u0000i\f0z; (26)\n\u000bHin= [ep\nin^y\u0000es\nin(\f0^x\u0000q^z)=k0]ei!t\u0000iqx\u0000i\f0z; (27)\nwhere the superscripts sandpare used to denote the polarizations. The reflected fields are ex-\npressed as\nEre= [es\nre^y\u0000ep\nre(\f0^x+q^z)=k0]ei!t\u0000iqx+i\f0z; (28)\n\u000bHre= [ep\nre^y+es\nre(\f0^x+q^z)=k0]ei!t\u0000iqx+i\f0z: (29)\nAt the interface of the vacuum side with z= 0+, the in-plane electric and magnetic fields can be\n15written as\nEk=[(es\nin+es\nre)^y+ (ep\nin\u0000ep\nre)\f0=k0^x]ei!t\u0000iqx; (30)\n\u000bHk=[(ep\nin+ep\nre)^y\u0000(es\nin\u0000es\nre)\f0=k0^x]ei!t\u0000iqx: (31)\nFor the case of the s-polarized incoming field, that is, ep\nin= 0, the interface conditions give\n\u0000ep\nre\f0=k0=Ex(z= 0); (32)\nes\nin+es\nre=Ey(z= 0); (33)\n\u0000(es\nin\u0000es\nre)\f0=k0=\u000bHx(z= 0); (34)\nep\nre=\u000bHy(z= 0): (35)\nFrom Eqs. (32) and (35), we have\nc2=c1=\u0000(u1;1k0+u4;1\f0)=(u1;2k0+u4;2\f0): (36)\nThe reflection coefficient rss=es\nre=es\nincan be obtained from Eqs. (33) and (34) as\nrss=(u2;1\f0+u3;1k0) + (u2;2\f0+u3;2k0)c2=c1\n(u2;1\f0\u0000u3;1k0) + (u2;2\f0\u0000u3;2k0)c2=c1: (37)\nFrom Eqs. (32) and (34), we can obtain rps=ep\nre=es\ninas\nrps= (1\u0000rss)u1;1+u1;2c2=c1\nu3;1+u3;2c2=c1: (38)\n16For the case of the p-polarized incoming field, that is, es\nin= 0, the interface conditions give\n(ep\nin\u0000ep\nre)\f0=k0=Ex(z= 0); (39)\nes\nre=Ey(z= 0); (40)\nes\nre\f0=k0=\u000bHx(z= 0); (41)\nep\nin+ep\nre=\u000bHy(z= 0): (42)\nFrom Eqs. (40) and (41), we have\nd2=d1\u0011c2=c1=\u0000(u2;1\f0\u0000u3;1k0)=(u2;2\f0\u0000u3;2k0): (43)\nNoticec2=c1here is different from that in Eq. (36) and we denote it as d2=d1instead. The reflection\ncoefficientrpp=ep\nre=ep\ninobtained from Eqs. (40) and (41) is expressed as\nrpp=(u4;1\f0\u0000u1;1k0) + (u4;2\f0\u0000u1;2k0)d2=d1\n(u4;1\f0+u1;1k0) + (u4;2\f0+u1;2k0)d2=d1: (44)\nFrom Eqs. (39) and (41), we obtain rsp=es\nre=ep\ninas\nrsp= (1\u0000rpp)u3;1+u3;2d2=d1\nu1;1+u1;2d2=d1: (45)\nNear-field radiative heat transfer\nFrom the fluctuating electrodynamics, the radiative heat current with vacuum gap dis given by\nJ=Z1\n0d!\n2\u0019\u0016h!(N1\u0000N2)Z1\n0dq\n2\u0019qZ2\u0019\n0d\u001e\n2\u0019Z(!;q;\u001e ); (46)\n17whereNi= 1=[e\u0016h!=(kBTi)\u00001]withi= 1;2is the Bose-Einstein distribution function. The photonic\ntransmission coefficient Z(!;q;\u001e )reads\nZ=8\n>><\n>>:Tr[(I\u0000R\u0003\n2R2)D(I\u0000R\u0003\n1R1)D\u0003]; q<!=c\nTr[(R\u0003\n2\u0000R2)D(R\u0003\n1\u0000R1)D\u0003]e\u00002j\f0jd; q>!=c(47)\nwhereqand\f0=p\n(!=c)2\u0000q2are the in-plane and out-of-plane wave vectors, respectively. The\nidentity matrix is denoted as I. The reflection coefficient matrix for the interface between vacuum\nand the FMI iis\nRi=2\n64ri\nssri\nsp\nri\npsri\npp3\n75: (48)\nThe Fabry-Perot-like matrix reads D= (I\u0000R1R2e2i\f0d)\u00001.\nThe heat transfer coefficient, which is defined as \u0014(T)\u0011lim\u0001T!0J=\u0001T, is expressed as\n\u0014(T) =Z1\n0d!\n2\u0019\u0016h!N0Z1\n0dq\n2\u0019qZ2\u0019\n0d\u001e\n2\u0019Z(!;q;\u001e ); (49)\nwhere the derivative of the Bose-Einstein distribution with respect to the temperature is expressed\nas\nN0\u0011@N=@T =\u0016h!e\u0016h!=(kBT)\nkBT2[e\u0016h!=(kBT)\u00001]2: (50)\nFor the case of \u001e= 0, the reflection coefficient for the s-polarized mode can be expressed as\nrss=\f0\u0000(\f1\u0016r\u0000iq\u0016i)=(\u00162\nr\u0000\u00162\ni)\n\f0+ (\f1\u0016r\u0000iq\u0016i)=(\u00162\nr\u0000\u00162\ni); (51)\nand the other reflection coefficients vanish. The photonic transmission coefficient can be expressed\nas,\nZ(!;q;\u001e = 0) =4[Im(rss)]2e\u00002j\f0jd\nj1\u0000r2\nsse\u00002j\f0jdj2; (52)\n18from which one can obtain the nonreciprocal dispersion of the symmetric and asymmetric SMP\nmodes between two FMI slabs.\nPermeability tensor components of uniaxial FMI\n(a) (b)\n𝜔𝑑\n𝜔𝑢\nFigure 5: (a-b) The real part of diagonal term of permeability tensor with different azimuthal angle.\nThe damping factor is \u000b=0.01.\nIn our calculation, we use the parameters of YIG and rewrite the permeability tensor as below:\n\u0016=2\n66664\u0016r0\u0000i\u0016i\n0 1 0\ni\u0016i0\u0016r3\n77775(53)\nAfter a\u001erotation inx\u0000yplane, the rotated permeability tensor regards as:\n\u0016(\u001e) =2\n66664\u0016rcos(\u001e)2+ sin(\u001e)2cos(\u001e) sin(\u001e)(\u0016r\u00001)\u0000i\u0016icos(\u001e)\ncos(\u001e) sin(\u001e)(\u0016r\u00001) cos(\u001e)2+\u0016rsin(\u001e)2\u0000i\u0016isin(\u001e)\ni\u0016icos(\u001e) i\u0016isin(\u001e) \u0016r3\n77775(54)\n19Fig. 5 shows the real part of \u0016xxand\u0016yywith different azimuthal angle and \u0016zz=\u0016r=\u0016xx(\u001e) = 0 .\nIt indicates value of !uand!dpresented in the main page at the \u0016-near-zero frequency.\nReferences\n(1) Planck, M.; Masius, M. 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Nano/Microscale Heat Transfer 2nd edition ; Springer, Cham, 2020.\n(54) Latella, I.; Ben-Abdallah, P. Giant Thermal Magnetoresistance in Plasmonic Structures. Phys.\nRev. Lett. 2017 ,118, 173902.\n(55) Iizuka, H.; Fan, S. Significant Enhancement of Near-Field Electromagnetic Heat Transfer in\na Multilayer Structure through Multiple Surface-States Coupling. Phys. Rev. Lett. 2018 ,120,\n063901.\n25Graphical TOC Entry\nThrough the ‘twist angle’ between two fer-\nromagnetic insulators, we introduce a large\nnear-field thermal switch and the nonmono-\ntonic twist manipulation for heat flux.\n26"    },    {        "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": "2303.01189v1.Using_vibrating_wire_in_non_linear_regime_as_a_thermometer_in_superfluid___3_He_B.pdf",        "content": "Using vibrating wire in non-linear regime as a thermometer\nin super\ruid3He-B\nV. V. Zavjalov∗\nMarch 3, 2023\nAbstract\nVibrating wires are common temperature probes in3He experiments. By measuring\nmechanical resonance of a wire driven by AC current in magnetic \feld one can directly\nobtain temperature-dependent viscous damping. This is easy to do in a linear regime where\nwire velocity is small enough and damping force is proportional to velocity. At lowest\ntemperatures in super\ruid3He-B a strong non-linear damping appears and linear regime\nshrinks to a very small velocity range. Expanding measurements to the non-linear area can\nsigni\fcantly improve sensitivity. In this note I describe some technical details useful for\nanalyzing such temperature measurements.\nVibrating wire in the linear regime\n1mm\nvdlB\nI\n+−\nVL\nFigure 1: Wire loop driven by current\nin magnetic \feld. On the right image\nof a real device is shown (wire thick-\nnessd= 13\u0016m, wire projection length\nL= 2:7 mm).First consider a wire loop with current Imoving in mag-\nnetic \feld Bwith velocity v?B. Geometry of this\nsystem is shown on Fig 1. Force dFacting on a piece of\nwiredland EMF voltage dVacross it are\ndF=I[dl\u0002B]; dV =v\u0001[dl\u0002B]; (1)\nwhere positive direction of current is along dland posi-\ntive voltage produces current in the same direction (thus\npotential is decreasing along dl). By integrating along\nthe wire one can \fnd total force and voltage:\nF=ILB; V =vLB; (2)\nwhereLis projection of the wire loop to a plane perpen-\ndicular to magnetic \feld and vis mean velocity along\nthe wire length. Also from (1) one can \fnd mechanical\npower produced by the force:\nW=Z\nv\u0001dF=IV: (3)\nIn the linear regime the wire driven by AC current can be represented as a linear oscillator:\nx=\u0000!2\n0x\u0000\u000e_x+\rcos!t; \r =ILB=mw (4)\nwherexis average displacement, average velocity v= _x, and parameters mw,!0,\u000eare e\u000bective\nmass of the wire, resonance frequency, and resonance width. \ris force divided by mass mw.\n∗e-mail: v.zavjalov@lancaster.ac.uk\n1arXiv:2303.01189v1  [cond-mat.other]  2 Mar 2023Non-linear regime in super\ruid3He-B\nBehaviour of a vibrating wire in super\ruid3He-B at low temperatures was investigated long time\nago in Lancaster [1{3]. There is a threshold velocity at which wire can emit quasiparticles and\ndissipate extra energy. This threshold is called \"pair-breaking velocity\", it is some fraction of\nLandau velocity \u0001 =pF, usually 1/3 or smaller depending on wire geometry. At zero pressure and\nlow temperatures typical value is less then 10 mm/s. Below the pair-breaking velocity damping is\ndetermined by quasiparticle scattering, this is the non-linear regime we are interested in. Force Fv\nacting on a wire moving with velocity vhas been derived by applying spectrum of Bogolubov\nquasiparticles to some scattering model. There are a few results: for a simple 1D model, for\nspecular and di\u000busive 3D scattering. At the moment we are not interested in exact expression,\nbut there is an important result that the force can be written via a temperature-independent\nfunction\rvof reduced velocity v=v 0:\nFv( _x) =\u0000mw\u000e0v0\rv\u0012v\nv0\u0013\n; (5)\nwhere temperature-dependent parameters \u000e0andv0are\n\u000e0=p2\nFvFN(0)\n\u001awdexp\u0012\n\u0000\u0001\nkT\u0013\n; v 0=kT\npF; (6)\npF,vF,N(0) are3He Fermi momentum, Fermi velocity, and density of states at Fermi level,\n\u0001 is super\ruid gap, \u001awanddare density and diameter of the wire. Asymptotic behaviour of\nfunction\rvat small velocity ( v\u001cv0) should be linear, which corresponds to the linear regime:\nFv( _x) =\u0000mw\u000e0vs0with some temperature-independent dimensionless calibration factor s0. We\nalso assume that function \rvis already averaged along the wire loop and written for the average\nvelocityv.\nNow the equation of motion is:\nx=\u0000!2\n0x\u0000\u000ei_x\u0000\u000e0v0\rv\u0012_x\nv0\u0013\n+\rcos!t; (7)\nwhere intrinsic damping of the wire is called \u000eiand damping caused by interaction with helium\nis described by the term with \rv.\nTo \fnd response at frequency !we rewrite the equation of motion in van der Pol coordinates:\nu=xcos!t\u0000_x\n!sin!t; v =\u0000xsin!t\u0000_x\n!cos!t: (8)\nand average over period 0 < !t < 2\u0019. During this calculation we need to average function\n\rv( _x=v 0) multiplied by sin !tand cos!t. This can be done by making substitution\nu!=v 0=ccos\u001e; v!=v 0=csin\u001e; (9)\nshifting averaging period by \u001e, and obtaining integrals with one parameter c:\nh\rv( _x=v 0) sin!ti=h\rv(\u0000u!=v 0sin!t\u0000v!=v 0cos!t) sin!ti= (10)\n=u!\ncv0Z2\u0019\n0\rv(\u0000csin!t) sin!td!t\n2\u0019=\u0000u\n2!\nv0S(c); (11)\nand similarly\nh\rv( _x=v 0) cos!ti=\u0000v\n2!\nv0S(c) (12)where we introduce function S(c) of positive dimensionless argument c=!\nv0p\nu2+v2as\nS(c) =\u00002\ncZ2\u0019\n0\rv(\u0000csin!t) sin!td!t\n2\u0019(13)\npoints:δ(v/v0)−δi\nδ(v= 0)\nv0= 1.255mm/s\n0.20.40.60.81.0\n0.00 1 2 3 4 5\nFigure 2: Solid line: analyti-\ncal function S(c) for 1D scattering\nmodel (15). Points: an example of\nexperimental data scaled by choos-\ningv0= 1:255 mm/s (see Fig. 3 and\n(23) for details). Deviation at c>4\nis the pair-breaking regime.If function \rvis known then function Scan be calculated\nanalytically or numerically. Normally only a limited range\nof reduced velocities is needed and Scan be easily tabulated\nor approximated by some smooth function. For example, if\ndamping is linear, \rv(x) =xthenS(c) = 1. For 1D scattering\nmodel [1]Scan be written via special functions:\n\rv(x) = sign( x)(1\u0000exp(\u0000jxj)); (14)\nS(c) =2\nc\u0012\nI1(c)\u0000L\u00001(c) +2\n\u0019\u0013\n; (15)\nwhereIn(x) is modi\fed Bessel function of \frst kind and Ln(x)\nis modi\fed Struve function. For an arbitrary expansion\n\rv(x) =a1x+ sign(x)a2x2+a3x3+ sign(x)a4x3;(16)\nS(c) =a1+a28c\n3\u0019+a33c2\n4+a432c3\n15\u0019: (17)\nNote that\rvshould be an odd function. In3He-B it contains\nquadratic term and have non-smooth second derivative at zero\nvelocity. This comes from the physical model where scattering\nchannels open or close when velocity sign changes. Quadratic\nterm in\rxproduces linear term in S(c). As it will be clear\nlater this leads to a linear shift of measured damping with\ndriving force amplitude in AC measurements and makes usual \"linear\" approach problematic.\nOn Fig. 2 analytical function S(c) for 1D scattering model (15) is shown together with\nproperly scaled experimental data. One can see that shape of the function is very close to the\nexperiment. For practical purposes it is convenient to approximate function S(c) obtained in\nexperiment with expression\nS(c) =s0\n1 +s1c+s2c2(18)\n.\nAfter averaging equation of motion, \fnding equilibrium h_ui=h_vi= 0, and switching to\na usual complex notation we can write expression for complex velocity (bold font is used for\ncomplex values) as\nv=i!\r\u0012\n!2\n0\u0000!2+i\u000ei!+i\u000e0!S\u0012jvj\nv0\u0013\u0013\u00001\n; (19)\nor in terms of current and voltage:\nV=I(LB)2\nmwi!\u0012\n!2\n0\u0000!2+i!\u000ei+i!\u000e0S\u0012jVj\nv0LB\u0013\u0013\u00001\n: (20)\nThis is a non-linear analog of Lorentzian formula which can be used for \ftting response of a\nvibrating wire. Voltage (or velocity) enters both sides of the expression, but is can be calculated\niteratively by starting with V= 0 and calculating next-step value using the previous one in the\nright-hand side.Obtaining function S(c)in experiment\nConsider a vibrating wire connected to an AC current source and a lock-in ampli\fer. First a\n\"frequency sweep\" is done: voltage components are measured as a function of frequency at a\nconstant current amplitude. If amplitude is small and the wire is close to the linear regime, then\ndata can be \ftted with Lorentzian curve using a few parameters: complex amplitude, resonance\nfrequency, damping and a complex o\u000bset. If we are not in the linear regime then after measuring\nfunctionSthe \ft can be corrected by using nonlinear expression (20) to iteratively improve the\nanalysis. As a result we have a function\nV=A0i!\n!2\n0\u0000!2+i\u000e!+V0: (21)\nwith complex parameters A0andV0, and real parameters !0and\u000e. We can not separate\nintrinsic dumping here and use some total e\u000bective damping \u000ewhich is not very important for\nthe following analysis.\nVoltage o\u000bset always exists because of parasitic coupling in the electric circuit. Usually it is\nproportional to the drive current and weakly depends on frequency. It is reasonable to measure\no\u000bset once and then subtract it from all data. When \ftting the frequency sweep an o\u000bset\nparameter V0is still needed to compensate some time-dependent drifts and non-linearities in\nthe circuit, but in this case just a constant o\u000bset (without any frequency dependence) can be\nused here.\nWhen we know A0andV0parameters, we can do an \"amplitude sweep\": set generator\nfrequency close to the resonance and start increasing current amplitude. We assume that o\u000bset\nand driving force are changing proportionally and write (20) as\n!2\n0\u0000!2+i\u000ei!+i\u000e0!S\u0012jVj\nv0LB\u0013\n=i!A0I\nVI0\u0000V0I: (22)\nNote that currents IandI0here can be just real numbers in arbitrary units, because multiplying\nthem by a complex factor will not change the result.\nBy taking real part of this expression we can \fnd new value of the resonance frequency (it\ncan change because of other non-linear e\u000bects). From the imaginary part we can \fnd an e\u000bective\ndamping\u000e:\n\u000e(jVj) =\u000ei+\u000e0S\u0012jVj\nv0LB\u0013\n==\u001a\niA0I\nVI0\u0000V0I\u001b\n: (23)\nOn Fig. 3A. an example of measured \u000eat zero pressure and seven di\u000berent temperatures\nand magnetic \felds is shown as a function of velocity jvj. A NbTi wire with d= 4:5\u0016m and\nL= 1:49 mm is used. Intrinsic width of NbTi wire partially originates from motion of vortices\nin the superconductor and depends on magnetic \feld as\n\u000ei(B) =\u000ei\n0+\u000ei\n2B2: (24)\nThis \feld dependence can be obtained from measurements of the wire resonance in vacuum\nor by using two \ftting parameters in \u000e(jvj). This is done on Fig. 3A: seven datasets are \ftted\ntogether using model with 7+5 parameters: temperature of each measurement, intrinsic damping\n\u000ei\n0and\u000ei\n2, and function Sdescribed via parameters s0,s1,s2introduced in (18). Because v0\nand\u000e0have di\u000berent dependence on temperature (6) by this kind of \ftting it could be possible\nto extract temperature without external calibration. Unfortunately, temperature and s0have\nbig mutial covariance and accuracy of this self-calibration for our data is poor. For obtaining\ngood temperature calibraition it's better to get value of s0from some external calibration or use0 1 2 3 4 520\n0406080100120140100\n10\n1pair-breaking\n2.19\n1.6410.9\n8.20\n6.56\n5.47\n4.38\n3.28\n2.73(RMS)I, nAA B\n3760 3780 3770Vy/IVx/I\n180 mT\n|v|, mm/s113 mT136 mT102 mT\n0.167Tc0.172Tc0.174Tc0.185Tc0.199Tc0.209Tc\nω/2π, Hz0.224Tcδ/2π, Hz\nV/I, Ω, shiftedFigure 3: A: measured \u000e(jvj) at seven di\u000berent temperatures and magnetic \felds, at zero pressure.\nBlack lines show the \ftting model with same intrinsic damping and same function S(c) for all data.\nTemperatures (in Tcunits) are shown in assumption that s0= 1.B. Frequency sweeps at nine di\u000berent\ndrive currents are \ftted together with the non-linear equation (20) using intrinsic damping and function\nS(c) from Fig. 3A, and usual set of free parameters for Lorentzian \ftting: complex amplitude, complex\no\u000bset,!0and\u000e(v= 0)\ntheoretical value s0= 1. But regardless of this choice we will have an accurate model which can\nbe used to extrapolate \u000etov= 0 and remove all e\u000bects of the non-linear regime. In all following\n\fgures obtaind model will be used in this way. Temperature calibration can be done as a next\nindependent step.\nOn Fig. 3B an example of frequency sweeps at nine di\u000berent drive currents is shown. All data\nare \ftted together with equation (20) using intrinsic damping and function S(c) obtained from\nFig. 3A and usual set of free parameters for Lorentzian \ftting: complex amplitude, complex\no\u000bset,!0and\u000e(v= 0). On Fig. 4A damping for same frequency sweeps is obtained by \ftting\nthem separately with either Lorentzian formula or non-linear equation (20). One can see that even\nat small currents Lorentzian \ft gives noticeably smaller damping then the \"linear\" value \u000e(v= 0).\nOn Fig. 4B an example of a \"tracking mode\" measurement is shown. This is similar to the\n\"amplitude sweep\" described above: damping \u000eis extracted from a single measurement of voltage\ncomponents using (23). Normally this is the way of measuring temperature as a function of timeLorentzian fitnon-linear fit\ncurrent, nA RMSpair breaking\nheating events\n268\n4\n00 2 4 6 8 10 12\ntime, h current, nA RMS0024624\n0δ/2π, Hz\nδ/2π, Hz\n0.4 0.8 1.2A B6\nuncorrectedcorrectedFigure 4: A: Damping parameter \u000eobtained by usual Lorentzian \ft of frequency sweeps from Fig. 3B\n(lower points); \u000e(v= 0) found by \ftting same data with the non-linear formula (20) (upper points).\nBlack horizontal line is \u000e(v= 0) = 6:213 found on Fig. 3B by \ftting all sweeps together. B: An example\nof measurement of \u000eas a function of time in \"tracking mode\". Current (lower plot) is ramped up and\ndown to reach pair-breaking regime and then kept constant with a few step adjustments. Measured\nvalue of\u000eand corrected value \u000e(v= 0) are shown on the upper plot. One can see that corrected value\ndoes not depend on drive current (except the pair-breaking regime) and can be used for temperature\nmeasurement. Small peaks are random heating events in helium caused by natural radioactivity.\nat constant drive, but here a few changes in drive have been done: in the beginning current was\nramped up and down to get information about non-linear regime and reach pair-braking velocity.\nThen current was increased to do measurements in the non-linear regime where signal-to-noise\nratio is better. At the end current have been adjusted again. On the plot one can see measured\nvalue of\u000eand corrected value \u000e(v= 0). As expected, the corrected value does not depend on\ndrive current (except in the pair-breaking regime) and can be used for temperature measurement.\nSmall peaks are random heating events in helium caused by natural radioactivity.\nConclusion\nThis note shows importance of non-linear e\u000bects for vibrating-wire thermometry in super\ruid\n3He-B. Method of removing the non-linearity is described and tested on experimental data. It\nis possible to build a simple model with a few parameters (intrinsic damping and function S(c))\nwhich works at any temperature within ballistic regime (below 0 :3Tc) and any magnetic \feld.\nI would like to thank A.Shen, S.Loktev, D.Zmeev and S.Autti for useful discussions.\nReferences\n1. Fisher, S. N., Gu\u0013 enault, A. M., Kennedy, C. J. & Pickett, G. R. Beyond the two-\ruid model:\nTransition from linear behavior to a velocity-independent force on a moving object in3B.Phys. Rev. Lett. 63,2566{2569. https://link.aps.org/doi/10.1103/PhysRevLett.63.\n2566 (23 1989).\n2. Fisher, S. N., Pickett, G. R. & Watts-Tobin, R. J. A microscopic calculation of the force on\na wire moving through super\ruid3He-B in the ballistic regime. Journal of Low Temperature\nPhysics 83,225{235. https://doi.org/10.1007/BF00682120 (1991).\n3. Enrico, M. P., Fisher, S. N. & Watts-Tobin, R. J. Di\u000buse scattering model of the thermal\ndamping of a wire moving through super\ruid3He-B at very low temperatures. Journal of\nLow Temperature Physics 98,81{89. https://doi.org/10.1007/BF00754069 (1995)."    },    {        "title": "2104.10723v1.On_absorbing_set_for_3D_Maxwell__Schrödinger_damped_driven_equations_in_bounded_region.pdf",        "content": "arXiv:2104.10723v1  [math.AP]  21 Apr 20211\nOn absorbing set for 3D Maxwell–Schr ¨odinger\ndamped driven equations in bounded region\nA. I. Komech1\nFaculty of Mathematics of Vienna University\nInstitute for Transmission Information Problems of RAS, Mo scow, Russia\nMechanics-Mathematics Department, Moscow State Universi ty\nalexander.komech@univie.ac.at\nAbstract\nWe consider the 3D damped driven Maxwell–Schr¨ odinger equa tions in a bounded region under suitable\nboundary conditions. We establish new a priori estimates, w hich provide the existence of global finite energy\nweak solutions and bounded absorbing set. The proofs rely on the Sobolev type estimates for magnetic\nSchr¨ odinger operator.\nContents\n1 Introduction 1\n2 Damped driven Maxwell–Schr ¨odinger equations 2\n3 Boundary conditions 2\n4 Hamiltonian structure 3\n5 Comments on previous results 3\n6 Sobolev type estimates for magnetic Schr ¨odinger operator 4\n7 A priori estimates 5\n8 Absorbing set 5\nA Proof of the Sobolev type estimates for magnetic Schr ¨odinger operator 7\n1 Introduction\nThe Maxwell–Schr¨ odinger, coupled equations form a fundam ental dynamical system of Quantum Theory.\nThese equations describe crucial phenomena of the matter-radiation interaction which is in the center of Quan-\ntum Theory and its applications. In particular, in the appli cations to the design and optimal control of quantum\nhigh-frequency electronic devices: laser, maser, klystro n, magnetron, traveling wave tube, synchrotron, elec-\ntron microscope, and others. The importance of these questi ons was pointed out in early paper by Kapitza [ 8].\nThus, a rigorous investigation of the long-time asymptotic s for solutions of these equations is indispensable for\nphysical applications.\nHowever, the mathematical theory of these nonlinear evolut ionary equations is currently in an initial stage.\nRespectively, applications of these equations rely on the p erturbation theory which cannot provide the long-\ntime behaviour of solutions of these equations. On the other hand, almost all applications require to know the\nlong-time behaviour.\nTill now the design and control of quantum devices uses mainl y the quasiclassical approximation, which\ntreats electrons as classical particles. For example, in th e fundamental monograph [ 9], the words ‘quantum’\n1The research supported by the Austrian Science Fund (FWF) un der Grant No. P28152-N35.2\nand ‘Schr¨ odinger equation’ were not even mentioned. The mo st important exception is the study of the laser\nand maser action, based on 1D coupled Maxwell–Schr¨ odinger equations [ 6,7]. However, these studies are not\nrigorous, which was one of our motivations for the present in vestigation.\n2 Damped driven Maxwell–Schr ¨odinger equations\nWe consider the coupled damped driven Maxwell–Schr¨ odinge r equations (MS) in a bounded domain V⊂R3\nwith a smooth boundary Γ:=∂V. We choose the units where e=−1 and m=c=¯h=1. Then in the Coulomb\ngauge div A(x,t)≡0 the equations read (cf.[ 1,2,10,11])\n\n\n¨A(x,t) = ∆A(x,t)−σ˙A(x,t)+Pj(·,t),∆A0(x,t):=−ρ(x,t)\ni˙ψ(t) = ( 1−iε))H(t)ψ(t)−iγE(t)ψ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, x∈V. (2.1)\nHere σ>0 is the electrical conductance of the medium, ε,γ>0 are the absorption coefficients, and Pdenotes\nthe orthogonal projection onto free-divergent vector field s from the Hilbert space L2(V)⊗R3. Further, E(t):=\n/an}bracketle{tψ(t),H(t)ψ(t)/an}bracketri}ht,where H(t)is the Schr¨ odinger operator\nH(t):=1\n2D2(t)+φ(x)+A0(x,t), D(t):=−i∇+A(x,t)+Ap(x,t). (2.2)\nHere Ap(x,t)is an external ‘pumping potential’, and φ(x)stands for a static external potential (in the case of an\natom, φ(x)is the nucleus potential). Finally, the charge and current d ensities are expressed in the wave function\nand the Maxwell potentials as\nρ(x,t)=|ψ(x,t)|2, j(x,t)=Re[ψ(x,t)D(t)ψ(x,t)]. (2.3)\nRemark 2.1. We introduce in the Schr¨ odinger equation of the system ( 2.1) the novel specific nonlinear damping\nterm−iγE(t)ψ(t)which plays the key role in our approach.\n3 Boundary conditions\nWe choose the boundary conditions modelling ideally conduc ting diamagnetic materials (like cooper, silver,\ngold, etc). In such materials the electric and magnetic field should vanish as well as the charge and current\nsurface densities. Hence, the tangential component of the e lectric field\nE(x,t)=−˙A(x,t)−∇A0(x,t),A(x,t):=A(x,t)+Ap(x,t)\nvanishes on the boundary Γ=∂Vas well as the normal component of the magnetic field B(x,t) =rotA(x,t).\nMore precisely, we assume that\nA0(x,t)=0, n(x)×˙A(x,t)=0,n(x)·rotA(x,t)=0, x∈Γ,t>0, (3.4)\nwhere n(x)is the outward normal to the resonator boundary at the point x∈Γ. We slightly reinforce the middle\nboundary conditions assuming\nn(x)×A(x,t)=0, x∈Γ,t>0. (3.5)\nThen the second conditions of ( 3.4) hold by differentiation. Moreover, then the third conditi on follows from\n(3.5) in local orthogonal coordinates. Indeed, let a point y∈∂Vandx3=0 on the tangent plane TyΓ. Then\n(3.5) means that\nA1(y,t)=A2(y,t)=0⇔A(y,t)=C(y)n(y), y∈Γ. (3.6)\nThese identities imply that\n∂kAj(y,t)=0, y∈Γ,k,j=1,2, (3.7)\nifA∈C1(V)⊗R3. In particular, ∂1A2(y,t)−∂2A1(y,t)=0 which implies the last condition of ( 3.4).\nFinally, for the electronic wave function we assume the Diri chlet boundary condition\nψ(x,t)=0, x∈∂V,t>0, (3.8)\nwhich ensures the absence of electronic current on the bound ary: j(x,t)=0 for x∈∂V.3\n4 Hamiltonian structure\nThe Hamiltonian functional is defined by\nH(A,Π,ψ,t)=1\n2[c2/bardblΠ/bardbl2+/bardblrotA/bardbl2]+1\n2/an}bracketle{tψ,H0(t)ψ/an}bracketri}ht, H0(t):=1\n2D2\n0(t)+φ(x)+1\n2A0(x). (4.9)\nHere D0(t):=−i∇+A(x)+Ap(x,t)andA0(x):=(−∆)−1ρ(·) =(−∆)−1|ψ(·)|2, where(−∆)−1is specified\nwith the Dirichlet boundary conditions for A0(x,t)from ( 3.4). The system ( 2.1) under the boundary conditions\n(3.4), (3.8) can be formally written as\n˙A(t)=HΠ,˙Π(t)=−HA−σΠ(x,t),i˙ψ(t)=( 1−iε)Hψ−iγE(t)ψ(t). (4.10)\n5 Comments on previous results\n•The Maxwell–Schr¨ odinger system of type ( 2.1) in a bounded region was not considered previously. Such\nsystem was considered in [ 10,11] for the case of the infinite space V=Rdwith d=1,2,3 for σ=ε=γ=0,\nc=1 and zero pumping Ap(x,t)≡0:\n\n\n¨A(x,t) = ∆A(x,t)+Pj(·,t),∆A0(x,t)=−ρ(x,t)\ni˙ψ(t) = H(t)ψ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,x∈Rd. (5.11)\nFor this system the existence of global solutions for all fini te energy initial states was proved for the first\ntime by Guo, Nakamitsu and Strauss [ 10]. Their approach relies on application of the Gagliardo–Ni renberg\ninterpolation inequality. The uniqueness of the solution w as not proved.\nThe complete result on the well-posedness in the energy spac e was established by Bejenaru and Tataru\n[11] providing strong a priori estimates. The methods [ 11] rely on microlocal analysis of pseudodifferential\noperators with “rough symbols”. In particular, these metho ds provide Lemma 11 of [ 11]: For each 0 ≤s≤2 the\noperator 1 −[∇−iA]2is a diffeomorphism Hs(R3)→Hs−2(R3)which depends continuously on A∈H1(R3).\nThis lemma is a refinement of Proposition A.I of [ 12].\n•Dissipative autonomous evolutionary PDEs. The theory of attractors and long-time behaviour of solutio ns\nof such equations originated in the works of Ball, Foias, Hal e, Henry, Temam, and was developed further by\nBabin and Vishik, Chepyzhov, Haraux, Ilyin, Miranville, Pa ta, Zelik, and others for the Navier-Stokes, reaction-\ndiffusion, Ginzburg–Landau, damped wave and nonlinear Sch r¨ odinger, and sine-Gordon equations [ 22]–[40].\n•Hamiltonian autonomous evolutionary PDEs. My team has a long time experience working with the theory\nof global attractors for nonlinear Hamiltonian evolutiona ry PDEs. I initiated this theory in 1990. The theory was\ninspired by Bohr’s transitions between quantum stationary states and resulted in more than 50 papers including\njoint papers in collaboration with H. Spohn, V . Buslaev and o thers. The global attraction to a compact attractor\nwas established for a list of nonlinear Hamiltonian PDEs, se e the surveys [ 3,4,5]. The proofs rely on a novel\napplication of subtle tools of Harmonic Analysis: the Wiene r Tauberian theorem, the Titchmarsh convolution\ntheorem, the theory of quasimeasures, and others.\nTao established in [ 38] the existence of the global attractor for radial solutions to nonlinear defocusing\nSchr¨ odinger equation without damping in Rnwith n≥11.\n•Damped driven nonlinear wave and Ginzburg-Landau equation s.Absorbing sets and global attractors\nwere constructed i) for damped driven nonlinear wave equati ons by Haraux [ 28,29] for almost periodic ex-\nternal force, see also Ghidaglia and Temam [ 27], Mora and Sol` a-Morales [ 31], Babin, Chepyzhov and Vishik\n[22]–[24], and others, and ii) for the damped driven Ginzburg-Landau equations by Ghidaglia and Heron [ 35]\n(see also [ 22,32]) in the case of a bounded region V⊂Rnwith n=1,2.\n•Damped driven nonlinear Schr ¨odinger equations. The theory of global attractors was developed by\nGhidaglia [ 34] on a bounded interval V⊂R, by Wang [ 39] on the circle V=R/Z, by Abounouh [ 33] on\na bounded region V⊂R2, and by Laurenc ¸ot [ 37] on V=RNwith N≤3. These results on attraction were\ndeveloped in [ 36,39]. In these papers the pumping term does not depend on time, an d in [ 34] the pumping term\nis time-periodic. The main achievement in these papers is th e construction of a compact global attractor in the\nenergy space H1relying on the Ball ideas [ 23].4\nA compact global attractor in the energy space for 2D weakly d amped driven nonlinear Schr¨ odinger equa-\ntion with general nonlinearity was constructed in joint pap er of the PI with E. Kopylova [ 40] for general\nbounded region V⊂R2and almost periodic driving.\n•The Maxwell–Klein–Gordon and other coupled equations. For various coupled equations the results on\nwell-posedness, existence of solitary waves and their effe ctive dynamics were obtained by P. D’Ancona, M.\nEsteban, S. Klainerman, M. Machedon, S. Selberg, E. S´ er´ e, D. Stuart, and others, [ 12]– [21].\n6 Sobolev type estimates for magnetic Schr ¨odinger operator\nWe denote the spaces Lp=Lp(V),Hs=Hs(V),o\nH1=o\nH1(V), and/bardbl·/bardbl is the norm in L2. Denote X=[H1⊗\nR3]⊕[L2(R3)⊗R3]⊕o\nH1the Hilbert space of states X= (A,Π,ψ)satisfying the boundary conditions ( 3.4)\nand div A(x)=divΠ(x)=0 for x∈V.\nWe will prove below in Lemma A.1the bound for magnetic potential\n/bardblA/bardbl2\nL2≤C/bardbl∇A/bardbl2, A∈A. (6.12)\nThis bound holds since the Laplacian ∆under the boundary conditions ( 3.5) is nonnegative and symmetric on a\ndense domain D⊂A, and hence, it admits the selfadjoint extension. Finally, t he spectrum is discrete and zero\nis not an eigenvalue. In Lemma A.2we prove the equivalence of norms for magnetic Schr¨ odinger operator\nb1(/bardblA(t)/bardbl2\nH1)/bardblψ/bardbl2\nH1≤[/bardblD(t)ψ/bardbl+/bardblψ/bardbl]2≤b2(/bardblA(t)/bardbl2\nH1)/bardblψ/bardbl2\nH1, ψ∈o\nH1, (6.13)\nwhere D(t):=i∇−A(x,t),b1(r)>0 (respectively b2(r)>0) is a decreasing (respectively an increasing)\nfunction of r≥0. Similarly,\nb1(/bardblA(t)/bardbl2\nH1)/bardblψ/bardbl2\nH2≤[/bardblH(t)ψ/bardbl+/bardblψ/bardbl]2≤b2(/bardblA(t)/bardbl2\nH1)/bardblψ/bardbl2\nH2, ψ∈H2∩o\nH1. (6.14)\nThe bounds ( 6.13) and ( 6.14) extend Lemma 11 of [ 11] to the case of bounded region. For the proof of ( 6.13)\nwe will show that the difference of D2(t)with−∆is a relatively compact operator.\nWe will assume that the potential φ(x)is bounded,\nsup\nx∈V|φ(x)|<∞. (6.15)\nHence, we can assume that it is positive,\nφ(x)≥κ>0, x∈V (6.16)\nsince the potential is defined up to an additive constant. Hen ce,\nE(t)=/an}bracketle{tψ,H(t)ψ/an}bracketri}ht≥/bardbl D(t)ψ/bardbl2+κ/bardblψ/bardbl2+/an}bracketle{tρ,(−∆)−1ρ/an}bracketri}ht. (6.17)\nNow ( 6.13) implies\nE(t)≥κ1(/bardblA(t)/bardbl2\nH1)/bardblψ/bardbl2\nH1, (6.18)\nwhereκ1>0 is a decreasing function. Hence, the standard Sobolev esti mates together with ( 6.17) imply from\n(6.12) and ( 6.13) that\n\n\n/bardblA/bardbl2\nLp≤C∑k/bardbl∇kA/bardbl2, A∈A\n/bardblψ/bardbl2\nLp≤b(/bardblA(t)/bardblH1)E(t),ψ∈o\nH1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, p∈[2,6]. (6.19)\nHere the last bound extends Lemma 11 of [ 11] to the case of bounded region.5\n7 A priori estimates\nFirst, we obtain a priori estimates for sufficiently smooth s olutions(A(x,t),Π(x,t),ψ(x,t))of the Maxwell–\nSchr¨ odinger system ( 2.1), where all functions are C∞(R4). We plan to get rid of this smoothness assumption\nand establish the same estimates for all finite energy soluti ons. We will assume for the pumping field Ap(x,t)\nthat it is almost periodic and\nsup\nx∈V,t∈R[|Ap(x,t)|+|∇Ap(x,t)|+|˙Ap(x,t)|]<∞. (7.20)\nDifferentiating the charge Q(t):=/bardblψ(t)/bardbl2we have\n˙Q(t) =/an}bracketle{t˙ψ(t),ψ(t)/an}bracketri}ht+/an}bracketle{tψ(t),˙ψ(t)/an}bracketri}ht\n=/an}bracketle{t(−i−ε)H(t)ψ(t),ψ(t)/an}bracketri}ht+/an}bracketle{tψ(t),(−i−ε)H(t)ψ(t)/an}bracketri}ht−2γE(t)Q(t)\n=−2εE(t)−2γE(t)Q(t)≤−2εκQ(t)−2γκQ2(t), (7.21)\nwhere we used ( 6.17). Hence,\nQ(t)≤Q(0). (7.22)\nNow differentiating the energy E(t):=H(Π(t),A(t),ψ(t),t)and using ( 2.1) and ( 6.17), (7.20), we get\n˙E(t) =/an}bracketle{tHA,˙A/an}bracketri}ht+/an}bracketle{tHΠ,˙Π/an}bracketri}ht+/an}bracketle{tHψ,˙ψ/an}bracketri}ht+Ht\n=/an}bracketle{tHA,HΠ/an}bracketri}ht+/an}bracketle{tHΠ,−HA−σΠ+/an}bracketle{tHψ,(−i−ε)Hψ−γE(t)ψ(t)−/an}bracketle{tD(t)ψ,˙Apψ/an}bracketri}ht\n≤ − σ/bardblΠ(t)/bardbl2−ε/an}bracketle{tH(t)ψ(t),H(t)ψ(t)/an}bracketri}ht−γE2(t)+CpE(t)+Cp/bardblD(t)ψ(t)/bardbl/bardblψ(t)/bardbl≤C1<∞,(7.23)\nsinceHψ=H(t)ψ(t). Hence, ( 2.2) and ( 6.13), and ( 6.14) imply a priori estimate\n/bardbl∇A(t)/bardbl2+/bardblΠ(t)/bardbl2+/bardblψ(t)/bardbl2\nH1+ε/integraldisplayt\n0b1(/bardblA(s)/bardblH1)/bardblψ(s)/bardbl2\nH2ds≤C(t+1),/bardblψ(t)/bardbl≤C<∞,t>0.(7.24)\n8 Absorbing set\nThe estimates ( 7.24) are insufficient to prove the existence of a bounded absorbi ng set. We will follow the ideas\nof Haraux [ 28] (see also [ 22]) introducing the functional\nΦ(A,Π,ψ,t)=H(A,Π,ψ,t)+η/an}bracketle{tΠ,A/an}bracketri}ht (8.25)\nwith a small η>0. Differentiating Φ(t):=Φ(A(t),Π,ψ(t),t)and using ( 7.23), we obtain\n˙Φ(t) = ˙E(t)+η/an}bracketle{tΠ(t),Π(t)/an}bracketri}ht+η/an}bracketle{t˙Π(t),A(t)/an}bracketri}ht\n≤−σ/bardblΠ(t)/bardbl2−ε/an}bracketle{tH(t)ψ(t),H(t)ψ(t)/an}bracketri}ht−γE2(t)+CpE(t)+η/an}bracketle{tΠ(t),Π(t)/an}bracketri}ht\n+η/an}bracketle{t∆A(t)−σΠ(t)+j(t),A(t)/an}bracketri}ht (8.26)\nThe most problematic term\n/an}bracketle{tj(t),A(t)/an}bracketri}ht=−Re/an}bracketle{tψ(t)D(t)ψ(t),A(t)/an}bracketri}ht (8.27)\ncan be estimated using the Sobolev-type estimates ( 6.19):\n|/an}bracketle{tj(t),A(t)/an}bracketri}ht| ≤ C/bardblA(t)/bardblL6/bardblψ(t)/bardblL3·/bardblD(t)ψ(t)/bardbl≤C1/bardbl∇A(t)/bardblE1/2(t)/bardblD(t)ψ(t)/bardbl\n≤δ/bardbl∇A(t)/bardbl2+C2\nδE2(t), (8.28)6\nwhere the last inequality holds by ( 6.17). For the remaining terms similar estimates follow from the first\nestimate ( 6.12) and from ( 7.20):\n|/an}bracketle{tΠ(t),A(t)/an}bracketri}ht|≤ δ/bardbl∇A(t)/bardbl2+1\nδ/bardblΠ(t)/bardbl2. (8.29)\nNow ( 8.26) implies that for any δ>0\n˙Φ(t)≤−η(1−3δ)/bardbl∇A(t)/bardbl2−(σ−η−η\nδ)/bardblΠ(t)/bardbl2−(γ−C2η\nδ)E2(t)−ε/bardblH(t)ψ(t)/bardbl2+Cp. (8.30)\nIt remains to choose δ,η>0 such that\nmin(η(1−3δ),σ−η−η\nδ,γ−C2η\nδ)>0. (8.31)\nThen ( 8.30) and ( 4.9) imply that\n˙Φ(t)≤−αH(t)−ε/bardblH(t)ψ(t)/bardbl2+Cp, t>0, (8.32)\nwhere α>0 and Cp∈Rdo not depend on the solution. However, ( 6.12) implies that for sufficiently small\nη>0 we have\ncH(A,Π,ψ,t)≤Φ(A,Π,ψ,t)≤CH(A,Π,ψ,t) (8.33)\nwith c,C>0. Hence, ( 8.32) and ( 6.14) imply that for small η>0\n˙Φ(t)≤−βΦ(t)−ε/bardblH(t)ψ(t)/bardbl2+Cp, t>0, (8.34)\nwhere β>0 and Cp∈Rdo not depend on the solution. Now the integration yields\nΦ(t)+ε/integraldisplayt\n0e−β(t−s)/bardblH(s)ψ(s)/bardbl2ds≤Φ(0)e−βt+Cp\nβ, t>0. (8.35)\nHence, ( 4.9) and ( 6.17), (8.33) imply that for sufficiently small η>0 a priori estimate ( 7.24) refines to\n/bardbl∇A(t)/bardbl2+/bardblΠ(t)/bardbl2+/bardblD(t)ψ/bardbl2+/bardblψ/bardbl2+ε/integraldisplayt\n0e−β(t−s)/bardblH(s)ψ(s)/bardbl2ds≤C[Φ(0)e−βt+Cp\nβ],t>0.(8.36)\nNow ( 6.13) and ( 6.14) imply that\n/bardbl∇A(t)/bardbl2+/bardblΠ(t)/bardbl2+b1(M)/bardblψ/bardbl2\nH1+εb1(M)/integraldisplayt\n0e−β(t−s)/bardblψ(s)/bardbl2\nH2ds≤C[Φ(0)e−βt+Cp\nβ],t>0.(8.37)\nwhere\nM:=sup\nt≥0/bardblA(t)/bardblH1≤Csup\nt≥0/bardbl∇A(t)/bardblH1<∞\nby (8.36). Let us write the system ( 2.1) as\n˙X(t)=F(X(t),t), X(t)=( A(t),Π(t),ψ(t)). (8.38)\nCorollary 8.1. The bounds ( 8.37) imply for solutions X (t)of (8.38)\n/bardblX(t)/bardbl2\nX+ε1/integraldisplayt\n0e−β(t−s)/bardblψ(s)/bardbl2\nH2ds≤C(Φ(0)e−βt+Cp\nβ), t>0, (8.39)\nwhere ε1,β>0. Hence, for any R >0the setB:={Y∈X:/bardblY/bardbl2\nX≤C(1+Cp\nβ)}absorbs the ball {Y∈X:\n/bardblY/bardblX≤R}for large times t >tR.7\nA Proof of the Sobolev type estimates for magnetic Schr ¨odinger operator\nLet a point y∈Γ:=∂Ωandx3=0 on the tangent plane TyΓ. Then ( 3.7) together with div A(y,t)≡0 for y∈Ω\nimplies that\n∂3A3(y,t)=0 (A.40)\nifA∈C1(Ω)⊗R3. Hence, the boundary conditions ( 3.6), (A.40 ) can be written as\nA/bardbl(x)=0, ∇nAn(x)=0, x∈Γ, (A.41)\nwhere A/bardbl(x)is the tangential to the boundary projection, while An(x)is the normal to the boundary projection\nofA(x). These boundary conditions and the Stokes formula imply tha t for A∈C2(Ω)⊗R3\n/an}bracketle{tA(t),∆A(t)/an}bracketri}ht=/integraldisplay\nΓA(x,t)·∇nA(x,t)dx−∑\nk/bardbl∇kA(t)/bardbl2=−∑\nk/bardbl∇kA(t)/bardbl2, (A.42)\nsince the field A(x)is orthogonal to the boundary Γwhile ∇nA(x)is parallel to the boundary at any point x∈Γ\ndue to the boundary conditions ( A.41 ). Hence, the variational derivative\nDA∑\nk/bardbl∇kA/bardbl2=−2∆A (A.43)\nifA∈C2(Ω)⊗R3and satisfies the boundary conditions ( A.41 ).\nLemma A.1. The Laplacian Λ=−∆is symmetric and nonnegative on the dense domain D 0=A(Ω)∩\nC∞(Ω)⊗R3in the Hilbert space of vector fields X :=L2(Ω)⊗R3, and the identity holds\n/an}bracketle{tA,ΛA/an}bracketri}ht=∑\nk/bardbl∇kA/bardbl2, A∈D0. (A.44)\nThe operator Λadmits a selfadjoint extension with a domain D ⊂A, and\n/an}bracketle{tA,ΛA≥δ/bardblA/bardbl2, A∈D, (A.45)\nwhere δ>0.\nProof. The Green formula implies that for any vector fields A1,A2∈D0\n/an}bracketle{tΛA1,A2/an}bracketri}ht−/an}bracketle{tA1,ΛA2/an}bracketri}ht=−/integraldisplay\nΓ[∇nA1(x)·A2(x)−A1(x)·∇nA2(x)]dx=0 (A.46)\nwhich follows similarly to ( A.42 ). Moreover, the operator Λis nonnegative on the domain D0by (A.42 ), and\nhence, it admits a selfadjoint extension by the Friedrichs t heorem. The identity ( A.44 ) follows from ( A.42 ) .\nFinally, Λis an elliptic operator in the bounded region Ω, and the boundary conditions ( A.41 ) satisfy the\nShapiro–Lopatinski condition. Hence, the spectrum of Λis discrete. Namely, Λ−zis invertible for z<0 and\nthe resolvent R(z) = ( Λ−z)−1is a compact selfadjoint operator in Xby the elliptic theory and the Sobolev\nembedding theorem. Finally, the spectra of Λ−zand of Λdiffer by the shift.\nNow to prove ( A.45 ) it suffices to check that λ=0 is not an eigenvalue. Indeed, ΛA=0 implies A∈Dby\nthe elliptic theory, and hence ( A.44 ) implies\n/an}bracketle{tA,ΛA/an}bracketri}ht=∑\nk/bardbl∇kA/bardbl2=0. (A.47)\nHence, A(x)≡a∈R3forx∈Ω. Finally, the boundary conditions ( A.41 ) imply that a=0.\nTo prove ( 6.13) let us rewrite it equivalently as\n/an}bracketle{tψ,Λψ/an}bracketri}ht≤C2(/bardblA/bardblH1)/an}bracketle{tψ,Lψ/an}bracketri}ht, ψ∈o\nH1, (A.48)\nwhere Λ:=−∆+1 and L:=(i∇+A(x))2+1. Now ( A.48 ) follows from the next lemma with δ<1.8\nLemma A.2. Let A∈A. Then the difference T :=L−Λadmits the estimate\n|/an}bracketle{tψ,Tψ/an}bracketri}ht|≤ δ/an}bracketle{tψ,Λψ/an}bracketri}ht+Cδ(/bardblA/bardblH1)/an}bracketle{tψ,ψ/an}bracketri}ht, ψ∈o\nH1(A.49)\nfor any δ>0, where C δ(·)is a continuous increasing function on [0,∞).\nProof. The difference Treads\nT=2iA(x)∇+A2(x), (A.50)\nwhere we have used that div A(x)≡0. Further, it suffices to prove ( A.49 ) for ψ∈D0:=C∞\n0(Ω)which is dense\nin the spaceo\nH1by its definition. Hence, the operators ΛandTcan be considered as the operators on entire\nspaceR3. In particular, let us extend A(x)by zero outside Ωand define the powers Λsfors∈Rby the Fourier\ntransform, Then ( A.49 ) can be reduced by the substitution ψ=Λ−1/2ϕto the estimate\n|/an}bracketle{tϕ,Λ−1/2TΛ−1/2ϕ/an}bracketri}ht|≤ δ/bardblϕ/bardbl2+Cδ(/bardblA/bardblH1)/bardblΛ−1/2ϕ/bardbl2, ϕ∈L2(R3). (A.51)\nThis reduction is not equivalent, but ( A.51 ) implies ( A.49 ) since every function ψ∈D0admits the representa-\ntionψ=Λ−1/2ϕwith ϕ∈L2(R3).\nIn the case A(x)∈C∞\n0(R3)the operator Λ−1/2TΛ−1/2is the PDO of order −1, and estimates of type ( A.51 )\nin this case follows from the interpolation inequality for t he Sobolev norms. However, the constant Cδis known\nto depend on some derivatives of the symbol of the compositio n. So we should find another arguments to prove\nthat this constant depends only on the norm /bardblA/bardblH1(Ω).\nFor this purpose we note that the operator Λ−1/2is the multiplication by /an}bracketle{tξ/an}bracketri}ht−1/2in the Fourier transform.\nHence, it is the convolution with a distribution S(x)∈L1\nloc(R3)which is a radial smooth function for x/ne}ationslash=0 and\nasymptotically homogeneous at the origine\nS(x)∼C|x|−2,|x|→0. (A.52)\nHence,\nS∈Lq\nloc(R3), q∈[1,3\n2). (A.53)\nThus, Λ−1/2=S∗, and hence,\nΛ−1/2TΛ−1/2=2iS∗A(x)∇S∗+S∗A2(x)S∗, (A.54)\nLet us estimate the first term on the right hand side. The secon d can be bounded similarly.\nWe denote by ∇S∗the composition of ∇with S∗. This composition is the bounded operator in L2(R3)since\nit is the multiplication by −iξ/an}bracketle{tξ/an}bracketri}ht−1/2in the Fourier transform. Thus,\n/bardbl∇S∗ϕ/bardbl≤C/bardblϕ/bardbl. (A.55)\nFurter, the multiplication by A∈L6maps continuously L2(R3)intoL3/2(R3)by the H¨ older inequality.\n/bardblA(x)∇S∗ϕ/bardblL3/2(R3)≤C/bardblϕ/bardbl. (A.56)\nMoreover, supp A⊂Ω, and hence, the convolution S∗[A(x)∇S∗ϕ]∈L3−α′(Ω)with sufficiently small α′>0\nby (A.53 ) and the Young theorem on the convolution. Similarly,\n/bardblΛαS∗[A(x)∇S∗ϕ]/bardblL3(Ω)≤C/bardblϕ/bardbl (A.57)\nfor small α>0. This follows by the same Young theorem since ΛαS∗forα<1/2 is the operator of convolution\nwith the distribution Sα∈L1\nloc(R3)which admits the asymptotics\nSα(x)∼C|x|−2−2α,|x|→0. 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Wang, An energy equation for the weakly damped driven nonlinear Schr¨ odinger equations and its\napplication to their attractors, Physica D 88(1995), 167–175.\n[40] A. Komech, E. Kopylova, On global attractors for 2D damp ed driven nonlinear Schr¨ odinger equations,\nsubmitted to SIAM J. Appl. Anal. , 2020."    },    {        "title": "1304.3798v1.Parametric_survey_of_longitudinal_prominence_oscillation_simulations.pdf",        "content": "arXiv:1304.3798v1  [astro-ph.SR]  13 Apr 2013Astronomy& Astrophysics manuscriptno.ms c∝circleco√yrtESO 2021\nJuly29,2021\nParametricsurvey of longitudinal prominence\noscillationsimulations\nQ.M. Zhang1,2,P.F.Chen1,3, C.Xia4, R.Keppens4,and H. S.Ji2\n1School of Astronomyand Space Science, NanjingUniversity, Nanjing 210093, China\n2Key Laboratory for Dark Matter and Space Science, Purple Mou ntain Observatory, CAS,\nNanjing 210008, China\ne-mail:zhangqm@pmo.ac.cn\n3Key Labof Modern Astronomyand Astrophysics, Ministryof Ed ucation, China\n4Centre for mathematical Plasma Astrophysics, Department o f Mathematics, KU Leuven,\nCelestijnenlaan 200B, 3001 Heverlee, Belgium\nReceived; accepted\nABSTRACT\nContext.Longitudinal filamentoscillationsrecentlyattractedmor e andmoreattention,whilethe\nrestoringforce and the damping mechanisms are stillelusiv e.\nAims.In this paper, we intend toinvestigate the underlying physi cs for coherent longitudinal os-\ncillations of the entire filament body, including their trig gering mechanism, dominant restoring\nforce, anddamping mechanisms.\nMethods. With the MPI-AMRVAC code, we carry out radiative hydrodynam ic numerical sim-\nulations of the longitudinal prominence oscillations. Two types of perturbations, i.e., impulsive\nheating at one leg of the loop and an impulsive momentum depos ition are introduced to the\nprominence, which then starts to oscillate. We study the res ulting oscillations for a large param-\neter scan, including the chromospheric heating duration, i nitial velocity of the prominence, and\nfieldline geometry.\nResults.Itisfoundthatbothmicroflare-sizedimpulsiveheatingato ne legoftheloopandasud-\ndenly imposed velocity perturbation can propel the promine nce to oscillate along the magnetic\ndip. An extensive parameter survey results in a scaling law, showing that the period of the oscil-\nlation, which weakly depends on the length and height of the p rominence, and the amplitude of\nthe perturbations, scales with/radicalbig\nR/g⊙, whereRrepresents the curvature radius of the dip, and g⊙\nis the gravitational acceleration of the Sun. This is consis tent with the linear theory of a pendu-\nlum,whichimpliesthatthefield-alignedcomponent ofgravi tyisthe mainrestoringforceforthe\nprominence longitudinal oscillations, as confirmed by the f orce analysis. However, the gas pres-\nsure gradient becomes non-negligible for short prominence s. The oscillation damps withtime in\nthepresenceofnon-adiabaticprocesses.Comparedtoheatc onduction,theradiativecoolingisthe\ndominant factor leading tothe damping. A scaling lawfor the damping timescale isderived, i.e.,\nτ∼l1.63D0.66w−1.21v−0.30\n0, showing strong dependence on the prominence length l, the geometry\nofthemagneticdip(characterizedbythedepth Dandthewidth w),andthevelocityperturbation\namplitude v0.Thelargertheamplitude,thefastertheoscillationdamps .Itisalsofoundthatmass\ndrainage significantlyreduces the damping timescale when t heperturbation is toostrong.\n1Zhang et al.:Simulatingprominence oscillations\nKey words. Sun: filaments, prominences – Sun: oscillations – Methods: n umerical –\nHydrodynamics\n1. Introduction\nSolarprominences,orfilamentswhenappearingonthesolard isk,arecoldanddenseplasmassus-\npendedinthecorona(Tandberg-Hanssen1995;Labrosseetal .2010;Mackayetal.2010).Theyare\nformedabovethemagneticpolarityinversionlines.Theden sermaterialisbelievedtobesupported\nbythemagnetictensionforceofthedip-shapedmagneticfiel dlines(Kippenhahn&Schl¨ uter1957;\nKuperus&Raadu1974;Guoetal.2010;Zhangetal.2012;Xueta l.2012;Su&vanBallegooijen\n2012).Thesefascinatingphenomenaattractedalotofsimul ationeffortsfromdifferentaspects,such\nas their formation, oscillations, and eruptions. With resp ect to the formation, the chromospheric\nevaporationplus coronal condensationmodel has been studi ed widely with one-dimensional(1D)\nsimulations(e.g.,M¨ ulleretal.2004;Karpenetal.2005,2 006;Karpen&Antiochos2008;Antolin\net al. 2010; Xia et al. 2011; Luna et al. 2012b), where no back- reaction on the field topology is\naccounted for. It was then for the first time extended to 2.5D b y Xia et al. (2012) who simulated\nthe in situ formationof a filament in a sheared magneticarcad e and showed that the condensation\nself-consistentlyformsmagneticdipswhileensuringforc e-balancestates.Thisfindingstrengthens\nthehithertoinvariably1Danalysisperformedforprominen ceformationandevolutions,asadopted\nby many authors to date. Once a prominence is formed, it might be triggered to deviate from its\nequilibriumpositionandstart tooscillate.\nObservationsdemonstratethat prominencesare hardlystat ic. Besidessmall-amplitudeoscilla-\ntions (Okamoto et al. 2007; Ning et al. 2009), large-amplitu deand long-periodprominenceoscil-\nlations have been observed(e.g., Eto et al. 2002; Isobe & Tri pathi2006; Gilbert et al. 2008; Chen\net al. 2008; Tripathi et al. 2009; Hershaw et al. 2011; Bocchi alini et al. 2011). The observations\nof the prominenceoscillations led to the comprehensiveres earch topic of prominenceseismology\n(Blokland&Keppens2011a,2011b;Arregui&Ballester2011; Arreguietal.2012;Luna&Karpen\n2012; Luna et al. 2012a), and the long-term oscillations wer e considered as one of the precursors\nforcoronalmassejections(CMEs;Chen,Innes,&Solanki200 8;Chen2011).Ofparticularinterest\nin this paperare the longitudinaloscillationsalongthe ax is of prominences/filaments,whichwere\nfirstpresentedinthesimulationresultsofAntiochosetal. (2000)discoveredfromH αobservations\nbyJingetal.(2003).Thephenomenonwasfurtherinvestigat edbyJingetal.(2006)andVrˇ snaket\nal. (2007). Such large-amplitude oscillations are trigger ed by small-scale solar eruptions near the\nfootpointsof the main filaments, such as mini-filament erupt ions, subflares, and flares. The initial\nvelocitiesof the oscillationsare 30–100kms−1. The oscillation periodrangesfrom40 min to 160\nminandthedampingtimesare ∼2–5timestheoscillationperiod(Jinget al.2006).\nUnlike the transverse oscillations whose restoring force i s known to be the magnetic tension\nforce,thedominantrestoringforceforthelongitudinalos cillationsstillawaittobeclarified.Jinget\nal. (2003)proposedseveralcandidatesforthe restoringfo rce,i.e.,gravity,the pressureimbalance,\nand the magnetictension force. Vrˇ snak et al. (2007) sugges ted that the restoringforce is the mag-\nnetic pressure gradientalong the filament axis. With radiat ivehydrodynamicsimulations, Luna &\nKarpen (2012) and Zhang et al. (2012) suggested that the grav ity component along the magnetic\nfieldisthemainrestoringforce.Li&Zhang(2012),ontheoth erhand,suggestedthatbothgravity\n2Zhang et al.:Simulatingprominence oscillations\nand magnetic tension force contribute to the restoring forc e. As for the damping mechanism, it\nreally depends on the oscillation mode. For the vertical osc illations, Hyder (1966) proposed that\nthe magneticviscositycontributesto the decay.Forthe hor izontaltransverseoscillations, Kleczek\n&Kuperus(1969)proposedthattheinducedcompressionalwa veinthesurroundingcoronaactsto\nseemingly dissipate the oscillatory power. More damping me chanisms have been proposed, such\nasthermalconduction,radiation,ion-neutralcollisions ,resonantabsorption,andwaveleakage(see\nArregui et al. 2012 and Tripathi et al. 2009 for reviews). For the longitudinal oscillations, Zhang\net al. (2012) found that non-adiabatic terms such as the radi ation and the heat conduction con-\ntribute to the damping, but they might not be su fficient to explain the observed shorter timescale.\nIn their simulations the chromospheric heating is switched off, so that the prominence mass was\nnearly fixed. On the contrary, Luna & Karpen (2012) studied th e prominence oscillations while\nkeeping the chromospheric heating and the resulting chromo spheric evaporation. As a result, the\nprominence was growing in length and mass during oscillatio ns. They found that there are two\ndampingtimescales, a shortone forthe initial stage anda lo ngerone later. The analyticalsolution\nindicates that the mass accumulation can explain the fast da mping of the initial state. As for the\nlater slower damping, they suggested non-adiabatice ffects such as radiation and heat conduction.\nA quantitative survey is in order to clarify how di fferent geometrical and physical parameters of\ntheprominenceaffectthedampingtimescale.\nWithin the framework of gravity serving as the restoring for ce for the filament longitudinal\noscillations, in this paper we try to do a parameter survey, a iming to clarify how the geometry\nof the magnetic field a ffects the oscillation period and how the combined e ffects of radiation and\nheat conduction contribute to the damping of the oscillatio ns. We describe the numerical method\nin Section2. Aftershowingthe e ffectsofthe perturbationtypein Section3, we displaythe res ults\nofourparametersurveyinSection4. Discussionsandsummar yarepresentedinSections5and6.\n2. Numericalmethod\nHigh-resolution observations indicate that a filament /prominence is made of many thin threads\nwhich are believed to be aligned to the individual magnetic t ubes (Lin et al. 2005). Since the\nmagneticfieldinsidethefilamentisquitestrong(Schmieder &Aulanier2012)andtheplasmabeta\nis very low (β∼0.01−0.1) (Antiochos et al. 2000; DeVore & Antiochos 2000; Aulanier et al.\n2006), plus that the thermal conduction is strongly prevent ed across the field lines, the dynamics\ninside differentmagnetictubescan be consideredto be independent.Th erefore,the formationand\nevolution of a filament thread can be treated as a 1D hydrodyna mic problem. Following Xia et\nal. (2011), the 1D radiativehydrodynamicequations,shown as follows,are numericallysolvedby\nthe state-of-the-art MPI-Adaptive Mesh Refinement-Versat ile Advection Code (MPI-AMRVAC;\nKeppenset al.2003, 2012).\n∂ρ\n∂t+∂\n∂s(ρv)=0, (1)\n∂\n∂t(ρv)+∂\n∂s(ρv2+p)=ρg∝bardbl(s), (2)\n∂ε\n∂t+∂\n∂s(εv+pv)=ρg∝bardblv+H−nHneΛ(T)+∂\n∂s(κ∂T\n∂s), (3)\n3Zhang et al.:Simulatingprominence oscillations\nwhereρisthe mass density, Tis the temperature, sis the distancealongthe loop, vis the velocity\nofplasma, pisthegaspressure, ε=ρv2/2+p/(γ−1)isthetotalenergydensity, nHisthenumber\ndensityofhydrogen, neisthenumberdensityofelectrons,and g∝bardbl(s)isthecomponentofgravityat\na distance salong the magnetic loop, which is determined by the geometry of the magnetic loop.\nFurthermore,γ=5/3 is the ratio ofthe specific heats, Λ(T)is the radiativeloss coe fficientforthe\noptically thin emission, H(s) is the volumetric heating rate, and κ=10−6T5/2ergs cm−1s−1K−1\nis the Spitzer heat conductivity. As done in previous works m entioned in§1, we assume a fully\nionizedplasmaand adoptthe one-fluidmodel.Consideringth e heliumabundance( nHe/nH=0.1),\nwe takeρ=1.4mpnHandp=2.3nHkBT, wherempis the proton mass and kBis the Boltzmann\nconstant. Note that the above equations are di fferent from those in Luna & Karpen (2012) in that\na uniform cross section is assumed here for the flux tube for si mplicity, where expanding flux\ntubes based on given, immobile 3D magnetic fields are adopted in Luna & Karpen (2012). The\nradiativehydrodynamicequations(1–3)arenumericallyso lvedbytheMPI-AMRVACcode,where\ntheheatconductiontermissolvedwithanimplicitschemese paratelyfromotherterms.Toinclude\nthe radiative loss, we take the second-orderpolynomialint erpolation to compile a high resolution\ntable based on the radiativeloss calculationsusing update delement abundancesand better atomic\nmodelsoverawide temperaturerange(Colganet al. 2008).Th ecorrespondingvaluesin thistable\nare systematically ∼2 times larger than the previous radiative loss function ado pted by Luna &\nKarpen(2012).\nIt is often believed that a prominence is hosted at the dip of a magnetic loop, supported by\nthe magnetic tension force. Therefore, we adopt a loop geome try with a magnetic dip, which is\nsymmetric about the midpoint, as shown in Fig. 1. The loop con sists of two vertical legs with a\nlengthof s1,twoquarter-circularshoulderswitharadius r(thelengthofeacharc, s2−s1,isπr/2),\nand a quasi-sinusoidal-shaped dip with a half-length of w. The height of the dip is expressed as\ny=D−Dcos(πx/2w) if the local coordinates ( x,y) are centered at the midpoint of the dip. The\ndip has a depth of Dbelow the apex of the loop. Such a geometry determines the fiel d-aligned\ncomponentof the gravity, whose distribution along the left half of the magnetic loop is expressed\nasfollows:\ng∝bardbl(s)=−g⊙, s/lessorequalslants1;\n−g⊙cos/parenleftBigπ\n2s−s1s2−s1/parenrightBig\n, s1<s/lessorequalslants2;\ng⊙πD\n2(L/2−s2)sin/parenleftbigg\nπs−s2\nL/2−s2/parenrightbigg\n,s2<s/lessorequalslantL/2,(4)\nwhere the gravity at the solar surface g⊙=2.7×102m s−2, the total length of the loop L, the\nlengthof each vertical segment s1=5 Mm, and s2=s1+πr/2Mm. The total lengthof the dip is\n2w=L−2s2. The field-alignedcomponentof the gravity in the right half is symmetric to the left\nhalf.Theparameter h=s1+r−Dgivestheheightofthecentraldipabovethelowerboundary.\nOur simulations start from a thermal and force-balanced equ ilibrium state where the back-\ngroundheatingis balancedby radiativeloss andthermalcon duction,andthe plasma in theloopis\nquiescent. The simulations are divided into three steps: (1 ) Prominence formation: A prominence\nformsand growsnear the center of the magnetic dip as chromos phericmaterial is evaporatedinto\nthe corona and condensates due to thermal instability after chromospheric heating is introduced\nnear the footpoints of the loop; (2) Prominence relaxation: The prominence relaxes to a thermal\nandforce-balancedequilibriumstateasthelocalizedheat ingishaltedandthechromosphericevap-\n4Zhang et al.:Simulatingprominence oscillations\nFig.1.Magnetic loop used for the 1D radiative hydrodynamicsimula tions of the prominence os-\ncillations.Notethatthe horizontalandtheverticalsizes are notto scale.\noration ceases; (3) Prominence oscillation subjected to pe rturbations: The prominence starts to\noscillate with a damping amplitude after perturbationsare introduced. In step 1, which lasts for a\ntime interval of∆t1, the heating term H(s) in Eq. (3) is composed of two terms, i.e., the steady\nbackgroundheating H0(s) and the localized chromosphericheating H1(s), which are expressed as\nfollows:\nH0(s)=/braceleftBiggE0exp(−s/Hm),s<L/2;\nE0exp[−(L−s)/Hm],L/2/lessorequalslants<L;(5)\nH1(s)=E1, s/lessorequalslantstr;\nE1exp[−(s−str)/λ],str<s/lessorequalslantL/2;\nE1exp[−(L−str−s)/λ],L/2<s/lessorequalslantL−str;\nE1, s>L−str;(6)\nwhere the quiescent heating term H0is adopted to maintain the hot corona with the amplitude\nE0=3×10−4ergs cm−3s−1and the scale-height Hm=L/2, and the localized heating term H1is\nadoptedto generatechromosphericevaporationintothecor onawiththe amplitude E1=10−2ergs\ncm−3s−1, the transition region height str=6 Mm, and the scale height λ=10 Mm. The heating\nis taken to be symmetricin orderto forma static prominencen earthe magneticdip center,so that\nwecaneasilycontrolthemannerhowtheprominenceistrigge redtooscillate.Ourmethodologyis\ndifferentfromLuna& Karpen(2012)who usedasymmetricheatingw hichspontaneouslyleadsto\nthe oscillation oncethe prominenceis formed.In step 2, H1is switched off. Owing to the absence\nof the chromospheric evaporation, the gas pressure inside t he magnetic loop drops down, so the\ncompressedprominenceexpandsuntilanewequilibriumisre ached,whichroughlytakeslessthan\n2.4 hr. In step 3, a perturbation is introduced to the promine nce in order to trigger its oscillation.\nNotethat H0remainsthroughoutthesimulations.\nFrom the observationalpoint of view, there might be two kind s of perturbations.The first one\nis an impulsive momentum injected to the magnetic loop as the magnetic reconnection near the\nfootpoints rearrangesthe magnetic loop rapidly. The secon d is impulsive heating due to subflares\n(e.g., Jing et al. 2003, Vrˇ snak et al. 2007, Li & Zhang 2012) o r microflares (Fang et al. 2006)\nnear the footpoints of the magnetic loop where a large amount of magnetic energy is impulsively\nreleasedthroughmagneticreconnection.Thegaspressurei sgreatlyincreasedthatcouldpropelthe\nprominencetooscillatealongthedip-shapedfieldlines.In our1Dsimulations,weseparatethetwo\n5Zhang et al.:Simulatingprominence oscillations\neffectstosee theirdifference.Inonecase,avelocityperturbationwiththefollow ingdistributionis\nimposedtothe prominence,\nv(s)=0, s<spl−δ;\nv0(s−spl+δ)/δ,spl−δ/lessorequalslants/lessorequalslantspl;\nv0, spl/lessorequalslants/lessorequalslantspr;\nv0(−s+spr+δ)/δ,spr/lessorequalslants/lessorequalslantspr+δ;\n0, s>spr,(7)\nwheresplandsprare the coordinates of the left and right boundaries of the pr ominence,δ=10\nis the buffer zone which allows that the perturbation velocity varies s moothly in space, and v0is\nthe perturbation amplitude. In the other case, impulsive he ating (H2), as described as follows, is\nintroducednearthe right-handfootpointofthe magneticlo op,\nH2(s)=E2exp−(s−speak)2\ns2\nscale−(t−tpeak)2\nt2\nscale, (8)\nwhere the heating spatial scale sscale=2.5 Mm, the peak location speak=245 Mm, the heating\ntimescale tscale=5min,andthepeaktime tpeak=15min.Theheatingrampsuptothepeakfor15\nminandthenfadesdownto 0.\nAsfortheboundaryconditions,allvariablesatthetwofoot pointsofthemagneticlooparefixed,\nwhichisjustifiedbecausethedensityinthelowatmospherei smorethanfourordersofmagnitude\nhigherthanthatinthecorona.Thesameapproachhasbeenado ptedbyOfman&Wang(2002)and\nXia et al. (2011), assuming that the coronaldynamicshas lit tle effect on the low atmosphere. The\napproach was verified by Hood (1986) with the parameters bein g far from the marginal stability.\nThe violation of the rigid wall conditions in certain cases w as discussed by van der Linden et al.\n(1994).\n3. Effectsoftheperturbationtype\nInordertocheckhowthetwotypesofperturbationsasdescri bedin§2influencethecharacteristics\noftheprominenceoscillations,weperformsimulationsoft heoscillationswhichareexcitedbythe\ntwotypesofperturbationswhilekeeping ∆t1=7.2hr,r=20Mm,D=10Mm,and L=260Mm.\nIn case A, the prominence oscillation is triggered by a veloc ity perturbation over the whole\nprominencebody.With v0=−40kms−1(theminusmeansthatthevelocityistowardtheleft),the\ntemporal evolutionof the plasma temperature distribution along the magnetic loop is displayed in\nthe left panel of Fig. 2. It is seen that in responseto the pert urbation,the prominence,signified by\nthe low temperature, starts to oscillate around the equilib rium position. The oscillation amplitude\ndecays with time. Fitting the trajectory of the mass center o f the oscillating prominence with a\ndecayedsinefunction\ns=s0+A0sin(2π\nPt+φ0)exp(−t/τ), (9)\nwe find the initial amplitude A0=34.9 Mm, the oscillation period P=84.3 min, and the damp-\ning timescaleτ=272 min. Assuming that the prominence thread has a cross-sec tion area of\n∼3.14×1014cm2(Lin et al. 2005), the initial kinetic energyof the oscillat ing prominencethread\nis estimated to be∼7.2×1023ergs. It is noted that the single decayed sine function, as us ed for\n6Zhang et al.:Simulatingprominence oscillations\n50100150200250\ns (Mm)01234Time (hr)\n0.01 0.43 0.85 1.26 1.68 2.10T (MK)\n50100150200250\ns (Mm)01234Time (hr)\n0.01 0.91 1.81 2.71 3.62 4.52T (MK)\nFig.2.Comparison of the evolutions of the temperature of the loop b etween the two types of\nperturbations. The left panel corresponds to the case with v elocity perturbations with v0=−40\nkm s−1and the right panel to the case with localized heating pertur bations with E2=0.24 ergs\ncm−3s−1.\nfitting the Hαobservations(Jing et al. 2003; Vrˇ snak et al. 2007; Zhang et al. 2012), fits the simu-\nlatedobservationsverywell.Onthecontrary,acombinatio nofBesselfunctionandanexponential\ndecay functionis necessary to fit the initial overtonein the simulationsof Luna& Karpen (2012),\nwhichresultsfromthecontinualmassaccumulation.\nIn case B, the prominenceoscillation is triggered by the imp ulsive heating which is deposited\nnear the right leg of the magnetic loop in order to mimic a micr oflare near the prominence.To do\nthat, an impulsive heating term H2(s) in Eq. (8) is added to the heating term Hin Eq. (3), where\nspeak=245Mmmeaningtheheatingisconcentratedataheightof15Mm abovetherightfootpoint\nofthemagneticloop.\nTherightpanelofFig.2depictsthetemporalevolutionofth etemperaturedistributionalongthe\nmagnetic loop with E2=0.24 ergs cm−3s−1. With the typical cross-section area of a prominence\nthread being∼3.14×1014cm2, the corresponding total energy deposited into the single m agnetic\nloopEheatingis 1.8×1025ergs. This value is reasonable since observations indicate that the total\nenergy of a microflare is 1026–1027ergs or even more (e.g., Shimizu et al. 2002; Hannah et al.\n2008;Fangetal.2010),andseveralpercentofthereleasede nergygoesintooneprominencethread.\nFromanotherpointofview,undertheframeworkofmagneticr econnectionmodelformicroflares,\nthe magnetic energy release rate is estimated to be B2vin/(4πL). With the magnetic field B∼20\nG, the reconnectioninflow speed vinbeing about 0.1 times the Alfv´ en speed which is about 1000\nkm s−1(Jiang et al. 2012), and the spatial size L=10′′, the energy release rate is estimated to be\n∼0.88 ergs cm−3s−1, which is of the order adopted here. Fitting the trajectory o f the oscillating\nprominence with the damped sine function as shown in Eq. (9) y ieldsA0=35.8 Mm,P=84.3\nmin, andτ=268 min. The corresponding initial velocity is also -40 km s−1. This indicates that\na typical microflare near the leg of the magnetic loop hosting a prominence thread can excite the\nprominencelongitudinaloscillations with an initial velo city of tens of km s−1. The corresponding\nkinetic energy is only ∼7.2×1023/1.8×1025, i.e.,∼4% of the deposited thermal energy. The\nremaining∼96%oftheenergydepositcontributestothe heatingofthech romosphere.\n7Zhang et al.:Simulatingprominence oscillations\n4. Parametersurvey\nThe results in§3 reveal that the oscillation period does not strongly depen d on the two types of\nperturbations,i.e., impulsive momentumand localized hea ting at one footpointused in our inves-\ntigation. Note that we concentrate on the oscillation chara cteristics which follow the small tran-\nsient/excitationphase,alreadyobtainedfromsimpledecayingsi nusoidalfitting.Asmalldi fference\nin the decay timescale exists between the two perturbation t ypes. With the same initial velocity,\nthe decay timescale is 4 minutes shorter in the case of impuls ive heating than that in the case of\nimpulsivemomentum.However,therelativevariation,1.4% ,isverysmall.Therefore,wecancon-\nclude that the oscillation is basically intrinsic and the ch aracteristics of the oscillation depend on\nthe prominence itself and the geometry of the magnetic loop i n our case where there is no mass\naccumulationand the oscillationsare excited by either imp ulsivemomentumor localized heating.\nThe prominence feature is only characterized by the thread l ength (l), and the geometry of the\nmagnetic loop is characterized by r,D, andwas depicted in Fig. 1. Among the three geometrical\nparameters, h=s1+r−Ddeterminestheheightoftheprominence, Dandwdeterminethecurva-\nture of the magneticdip. If other parametersare fixed, the le ngthof the prominenceis determined\nby the duration of the chromospheric evaporation in step 1, i .e.,∆t1, as described in§2. Besides,\nthedecaytimescalemightvarywith theperturbationamplit ude,thereforeanotherparameteristhe\ninitial perturbation velocity v0. In this section, we perform a parameter survey to investiga te how\neachindividualoneamongthefiveparameters( ∆t1,r,D,w,andv0)changestheoscillationperiod\nandthedecaytimescale.Foreachparameter,severalcasesw ithdifferentvaluesaresimulatedwith\nother parameters fixed. In our simulations, we set r=10 Mm,D=5 Mm,w=110 Mm, and\nv0=−20 km s−1when varying∆t1. We set∆t1=7.16 hr,D=5 Mm,w=90 Mm, and v0=−20\nkms−1when varying r. We set∆t1=7.16hr,D=5 Mm,r=10Mm, and v0=−20 kms−1when\nvaryingw. We set∆t1=7.16 hr,r=20 Mm,w=93.6 Mm, and v0=−20 km s−1when varying\nD. We set∆t1=7.16 hr,r=20 Mm,w=93.6 Mm, and D=10 Mm when varying v0. Since the\noscillationcharacteristicsarefoundnearlyinsensitive tothe perturbationtype,we use the velocity\nperturbationtoexcitetheoscillationsinthesurvey.\n4.1. Lengthand massof theprominence\nAfterfinishingthefirsttwostepsofthesimulationsasdescr ibedin§2,wegetaquasi-staticpromi-\nnence.The dependenceof the prominencelength lon∆t1,h,D, andwis shownin the fourpanels\nofthe upperrowof Fig. 3. It can be seen that l, whichfits into the scalinglaw l∼∆t0.70\n1, increases\nwiththedurationoftheheatingtime ∆t1. Itisunderstandablesincemorechromosphericplasmais\nevaporatedintothecoronawhen ∆t1increases.Thelength ldecreaseswith hasl∼h−0.37,whichis\nprobablybecauseit takesa longertime forthemoretenuousc oronato condensateastheheightof\nthemagneticdipincreases,andthereforethee ffectiveheatingtimeisshorter.Thelength ldecreases\nwithDasl∼D−0.21,whichcanbeunderstoodastheprominencebecomesmorecomp ressedasthe\nmagnetic dip becomes deeper. However, the length of the prom inencedoes not vary considerably\nwithw. Of course, wshould not be too small, otherwise thermal instability woul d not occur. The\nlengthsof these simulated prominencethreadsare consiste nt with the reportedvalues, i.e., tens of\nMm(Linet al.2005).\n8Zhang et al.:Simulatingprominence oscillations\nFig.3.Scatter-plots of the total length l(upper panels) and mass M(lower panels) of the promi-\nnencesattheendofrelaxationstepasfunctionsof ∆t1,h,D, andw.\nThe dependenceof the prominencemass Mon∆t1,h,D, andwis shown in the fourpanels of\nthe lower row of Fig. 3. It can be seen that the dependence of Mon∆t1,h, andwis similar to l.\nTheir difference is that ldecreases with DwhereasMdoes not change with D, which means that\ntheplasmanumberdensity(1010–1011cm−3,andthecorrespondingdensityis10−14–10−13gcm−3)\nis higher in the prominence with a deeper magnetic dip. A scal ing law is obtained by fitting the\ndatapoints,whichis M∼∆t0.98\n1h−0.34.\nItisnotedthattheaboveresultsarederivedforadippedmag neticloopfilledviachromospheric\nevaporationwith a limited lifetime, where the prominencet hread can sustain in the corona.In the\ncaseofmagneticloopswithoutadip(e.g.,Mendoza-Brice˜ n oetal.2005)orwithashallowdipand\nasymmetric heating (e.g., Karpen et al. 2006), condensatio ns repetitively form, stream along the\nmagnetic field, and ultimately disappear after falling back to the nearest footpoint. Therefore, the\nmassandlengthoftheprominenceevolvedynamically,witho utreachinganequilibriumvalue.\n4.2. The oscillationperiodand decaytimescale\nAs the velocity perturbation is introduced to the quasi-sta tic prominence, the prominence starts\nto oscillate. Fitting the trajectory of the oscillating pro minence with the damped sine function\nshownin Eq.(9), we getthe oscillationperiod( P)andthe decaytimescale ( τ) foreachcase inthe\nparametersurvey.\nThe variationsof Palongwith the parameters l,h,D,w, andv0are shown in the upperrow of\nFig. 4. It is seen that Pincreases slightly with landv0, and decreases slightly with h. However, it\nincreasesseriouslywith wanddecreaseswith D.Tofit thevariationswithascalinglaw,weobtain\nP∼l0.16h−0.05D−0.54w0.91v0.05\n0. Therefore,the periodof prominencelongitudinaloscilla tions relies\ndominantlyonthe geometryofthedip,especiallyits curvat ure.It isnotedthat therangeof Pisin\nagreementwith thereportedvaluesin previousstudies(e.g .,Jingetal. 2006).\nThevariationsofτalongwiththefiveparametersareshowninthelowerrowofFig .4.Itisseen\nthatτincreasessignificantlywith landD,anddecreaseswith wandv0.Itisnotedthatinthecases\nof|v0|=70 and 80 km s−1, part of the prominencemass drains down to the chromosphere ,which\n9Zhang et al.:Simulatingprominence oscillations\nFig.4.Scatter-plotsoftheperiod P(upperpanels)anddampingtime τ(lowerpanels)ofthepromi-\nnences in the oscillation step as functionsof l,h,D,w, andv0. The values of Pandτin the cases\n|v0|=70 and 80 km s−1that cause mass drainage at the footpoint of the coronal loop are marked\nwithtrianglesinthe rightpanels.\nis why the triangles in the lower-right panel of Fig. 4 do not f ollow the trend of the data points\ndenotedby the diamondswhere |v0|<70 km s−1. The decay timescale does not vary significantly\nwithh.Tofit thevariationswithascalinglaw,weobtain τ∼l1.63h−0.18D0.66w−1.21v−0.30\n0,wherethe\ncaseswithprominencedrainagearenotincludedinthefittin g.Thevaluesofτarealsointhesame\norderofmagnitudeastheobservedones.\n5. Discussions\n5.1. Restoring force\nForanoscillatingphenomenon,themostimportantthingist hedeterminationoftherestoringforce,\nwhichdirectlydecidestheoscillationperiod.Inour1Dhyd rodynamicsimulations,theonlyforces\nexerted on the prominenceare the gravity and the gas pressur e gradient, both are restoring forces\nforthe longitudinaloscillations.In orderto comparethei rimportance,we calculate the two forces\nin the case with∆t1=7.16 hr,v0=−40 km s−1,r=20 Mm,D=10 Mm, and w=93.6 Mm.\nThe two forces are calculated when the prominence is the furt hest from the equilibrium position.\nDespite that the plasma in prominences is hundreds of times d enser than the ambient corona, it\nis not an ideal rigid body. For oscillations with higher mode s as studied by Luna et al. (2012a),\nthe pressure gradient changes rapidly along the prominence thread. For the fundamental-mode\noscillations in this paper, the prominence oscillates as a w hole and the pressure gradient changes\nslightly along the thread. Therefore, for simplicity, we co mpare the overall magnitude of the two\nforces by a simple calculation instead of as point-to-point one in the simulations. The integral of\nthe gravity force is quantified between the two ends of the pro minence,i.e., Fg=/integraltextright\nleftρ|g∝bardbl|ds=/integraltextright\nleftρg⊙πD\n2w|sin(π(s−L/2)\nw)|ds, where a unit area is assumed for the cross section. The integ ral of\npressure gradient force over the prominence is expressed as Fp=/integraltextright\nleft|∂p/∂s|ds=|pright−\npleft|. Theleft andrightboundariesoftheprominencearedefinedt obe wherethedensitydropsto\n10Zhang et al.:Simulatingprominence oscillations\nFig.5.Temporal variation of Fg/Fpwhen the displacement of the prominencereaches maximum\nduringeachhalf-cyclein thecase of r=20Mm and D=10Mm. Thevelocityperturbationis-40\nkms−1.\n7×10−14gcm−3.Figure5displaysthetemporalevolutionoftheratio Fg/Fp,fromwhichitisseen\nthatthegravitationalforceis generally ∼10timeslargerthanthegaspressuregradientforce.\nSincethegravityisthedominantrestoringforce,theovera llmotionoftheprominencecanalso\nbedescribedforsimplicityas\nMd2x\ndt2=Mg∝bardbl=−Mg⊙πD\n2wsin(πx\nw), (10)\nwherex=s−L/2 is the displacement of the prominence from the equilibrium position. It is not\neasyto solvethisequationanalytically.However,iftheos cillationamplitudeismuchsmallerthan\nthehalfwidthofthewholemagneticdip( w),we gettheapproximationsin( πx/w)≈πx/w.So,the\naboveequationissimplifiedto be\nMd2x\ndt2=−Mg⊙πD\n2wπx\nw, (11)\nwiththe solution x=A0sin(2π\nPt+φ).Thecorrespondingperiodis\nP=/radicalBigg\n8w2\ng⊙D. (12)\nSuchaperiodcanalsobereadilyobtainediftheprominencei stakeninanalogytoapendulum\nwhoseperiodis\nP=2π/radicalBigg\nR\ng⊙, (13)\nwhereRis the curvature radius of the dipped magnetic loop. With the shape of the loop being\ny=D−Dcos(πx/2w),thecurvatureradiusattheloopcenterisapproximatedto beR=2w2/(Dπ2).\nSubstituting RintoEq.(13),weget P=/radicalbig\n8w2/(g⊙D),thesameasEq.(12).Figure6comparesthe\noscillation periods obtained from the hydrodynamic simula tions (diamonds ) and those estimated\nfrom Eq. (12) ( solid line ) when the two parameters, Dandw, are changed. It is revealed that\nEq. (12) is a very good approximation for estimating the peri od of the prominence longitudinal\noscillation. Of course, it should be kept in mind that the der ivation of Eq. (12) is based on the\n11Zhang et al.:Simulatingprominence oscillations\nassumption that the dipped magnetic loop has a sinusoidal sh ape. More generally, the oscillation\nperiodisrelatedtothelocalcurvatureradius Rbytheformula P=2π/radicalbig\nR/g⊙,asalsodemonstrated\nbyLuna&Karpen(2012).\nRecently, Luna et al. (2012a) extended the theoretical anal ysis of longitudinal prominenceos-\ncillations by including the e ffect of the pressure gradient force. They found that the ultim ate fun-\ndamental frequencyof the oscillations is found from ω2\nfund=ω2\ng+ω2\ns, whereωgandωsstand for\nthe gravity-driven and pressure-driven frequencies, resp ectively. The ratio of the two frequencies\nω2\ng/ω2\ns=Rlim/R, whereRlimdenotes the critical value of the curvature radius ( R) of the magnetic\ndip. IfR≪Rlim, then gravity dominatesover pressure in the restoring forc e of longitudinaloscil-\nlations.Theypointedoutthatthereportedvaluesofthecur vaturearesmallcomparedwith Rlim,so\nthatitisreasonabletoignorethee ffectofthepressureterminmostcases.Inourparametersurve y,\nRlim=0.175(L−l)lranges from 760 to 2100 Mm and the ratio R/Rlimranges from 0.1 to 0.5.\nHence,theirtheoreticalresultsofgravitybeingthemainr estoringforceforthe fundamentalmode\ninthisparameterrangearethusconfirmedbyoursimulations .\nForaprominenceabovethesolarlimb,alltheparametersinE q.(12)canberoughlymeasured.\nCombined with the results in this paper, the comparison betw een simulations and observations\nin Zhang et al. (2012) implies that Eq. (12) is a good approxim ation to estimate the oscillation\nperiod. For the prominence longitudinal oscillations on th e solar disk, i.e., filament longitudinal\noscillations, only the oscillation period can be unambiguo usly measured. Eq. (12) then provides\na diagnostic tool for inferring the geometry of the dipped ma gnetic loop. Especially, when wcan\nbe roughly estimated from force-free magnetic extrapolati ons, the depth of the dip, D, can be\ndetermined.Atleast,we canestimatethecurvatureradiuso fthedippedmagneticfield, R,through\nEq. (13). After the determination of R, Luna & Karpen (2012) further proposed an approximate\nmethodtoestimate themagneticfieldin theprominence.\nBesides the dominant dependence on the geometric parameter s, the oscillation period also\nweaklychangeswiththelengthandtheheightoftheprominen ce,aswellaswiththeinitialvelocity.\nThesecanbeunderstoodasfollows:(1)Dependenceonthepro minencelength:Astheprominence\nthreadisshorter,theratioofthegaspressuregradienttot hegravitywouldincreaseasindicatedby\noursimulations,therefore,thegaspressuregradientwoul dcontributetotherestoringforce,result-\ningina shorteroscillationperiod;(2)Dependenceonthepr ominenceheight:AsseenfromFig. 3,\nwith other parametersthe same, a high prominencehas a short er length.Therefore,with the same\nreason as in (1), the oscillation period would be smaller; (3 ) Dependence on the initial velocity:\nSince sin(πx/w) is always smaller than πx/win Eq. (10), the nonlinear term would naturally lead\ntoa longperiodastheoscillationamplitudeincreases.\n5.2. Dampingmechanisms\nIf the energy dissipation terms such as the radiative coolin g and the heat conduction are removed\nfrom Eq. (3), as we did in a test simulation, we found that the p rominence oscillation does not\ndampatall.Whenthetwonon-adiabatictermsarekept,thepr ominenceoscillationalwaysdamps.\nInordertosee theimportanceofthe twoterms,we calculatet hetimeintegrationsofradiativeloss\n(ER) and thermal conduction( EC) of the whole system after subtracting the correspondingva lues\nwhen the prominence is static at the center of the dip. Here ERandECare the integrals of the\n12Zhang et al.:Simulatingprominence oscillations\nFig.6.Comparisonoftheperiodsoftheprominenceoscillationsfr omsimulations( diamonds )and\ntheoreticalanalysis( solidline)asafunctionofthedepthofthemagneticdip D(leftpanel)andthe\nwidthofthedip w(rightpanel ).Note thatbothaxesarein logarithmicscale.\nFig.7.Temporalvariationsof ER/ECin the oscillation step in the cases of v0=−40, -50, and -60\nkms−1.\nradiative and the conductive terms in the energy equation Eq . (3), where the integrals are taken\nin the whole corona above the two footpoints. The evolutions of the ratio ( ER/EC) in the cases of\nv0=−40,-50,and -60km s−1are displayedin Fig. 7. It is seen that the ratio is alwayslar gerthan\nunity. Especially in the early stage of the oscillation when the amplitude is still large, ERis even\none order of magnitude larger than EC. It is also revealed that as the initial velocity increases, ER\nbecomes more and more important in most of the lifetime of the oscillation. Our results support\ntheconclusionsofTerradaset al.(2001,2005) thatradiati velossisresponsibleforthedampingof\ntheslowmodeofprominenceoscillationsinthedip-shapedm agneticconfigurations,whichseems\nto be different from the case of slow-mode waves propagating in the cor onal loops where heat\nconductioncontributesmoretothe damping(DeMoortelet al . 2002a, 2002b).\nThe role of the radiative cooling can be understood in a simpl e model as follows: Since there\nare two segmentsof the coronain the magneticloop,as thepro minenceoscillates, onepart would\nbe attenuated and the other be compressed.Suppose that the t otal length of the coronalpart of the\n13Zhang et al.:Simulatingprominence oscillations\nmagnetic loop is unity, which includes the part x, which is to the left of the prominence, and the\notherpart 1−x, which is to the rightof the prominence.Hence, the densitie sof the coronaon the\ntwo sidesareproportionalto1 /xand1/(1−x), respectively.Thetotal optically-thinradiativeloss\nof the coronal part is proportional to x−2+(1−x)−2, which is the minimum when x=0.5, i.e.,\nwhen the prominence is situated at the equilibrium position . Whenever the prominence deviates\nfrom the loop center, the cooling becomes larger, dissipati ng the kinetic energy of the oscillating\nprominence. The model is best illustrated by the relationsh ip between the damping timescale ( τ)\nand the initial amplitude of the oscillation, i.e., A0in Eq. (9). As A0increases, one of the two\ncoronalpartsismoreseverelycompressed,sotheradiative coolingx−2+(1−x)−2deviatesfurther\naway from the minimum value, i.e., it becomes larger. As a res ult, the oscillation decays more\nrapidly.\nBasedonthesinusoidalfunction, A0∝v0P.SubstitutingEq.(12)intoit,weget A0∝v0wD−1/2.\nWiththis,itisnotdi fficulttounderstandthepositivecorrelationbetweenthedec aytimescaleτand\nDandthe negativecorrelationbetween τandwas revealedby the lower rowof Fig. 4. Along this\nline ofthought,the dependenceofthe decaytimescale on the prominencelengthcanbe explained\nasfollows:Astheprominencethreadislonger,thecoronalp artofthemagneticloop,whichradiates\noutthethermalenergy,isshorter.Moreimportantly,thelo ngerthread,withthesameinitialvelocity,\nhas a larger kinetic energy. Therefore, it takes a longer tim e for the compressed coronal part to\nradiateit out.\nItisseenfromthefirstsixcases(i.e., |v0|from10kms−1to60kms−1)inthelower-rightpanel\nof Fig. 4 that the decay timescale decreases with the initial perturbation velocity nearly linearly.\nHowever, when v0is larger than 70 km s−1, part of the prominence would overpass the magnetic\nloop apex and drain down. The critical velocity for the promi nence to reach the loop apex can\nbe roughly estimated as vcriti∼/radicalbig\n2g⊙D=23√D/Mm km s−1. Therefore, the value of vcritiis\n73 km s−1in the case of D=10 Mm. As revealed from our simulations, even when v0=-70 km\ns−1, mass drainagealready happens,althoughthe amountof the d rainageis much less than that in\nthe case of v0=-80 km s−1. The temperature evolution along the loop in the case of v0=-80 km\ns−1is presented in Fig. 8. It is seen that part of the prominence f alls down to the left leg of loop,\nleadingtothedrainageoftheprominencemassandkineticen ergyaswell,whiletheremainingpart\ncontinuestooscillatealongthedip.Theoscillationperio dandthedecaytimescaleinthecaseswith\nmass drainage are marked as triangles in Fig. 4. Their period s,∼90.6 min, are slightly below the\ntrenddefinedby othercases withoutmassdrainage( diamonds ),whichis consistentwith the weak\npositive correlationbetween Pand the prominencelength l. However,the dampingtimescales are\ngreatlyreduced,comparedto the trenddefinedby othercases withoutmass drainageas seen from\nthe lower-rightpanel of Fig. 4. Such a result, namely that ma ss drainagewould greatly reducethe\ndecay timescale, might explain the mismatch between the sim ulation and the observation of the\ndecayofa prominenceoscillationreportedinZhanget al.(2 012).\n6. Summary\nIn this paper, we carry out 1D hydrodynamicsimulations of lo ngitudinal prominence oscillations\nusing the MPI-AMRVAC code, extending earlier numerical sim ulations of prominence formation\n(Xia et al. 2011) and of prominence oscillations (Luna & Karp en 2012; Zhang et al. 2012). The\n14Zhang et al.:Simulatingprominence oscillations\n50100150200250\ns (Mm)01234Time (hr)\n0.01 0.48 0.95 1.43 1.90 2.37T (MK)\nFig.8.Temporal evolution of the temperature along the magnetic lo op when the initial velocity\nperturbationis aslargeas v0=-80kms−1. Note thatthe prominenceoverpassesthe magneticloop\napexanddrainsdowntothechromosphereat theleft footpoin taroundt=0.8hr.\nsimulations are divided into three steps: First, a prominen ce forms and grows near the center of\nthe dip-shaped coronal loop due to chromosphericheating an d the subsequent thermal instability.\nThen, it relaxes to a quiescent state after the chromospheri c heating is switched o ff. Subjected\nto two kinds of perturbations that mimic subflares, the promi nence starts to oscillate along the\ndip. Within the framework of the evaporation-condensation model, we obtained scaling-laws for\nthe prominence length ( l) and mass ( M), which are expressed as l∼∆t0.70\n1h−0.37D−0.21andM∼\n∆t0.98\n1h−0.34, where∆t1is the time durationof the chromosphericheating and evapor ation,his the\nprominence height, Dis the depth of the magnetic dip. It is found that lis insensitive to the half\nlength of the magnetic dip ( w) oncewis large enough, say, 60 Mm; Mis insensitive to Dand\nw. Both transient heating at one leg of the loop and an impulsiv e velocity perturbation applied\nto the prominence as a whole are capable of driving a coherent oscillation along the dip. The\noscillation properties are found insensitive to the pertur bation type in the regimes studied. In the\ncase of the transient heating, ∼4% of the deposited energy is converted into the kinetic ener gy of\ntheprominence.Thelongitudinaloscillationsaresustain edmainlybythetangentialcomponentof\ngravity,exceptwhentheprominenceisshortandthegaspres suregradientbecomesalsoimportant.\nBoth simulationsandlinearanalysisrevealthat theperiod ofoscillation( P) is2π/radicalbig\nR/g⊙, whereR\ndenotesthecurvatureradiusofthedip,asalsofoundbyLuna &Karpen(2012).Otherparameters,\nsuchasthelengthandtheheightoftheprominence,aswellas theperturbationvelocity,alsoa ffect\nP,thoughslightly.Thelongitudinaloscillationsdampinth epresenceofnon-adiabatice ffects,i.e.,\nradiative loss and thermal conduction (Soler et al. 2009), a mong which the radiative loss plays a\nleadingrole.Withtheparametersurvey,weobtainedascali ng-lawforthedecaytimescale τ,which\nisexpressedasτ∼l1.63D0.66w−1.21v−0.30\n0,wherev0istheinitialvelocityperturbation.Wealsofound\nthat prominencemass drainage, once it happens, significant ly reduces the decay timescale, which\nmayexplainthe mismatchingbetween thesimulationsandthe observationsdisclosed byZhanget\nal.(2012).\nIt is worthmentioningthe limitationof the applicationsof the aboveresults. Accordingto this\npaper,themassofaprominencethreadisinsensitivetothed epthDandthewidth wofthemagnetic\ndip.Thisisbasedontheprominenceformationdirectlyviac hromosphericevaporationwithafixed\n15Zhang et al.:Simulatingprominence oscillations\nlifetime∆t1.AccordingtoXiaetal.(2011),theprominencewouldgrowvi asiphonflowevenwhen\nthe localized heating is switched o ff, though the growth speed is much slower. Recently, Luna et\nal.(2012a)pointedoutthattherestoringforceofthelongi tudinaloscillationsdependsonthedepth\nof the magnetic dip. For shallow dips, gas pressure plays an i mportant role, while gravity is the\nmain factor for deep dips. Besides, Li & Zhang (2012) suggest ed that magnetic tension may also\ncontribute to the restoring force. As for the damping mechan isms, several other e ffects might be\ntakenintoaccountinthefuturesimulations,suchasthewav eleakageandplasmaviscosity(Ofman\n& Wang 2002). 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Silva1,∗\n1Quantum Electromagnetics Division,\nNational Institute of Standards and Technology, Boulder, C O 80305, U.S.A.†\n2Department of Physics and Astronomy,\nUppsala University, Box 530, 751 20 Uppsala\n3Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften, Garching, Germany\n4Physik-Department, Technische Universit¨ at M¨ unchen, Ga rching, Germany\n(Dated: March 22, 2018)\n1Abstract\nUnderstanding the evolution of spin-orbit torque (SOT) wit h increasing heavy-metal thickness\nin ferromagnet/normal metal (FM/NM) bilayers is critical f or the development of magnetic mem-\nory based on SOT. However, several experiments have reveale d an apparent discrepancy between\ndamping enhancement and damping-like SOT regarding their d ependence on NM thickness. Here,\nusing linewidth and phase-resolved amplitude analysis of v ector network analyzer ferromagnetic\nresonance (VNA-FMR) measurements, we simultaneously extr act damping enhancement and both\nfield-like and damping-like inverse SOT in Ni 80Fe20/Pt bilayers as a function of Pt thickness. By\nenforcing an interpretation of the data which satisfies Onsa ger reciprocity, we find that both the\ndamping enhancement and damping-like inverse SOT can be des cribed by a single spin diffusion\nlength (≈4nm), and that we can separate the spin pumping and spin memor y loss contribu-\ntions to the total damping. This analysis indicates that les s than 40% of the angular momen-\ntum pumped by FMR through the Ni 80Fe20/Pt interface is transported as spin current into the\nPt. On account of the spin memory loss and corresponding redu ction in total spin current avail-\nable for spin-charge transduction in the Pt, we determine th e Pt spin Hall conductivity ( σSH=\n(2.36±0.04)×106Ω−1m−1) and bulk spin Hall angle ( θSH= 0.387±0.008) to be larger than\ncommonly-cited values. These results suggest that Pt can be an extremely useful source of SOT if\nthe FM/NM interface can be engineered to minimize spin loss. Lastly, we find that self-consistent\nfitting of the damping and SOT data is best achieved by a model w ith Elliott-Yafet spin relax-\nation and extrinsic inverse spin Hall effect, such that both th e spin diffusion length and spin Hall\nconductivity are proportional to the Pt charge conductivit y.\n∗thomas.silva@nist.gov\n†Contribution of the National Institute of Standards and Technolo gy; not subject to copyright.\n2I. INTRODUCTION\nThe use of nonmagnetic metals with strong spin-orbit coupling (SOC) to generate pure\nspin currents via spin-orbit effects is currently an area of intense f ocus, driven largely by the\npromise of efficient electrically-controllable magnetic memory. For th is application, the spin\ncurrent or spin accumulation generated by SOC in a non-magnetic lay er can be used to exert\na torque on an adjacent ferromagnetic (FM) layer—so called spin-o rbit torque (SOT)—in\norder to excite magnetization dynamics or cause switching. Centra l to this field of study\nis proper characterization of the spin-to-charge conversion tha t occurs in heavy metal films\nsuch as Pt, Ta, W, and Au. There are many techniques for measurin g this conversion,\nincluding ferromagnetic resonance (FMR) spin pumping1, non-local spin valves2,3, thermal\nspin injection via the spin Seebeck effect4, spin Hall magnetoresistance5, spin torque FMR6,\nand harmonic analysis of Hall effect voltage measurements7. Several groups, using various\ntechniques8–12, have uncovered a discrepancy when comparing the excess dampin g and the\nspin-to-charge conversion by inverse spin Hall effect (iSHE) contr ibuted by the normal metal\n(NM) layer. Specifically, the FM damping exhibits a steep increase with the introduction\nof only a very thin ( <2nm) NM film13–15. Meanwhile, the measured SOT, characterized\nby either spin-to-charge conversion via DC iSHE or harmonic Hall tec hnique, develops over\na much longer length length scale16,17. Magneto-optical measurements also demonstrate an\ninterfacial spin accumulation in Pt due to SHE with a spin diffusion length of about 10nm18.\nSpin memory loss (SML)10,19and proximity-induced magnetic moments at the FM/NM\ninterface15have been invoked to explain the large damping enhancement caused by thin NM\nfilms even when the NM thickness is less than its spin diffusion length. In this model, spin\nloss at the FM/NM interface acts as an additional parallel spin relaxa tion pathway to that\nof spin pumping and diffusion into the Pt bulk. From damping measureme nts alone, the\nrelative contributions of these mechanisms is not resolvable. In this work, we show that a\nself-consistent fit of Gilbert damping and damping-like iSOT versus Pt thickness—where\nboth sets of data are described by the same spin diffusion length λs—makes it possible to\nseparate these sources of damping. Furthermore, this data ana lysis methodology allows for\nunambiguous determination of the spin-mixing conductance G↑↓at the FM/NM interface.\nWe therefore can ascertain the spin Hall conductivity (or spin Hall a ngle) without having\nto refer to spin transport parameters G↑↓andλsdetermined from measurements performed\n3on dissimilar samples or theoretical idealized values. For our samples o f Pt deposited on\nNi80Fe20(or Permalloy, Py), only 37 ±6% of the total damping enhancement from the Pt\nfilm is attributable to spin pumping into the Pt layer when dPt≫λs.\nII. EXPERIMENTAL TECHNIQUE\nThe data presented in this work are based on the spectroscopic an d complex amplitude\ninformation encoded in VNA-FMR spectra, which yield a measure of th e damping and SOT,\nrespectively. FMR damping extracted from a spectral linewidth ana lysis20has been used\nextensively to study the damping enhancement due to the spin pump ing effect into an NM\nadjacent to the FM layer21–24. If such spectra are measured inductively with phase-sensitive\nVNA-FMR, it is also possible to analyze the phase and amplitude informa tion of those\nspectra to quantitatively extract the field-like (FL) and damping-lik e (DL) SOT conductiv-\nities, as we have previously described25. These conductivities, σSOT\nFLandσSOT\nDL, relate the\nAC charge currents produced in the NM layer via iSHE or inverse Rash ba-Edelstein effect\n(iREE) in response to driven magnetization dynamics in the FM layer. D irect coupling to\nthe magnetization dynamics via Faraday’s law also drives AC charge cu rrents in the NM\nlayer, quantified by σF\nFL. The superposition of these charge currents presents a complex\ninductive load to the microwave coplanar waveguide (CPW) used in VNA -FMR measure-\nments, altering the amplitude and phase of the transmitted microwa ve signal. By Onsager\nreciprocity, σSOT\nFLandσSOT\nDLmeasured inductively viainverse spin-chargeconversion processe s\nare equivalent to the spin torque efficiency per unit applied electric fie ld used by Nguyen et\nal. in Ref. 17 to describe the forward SOT process25.\nA. Samples\nTo study the Pt-thickness dependence of damping and damping-like iSOT, we prepared\ntwo sample sets, with sputter-deposited metal multilayers consist ing of substrate/Ta(1.5)/\nPy(dPy)/Pt(dPt)/Ta(3), where thicknesses are indicated in nanometers and are c alibrated\nwith X-ray reflectivity measurements. In the first sample set, the thickness dPywas varied\nfrom 1.5nm to 10nm while dPt= 6nm was fixed. In the second set, the thickness dPtwas\nvaried from 2nm to 20nm with fixed dPy= 3.5nm. For each sample, an identical control\n4sample was prepared, where Pt is substituted with Cu. The Cu thickn esses were chosen to\nmatch the sheet resistance of the corresponding Pt layer, so as t o control for Faraday effect\ninduced currents in the NM layer.\nIII. RESULTS AND DISCUSSION\nA. Py thickness series\nFrom the Py thickness series we focus on three quantities: (1) the FM contribution to\nthe sample inductance ( LFM, as in Ref. 25), (2) the effective magnetization Meff, and (3) the\nGilbert damping parameter α. FromLFMas a function of Py thickness (Fig. 1), we are able\nto extract the dead layer thickness, and therefore determine th e effective magnetic thickness\nof the FM layer. From Meff, we are able to determine the saturation magnetization Ms(Fig.\n2). Lastly, from the Gilbert damping as a function of Py thickness, w e can separate the\nintrinsic and interfacial contributions to α(Fig. 4). This is a critical first step to determine\nthe spin pumping and SML contributions to the total damping.\n1. Ferromagnetic dead layer measurement\nIninductiveVNA-FMRmeasurements, theFMlayercontributesafr equency-independent\ninductance to the S21measurement according to25,26:\nLFM=µ0ldFM\n4Wwgη2(z,Wwg) (1)\nwhereµ0is the permeability of free space, lis the sample length along the CPW signal\npropagation direction, dFMis the deposited FM thickness, Wwgis the CPW signal line\nwidth, and η(z,Wwg)≡(2/π)arctan(Wwg/2z) is the spacing loss, ranging from 0 to 1, due\nto a finite distance zbetween sample and CPW. When plotted vs. dFM, theLFM= 0\nintercept indicates the magnetic dead layer thickness. From the da ta in Fig. 1, we find\nddead= (0.5±0.1)nm for Py/Pt samples. Also shown are the data for Py/Cu contro l\nsamples, which exhibit a similar dead layer thickness of (0 .41±0.04)nm, suggesting that\nthe Py dead layer exists primarily at the Ta/Py interface.\n52. Determination of Ms\nTheeffective magnetization Meffasafunctionofappliedmicrowave frequency isextracted\nfrom the FMR spectral fits and the Kittel FMR condition for magnet ization oriented out of\nthe film plane27:\nHres=ω\nγµ0+Meff (2)\nwhereHresis the center field of the resonant absorption line, ωis the applied microwave fre-\nquency, and γ=gµB//planckover2pi1is the gyromagnetic ratio. Assuming the Py has no bulk anisotropy,\nMeffis determined by the saturation magnetization Msof the material, and the interfacial\nanisotropy energy Kintaccording to Ref. 28:\nµ0Meff=µ0Ms−2Kint\nMs/parenleftbigg1\ndFM−ddead/parenrightbigg\n(3)\nTherefore, a linear fit of Meffvs. inverse effective FM thickness (Fig. 2) provides a measure-\nment of the saturation magnetization Ms. We find µ0Ms= (1.0671±0.0001)T, comparable\nto previous findings28. Similarly, for Py/Cu we find µ0Ms= (1.0453±0.0004)T.\n/s32/s33\n/s34/s33\n/s35/s33\n/s36/s33\n/s37/s33\n/s38/s33\n/s33/s39/s40/s41/s42/s43/s44/s45\n/s38/s33/s46/s32/s35/s37/s33\n/s47/s48/s49/s42/s43/s50/s51/s45/s42/s52/s53/s43/s38/s54/s34/s45/s55/s48/s49/s43/s47/s48/s49/s45/s55/s48/s56/s43/s32/s45/s55/s52/s53/s43/s36/s45\n/s42/s52/s53/s43/s38/s54/s34/s45/s55/s48/s49/s43/s47/s48/s49/s45/s55/s57/s58/s43/s36/s54/s36/s45/s55/s52/s53/s43/s36/s45\nFigure 1. Py-thickness dependent zero-frequency inductan ce for both Py/Pt and Py/Cu control\nsamples.\n6/s32/s33/s34\n/s34/s33/s35\n/s34/s33/s36\n/s34/s33/s37\n/s34/s33/s38\n/s34/s33/s39/s40/s34/s41/s42/s43/s44/s44/s41/s45/s46/s47\n/s34/s33/s36/s34/s33/s38/s34/s33/s48/s34/s33/s49/s34/s33/s34\n/s32/s50/s45/s51/s52/s53/s41/s54/s41/s51/s51/s43/s55/s51/s47/s41/s45/s56/s57/s47/s54/s32/s41/s46/s55/s45/s32/s33/s39/s47/s50/s52/s53/s45/s51/s52/s53/s47/s50/s52/s58/s45/s38/s47/s50/s46/s55/s45/s59/s47\n/s41/s46/s55/s45/s32/s33/s39/s47/s50/s52/s53/s45/s51/s52/s53/s47/s50/s60/s61/s45/s59/s33/s59/s47/s50/s46/s55/s45/s59/s47\n/s41\nFigure 2. Meffvs. inverse effective FM thickness ( dPy−ddead) for Py( dPy)/Pt(6) and\nPy(dPy)/Cu(3.3). Dead layer thickness is determined from Fig. 1.\n3. Determination of intrinsic Gilbert damping constant\nThe total Gilbert damping due to intrinsic and interfacial contributio ns can be described\nby:\nα=αint+G↑↓\neff/parenleftbiggγ/planckover2pi12\n2MsdFMe2/parenrightbigg\n(4)\nwhereγ=gµB//planckover2pi1is the gyromagnetic ratio, gis the spectroscopic g factor, µBis the Bohr\nmagneton, /planckover2pi1is Planck’s constant divided by 2 π, andeis the electron charge. MsanddFM\nfor the Py layer are determined as described above. For the thin FM layers studied here, we\ncan ignore the contribution from radiative damping29. When plotted vs. 1 /(dPy−ddead), we\ncan extract αintas the infinite-thickness limit of the measured damping. We calculate t he\nintercept of the data in Fig. 4 using linear regression in order to fix αint= 0.0041±0.0001,\nin good agreement with a previous systematic study of damping in mag netic alloys28.\nFortheinterfacialcontributiontothedamping(secondterminEq. (4)), thefullmodelwe\nuse for the effective spin-mixing conductance G↑↓\neffincludes contributions from spin pumping\ninto Pt via the spin-mixing conductance G↑↓\nPy/Pt, spin pumping into the Ta seed layer via\nthe spin-mixing conductance G↑↓\nPy/Ta, and spin memory loss (SML). (In all instances where\nwe invoke the spin-mixing conductance, it is to be understood that w e are only considering\n7Figure 3. Circuit model for angular momentum flow sourced by F MR excitation in Ta/Py/Pt\ntrilayer. Spin current is drawn into parallel resistance ch annels provided by spin pumping into the\nTa seed and Pt spin sink layers, as well as spin memory loss.\nthe real part of said quantity).\nG↑↓\neff=G↑↓\nPy/Pt\n1+2λs,PtG↑↓\nPy/Pt\nσPt(dPt)tanh/parenleftbiggdPt\nλs,Pt/parenrightbigg+G↑↓\nPy/Ta\n1+2λs,TaG↑↓\nPy/Ta\nσTa(dTa)tanh/parenleftbiggdTa\nλs,Ta/parenrightbigg+G↑↓\nPy/Pt∆SML(5)\nThis model is depicted as a network of series and parallel conductan ce channels for the\nflow of angular momentum, treating FMR as an angular momentum pot ential source, as\ndepicted in Fig. 3 (also see Ref. 30). The first two terms of Eq. (5) r epresent spin pumping\ninto the Pt and Ta layers, respectively. Within those layers, spin is pu mped through series\nresistances set by the interfacial spin-mixing conductance, and t hickness-dependent spin\nresistance (which accounts for the exponential spin accumulation profile in the NM layer,\nas a solution to the spin diffusion equation, subject to the boundary condition that no spin\ncurrent can flow through the distant interface). The final term r epresents a spin memory\nloss channel, where the phenomenological parameter ∆ SMLcan be arbitrarily large. By\nmultiplying Eq. (5) by the bracketed term in Eq. (4), conductances are converted to the\nunitless damping parameters αsp,Pt(Ta)(due to spin pumping into Pt (or Ta)) and αSML(due\nto spin memory loss).\nTaken together, Eqs. (4) and (5) describe both the NM and FM thic kness dependencies\n8/s32/s33/s32/s34\n/s32/s33/s32/s35\n/s32/s33/s32/s36\n/s32/s33/s32/s37\n/s32/s33/s32/s38\n/s32/s33/s32/s32/s39\n/s32/s33/s40 /s32/s33/s35 /s32/s33/s37 /s32/s33/s32\n/s38/s41/s42/s43/s44/s45/s46/s47/s46/s43/s43/s48/s49/s43/s50/s46/s42/s51/s52/s50/s47/s38/s46/s53/s49/s42/s38/s33/s34/s50/s41/s44/s45/s42/s43/s44/s45/s50/s41/s44/s54/s42/s40/s50/s41/s53/s49/s42/s36/s50\n/s46/s53/s49/s42/s38/s33/s34/s50/s41/s44/s45/s42/s43/s44/s45/s50/s41/s55/s56/s42/s36/s33/s36/s50/s41/s53/s49/s42/s36/s50\n/s46\nFigure 4. Total Gilbert damping vs. inverse effective FM thick ness (dPy−ddead) for Py(dPy)/Pt(6)\n(circles) and Py( dPy)/Cu(3.3) (squares). Dead layer thickness is determined fr om Fig. 1.\nof the damping. As a part of our self-consistent fitting routine (de scribed in Section IIIC,\nand using the previously determined value for αint, we fit the Py thickness dependence of α\nsimultaneously with the Pt thickness dependence (Fig. 7(b)), with G↑↓and ∆ SMLas fit pa-\nrameters. The result of that simultaneous fit is shown in Fig. 4. Also s hown for comparison\nare damping data for the Py/Cu controls, which exhibit a drastically r educed spin pumping\ncontribution (slope), and slightly increased intrinsic contribution ( αint= 0.0054±0.0001).\nB. Pt thickness series\nFor samples where the Pt thickness is varied, the measured values f orσSOT\nFLandσSOT\nDL—\nextracted from our quantitative VNA-FMR complex amplitude analys is25—are shown as a\nfunction of NM thickness in Fig. 5. Two corrections must be made to t hese values in order\nto extract the iSOT due to Pt. First, we subtract the values for σFLandσDLobtained from\nthe Cu control samples (blue squares) from those of the Pt sample s (red circles). Since\nwe used Cu thicknesses to match the sheet resistance of the Pt sa mples, this removes the\nFaraday contribution. This subtraction also removes any FL or DL iS OT due to the Ta seed\nand capping layers. While we do not completely understand the Cu thic kness dependence\nofσDL, the DL signal is essentially eliminated for Py in isolation, without seed o r capping\n9(a) (b)\nFigure 5. Measured quantities for FL and DL conductivities, for both Pt and Cu control samples,\nextracted fromcomplex inductanceanalysis of VNA-FMR data25. (a)σFLas afunctionof either Pt\n(top axis) or Cu thickness (bottom axis). Linear dependence on NM thickness at large thicknesses\nindicates dominance of σF\nFLterm. (b) Same as (a), but for σDL.\nlayers, which suggests details of the iSOT from cap and/or seed laye rs are responsible for\nthe peculiar behavior (see discussion and measurements in Section I IID). In Fig. 6(a), σFL\nandσDLafter Cu reference subtraction are plotted.\nSecond, we correct for shunting effects of the iSOT currents. Th e data of Fig. 6(a) atten-\nuate as the Pt thickness is increased. This is attributed to the decr easing sheet resistance\nof the metallic stack, which effectively shunts the AC iSOT currents, therefore producing a\nweaker inductive response. This is functionally similar to the current divider effect observed\nin DC voltage iSHE spin pumping experiments8,9,31,32. However, in our AC iSOT experi-\nments with the sample placed on a CPW with characteristic impedance o f 50Ω, the sample\nsheet resistance acts as a shunt path in parallel with the CPW chara cteristic impedance\n(inset of Fig. 6(b)). We therefore multiply the σFLandσDLresults of Fig. 6(a) by the shunt\nfactor (1+ Z0/R/square), where R/squareis the measured sheet resistance of the multilayer stack (Fig.\n6(b)).\nAfter application of the shunting correction, the final results for σSOT\nFLandσSOT\nDLare\n10Z0\nISOT(a) (b)\nFigure 6. (a) FL and DL iSOT conductivities, after subtracti on of Cu control samples. (b)\nMeasured sheet resistance of metallic layers, as a function of Pt thickness. Inset: the sample sheet\nresistance acts as a parallel shunting path to the signal gen erating component of ISOT, which flows\nthrough the characteristic impedance Z0andR/square.\npresented in Fig. 7(a). These results are shown adjacent to the d ependence of the measured\nGilbert damping parameter αon Pt thickness in Fig. 7(b) to compare their evolution with\ndPt. The DL conductivity increases monotonically with Pt thickness. Mea nwhile, the FL\nconductivity remains more or less constant, consistent with the pr esumption of an interfacial\nsource of spin-charge conversion such as iREE, where additional P t beyond 2nm does not\nincrease the charge signal further. From Fig. 7(b), it is clear that if the enhanced damping\n(second term in Eq. (4)) were ascribed entirely to spin pumping into t he Pt, the length scale\nnecessary to capture the rapid increase in αabove the intrinsic value must be much shorter\nthan the length scale over which σDLis seen to increase in 7(a). In other words, using only\nthe data for Pt-thickness dependence of damping (Fig. 7(b)), it is impossible to separate the\ndifferent contributions to G↑↓\neff. Several other groups have observed this apparent discrepanc y\nwhen comparing damping with DC voltages measured by iSHE8–10. In this work, we are\nable to resolve the discrepancy through a self-consistent fit of bo th the damping data and\nσSOT\nDLversus Pt thickness.\nAlthough we measure only a 3% enhancement of damping as dPtincreases from 2nm to\n20nm, given the high signal-to-noise ratio of the damping data, it can be fit with Eq. (4)\n11(a) (b)\nFigure 7. (a) Final values for σSOT\nDLandσSOT\nFLfor Py(3.5)/Pt( dPt). TheFL torque remains constant\nover the range of studied thicknesses, whereas the DL torque increases with a characteristic length\nscale. (b) Gilbert damping for the same sample series (error bars are smaller than symbols). Color\ncoding indicates different contributions to the Gilbert damp ing. Both the SOT conductivity and\ndamping are fit to four different models, where spin relaxation is either EY or DP, and the spin\nHall effect arises from intrinsic (int) or extrinsic (ext) pro cesses. In both cases EY + ext (black\nsolid line) provides the best fit, as determined by a χ2analysis.\nand (5) by use of the same spin diffusion length that describes the be havior of σSOT\nDL, as\ndiscussed in detail later. Because of the better dynamic range of t heσSOT\nDLdata, we use it\nas the basis for establishing λsby fitting with a model provided by Haney et al.33:\nσSOT\nDL=σSH\n\n(1−e−dNM/λs)2\n(1+e−2dNM/λs)|˜G↑↓|2+Re(˜G↑↓)tanh2/parenleftbiggdNM\nλs/parenrightbigg\n|˜G↑↓|2+2Re(˜G↑↓)tanh2/parenleftbiggdNM\nλs/parenrightbigg\n+tanh4/parenleftbiggdNM\nλs/parenrightbigg\n\nǫ(6)\nwhere˜G↑↓=G↑↓2λstanh(dNM/λs)/σ,σrepresents the NM charge conductivity, and ǫ≡\nαsp,Pt/(αsp,Pt+αSML) represents the fraction of spin current pumped out of the FM th at is\navailable for spin-charge conversion in the Pt layer, as determined b y the the spin current\ndivider model applied to the first and last terms of Eq. (5).\n12C. Self-consistent fit routine of damping and SOT\nTo perform the self-consistent fits of σSOT\nDLandα, an initial fit of σSOT\nDLis performed to\nextract the Pt spin diffusion length λs,Pt. This is then used as a fixed parameter in Eq. (4)\nand (5) when fitting α. With this constraint on λs,Pt, the Pt and Py thickness series (Figs.\n7(b) and 4, respectively) are fitted simultaneously with Eq. (4) and (5) to determine G↑↓\nPy/Pt\nand ∆ SML. These are then put back into Eq. (6) to re-fit σSOT\nDLand extract refined values for\nσSHandλs,Pt. This process is iterated until the change in fit parameters is less th an 0.01%.\nOur self-consistent data analysis is tantamount to enforcing Onsa ger reciprocity on the\nspin-to-charge interconversion processes of spin pumping and sp in torque34. If the enhanced\ndamping of Fig. 7(b) were ascribed purely to spin pumping, it would imply that the Pt\nalready draws a maximum amount of spin current from the precessin g FM at thicknesses of\nonly≈2nm. By contrast, a damping-like torque conductivity that continu es to increase for\nthicknesses up to 10nm (Fig. 7(a)) suggests that the Pt layer can continue to generate (or\ndraw) increasingly larger spin current for thicknesses well beyond 2nm. The use of unequal\nlength scales to describe diffusive spin current flow due to spin pumpin g and spin-orbit\ntorque generation would violate the reciprocity of spin-to-charge interconversion.\nEquations (5) and (6) can be used either with an Elliott-Yafet (EY)35,36or D’yakonov-\nPerel’ (DP)37spin relaxation model. In the EY case, the spin diffusion length is a func tion\nof the charge conductivity: λs(σ(dNM)) = (σ(dNM)/σbulk)λmax\ns. The thickness-dependent\nconductivity and bulk conductivity σbulkare both determined by four-probe resistance mea-\nsurements (see Section VA). By contrast, for DPspin relaxation, λsisindependent of charge\nconductivity13. Additionally, the spin Hall conductivity in Eq. (6) can be attributed t o in-\ntrinsic or extrinsic SOC. For intrinsic spin Hall, σSHis independent of charge conductivity,\nwhile for extrinsic SHE, σSH(dNM) =θSHσ(dNM), where θSHis fixed. Fits using the four\ncombinations of these models are shown in Fig. 7, with results collecte d in Table II. To\ndistinguish between the quality of fit for the different models, we utiliz e aχ2test.\nTheχ2values for each fit of the SOT and damping data is calculated as χ2≡/summationtextn\ni(yi−\nfi)2/σ2\ni, whereyiis the measured value, fiis the calculated value based on the fit model,\nandσ2\niis the measured variance, for each of nmeasurements. Results are shown in Table\nI. Using the cumulative distribution function (CDF) of a χ2distribution for each fit, with\nν=n−pdegrees of freedom, and pfit parameters, we also calculate the joint probability\n13Fit model χ2(SOT)χ2(damping) Joint Probability\nEY + ext 0.668 3.696 0.89\nEY + int 3.123 12.032 0.11\nDP + ext 8.149 5.715 0.07\nDP + int 8.819 6.101 0.05\nTable I. χ2values for SOT fit (Fig. 7(a)) and simultaneous damping fit (Fi gs. 7(b) and 4). The\njoint probability represents the confidence with which we ca n reject the null hypothesis.\nwith which we can reject the null hypothesis in which there is no relatio nship between our\nmeasurements and the given model. The CDF is determined by\nCDF(χ2) =χ2/integraldisplay\n0tν/2−1e−t/2\nΓ(ν/2)2ν/2dt (7)\nwhere Γ( x) = (x−1)!. The joint probability is calculated as the product of (1 −CDF(χ2))\nfor the two fits. The EY/extrinsic model provides the highest confi dence that we can reject\nthe null hypothesis. Because this analysis reveals EY spin relaxation with extrinsic SHE as\nthe best fit to our data, we focus on the fitted parameters from t hat model combination in\nthe discussion below.\nBy choosing to enforce reciprocity, we find that the fraction of sp in current absorbed by\nthe Pt layer (which produces the damping-like AC charge currents) reaches a maximum of\n(37±6)% for the thickest Pt layers. This is comparable to previous finding s of large SML\nat Co/Pt interfaces38and Pt/Cu interfaces19. The different contributions to the total mea-\nsured damping are represented as shaded areas in Fig. 7(b), with a color code to match Fig.\n3. Note that only the contribution from spin pumping into Pt is Pt-thic kness dependent.\nThe self-consistent fit also results in a spin diffusion of length of λmax\ns,Pt= (4.2±0.1)nm,\nG↑↓=(1.3±0.2)×1015Ω−1m−2, which is in good agreement with the maximum theoret-\nical value for Pt of G↑↓= 1.07×1015Ω−1m−239, given the estimated error, and σbulk\nSH=\n(2.36±0.04)×106Ω−1m−1. This corresponds to a spin Hall angle of 0 .387±0.008. While\nthisθSHisamongthelargestreportedforPt40,41, itisanecessary logicalconclusion thatwith\nless spin current driven into the NM (on account of SML), a larger sp in-to-charge conversion\nefficiency is required to fit the data than would be otherwise obtained if the SML were negli-\n14gible. Wefurthermorestress thatthephenomenological valuefor σSOT\nDL(theasymptotic value\nin Fig. 7(a)) is comparable to that measured with other techniques ( 5.8×105Ω−1m−1for\nAlOx(2)/Co(0.6)/Pt(3)7,4.8×105Ω−1m−1forTa(2)/Pt(4)/Co 50Fe50(0.5)/MgO(2)/Ta(1)41,\nand≈2.5×105Ω−1m−1for Ta(1)/Pt( dPt)/Co(1)/MgO(2)/Ta(1)17). This indicates consis-\ntency of the SOC strength of the Pt layers in each of these experim ents, and stresses the\nimportance of characterizing spin loss mechanisms to optimize SOT fo r magnetic switching.\nOurfinding thatthedataarebestfitwithanextrinsicSHEmodelisso mewhat surprising,\ngiventhatitconflictswithsomepreviousexperimentalwork17andtheoreticalexpectations42.\nQualitatively, both intrinsic and extrinsic SHE models are seen to desc ribe the data quite\nwell, given that the fit parameters can adjust to compensate for d ifferences in the models,\nas is seen by the various fits in Fig. 7(a). Nevertheless, the χ2analysis makes a clear\ndistinction. Finally, the value for σSHdetermined here is more than 5 times larger than\nthe 0K prediction by Guo et al. (using their result of σxy= 2.2×105(/planckover2pi1/e) Ω−1m−1, and\nsettingσSH= 2σxyto account for the total spin current due to both up and down spin s42).\nThis implies that the extrinsic effect dominates in our sputtered thin fi lm systems where\ninterfaces and crystal defects likely play a major role in determining the spin-orbit physics43.\nIt is possible that some amount of intrinsic SHE is present in addition to the extrinsic\neffect, asdiscussed by Sagasta, et al.3. In that work, theauthors show that the totaleffective\nspin Hall conductivity σeff\nSHcan be described by:\nσeff\nSH=σint\nSH+θSHσPt (8)\nwhereσint\nSHis the intrinsic spin Hall conductivity, and the second term describes the extrinsic\neffectaswehavemodeledithere. ThePtconductivitiesstudiedhere (from≈3×106Ω−1m−1\nto6×106Ω−1m−1)fallwithinthetransitionfromintrinsic-toextrinsic-regimes, asde scribed\nin Ref. 3. Therefore, depending on the details of the spin and momen tum scattering that\ngovernθSH, the extrinsic term in Eq. (8) can easily be the dominant effect. Furt hermore,\nwe see no evidence of a large interfacial source of spin Hall conduct ivity, as in Ref. 44, which\nwould manifest as a non-zero intercept of σSOT\nDLin the limit of dPt→0.\n15Fit model G↑↓/parenleftbig\n×1014Ω−1m−2)ǫ=αsp,Pt\n(αsp,Pt+αSML)λs(nm)σSH/parenleftbig\n×106Ω−1m−1)θSH=σSH\nσPt\nEY + ext 13±2 (0.37±0.06)4.2±0.12.36±0.040.387±0.008\nEY + int 5.6±0.1 (0.20±0.04)6.7±0.35.3±0.3 0.86±0.05\nDP + ext 3.0±0.3 (0.19±0.02)2.5±0.211.6±0.41.91±0.06\nDP + int 2.2±0.3 (0.13±0.02)3.7±0.313.5±0.52.22±0.08\nTable II. Comparison of fitted values for G↑↓,ǫ,λs,σSH, andθSHusing different models for the\nsource of spin relaxation (EY or DP) and SHE (intrinsic or ext rinsic). For EY models, the spin\ndiffusion length is reported as λmax\ns.\nD. Isolating the normal-metal layer contribution to sample inductance\nTo better understand the influence of the normal metal layers (T a seed, Pt or Cu spin\nsink, and Ta cap) on the perturbative inductance—and hence, the extracted FL and DL\nconductivities—that the sample contributes to a VNA-FMR measure ment, we measured\nseveral control samples. First, we inserted an AlO xlayer between the Py and the Pt in\norder to block spin pumping into the Pt45. To do so, 1nm of Al was sputter deposited\nonto the Py and subsequently oxidized for 10 minutes under 5Torr o f O2. The AlO xlayer\ndeposition and oxidation steps were repeated 1, 2, or 3 times, to en sure complete blocking of\nspin pumping. As can be seen in Fig. 8(b), the AlO xlayers effectively reduce the damping\nby blocking spin pumping. This reduction correlates strongly with a re duction in σDL,\nconfirming its signature as the damping-like conductivity.\nBy contrast, σFLactually changes sign with the introduction of the AlO xlayers (Fig.\n8(a)). The contribution to σFLby Faraday-type pickup in the Pt cannot be eliminated by\nthe AlO xbarrier, since the Pt can still inductively couple to the precessing ma gnetization\nin the Py. The Faraday contribution clearly adds a negative contribu tion toσFL, asσFL\nbecomes increasingly negative with thicker Pt and Cu layers, as in Fig. 5(a). Therefore, the\nAlOxbarrier might eliminate the σSOT\nFLcontribution at the top Py interface. Nevertheless,\neven for Py deposited directly on SiO 2(open square symbol), there remains a negative total\nσFL, perhaps due to the interface asymmetry that remains between t he top and bottom Py\ninterfaces.\nThe control samples also elucidate the impact of the Ta layers on our measurements. We\n16note that Eq. (5) does not explicitly include SML at the Ta interface. Using the data from\nFig. 8(b), we find that this simplification is justified. For these sample s, we measured a\ntotal damping of αtot= 0.0104±0.0002. If we set G↑↓\nPy/Ta= 7.4×1014Ω−1m−2(the Sharvin\nvalue for Ta39), and use our measured conductivity of σTa= (8.91±0.02)×105Ω−1m−1, we\nobtainαsp,Ta= 0.004 (the amount depicted in Fig. 3). Therefore, when damping path ways\ninto the Pt are blocked, the intrinsic damping plus spin pumping into the Ta accounts for\nall but 0.0023 of the total damping. Assigning this small amount of ex cess damping to SML\nat the Ta interface would reduce the contribution of SML at the Pt in terface by less than\n20% and the values for spin Hall conductivity and spin Hall angle in Pt by only 10%.\nFinally, we fabricated samples without any seed or capping layers. Fo r Py(5) deposited\ndirectly onto SiO 2,σDLisonly 5%of itsvalueforPy(3.5)/Pt(6) (see opencircle datapoint in\nFig. 8(b)). Theresidualdamping (beyondtheintrinsicvalue)andda mping-likeconductivity\nfor this sample could stem from the oxidized top surface or interfac ial asymmetries, as well\nas less-than-optimal Py crystal structure, since no Ta seed laye r was used.\nIn the cases of both σFLandσDL, some residual signal remains even when spin pumping\ninto the Pt is effectively blocked, or the seed and capping layers are e liminated entirely.\nTherefore, it is not surprising that even for our control samples in which Pt is replaced with\nCu (with its weak spin-orbit interaction), some weak sources of spin -to-charge conversion\n(interfacial or otherwise) persist.\nIV. CONCLUSION\nTo summarize, by use of simultaneously acquired damping and iSOT dat a, we are able\nto properly assign the portions of damping enhancement incurred b y a FM/NM bilayer\ndue to the parallel channels of SML and spin pumping into the NM. Thes e results suggest\nthat Pt is indeed a promising material for spintronic applications. Our data also validate\nprevious suggestions that interface engineering will be crucial for the optimization of SOT\nin multilayer systems10,38,40,41.\n17(a) (b)\nFigure 8. (a) FL and (b) DL conductivities for samples with Al Ox[×n] (where n= 1, 2, or 3)\nblocking layers inserted between Py(3.5) and Pt(6). Also sh own is a sample in which Py(5) is\ndeposited directly onto SiO 2(open symbols). Note that Py(3.5)/Py(6) (direct contact) w as re-\ngrown and re-measured as a part of the AlO xseries (duplicate data points for zero AlO xrepeats).\nThe lower data point for both σFLandσDLat zero AlO xlayers is that from the main text.\nV. APPENDIX\nA. Pt thickness-dependent resistivity\nTo extract the Pt contribution to the total measured stack resis tance, we have developed\na model for the metallic multilayer stack to account for different con ductivities in the bulk\nand at the metal interfaces. In this model, the interfacial conduc tivityσintat the Py/Pt\ninterfaces decays exponentially to the Pt bulk value, 1 /ρ0, with increasing distance from\nthe interface. Position-dependent conductivity through the Pt t hickness can therefore be\napproximated as the sum of bulk and interfacial contributions:\nσ(z) =1\nρ0/bracketleftbigg\n1−exp/parenleftbigg−z\nσintρ0λ/parenrightbigg/bracketrightbigg\n+σintexp/parenleftbigg−z\nσintρ0λ/parenrightbigg\n(9)\nwhereρ0isthebulk resistivity, σintistheinterfacial conductivity, and λis thebulk meanfree\npath. The length scale σintρ0λdescribes the effective thickness over which the conductivity\nis determined by σint. Whenσ(z) is integrated over the Pt thickness from z= 0 toz=dPt,\nwe obtain a final result for thickness-dependent resistivity:\n18/s32/s33/s34/s35\n/s32/s33/s34/s36\n/s32/s37/s34/s35\n/s32/s37/s34/s36\n/s32/s38/s34/s35\n/s32/s38/s34/s36/s39/s40/s41/s42/s43/s40/s44/s45/s46\n/s47/s36/s32/s35/s32/s36/s35/s36\n/s48/s49/s50/s40/s41/s51/s45/s46\nFigure 9. Thickness-dependentresistivity, measuredfor s ubtstrate/Ta(1.5)/Py(3.5)/Pt( dPt)/Ta(3)\nas a function of Pt thickness.\nρ(dPt) =ρ0/bracketleftbigg\n1+/parenleftbiggσintρ0λ\ndPt/parenrightbigg\n(ρ0σint−1)/bracketleftbigg\n1−exp/parenleftbigg−dPt\nσintρ0λ/parenrightbigg/bracketrightbigg\n+/parenleftbigg1\nRother/parenrightbigg/parenleftbiggρ0\ndPt/parenrightbigg/bracketrightbigg(10)\nwhereRotherrepresents the sheet resistances of any fixed-thickness metallic layers (here,\nPy and Ta). 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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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Phys. 17,\n113058 (2015)."    },    {        "title": "1711.00406v1.Tunable_magnetization_relaxation_of_Fe__2_Cr__1_x_Co__x_Si_half_metallic_Heusler_alloys_by_band_structure_engineering.pdf",        "content": "arXiv:1711.00406v1  [cond-mat.mtrl-sci]  1 Nov 2017Tunable magnetization relaxation of Fe 2Cr1−xCoxSi half-metallic Heusler alloys\nby band structure engineering\nShikun He,1, 2,∗Yifan Liu,3,∗Yuhong Zheng,3Qing Qin,4Zhenchao Wen,5Qingyun Wu,6\nYi Yang,2Yupu Wang,3YuanPing Feng,6Kie Leong Teo,3and Christos Panagopoulos1\n1Division of Physics and Applied Physics,\nSchool of Physical and Mathematical Sciences,\nNanyang Technological University, Singapore 637371\n2Data Storage Institute, Agency for Science, Technology and Research (A*STAR),\n2 Fusionopolis Way 08-01 Innovis, Singapore 138634\n3Department of Electrical and Computer Engineering,\nNational University of Singapore, Singapore 117582\n4Department of Materials Science and Engineering,\nNational University of Singapore, Singapore 117574\n5Center for Spintronics Research Network (CSRN),\nTohoku University, Sendai 980-8577, Japan.\n6Department of Physics, National University of Singapore, S ingapore 117542\n1Abstract\nWe report a systematic investigation on the magnetization r elaxation properties of iron-based half-metallic\nHeusler alloy Fe 2Cr1-xCoxSi (FCCS) thin films using broadband angular-resolved ferro magnetic resonance.\nBand structure engineering through Co doping ( x) demonstrated by first principles calculations is shown\nto tune the intrinsic magnetic damping over an order of magni tude namely, 1×10−2-8×10−4. Notably,\nthe intrinsic damping constants for samples with high Co con centration are amongst the lowest reported for\nHeusler alloys and even comparable to magnetic insulator yt trium iron garnet. Furthermore, a significant\nreduction of both isotropic and anisotropic contributions of extrinsic damping of the FCCS alloys was found\nin the FCCS films with x = 0.5 ~0.75, which is of particular impo rtance for applications. These results\ndemonstrate a practical recipe to tailor functional magnet ization for Heusler alloy-based spintronics at room\ntemperature.\nA. INTRODUCTION.\nControl of magnetization and its dynamics in nanomagnets by injecting an electrical current is\nincreasingly important [ 1,2]. Operations based on spin transfer torque (STT) are energy efficient\nwith superior scalability for high density device applicat ions compared to conventional methods\nemploying magnetic fields [ 3–6]. A promising application of STT is non-volatile magnetic r andom\naccess memory (MRAM) [ 7,8]. Key parameter here is magnetic damping; it determines the critical\nswitching current and speed of magnetization reversal [ 2,9]. Hence, a magnetic material with\ntunable low damping is needed. Magnetic relaxation is prima rily due to spin-orbit coupling (SOC)\nhowever, materials and device guidance from theoretical ca lculations is hindered by the complicity\nof inter- and intraband scattering, defects, disorder, int erface effects, and geometrical confinement\n[10,11]. A practical criterion for engineering magnetic damping i sα∝ξ2D(EF)[12,13], whereξis\nthe SOC parameter and D(EF)the density of states (DOS) at the Fermi level. Although Fe, C o and\nNi render the window for tunable SOC in 3d-based materials fo rmidably narrow, there have been\nencouraging developments through the correlation between damping and electronic structure, as\ndemonstrated in FeV and CoFe binary alloys [ 14,15]. Still, reducing and tuning magnetic damping\nin metallic materials can be difficult due to magnon scatterin g by high density conduction electrons\n[10]. Notably, the lowest reported damping for CoFeB, the materi al employed by industry for\nMRAM development, is approximately 0.004 [ 16,17].\nHalf-metallic Heusler alloy is a semiconductor for one spin projection and a metal for the other\n[18] hence, the band structure is expected to result in high spin polarization and low damping\n[19–23]. Both properties are favorable for STT [ 9]. However, except for a few Co-based Heusler\nalloys, the damping constant is larger than for Fe films (0.00 2), probably due to band broadening\nby structural defects, chemical disorder and magnon excita tion [13,24–28]. Furthermore, Heusler\nalloys usually exhibit large anisotropy in the ferromagnet ic resonance (FMR) linewidth [ 26,29],\nwhich may lead to large variation in performance among devic es. Therefore, tuning and reducing\nintrinsic damping, and a minimum extrinsic magnetic relaxa tion are required for reliable opera-\ntions. Fe-based Heusler alloys are expected to possess half -metallic band structure [ 30]. Also, these\nmaterials hold great promise for high interfacial perpendi cular anisotropy [ 31], an indispensable\n∗These authors contributed equally\n2property for next generation STT-MRAM, at a Fe-MgO interfac e [32,33]. However, a comprehen-\nsive magnetization relaxation study is still lacking for Fe -based Heusler alloys.\nHere, we tune the band structure of Fe 2Cr1−xCoxSi (FCCS) by Co doping ( x). Dynamic and\nstatic properties were investigated using angle-resolved broadband FMR. We show ordered states\n(B2andL21) exhibit considerably lower damping compared to disordere d states ( A2). Crucially,\nCo doping allows control of the intrinsic damping constant b y an order of magnitude ( 1×10−2−\n8×10−4). Furthermore, both isotropic and anisotropic extrinsic c ontributions to the in-plane\nmagnetization relaxation of quaternary FCCS can be signific antly reduced for xbetween 0.5 and\n0.75.\nB. EXPERIMENT\nThe samples were prepared in an ultra-high vacuum magnetron sputtering system with base\npressure 10−7Pa. Prior to deposition, the MgO (100) substrate was heated a t 600◦Cfor one hour.\nSubsequently, a 40nm Cr buffer layer was deposited followed b y one hour in-situ annealing at 700◦C\n. The quaternary Heusler alloy FCCS (30nm) layer with variab le Co doping, was deposited by co-\nsputtering Fe 2CrSi and Fe 2CoSi. The deposition speed of individual target, hence the n ominal\nvalue ofx, was tuned by adjusting the sputtering power. Total deposit ion rate was approximately\n0.3âĎń/s for smooth growth. An in-situ annealing procedure was performed at Tiafor 30 minutes\nto develop ordered B2orL21structures. A series of Tia(350-500◦C) was used for x=0.5 whereas\nTiawas fixed at 450◦Cfor all other samples. Finally, the FCCS layer was capped by R u (3nm) to\navoid oxidation.\nOur homebuilt angular-resolved FMR system equipped with an Agilent E8361C vector network\nanalyzer (VNA) can measure with frequency up to 40GHz. We use grounded coplanar waveguide\nwith a nominal impedance of 50 Ωand a center conductor of width 100 µm [16]. A customized\nsample holder is attached to a motorized stage for out-of-pl ane field rotation. In addition, an in-\nplane sample manipulator is used to rotate and mount the samp le face down on the waveguide\nautomatically. The accuracy of in-plane and out-of-plane fi eld orientations is better than 0.1 de-\ngree. Using angle-resolved broadband FMR, we determine the intrinsic damping by measuring with\nmagnetization perpendicular to the film plane, whereas the a nisotropy in magnetization relaxation\nwas investigated by varying the magnetization orientation within the film plane.\nC. RESULTS AND DISCUSSION\nStructure and chemical ordering\nAs shown in Fig. 1 (a), there are three structures in a FCCS Heusler alloy film ac cording to\ntheir atomic ordering: fully ordered full-Heulse L21structure, partially disordered B2and fully\ndisordered A2structures.\nFCCS(002) peaks of XRD θ−2θscans of Fe 2Cr0.5Co0.5Si films ( Fig. 1 (b)) indicate that or-\n3Figure 1. Structures of the films with stacks of MgO(001)-sub strate (30nm)/Cr (40nm)/Fe 2Cr1−xCoxSi\n(30nm)/Ru (3nm). (a) Schematics of the disordered A2, partially ordered B2and ordered L21structures.\n(b) XRD θ−2θscans for Fe 2Cr0.5Co0.5Si thin films with various in-situ annealing temperatures. ( c)φscans\nof FCCS, Cr and MgO of MgO(001)-substrate/Cr(40nm)/ Fe2Cr0 .5Co0.5Si(30nm)/ Ru(3nm) for the (111)\nplane. The in-situ annealing temperature is 450◦C. (d) XRD θ−2θscans for Fe 2Cr1−xCoxSi with various\nCo concentrations. The films were in-situ annealed at 450◦C. (e) Density of states (DOS) of FCCS alloys\ndetermined by first principle calculations.\ndering of the FCCS film evolves from A2toB2orL21forTia=400◦C. The Cr and FCCS layers\nexhibit epitaxial growth at 45◦with respect to MgO substrate (XRD phi-scan in Fig. 1 (c) and See\nSupplemental Material Fig. S4 for Transmission electron mi croscopy results). The degree of B2or-\ndering (SB2) andL21ordering ( SL21) was calculated using the Webster model [ 34]. Best crystalline\nstructure with SB2=0.95 and SL21=0.87 was obtained for Tia=450 (See Supplemental Material Fig.\nS1(a) for Tiadependence of ordering ). Correspondingly, the saturation magnetization Msincreases\nwithTiareaching maximum at Tia=450◦C(See Supplemental Material Fig. S2 for Tiadependence of\nMs). Furthermore, B2andL21phases were obtained for all Co concentrations with Tia=450◦Cas\ninferred from Fig. 1 (d). All samples exhibited high degree B2ordering ( SB2>0.74) and moderate\nSL21ordering (See Supplemental Material Fig. S1(b) for Co conce ntration dependence of ordering).\nB2 phase characterized by Y-Z atomic disorder in Fe-based X 2YZ full Heusler alloy is also favorable\nfor device applications since it smears the DOS spectral sha pes, does not degrade half-metallicity\nsignificantly and has little influence on damping [ 30,35,36]. Our first principles calculations (See\nSupplemental Material S3 for calculation methods) have sho wn that the band structures of L21\nFCCS were sucessfully tuned by Co concentration. As can be cl early seen from Fig. 1 (e), Fe 2CrSi\nand Fe 2Cr0.75Co0.25Si exhibit half-metallicity whereas for x/greaterorequalslant0.5, the minority band DOS at the\nFermi level is non-zero and increases with x. On the other hand, the total DOS at Fermi level\ndominated by the majority band decreases monotonically wit hx.\n4Dynamic properties of Fe 2Cr0.5Co0.5Si.\nRoom temperature out-of-plane FMR spectra were taken with ϕH=π/4.Fig. 2 (a) shows FMR\ndata for Fe 2Cr0.5Co0.5Si at different Tia. Stronger signals and narrower peaks are observed for\nTia> 400◦C. The narrowest spectrum is for Tia= 450◦C, in correspondence with the structure\nand magnetization measurements [ 37]. The resonance field HResand full width at half maximum\n(FWHM) △Hwere extracted by modified Lorentz fit (See Supplemental Mate rial S2 for fitting\nmethod). The frequency dependences of HResare shown in Fig. 2 (b). The Kittel equation describing\nthe resonance condition for out-of-plane configuration is g iven by\n2πf=γ(H−4πMeff) (1)\nindicating the linear relation between resonance frequenc y and field Here γ= g·e/2meis the gyro-\nmagnetic ratio determined by Lande g factor, electron charg eeand mass me.4πMeffthe effective\nsaturation magnetization 4πMeff= 4πMs−2K⊥\nu/Ms= 4πMs−Hk;K⊥\nuis the perpendicular\nanisotropy and Hkthe anisotropy field. The data are well fitted by straight line s. Using the slope,\nintercept of the fit, and Mswhich we measure using a vibrating sample magnetometer (See Supple-\nmental Material Fig. S2), we determined the value of g, 4πMeffandK⊥\nu. As seen from Fig. 2 (c),\nthe films possess a negative K⊥\nu(easy axis in the film plane). We find the corresponding Hkto be of\nthe order of kOe, considerably larger than the in-plane anis otropy field, which we discuss later. We\nbelieveK⊥\nuoriginates from interfacial stress and varies with Tiadue to the annealing temperature\ndependence of ordering and lattice constant [ 24].Fig. 2 (d) indicates the FMR linewidth increases\nlinearly with frequency for all samples. This allows the det ermination of damping constant α\nthrough:\n∆HOP=4π\nγαf+∆H0 (2)\nHere,αis the coefficient of the viscous phenomenological dissipati on term after Gilbert[ 38].∆H0is\nthe inhomogeneous broadening due to the dispersion of magne tic properties. The damping constant\nfor FCCS film with optimal L21andB2phase is (2.6±0.3)×10−3, i.e., 40 percent the value of\nA2phase ((6.5±0.2)×10−3,Fig. 2 (e). The results indicate the ordered half-metallic phase h as\nlower damping, suggesting atomic chemical disorder enhanc es damping [ 10]. Clearly, annealing at\nTia≥400◦Cimproves sample uniformity as inferred from the Tiadependence of ∆H0.\nCo doping dependence of magnetization relaxation of Fe 2Cr1−xCoxSi.\nThe out-of-plane FMR results with a magnetic field up to 2.1 T f or a series of FCCS samples\nare shown in Fig. 3 . All samples were annealed at Tia=450◦C. Again, we find linear frequency\ndependences of the linewidth. The perpendicular anisotrop y constant and damping values shown\ninFig. 3 (b) and (c) were determined using the same procedure describ ed earlier. K⊥\nuincreases\nwithxmay be attributed to the Co doping dependence of tetragonal d istortion [ 39], determined\n5Figure 2. Room temperature out-of-plane FMR results of Fe 2Cr0.5Co0.5Si for different in-situ annealing\ntemperature Tia. (a) Field dependence of the amplitude of FMR spectra for f= 10 GHz. The inset\nis an illustration of the measurement configuration. (b) Fre quency dependence of FMR resonance field.\n(c) Effective magnetization and perpendicular anisotropy v ersusTia. (d) Frequency dependence of FMR\nlinewidth.(e) Damping as a function of Tia\nby the reciprocal space mapping(See Supplemental Material Fig. S3 for structural informations).\nDamping for Fe 2CrSi (x=0) is the largest (1±0.03)×10−2. Forx=0.25, damping decreases\nby approximately 50% to (4.8±0.1)×10−3. Further increase of Co doping results in a nearly\nlinear reduction in damping. Fe 2CoSi (x= 1) exhibits the lowest damping (8±1)×10−4; this\nis considerably lower than 3d metals and remarkably close to the value for high quality Y 3Fe5O12\n(YIG) films [ 40,41].\nRelaxation frequency G =γMsαis directly related to the speed of relaxation and decreases by\n6/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s49/s48\n/s45/s49/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s51/s54/s57/s120/s61/s49/s120/s61/s48/s46/s55/s53/s120/s61/s48/s46/s53/s120/s61/s48/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41\n/s40/s99 /s41/s120/s61/s48/s46/s50/s53\n/s75\n/s117/s32/s40/s77/s101/s114/s103/s47/s99/s99/s41\n/s40/s100 /s41\n/s32/s120/s40 /s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s120\n/s120/s71/s32/s40/s49/s48/s55\n/s32/s72/s122/s41\n/s48/s49/s50/s51/s52/s53/s54/s55/s56/s40/s98 /s41 /s40/s97 /s41\n/s32/s68/s79/s83/s32/s97/s116/s32 /s69\n/s70\nFigure 3. Co concentration dependence of FMR results in Fe 2Cr1−xCoxSi. (a) FMR linewidth versus\nfrequency for various values of x. The solid lines are linear fits to the data. (b) Perpendicula r uniaxial\nmagnetic anisotropy versus x. (c) Damping constant αversusx. (d) Relaxation frequency Gand total\ndensity of states at the Fermi level versus x.\na factor of four with increasing x. The lowest relaxation frequency (30MHz) observed in Fe 2CoSi\n(Fig. 3 (d)) is comparable to that in FeV films [ 14]. The offset to smaller αandGforx=0.5 is\nbecauseTiawas optimized at this particular value of x. Hence, damping can be reduced further by\nfine tuning growth conditions.\nMagnetic damping depends on SOC and DOS G∝ξ2D(EF)[42]. Lowξin Heusler compounds\nbecause of quenching of orbital moments [ 13] and suppression of inter-band scattering [ 11] due to\nrelatively high spin-polarization band structure may acco unt for the overall low damping. On the\nother hand, the x-dependence of relaxation frequency is due to band structur e engineering as shown\ninFig. 1 (d), indicated by the simultaneous decrease of relaxation f requency and total DOS at the\nFermi level with x. A small misalignment may be due to band broadening by defect s and chemical\ndisorder.\n7Similarly, according to Du et al., the Fermi level of FCCS ing ots moves from the bottom of con-\nduction band to the top of valence band with increasing xfor the minority band.[ 43] The observed\nband structure tuning effect of Fe-based Heusler alloys also agrees with earlier investigations on\nCo-based Heusler compounds. [ 28]\nAnisotropic magnetization relaxation of Fe 2Cr1−xCoxSi.\nUsing out-of-plane FMR measurements, we determined the int rinsic damping by purposely sup-\npressing the extrinsic contributions causing non-linear f requency dependence [ 16,17,44]. In practi-\ncal applications, the relaxation mechanism for magnetizat ion in the film-plane is of primary impor-\ntance. Hence, we investigated the FCCS samples using in-pla ne magnetic field. The azimuthal angle\n(ϕH) dependence of resonance field shows four-fold symmetry ( Fig. 4 (a)), indicative of a dominant\nin-plane cubic magneto-crystalline anisotropy. Uniaxial anisotropy on the other hand, is negligible.\nThe in-plane magnetic easy axis is along [110] of the substra te as the minimum of resonance field\noccurs at ϕH=π/4andϕH= 3π/4, along which the lattice constant of FCCS matches MgO. We\nfit the data using the following equation derived from Smit-Be ljers formula [ 45,46]:\n2πf=γ√\nUV (3)\nwhereU=H·cos(ϕM−ϕH)+4πMeff+K4/bardbl/2Ms·[3+cos4( ϕM−π/4)]andV =H·cos(ϕM−ϕH)+\n2K4/bardbl/Ms·cos4(ϕM−π/4),ϕMandϕHare the orientation of magnetization and external field with\nrespect to [100] direction of MgO. As shown in Fig. 4 b, the in plane four-fold anisotropy energy\nincreases with Co composition. The corresponding anisotro py fieldH4/bardbl= 2K4/bardbl/Msobserved in\nFe2CoSi is approximately 190Oe. The ϕHdependence of FMR linewidth changes dramatically\nwithx,Fig. 5 (d). For Fe 2CrSi, we observe a nearly four-fold symmetry with minimum at ϕH= 0\n([100]), conversely, smallest linewidth occurs around ϕH=π/4, characteristic of Fe 2CoSi. Notably,\nthe linewidth anisotropy is much lower for FCCS films than for Fe2CrSi and Fe 2CoSi.\nThe anisotropy in the linewidth may be attributed to angle de pendent damping[ 47] or an ex-\ntrinsic origin[ 24]. Recent calculations show that anisotropy in intrinsic da mping is small at room\ntemperature[ 48,49]. Therefore, extrinsic linewidth broadening mechanisms, such as angle depen-\ndent inhomogeneous contribution and spin wave scattering s hould be considered. In general, the\nmeasured linewidth is the sum of intrinsic and extrinsic con tributions [ 47,50,51]:\n∆H= ∆Hint+|∂HRes\n∂4πMeff|∆4πMeff+|∂HRes\n∂HK|∆HK+∆HTMS (4)\nThe first term on the right-hand side is the intrinsic linewid th (Fig. 5 (a) ). For non-collinear\nmagnetization and external field ( ϕH∝negationslash=ϕM), the magnetization direction changes during field\nsweep, giving additional broadening to the intrinsic linew idth. In the present case of a relatively\nsmall in-plane anisotropy, the effect is calculated numeric ally to the first order using the resonance\nfield together with the equilibrium condition for magnetiza tion[47,50,52]:∆Hint=1\n|2π∂f/∂H |α\nγ(U+\nV)≈4πα\nγcos(ϕM−ϕH)f. The second and third terms are due to dispersion of the effect ive magnetization\nand anisotropy field, respectively ( Fig. 5 (b)). The partial derivatives are evalued numerically at th e\n8Figure 4. Room temperature in-plane FMR resonance fields and anisotropic energy. (a) In-plane FMR\nresonance field at f=10 GHz as a function of field direction for Fe 2Cr1−xCoxSi samples with different Co\nconcentration. Solid lines are calculated using Eqn. 3 . (b) Co concentration dependence of in-plane four-fold\nanisotropy energy. The schematic on the top right shows the c oordinates used in the measurements and the\norientations of DC and RF fields.\nresonance condition. The last term accounts for two magnon s cattering (TMS) [ 53,54] activated\nby defects such as dislocations (See Supplemental Material Fig. S4 for TEM results), representing\nrelaxation of the uniform mode to spin wave modes of non-zero wave vector ( Fig. 5 (c)). We fit\nthe data in Fig. 5 d by adopting a phenomenological form of TMS linewidth ∆HTMS=Γ1+\nΓ2cos2(2ϕM−φmax)[24,26]. Here, Γ1andΓ2are the isotropic and anisotropic TMS amplitude,\nrespectively. The anisotropic term correlates to defects o f preferential orientation [ 55].φmaxis the\norientation along which the 2D Fourier transformation of de fects potential shows a maximum. We\nfit the data for all Co dopings. The extracted parameters are s hown in Fig. 6 (a) and (c). Although\nthe anisotropic extrinsic damping of FCCS appears universa l in Hesuler alloys [ 29], when compared\nto Fe2CrSi and Fe 2CoSi, is considerably lower and almost vanishes for 0.5≤x≤0.75.\nTo study further the extrinsic relaxation and its anisotrop y, we measured the frequency depen-\ndence of FMR linewidth for magnetization along the in-plane easy MgO[110] and hard MgO[100]\naxes. Fig. 5 (e) depicts significant differences between the two magnetiz ation orientations observed\nonly for x=0 andx=1, in agreement with the angular dependence data. On the oth er hand, a\nnon-linearity as a signature of TMS, was observed in all samp les. We calculated the magnon wave\nvector (k0) at which the spin wave energy degenerates with the uniform m odesω(k0) = 2πffor a\ngiven propagation direction ϕwith respect to magnetization in the film plane [ 55,56]. The total\ncontribution to TMS was then estimated using [ 57,58] :\n9Figure 5. In-plane angular and frequency dependence of FMR l inewidth of Fe 2Cr1−xCoxSi. (a-c) Illustrations\nof magnetization relaxation due to spin-orbit coupling (a) , inhomogeneity (b) and scattering by magnon\nexcitation (c). (a) Intrinsic damping is caused by spin-orb it coupling. (b) Sample inhomogeneity broadens\nthe FMR linewidth as the spectrum is a superposition of local resonance. (c) The uniform precession mode\nof FMR can be scattered to degenerate spin-wave mode. (d) In- plane FMR linewidth for f=20 GHz as\na function of field direction for FCCS samples with different C o concentration; solid lines are fits to the\ndata using Eqn. 4 . (e) Frequency dependence of FMR linewidth for magnetizati on along [110] and [100]\ndirections. The solid lines are fits to the data using Eqn. 4 andEqn. 5 .\n∆HTMS=2K\nγ2∂f\n∂Hˆk0\ndf/dkdϕ (5)\nwheredf/dk was evaluated at k0and∂f/∂H calculated from the FMR resonance conditions.\nK= 0.16h2Aξis the only fitting parameter related to defects. The fits to th e frequency dependence\nof linewidths are shown by solid lines. The fitting parameter Kfor both magnetization orientation\nand their difference ∆K=K[100]−K[110]are indicative of the strength of TMS and its anisotropy,\nrespectively. The fitting results are plotted in Fig. 6 (b) and (d). The Co doping dependence of\nK[110]agrees with the trend obtained earlier from the angular depe ndence of the linewidth.\nThe anisotropy in TMS ( ∆K) is maximum for x=1 and negligible for xaround 0.7. Furthermore,\nwe obtain a relatively small anisotropy of TMS for x=0, as expected from the weak non-linearity\nof the two curves shown in the top panel of Fig. 5 (e). However, the results appear inconsistent\nwith the phenomenological fit shown in Fig. 6 (a). We attribute this to orientation dependent long\nrange inhomogeneity in the sample, apparent also from the la rge zero frequency broadening ( ∆H0)\ninFig. 3 (a) forx=0. Here, the phenomenological fit cannot separate the angul ar dependence of\ninhomogeneous broadening from four-fold TMS hence, overes timating the TMS contribution for\nx=1. Nevertheless, both the frequency dependence and angula r dependence of the linewidths are\n10/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s50/s48/s52/s48/s54/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48/s53/s48/s49/s48/s48\n/s40/s100/s41 /s40/s99/s41\n/s75 /s40/s49/s48/s45/s57\n/s79/s101 /s99/s109/s50\n/s41\n/s75\n/s91/s49/s48/s48/s93/s45/s75\n/s91/s49/s49/s48/s93/s40/s98/s41\n/s75 /s40/s49/s48/s45/s57\n/s79/s101 /s99/s109/s50\n/s41\n/s120/s40/s97/s41\n/s40/s79/s101/s41\n/s120/s75\n/s91/s49/s49/s48/s93/s32/s40/s79/s101/s41\nFigure 6. Isotropic and anisotropic TMS contributions in Fe 2Cr1−xCoxSi. (a, c) Isotropic and anisotropic\ncomponents of the TMS linewidth determined by the phenomeno logical method of angular dependent\nlinewidth. (b, d) Co concentration dependence of Kfor magnetization along [110] and anisotropy of K.\nin agreement, confirming that the isotropic and anisotropic extrinsic damping constants of FCCS\nare significantly reduced for x=0.5-0.75. Therefore, in FCCS with 0.5≤x≤0.75, low damping\nand nearly isotropic response are promising for practical a pplications.\nIn summary, we studied the structure order and composition d ependence of magnetization re-\nlaxation in Fe 2Cr1−xCoxSi Heusler alloy thin films. We show the ordered phase has a low intrinsic\ndamping, tunable by Co doping. Notably, the anisotropic ext rinsic relaxation of the quaternary\nalloys can be significantly reduced for FCCS films with x=0.5~0.75. Furthermore, perpendicu-\nlar and in-plane magnetic anisotropy show nearly linear dep endence on Co doping. The present\nwork demonstrates the tunable nature of both static and dyna mic magnetic properties of Fe-based\nHeusler alloys. The engineered low damping constant in Fe 2Cr1−xCoxSi thin films encourages the\nutilization of Heusler alloys in room temperature spintron ics.\nACKNOWLEDGMENTS\nThe work was supported by the Ministry of Education (MOE), Ac ademic Research Fund (AcRF)\nTier 2 Grant (MOE2014-T2-1-050), The A*STAR Pharos Fund (15 27400026), and the National Re-\nsearch Foundation (NRF), NRF-Investigatorship (NRF-NRFI 2015-04). The authors acknowledge\nthe Singapore Synchrotron Light Source (SSLS) for providin g the facilities to perform x-ray ex-\nperiments. The Laboratory is a National Research Infrastru cture under the National Research\n11Foundation Singapore.\n[1] L. Berger, Phys. Rev. B 54, 9353 (1996) .\n[2] J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 159, L1 (1996) .\n[3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa,\nF. Matsukura, and H. Ohno, Nat Mater 9, 721 (2010) .\n[4] K. Wang, J. Alzate, and P. K. 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Patton, Journal of Applied Physics 101, 083901 (2007) .\n13"    },    {        "title": "2201.05168v1.Damping_of_Alfvén_waves_in_MHD_turbulence_and_implications_for_cosmic_ray_streaming_instability_and_galactic_winds.pdf",        "content": "DRAFT VERSION JANUARY 17, 2022\nTypeset using L ATEX default style in AASTeX63\nDamping of Alfv ´en waves in MHD turbulence and implications for cosmic ray streaming instability and galactic winds\nALEX LAZARIAN1, 2AND SIYAO XU3\n1Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA; lazarian@astro.wisc.edu\n2Centro de Investigaci ´on en Astronom ´ıa, Universidad Bernardo O’Higgins, Santiago, General Gana 1760, 8370993,Chile\n3Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA; sxu@ias.edua\nABSTRACT\nAlfv ´enic component of MHD turbulence damps Alfv ´enic waves. The consequences of this effect are impor-\ntant for many processes, from cosmic ray (CR) propagation to launching outflows and winds in galaxies and\nother magnetized systems. We discuss the differences in the damping of the streaming instability by turbulence\nand the damping of a plane parallel wave. The former takes place in the system of reference aligned with the\nlocal direction of magnetic field along which CRs stream. The latter is in the reference frame of the mean mag-\nnetic field and traditionally considered in plasma studies. We also compare the turbulent damping of streaming\ninstability with ion-neutral collisional damping, which becomes the dominant damping effect at a sufficiently\nlow ionization fraction. Numerical testing and astrophysical implications are also discussed.\n1.PROPAGATION OF ALFV ´EN WA VES IN MHD TURBULENCE\nAstrophysical media are turbulent and magnetized (see a collection of relevant reviews in Lazarian et al. 2015a). The propa-\ngation of Alfv ´en waves in turbulent magnetized media is an important astrophysical problem that influences fundamental astro-\nphysical processes (see e.g., Uhlig et al. 2012, Wiener, Oh & Guo 2013, van der Holst et al. 2014, Lynch et al. 2014). This\nreview focuses on the damping of Alfv ´en waves in MHD turbulence. The Alfv ´en waves can arise from instabilities induced by\ncosmic rays (CRs), e.g. from the streaming of CRs (Lerche 1967, Kulsrud & Pearce 1969, Wentzel 1969, Skilling 1971), and the\ngyroresonance instability related to the compression of magnetic field and CRs (see Lazarian & Beresnyak 2006). They can also\nbe generated by large scale perturbations of magnetic field (see Konigl 2009 and ref. therein, Suzuki 2013).\nTurbulent damping of Alfv ´en waves causes heating of, e.g. coronal gas in Solar atmosphere (e.g. Arber, Brady & Shelyag\n2016, Reep & Russell 2016). In the case of the streaming instability, turbulent damping suppresses its growth and affects the\nstreaming speed of CRs. As a result, turbulent damping of streaming instability is important for studies on the diffusion and\nacceleration of CRs in shocks, galaxies, and galaxy clusters (Bell 1978, Kulsrud 2005, Ensslin et al. 2011, Blasi et al. 2012,\nWiener et al. 2013, Badruddin, & Kumar, A. 2016, Xu & Lazarian 2022), stellar wind launching (e.g. Suzuki & Inutsuka 2005,\nvan Ballegooijen, & Asgari-Targhi 2016), and galaxy evolution (e.g., Hopkins et al. 2021).\nIt should be noted that the well-known study of Alfv ´en wave damping by turbulent plasmas performed by Silimon & Sudan\n(1989) employed an unrealistic model of isotropic MHD turbulence. Later, turbulent damping of Alfv ´en waves was mentioned\nas a process for suppressing CR streaming instability in Yan & Lazarian (2002, henceforth YL02). This process was quantified\nby Farmer & Goldreich (2004, henceforth FG04), where the Goldreich & Sridhar (1995; henceforth, GS95) model of Alfv ´enic\nturbulence with scale-dependent anisotropy was adopted. The limitation of the aforementioned study was that for the calculations\nit was assumed that turbulence is injected isotropically with the turbulent velocity uLexactly equal to the Alfv ´en velocityVA,\ni.e. Alfv ´en Mach number MAequal to unity. In addition, only turbulent damping of streaming instability was considered.\nFollowing the study in Lazarian (2016), we will seperately discuss the turbulent damping of Alfv ´en waves that are generated\nby streaming instability and by large-scale magnetic perturbations. We will demonstrate the strong dependence of turbulent\ndamping on MAin various turbulence regimes and astrophysical media with different levels of medium magnetization. In §2\nwe provide the derivation of the Alfv ´enic turbulent scaling. In §3 we describe the turbulent damping of Alfv ´en waves generated\nby streaming instability in the reference system aligned with the local direction of turbulent magnetic field. In §4 we discuss the\nturbuelnt damping of Alfv ´en waves induced by large-scale magnetic perturbations in a global system of reference. We compare\nthe turbulent damping with ion-neutral collisional damping of streaming instability in a partially ionized medium in §5. The\nnumerical testing of the theoretical predictions is provided in §6. The discussion of the astrophysical implications on propagation\nof CRs in galaxies and launching of winds follows in §7. The summary is given in §8.\naHubble FellowarXiv:2201.05168v1  [astro-ph.GA]  13 Jan 20222\n2.DERIV ATION OF ALFV ´ENIC TURBULENT SCALING\nIn Alfv ´enic turbulence the relative perturbations of velocities and magnetic fields are related as follows:\n\u000eBl\nB=\u000eBl\nBLBL\nB=ul\nuLMA=ul\nVA; (1)\nwhereBlis the fluctuation of the magnetic field Bat scalel,BLis the fluctuation of the magnetic field at the driving scale\nLof turbulence. Correspondingly, ulis the turbulent velocity fluctuation at the scale landuLis the turbulent velocity at L.\nMA=uL=VAis the Alfv ´en Mach number.\nOne way to understand the non-linear interactions of Alfv ´en waves within the MHD turbulent cascade is to consider colliding\nAlfv ´en wave packets with parallel scales lkand perpendicular scales l?. The collision of a wave packet induces an energy change\n\u0001E\u0018(du2\nl=dt)\u0001t; (2)\nwhere the term in brackets manifests the change of the energy of a wave packet induced by its interaction with the oppositely\nmoving Alfv ´en wave packet. The time of this interaction is equal to the time of the passage of these wave packets through each\nother. As the size of the packet is lk, the interaction time is simply \u0001t\u0018lk=VA.\nThe rate of turbulent energy cascade is related to the rate of structure change of the oppositely moving wave packet. The latter\nisul=l?. As a result, Eq. (2) provides\n\u0001E\u0018ul\u0001_ul\u0001t\u0018(u3\nl=l?)(lk=VA); (3)\nThe fractional change of packet energy taking place per collision is \u0001E=E . This characterises the strength of the nonlinear\nturbulent interaction:\nf\u0011\u0001E\nu2\nl\u0018ullk\nVAl?: (4)\nIn Eq. (4),fis the ratio of the shearing rate of the wave packet, i.e. ul=l?, to its propagation rate, i.e. VA=lk.\nOne can identify two distinct cases. If f\u001c1, the shearing rate is significantly smaller than the propagation rate, and the\ncascade presents a random walk process. Therefore\n@=f\u00002(5)\nsteps are required for the energy cascade, and therefore the cascading time is\ntcas\u0018@\u0001t: (6)\n@>1corresponds to the weak turbulent cascade. Naturally, @cannot become less than unity. Therefore, the limiting case is\n@\u00191. This is the case of strong MHD turbulence.\nTraditionally, the wavevectors are defined in the system of reference related to the mean field. However, the system of reference\nrelated to a wave packet with given parallel and perpendicular dimensions is more relevant when dealing with strong MHD\nturbulence. We take this into account by considering Alfv ´en wave packets having the dispersion relation !=VAjkkj, where we\nusekk\u0018l\u00001\nkas the component of wavevector parallel to the local background magnetic field. As the result of interaction the\nincrease ofk?\u0018l\u00001\n?occurs. In the rest of the discussion we use lkandl?that are defined in the local frame of wave packets.\nIn weak turbulence, the decrease of l?whilelkdoes not change signifies the increase of the energy change per collision. This\nforces@to be of the order of unity. In this case one gets\null\u00001\n?\u0019VAl\u00001\nk(7)\nin strong turbulence, which signifies the cascading time being equal to the wave period \u0018\u0001t. Any further decrease of l?\ninevitably results in the corresponding decrease of lkand Eq. (7) is still satisfied. The change of lkentails the increase of the\nfrequencies of interacting waves. This is compatible with the conservation of energy condition above, as the cascade introduces\nthe uncertainty in wave frequency !of the order of 1=tcas.\nThe cascade of turbulent energy satisfies the relation (Batchelor 1953):\n\u000f\u0019u2\nl=tcas=const; (8)\nwhich for the hydrodynamic cascade provides\n\u000fhydro\u0019u3\nl=l\u0019u3\nL=L=const; (9)3\nwhere the relation for the cascading time tcas\u0019l=ulis employed.\nFor the weak turbulent cascade with @\u001d 1, we have (LV99)\n\u000fw\u0019u4\nl\nV2\nA\u0001t(l?=lk)2\u0019u4\nL\nVAL; (10)\nwhere Eqs. (8) and (6) are used. The isotropic turbulence injection at scale Lresults in the second relation in Eq. (10). Taking\ninto account that for the weak turbulence lkis constant, it is easy to see that Eq. (10) provides\nul\u0018uL(l?=L)1=2; (11)\nwhich is different from the hydrodynamic \u0018l1=3scaling.1\nIt was shown in LV99 that for turbulence with isotropic injection at scale LwithVL<VAthe transition to the strong regime\ncorresponding to@\u00191happens at the scale\nltrans\u0018L(uL=VA)2\u0011LM2\nA: (12)\nAs a result, the inertial range of weak turbulence is limited, i.e. [L;LM2\nA], and atltrans the turbulence transits into the regime of\nstrong MHD turbulence. At the transition, the velocity is\nutrans\u0019VAltrans\nL\u0019VAM2\nA; (13)\nwhich follows from @\u00191condition given by Eqs. (4) and (5).\nThe scaling relations for the strong turbulence with VL< VAcan be easily obtained. The turbulence is strong and cascades\nover one wave period, which according to Eq. (7) is equal to l?=ul. Substituting the latter in Eq. (8) one gets\n\u000fs\u0019u3\ntrans\nltrans\u0019u3\nl\nl=const: (14)\nThe latter energy cascading rate is analogous to that in an ordinary hydrodynamic Kolmogorov cascade. However, this cascading\ntakes place in the direction perpendicular to the local direction of the magnetic field.2\nThis strong MHD turbulence cascade starts at ltrans and its injection velocity is given by Eq. (13). This provides another way\nto obtain the Alfv ´enic turbulent scaling in strong turbulence regime (LV99)\nul\u0019VA\u0012l?\nL\u00131=3\nM4=3\nA; (15)\nwhich can be rewritten in terms of the injection velocity uL(see Eq. (15) )\n\u000eul\u0019uL\u0012l?\nL\u00131=3\nM1=3\nA: (16)\nSubstituting this in Eq. (7) we get the relation between the parallel and perpendicular scales of the eddies (LV99):\nlk\u0019L\u0012l?\nL\u00132=3\nM\u00004=3\nA: (17)\nThe relations Eq. (17) and (15) reduce to the GS95 scaling for transAlfv ´enic turbulence if MA\u00111.\nIn the opposite case we deal with superAlfv ´enic turbulence, i.e. with uL>VA. As a result, at scales close to the injection scale\nthe turbulence is essentially hydrodynamic as the influence of magnetic forces is marginal. Therefore, the velocity is Kolmogorov\nul=uL(l=L)1=3: (18)\n1Using the relation kE(k)\u0018u2\nkit is easy to show that the energy spectrum of weak turbulence is Ek;weak \u0018k\u00002\n?(LV99, Galtier et al. 2000).\n2There is an intuitive way of presenting the Alfv ´enic cascade in terms of eddies mixing the magnetic field in the direction perpendicular to the magnetic field\nsurrounding the eddies. The existence of such magnetic eddies is possible due to the fact that, as shown in LV99, the turbulent magnetic reconnection happens\nwithin one eddy turnover. As a result, the existence of magnetic field does not constrain magnetic eddies, if they are aligned with the magnetic field in their\nvicinity, i.e. with the local magnetic field. This eddy representation of MHD turbulence vividly demonstrates the importance of the local system of reference,\nwherel?andlkare defined.4\nThe magnetic field becomes more important at smaller scales and the cascade changes its nature at the scale\nlA=LM\u00003\nA; MA>1; (19)\nat which the turbulent velocity becomes equal to the Alfv ´en velocity (Lazarian 2006). The rate of cascade for l<lAis:\n\u000fsuperA\u0019u3\nl=l\u0019M3\nAV3\nA=L=const: (20)\nUnlike the case of subAlfv ´enic turbulence, the case of superAlfv ´enic turbulence can be reduced to the case of transAlfv ´enic\nturbulence, but with lAacting as the injection scale. At scales l<lA\nlk\u0019L\u0012l?\nL\u00132=3\nM\u00004=3\nA; (21)\nul\u0019uL\u0012l?\nL\u00131=3\nM1=3\nA: (22)\nThe relations for subAlfv ´enic and superAlfv ´enic tubulence that we obtain above coincide with the expressions first obtained in\nLazarian & Vishniac (1999) using a different approach. These expressions will be used below in our discussion on turbulent\ndamping of Alfv ´en waves.\n3.TURBULENT DAMPING OF STREAMING INSTABILITY\nLinear Alfv ´en waves undergo non-linear cascading when they propagate through Alfv ´enic turbulence. This process is of MHD\nnature and the non-linear damping of Alfv ´en waves does not depend on plasma microphysics. The interaction between CR-driven\nAflven waves and turbulence is similar to that of oppositely moving wave packets of turbulent cascade.\nThe Alfv ´en waves emitted parallel to the local magnetic field experiences the least distortions from the oppositely moving\neddies. Thus the least distorted Alfv ´en waves are those with the largest value of l?. Indeed, the larger l?, the longer time it\ntakes for the evolution of the oppositely moving wave packets. For instance, for strong GS95 turbulence the time corresponds to\nl?=vl\u0018l2=3\n?.\nThe case of Alfv ´en waves parallel to the local direction of magnetic field corresponds to streaming and gyroresonance insta-\nbilities. In what follows, we will focus on the streaming instability. The dispersion of magnetic field directions with respect\nto the mean magnetic field determines the corresponding l?. Naturally, the turbulent damping of Alfv ´en waves is different for\nweak turbulence and strong turbulence. Thus we will separately discuss turbulent damping of streaming instability in different\nturbulence regimes.\n3.1. Streaming instability and local system of reference\nThe streaming instability of CRs happens as CR particles moving in one direction scatter back from a magnetic field perturba-\ntion and thus increase the amplitude of the perturbation. The induced perturbations are Alfv ´en waves. If the Alfv ´en waves are\nseverely damped, the CR particles can stream freely along the magnetic field.\nPhysically, the generation of Alfv ´en waves takes place as CRs stream along the local magnetic field. During the process the\nsampling scale for the magnetic field is the CR Larmor radius rL. In this setting one should consider the process in the system of\nreference related to the local direction3of the wondering magnetic field (LV99, Cho & Vishniac 2000, Maron & Goldreich 2001,\nCho, Lazarian & Vishniac 2002).\nIn the direction parallel to the local magnetic field, the growth rate of the streaming instability is (see Kulsrud & Pearce 1969):\n\u0000cr\u0019\nBncr(>\r)\nni\u0012vstream\nVA\u00001\u0013\n; (23)\nwhere \nB=eB=mc is the nonrelativistic gyrofrequency, ncris the number density of CRs with gyroradius rL>\u0015=\rmc2=eB,\nand\ris the Lorentz factor. If the growth rate given by Eq. (23) is less than the rate of turbulent damping, the streaming instability\nis suppressed.\n3The fact that MHD turbulence is formulated in terms of the local quantities is required for describing the interaction of MHD turbulence with CRs. Indeed,\nperturbations in the local system of reference are exactly what CRs interact with.5\n3.2. Damping by SubAlfv ´enic strong turbulence\nOur first approach is based on calculating the distortion of Alfv ´en waves by MHD turbulence as the waves propagate along\nmagnetic field. The cause of the wave distortion is the field line wandering over angle \u0012x. This angle is determined by the\namplitude of magnetic field fluctuations \u000eBxthat are induced by turbulent eddies with perpendicular scale x. One can see that\nthe distortion induced during the time tis\n\u000ex\u0019VAtsin2\u0012x\u0019VAt\u0012\u000eBx\nB\u00132\nt; (24)\nwhere the fluctuation induced by turbulence evolves as\n\u0012\u000eBx\nB\u0013\nt\u0019\u0012ux\nVA\u0013 \u0012t\nx=ux\u0013\n: (25)\nIn the above expression uxdenotes the velocity corresponding to the magnetic field fluctuation \u000eBx. The timetin Eq. (25) is\nchosen to be less than the eddy turnover time x=ux. As a result, the ratio reflects the partial sampling of the magnetic perturbation\nby the wave. By using the velocity scaling of strong subAlfv ´enic turbulence for uxin Eq. (25), it is easy to rewrite Eq. (24) as\n\u000ex\u0019V3\nAM16=3\nAt3\nx2=3L4=3: (26)\nThe wave damping corresponds to the “resonance condition” \u000ex=\u0015, where\u0015is the wavelength. Inserting this in Eq. (26) we\nobtain the perpendicular scale of the “resonance” magnetic fluctuations that distort the Alfv ´en waves:\nx\u0019V9=2\nAt9=2M8\nA\n\u00153=2L2: (27)\nThe time required to damp the Alfv ´en waves is equal to the turnover time of the “resonant” eddy:\nt\u0019x\nul\u0019V2\nAt3M4\nA\n\u0015L: (28)\nThis provides the rate of non-linear damping of the Alfv ´en waves,\n\u0000subA;s\u0019t\u00001; (29)\nor\n\u0000subA;s\u0019VAM2\nA\n\u00151=2L1=2; (30)\nwhere the subsscript “s” denotes “strong turbulence”. For transAlfv ´enic turbulence, i.e. MA= 1, this result was obtained in\nFG04. The square of the Alfv ´en Mach number dependence presented in Eq. (30) means a significant change of the damping rate\ncompared to the transAlfv ´enic case.4\nIf the injection of turbulence is isotropic, the maximal perpendicular scale of strong subAlfv ´enic motions is xmax=LM2\nA.\nSubstituting this in Eq. (27) and using Eq. (29) and Eq. (30) to express t, we get\n\u0015max;s\u0019LM4\nA: (31)\nThe streaming CRs generate Alfv ´en waves at a scale comparable to the gyroradius rL. Thus it requires that\nrL<LM4\nA; (32)\nwhich is a notable limitation on CR energy if MAis small. The CRs with larger energies interact with weak turbulence as we\nwill discuss in Section 3.3.\nDue to the importance of turbulent damping of streaming instability, it is advantageous to provide another derivation of Eq. (30).\nThis alternative derivation is based on the picture of propagating wave packets that we used while obtaining Eq. (3). Consider\n4We note that in FG04 the injection scale for turbulence was defined not as the actual injection scale, but the scale at which the turbulent velocity becomes equal\nto the Alfv ´en one. Such scale does not exist for subAlfv ´enic turbulence.6\ntwo oppositely moving Alfv ´en wave packets with the perpendicular scale x0\u0018k0\u00001\n?. As we discussed earlier, each wave packet\ninduces the distortion \u00120\nxof the oppositely moving waves. Consider an Alfv ´en wave with wavenumber k\u00001\nk\u0018\u0015moving parallel\nto the local direction of magnetic field. Such a wave is mostly distorted when interacting with turbulent perturbations with the\nperpendicular wavenumber k?\u0018kksin\u00120\nx. The interactions are most efficient if they are “resonant”, i.e. a wave with k?interacts\nwith the oppositely moving packets and k0\n?=k?.5Thus the perpendicular scale of the “resonant” wave packet is determined by\nthe relationkksin\u0012x=k?, which results in\n\u0015\u0019xsin\u0012x\u0019x\u000eBx\nB: (33)\nUsing the scaling in Eqs. (15) and (25), we derive the “resonant” perpendicular scale x:\nx=L1=4\u00153=4M\u00001\nA: (34)\nThis can be used to determine the rate of damping defined as \u0000subA;s\u0019ux=x. This coincides with the earlier result given by Eq.\n(30). Then the maximal wavelength of the non-linearly damped Alfv ´en waves can be obtained from Eq. (33) if the scale ltrans is\nused instead of x, i.e.\n\u0015max;s\u0019\u0012utrans\nVA\u0013\nltrans\u0019LM4\nA: (35)\nNaturally, the latter coincides with the result given by Eq. (31). The minimal scale of non-linearly damped waves depends on the\nperpendicular scale of the smallest Alfv ´enic eddieslmin. The full range of rLfor which turbulent damping is essential can be\nobtained by using Eq. (33) and the scaling of strong turbulence given by Eq. (15):\nl4=3\nmin\nL1=3M4=3\nA<rL<LM4\nA: (36)\nThe value of lmin depends on the particular damping process of MHD turbulent cascade, which can be relatively large in a\nweakly ionized gas (see Xu & Lazarian 2017). Due to the differences of rLfor protons and electrons, Eq. (36) presents a\npossible situation when the streaming instability of CR electrons is not damped by turbulence, while it is damped for CR protons.\nWe note that the turbulent damping of streaming instability for rL<l4=3\nmin\nL1=3M4=3\nAis still present, although it is reduced. We can\nget an estimate of it by considering the distortion \u000ex\u001c\u0015given by Eq. (26) for the time period of the wave \u0015=VA. This time\nis significantly less than the period of the eddy at the scale lmin,teddy\u0019l2=3\nminL1=3=(VAM4=3\nA). The distortions act in a random\nwalk fashion with the time step given by teddy. The damping requires \u0015=\u000exsteps, which induces the damping rate\n\u0000sub;s;r L\u001clmin\u0019M12\nAVAr4\nL\nl2\nminL3: (37)\nThe latter clearly illustrates the inefficiency of damping when turbulence has the perpendicular scale larger than the “resonant”\nscale.\n3.3. Damping by subAlfv ´enic weak turbulence\nIn many instances the weak turbulence is not important. It has a limited inertial range and transfers to strong turbulence at\nsmaller scales. However, as we show below, this may not be true for wave damping by turbulence. For wavelengths longer than\n\u0015max;s the wave is non-linearly damped through interactions with the wave packets of the weak turbulence, having perpendicular\nscales given by Eq. (33). Naturally, the scaling of weak turbulence given by Eq. (11) should be used. This provides the relation\nbetween the Alfv ´en wave wavelength and the perpendicular scale of the “resonant” weak turbulence perturbation\n\u0015=l?\u0012l?\nL\u00131=2\nMA: (38)\nThis delivers the perpendicular scale\nl?\u0019\u00152=3L1=3M\u00002=3\nA: (39)\n5Simple estimates demonstrate that the interactions with smaller and larger turbulent scales are subdominant compared with the interaction with the “resonant”\nscale.7\nAccording to Eqs (5) and (6), the weak turbulence packets cascade @times slower compared to the case of strong turbulence.\nTaking into account that the parallel scale of weak turbulence wave packets is equal to the injection scale L, we have\n@\u0019\u0012VAl?\nulL\u00132\n: (40)\nThe rate of turbulent damping of the wave is therefore\n\u0000subA;w\u0019(@\u0001t)\u00001=@\u00001VA\nL; (41)\nwhich gives\n\u0000subA;w\u0019VAM8=3\nA\n\u00152=3L1=3; (42)\nwhere the subsscript “w” denotes “weak turbulence”. Note that compared to the case of damping given by Eq. (30) we now have\na stronger dependence on MA, as well as a different scaling with the wavelength \u0015.\nThe maximal wavelength of the Alfv ´en waves that is cascaded by the weak cascade we derive by substituting l?=L, i.e.\nusing the energy injection scale in Eq. (38). This gives:\n\u0015max;w\u0019LMA: (43)\nThus for CRs that generate Alfv ´en waves, their rLshould satisfy\nLM4\nA<rL<LMA (44)\nto interact with weak Alfv ´enic turbulence. The underlying assumption here is that LM4\nAis larger than the turbulent damping\nscalelmin. Otherwise the lower boundary in Eq. (44) is determined by lmin.\nWaves with\u0015>\u0015max;w can interact with the turbulent motions at the injection scale L. The cascade of such waves is induced\nby the largest wave packets at a rate @\u00001VA\nL, i.e.\n\u0000outer\u0019@\u00001VA\nL\u0019M2\nAVA\nL; (45)\nwhich does not depend on wavelength. Physically, this means that all waves in the range LMA<\u0015<L decay at the same rate\nthat is determined by the restructuring of the magnetic field at the injection scale.\nThe above expression is valid for \u0015<L . In the case of \u0015\u001dLthe rate is reduced due to the random walk, which results in a\nfactor (L=\u0015)2, i.e.\n\u0000outer;extreme\u0019@\u00001VA\nLL2\n\u00152\u0019M2\nAVA\nLL2\n\u00152: (46)\nThe latter result is relevant for the damping induced by turbulence injected at scales smaller than the wavelength.\nIn terms of the dependence of damping rate on \u0015for subAlfv ´enic turbulence, we observe that the dependence becomes stronger\nwith the increase of \u0015up to\u0015=LMA. For\u0015less thanLM4\nA, the waves interact with strong Alfv ´enic turbulence and the damping\nrate\u0000is proportional to \u0015\u00001=2. The scaling changes for waves longer than LM4\nAbut shorter than LMA. The scaling of \u0000gets\nto\u0015\u00002=3as Alfv ´en waves interact with weak turbulence. For smaller MAthe range for which weak Alfv ´enic turbulence damps\nAlfv ´en waves increases. A further increase of the wavelength, i.e. for \u0015fromLMAtoL, introduces a flat regime of damping,\ni.e., no dependence on \u0015. The damping at this regime is determined by the evolution of turbulence at the injection scale. In its\nturn, this regime proceeds untill \u0015gets of the order of L. Finally, if\u0015is much larger than L, the damping modifies further that\nit transfers to \u0015\u00002. In that regime the Alfv ´en waves have so large \u0015that they only feel the distortions that are introduced by\nturbulence at the outer scale. In comparison, the FG04 study considered only transAlfv ´enic turbulence and provided only \u0015\u00001=2\nscaling for all scales.\nIn terms of the dependence of turbulent damping rate on MA, it changes from M2\nAfor strong turbulence to M8=3\nAfor weak\nturbulence. The case of no damping naturally follows as MA!0. As for FG04 study, it only provided the result for MA= 1.8\n3.4. Damping by SuperAlfv ´enic turbulence\nAs we discussed earlier, if turbulence is superAlfv ´enic, at large scales the effects of magnetic field are marginal and turbulence\nis hydrodynamic-like. However, the turbulent velocity decreases with the decreasing scale and at a scale lAbecomes equal to\nthe Alfv ´en velocity. This scale can be considered as the injection scale of transAlfv ´enic turbulence. Therefore, the case of\nAlfv ´en wave damping by superAlfv ´enic turbulence at scales less than lAcan be related to the case of damping by transAlfv ´enic\nturbulence considered in FG04. Indeed, a simple substitution of LbylAprovides the required rate of magnetic structure evolution\non scales less than lA. This gives:\n\u0000super\u0019VA\nl1=2\nA\u00151=2=VAM3=2\nA\nL1=2\u00151=2: (47)\nTreatinglAas the effective injection scale and using Eq. (31), it is easy to obtain the maximal wavelength up to which our\ntreatment of the non-linear damping is applicable:\n\u0015max;super\u0019lA=LM\u00003\nA: (48)\nAssociating \u0015withrL, we define the corresponding range of rL\nl4=3\nmin\nL1=3MA<rL<LM\u00003\nA; (49)\nassuming that the minimal/damping scale of turbulent motions lminis less thanLM\u00003\nA.\nFor Alfv ´en waves with \u0015larger than that given by Eq. (48) and therefore for rL> LM\u00003\nA, the damping is induced by\nKolmogorov-type isotropic hydrodynamic turbulence. The characteristic damping rate in this case is expected to coincide with\nthe eddy turnover rate, i.e.\n\u0000hydro\u0019u\u0015\n\u0015\u0019VA\nl1=3\nA\u00152=3\u0019VAMA\nL1=3\u00152=3; (50)\nwhere we use Eq. (18).\nSimilar to the case of sub-Alfv ´enic turbulence, in superAlfv ´enic case, we observe the change of the rate of Alfv ´en wave\ndamping changing from \u0015\u00001=2for short wavelengths to \u0015\u00002=3for\u0015longer thanLM\u00003\nA. The turbulent damping rate of Alfv ´en\nwaves increases with MA.\n3.5. Other forms of presenting our results\nThe scaling of weak turbulence is different from that of strong turbulence that starts at the transition scale ltrans =LM2\nAof\nsubAlfv ´enic turbulence. However, what is the same in the two regimes of turbulence is the cascading rate. Indeed, the energy\ncascades at the same rate without accumulating at any scale and dissipates only at the small dissipation scale. Therefore, by\nexpressing the dissipation rate of Alfv ´en waves through the cascading rate of turbulence, we will demonstrate a higher degree of\nuniversality of the obtained expressions.\nThe cascading rate of the weak turbulence is given by Eq. (10) and we can write it as\n\u000fw\u0019V3\nAM4\nA\nL: (51)\nThis reflects the decrease of energy dissipation by M4\nAcompared to the case of transAlfv ´enic turbulence in FG04. If rL<LM4\nA,\nthe rate of Alven wave damping can be obtained by combining Eq. (51) and Eq. (30):\n\u0000subA;s\u0019\u000f1=2\nw\nV1=2\nAr1=2\nL: (52)\nThe peculiar feature of Eq. (52) is that if one formally substitutes instead of \u000fwthe cascading rate of strong turbulence, one will\nget the expression in FG04. This is exactly the universality of expressions that we sought. Nevertheless, this analogy is only\nformal as the cascading rate for weak turbulence is M4\nAtimes lower compared to the transAlfv ´enic case. Therefore, the obtained\ndamping rate for subAlfv ´enic turbulence is M2\nAtimes less than the case of trans-Alfv ´enic turbulence (see also Eq. (30)).\nFor wavelengths in the range LM4\nA< \u0015 < LM Athe weak turbulence is responsible for the Alfv ´en wave damping. Thus,\nexpressingMAfrom Eq. (51) and substituting it in Eq. (42) we can get\n\u0000subA;w\u0019\u000f1=3\nwL1=3\nVAr2=3\nL\u0019\u000f1=3\nwM4=3\nA\nr2=3\nL: (53)9\nThe expression given by Eq. (53) demonstrates a slower damping rate in comparison to Eq. (52). The decrease of damping rate\nby the factor M8=3\nA(see Eq. (42)) is significant. It is important to note that for MA\u001c1it provides the smooth transition to the\nregime of marginal Alfv ´en wave damping as the magnetic field perturbations get smaller and smaller. Naturally, this expression\nis very different from that in FG04, as the latter study did not consider the damping induced by weak Alfv ´enic turbulence.\nFor damping of Alfv ´en waves generated by CRs with larger rL, i.e.LMA<rL<L (see Eq. (38)) we obtain:\n\u0000outer\u0019\u000f1=2\nw\nL1=2V1=2\nA: (54)\nFor superAlfv ´enic turbulence at scales less than lA, by expressing MAfrom Eq. (20) and substituting it in Eq. (47), we obtain\n\u0000super\u0019\u000f1=2\nsuper\nV1=2\nAr1=2\nL: (55)\nFormally, the above expression coincides with the expression for the damping by subAlfv ´enic strong turbulence given by Eq.\n(52). Nevertheless, the subAlfv ´enic turbulence demonstrates the significant reduction of the cascading rate compared to the\ntransAlfv ´enic turbulence. On the contrary, the superAlfv ´enic strong MHD turbulence corresponds to a significant increase of\ndissipation rate in comparison with the transAlfv ´enic case. Thus, for the same injection scale Land the same injection velocity\nVL, the damping of Alfv ´en waves depends on the magnetization of media. At a lower magnetization, e.g., for superAlfv ´enic\nturbulence, the damping of Alfv ´en waves is more efficient than that at a higher medium magnetization, i.e. for the subAlfv ´enic\ncase.\nAs we discussed earlier, in superAlfv ´enic turbulence, the long Alfv ´en waves with \u0015larger thanlA=LM\u00003\nAinteract with\nhydrodynamic turbulence and the corresponding damping rate is\n\u0000hydro\u0019\u000f1=3\nhydro\nr2=3\nL; (56)\nwhere the hydrodynamic dissipation rate is \u000fhydro\u0019V3\nL=L.\nBelow we present a few more forms of presenting the damping rates that we obtained above. For instance, it could be sometimes\nuseful to rewrite the expressions given by Eq. (30) and (42) in terms of \u0015max;s given by Eq. (35). We remind the reader that the\nphysical meaning of \u0015max;s is the longest wavelength that still interacts with strong turbulent cascade. Then,\n\u0000subA;s\u0019VA\nL\u0012\u0015max;s\nrL\u00131=2\n; rL<\u0015max;s; (57)\nand\n\u0000subA;w\u0019VA\nL\u0012\u0015max;s\nrL\u00132=3\n; rL>\u0015max;s: (58)\nIt is easy to see that Eq. (57) demonstrates that the damping by strong MHD turbulence \u0000subA;s happens faster than the Alfv ´en\ncrossing rate of the injection scale eddies. In the case of weak turbulence, Eq. (58) demonstrates that \u0000subA;w is slower than the\nabove rate.\n4.TURBULENT DAMPING OF ALFV ´EN WA VES GENERATED IN THE GLOBAL SYSTEM OF REFERENCE\nThe turbulent damping of Alfv ´en waves generated by streaming CRs is an important special case of turbulent damping as the\nstreaming instability induces Alfv ´en waves that are aligned with the local direction of magnetic field. Another case arises if we\nconsider the damping of a flux of Alfv ´en waves generated by an extended source. The difference between the two cases is that\nin the latter setting the waves are generated irrespectively to the local direction of magnetic field. Therefore, such Alfv ´en waves\nshould be viewed in the global system of reference related to the mean magnetic field. As a result, our earlier treatment of the\nAlfv ´en wave damping by MHD turbulence should be modified.\n4.1. Case of Strong SubAlfv ´enic turbulence10\nConsider an Alfv ´en wave generated at an angle \u0012\u001d\u000eB=B with respect to the global mean magnetic field. In this situation it\nis natural to disregard the dispersion of angles that arises from magnetic wandering induced by turbulence.6To distinguish these\ntwo cases we use sin\u0012instead of sin\u0012xin Eq. (33). In this case the perpendicular scale of eddies that the waves interact with is\ngiven by:\nx\u0019\u0015\nsin\u0012: (59)\nFor strong turbulence the rate of the wave damping is equal to the turnover rate of subAlfv ´enic eddies. Therefore using Eq. (59),\nwe find\n\u0000subA;s;\u0012\u0019VAM4=3\nAsin2=3\u0012\n\u00152=3L1=3: (60)\nThis provides the non-linear damping rate of an Alfv ´en wave moving at the angle \u0012with respect to the mean field.\nUsing the expression of weak turbulent cascading rate \u000fw(see Eq. (10)), one can write:\n\u0000subA;s;\u0012\u0019\u000f1=3\nwsin2=3\u0012\n\u00152=3: (61)\nThe turbulent damping given by Eq.(61) is applicable to\nlminsin\u0012<\u0015<LM2\nAsin\u0012; (62)\nwherelminis the perpendicular damping scale, and LM2\nA=ltrans is the transition scale from strong to weak MHD turbulence.\nNaturally, the adopted approximation \u0012\u001d\u000eB=B fails if the wave is launched parallel to the mean magnetic field. The\ndirections of the local magnetic field deviates from the mean field and this makes the actual \u00120different from zero. In the global\nsystem of reference the dispersion is dominated by the magnetic field variations presented at the injection scale (see Cho et al.\n2002). Therefore\n\u00120\u0019BL\nB\u0019MA: (63)\nSubstituting this into Eq. (60) we get\n\u0000subA;s; 0\u0019\u000f1=3\nwM2=3\nA\n\u00152=3: (64)\nThe above expression is different from Eqs. (30) and (52). The difference stems from the different properties of Alfv ´en waves\ngenerated in the local system versus the global system of reference. The damping rate in Eq. (64) is applicable to the range of\nwavelength\nlminMA<\u0015<LM3\nA; (65)\nthe latter result trivially follows from Eqs.(62) and (63).\n4.2. The case of Weak SubAlfv ´enic turbulence\nFor weak subAlfv ´enic turbulence, in the case \u0012\u001d\u000eB=B , one should use Eq. (59) to relate \u0015to the scale of perpendicular\nmotions that the wave strongly non-linearly interacts with. To obtain the damping rate, Eq. (41) should be used:\n\u0000weak;global;\u0012\u0019VAsin\u0012M2\nA\n\u0015\u0019\u000f1=2L1=2sin\u0012\nV3=2\nA\u0015; (66)\nwhere we use the weak cascading rate \u000fw. The range of wavelength for this type of damping is\nLM2\nAsin\u0012<\u0015<LM Asin\u0012: (67)\nThe last inequality is obtained by substituting the maximal perpendicular eddy scale LMAforxin Eq.(59).\nIn the case of Alfv ´en wave propagation along the mean magnetic field, one should use Eq. (63) to get\n\u0000weak;global; 0\u0019VAM3\nA\n\u0015\u0019VA\u000f3=4L3=4\n\u0015V5=4\nA: (68)\nUsing Eqs.(63) and (67), we find the range of wavelength that is subject to the turbulent damping:\nLM3\nA<\u0015<LM2\nA: (69)\n6In the case\u0012\u0018\u000eB=B , one should average the final expressions over the \u0012dispersion that arises from magnetic field wandering.11\n4.3. Other cases\nAfter illustrating the difference of non-linear damping for waves generated in the local reference system of magnetic field and\nin the global reference system of the mean field, we can provide results for other cases. More detailed discussion was presented\nin Lazarian (2016). For instance, for superAlfv ´enic turbulence, there is\n\u0000super;global;\u0012\u0019VAMAsin2=3\u0012\n\u00152=3L1=3; (70)\nwhere in superAlfv ´enic turbulence angle \u0012varies from one turbulent eddy of size lAto another. As a result, the correspond-\ning averaging over such changing directions should be performed. For the random distribution of the relevant directions, the\ncorresponding geometric factor is hsin2=3\u0012i= 3=5.\nOn scales larger than lA, MHD turbulence is marginally affected by magnetic fields. As a result, no difference between local\nand global frames is present in terms of Alfv ´en wave damping. This difference also disappears for the damping by turbulent\nfluctuations at the injection scale.\n4.4. Summary of main results in Sections 3 and 4 on turbulent damping\nSome of our results for non-linear turbulent damping of Alfv ´en waves in different turbulence regimes are summarized in Table\n1. We show results relevant both to the damping of waves in the local system of reference, e.g. corresponding to the waves\ngenerated by streaming instability (fifth column in Table 1 with the name “Instability damping rate”), and the damping of waves\ngenerated by external sources parallel to the mean magnetic field (sixth column in Table 1 with the name “Wave damping rate).\nThe table illustrates that the rate of damping and the ranges of wavelengths for which damping is applicable are very different\nfor the two situations. At the first glance, this seems strange. However, the difference stems from the fact that in the case of\nstreaming instability the waves are aligned with the local magnetic field, while the waves generated by an extended source are\nsent parallel to the mean magnetic field.\nWe did not cover in Table 1 the general case of Alfv ´en waves generated at an arbitrary angle relative to the mean magnetic\nfield, as well as damping of Alfv ´en waves by outer-scale turbulence. It is also necessary to stress again the important role of\nweak turbulence for the suppression of streaming instability at low MA. While the weak turbulence is frequently disregarded due\nto its limited inertial range [LM2\nA;L], it can affect CR streaming for rLin the range [LM4\nA;LMA], which can be extensive for\nsufficiently small MA.\n5.ION-NEUTRAL COLLISIONAL DAMPING OF STREAMING INSTABILITY\nIn the presence of partial ionization, an additional effect of damping by ion-neutral collisions becomes important. This effect\nwas discussed originally by Kulsrud & Pearce (1965) for Alfv ´en waves. The damping of turbulent motions in partially ionized\ngas was recently summarized in Xu & Lazarian (2017).\nIn the presence of neutrals, a slippage between them and ions induces the dissipation. In a mostly neutral medium, at wave\nfrequencies !=VAkkless than the neutral-ion collisional frequency \u0017ni, both species move together and the dissipation is\nminimal. As the wave frequency increases, not all neutrals get the chance to collide with ions and the relative motions of ions\nand neutrals induce significant dissipation. For strongly coupled ions and neutrals, the ion-neutral collisional (IN) damping rate\nis (Kulsrud & Pearce 1969)\n\u0000IN=\u0018nV2\nAk2\nk\n2\u0017ni; (71)12\nwhere\u0018n=\u001an=\u001a, and\u001anand\u001aare the neutral and total mass densities. For weakly coupled ions and neutrals with !=VAikk>\n\u0017in, whereVAiis the Alfv ´en speed in ions and \u0017inis the ion-neutral collisional frequency, there is\n\u0000IN=\u0017in\n2: (72)\nWe note that both turbulent and wave motions are subject to the IN damping. Strong Alfv ´enic turbulence injected in the strong\ncoupling regime cannot cascade into the weak coupling regime due to the severe damping effect (Xu et al. 2015, 2016).\nIN damping is sensitive to the ionization fraction and becomes weak at a high ionization fraction. For strongly coupled ions\nand neutrals with VAkk<\u0017in,\u0000INis still given by Eq. (71). For decoupled ions with VAikk>\u0017in, there is (Xu et al. 2016)\n\u0000IN=\u0017ni\u001fV2\nAik2\nk\n2\u0002\n(1 +\u001f)2\u00172\nni+V2\nAik2\nk\u0003; (73)\nwhere\u001f=\u001an=\u001aiand\u001aiis the ion mass density. Furthermore, when neutrals are also decoupled from ions with VAikk> \u0017ni,\nthe above expression is reduced to Eq. (72). Because of the weak damping effect, Alfv ´enic cascade in a highly ionized medium\nis not dissipated by IN damping (Xu & Lazarian 2022).\nNaturally, to understand whether turbulent damping or IN damping is more important for damping the streaming instability,\n\u0000INshould be compared with the turbulent damping rate \u0000that we provided earlier. This comparison has been recently carried\nout in detail by Xu & Lazarian (2022). Here we selectively review some of their results.\nIn a weakly ionized interstellar medium, e.g., molecular clouds, CR-driven Alfv ´en waves are likely in the weak coupling regime\nwith\nVAi\nrL\u0017in\u00192\u0002103\u0010B0\n10\u0016G\u00112\u0010nH\n100cm\u00003\u0011\u00003\n2\u0010ne=nH\n10\u00004\u0011\u00001\n2\u0010ECR\n1GeV\u0011\u00001\n\u001d1; (74)\nwhereB0is the mean magnetic field strength, neandnHare number densities of electrons and atomic hydrogen, and ECRis\nthe CR energy. As already mentioned above, strong Alfv ´enic turbulence injected at a large scale in the strong coupling regime is\nseverely damped and its cascade cannot persist in the weak coupling regime. Therefore, there is\n\u0000<\u0000IN=\u0017in\n2: (75)\nSo the damping of streaming instability in a weakly ionized medium is dominated by IN damping.\nIn a highly ionized interstellar medium, e.g., the warm ionized medium, CR-generated Alfv ´en waves are still in the weak\ncoupling regime and have\nVAi\nrL\u0017ni= 7:6\u0002103\u0010B0\n1\u0016G\u00112\u0010ni\n0:1cm\u00003\u0011\u00003\n2\u0010ECR\n1GeV\u0011\u00001\n\u001d1: (76)\nTo have the turbulent damping dominate over IN damping, there should be\n\u0000\n\u0000IN=\u0000\n\u0017in\n2>1; (77)\nwhich can be rewritten as\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\u0010L\n100pc\u00111\n3\u0010ECR\n1GeV\u00111\n3(78)\nfor superAlfv ´enic turbulence, where niandnnare the number densities of ions and neutrals, and\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\u0010L\n100pc\u00111\n4\u0010ECR\n1GeV\u00111\n4(79)\nfor subAlfv ´enic turbulence. We see that the condition in Eq. (78) is naturally satisfied for superAlfv ´enic turbulence. In a highly\nionized medium, as the IN damping is weak, streaming instability is predominantly damped by the turbulent damping.13\nFigure 1. The damping time-scale \u0000\u00001of Alfv ´en waves that are injected at kk= 10 in 3D MHD turbulence, where the parallel direction is\nchosen with respect to the mean magnetic field. In one approach the Alfv ´en wave energy Ewdecays in the turbulent medium over the time scale\n\u001c1= ln(E(t1)=E(t2))=(t2\u0000t1). The values of \u001c1are given by triangular symbols. In the other approach the wave energy is continuously\ninjected atkk= 10 until it reaches a saturation level Ew. The corresponding damping time scale is given by \u001c2=Ew=\u000fdriving , where\u000fdriving\nis the wave energy injection rate. \u001c2is denoted by diamond symbols. The two measurements are both consistent with k\u00002=3scaling. From Cho\n& Lazarian 2022.14\n6.NUMERICAL TESTING OF TURBULENT DAMPING OF ALFV ´EN WA VES\nNumerical testing of Lazarian (2016) is essential in a variety of regimes. By using 3D MHD turbulence simulations (Cho et\nal. 2002), the results of numerical testing on turbulent damping of externally driven Alfv ´en waves are presented in Figure 1. The\nobserved scaling is consistent with Lazarian (2016) predictions, but inconsistent with FG04 prediction.\nThe reason for this difference arises from the global reference frame adopted in the numerical experiment. Launching of Alfv ´en\nwaves with respect to the local direction of magnetic field is complicated in turbulent fluid. Therefore, the testing presented in\nFigure 1 was carried out with Alfv ´en waves launched with respect to the mean magnetic field. This is the setting corresponding\nto turbulent damping of Alfv ´en waves generated in the global system of reference that we considered in §4. As a result, the\nnumerical simulations confirmed the scaling of inverse of damping rate \u0000\u00001, i.e., damping time scale, which is measured at\ndifferent\u0015as\u00152=3\u0018k\u00002=3\nk. This result is different from the prediction of \u0000\u00001\u0018k\u00001=2\nkof streaming instability in FG04 for\ntransAlfv ´enic turbulence and in Lazarian (2016) for the strong Alfv ´enic turbulence part of the cascade for a wide range of MA.\nNumerical testing on turbulent damping of streaming instability in the local reference frame requires a more complicated setup\nand has not been performed so far.\n7.ASTROPHYSICAL IMPLICATIONS\n7.1. Propagation of CRs\nFor decades the study on CR propagation was performed within a simple model, the so-called “leaky box model” (see Longair\n2011). In this model Galactic CRs propagate freely within the partially ionized disk of the Galaxy. The Alfv ´en waves experience\ndamping in the partially ionized gas (Kulsrud & Pearce 1969, Lithwick & Goldreich 2001, Xu et al. 2016, 2017) and thus the\nstreaming instability is suppressed. On the contrary, in fully ionized plasmas of the Galactic halo, the damping of Alfv ´en waves is\nsignificantly reduced and the streaming instability is present. Therefore, in this classical simplistic picture that ignores turbulence,\nGalactic CRs stream freely through the Galactic disk and are scattered backwards in the Galactic halo.\nThis classical “leaky box model” is problematic, as it is well known now that the Galactic disk is not fully filled with partially\nionized gas. In fact, a significant fraction of the Galactic disk material is warm ionized gas (McKee & Ostriker 1977, Draine\n2011). Therefore, CRs cannot zoom through the Galactic disk due to the streaming instability.\nFG04 quantified the idea of turbulent damping of streaming instability mentioned in Yan & Lazarian (2002) and came to a para-\ndoxical conclusion by applying their theory to the propagation of CRs in the Galaxy. By assuming homogeneous transAlfv ´enic\nturbulence in the Galaxy, they found significant turbulent damping of streaming instability and thus poor confinement of CRs.\nThis would entail problems with explaining e.g., the observed isotropy of CRs and their residence time in the Galaxy.\nIn Lazarian (2016) the gist of the “leaky box model” was preserved, but instead of damping by ion-neutral collisional friction,\nthe study appealed to the turbulent damping of streaming instability in the Galactic disk and proposed a “turbulent leaky box\nmodel”. Different from FG04, by considering inhomogeneous turbulence properties in the Galaxy and the strong MAdependence\nof turbulent damping, they found that the damping by weak subAlfv ´enic turbulence is marginal in the Galactic halo and thus CRs,\neven at high energies, can still be confined by streaming instability.\nIn a recent study by Xu & Lazarian (2022), they identified the important role of turbulent damping of streaming instability in\nthe warm ionized medium (WIM). Fig. 2 shows the diffusion coefficient Dof streaming CRs. The MAdependence comes from\nboth turbulent damping of streaming instability and wandering of turbulent magnetic field lines. In particular, the smaller Din\nsuperAlfv ´enic turbulence is caused by the tangling of turbulent magnetic fields, which results in an effective mean free path lA\nof the CRs streaming along turbulent magnetic fields (Brunetti & Lazarian 2007).\nTheMA-dependent diffusion of CRs is important for a realistic modeling of inhomogeneous CR diffusion in the Galaxy (Xu\n2021). The actual values of MAin the Galaxy can be measured from observations using a newly developed gradient technique\n(Lazarian et al. 2018, see also Xu & Hu 2021) or with more traditional magnetic field and turbulent velocity measurements.\n7.2. Launching of winds and heating\nWhile the damping of Alfv ´en waves by turbulence is an accepted process in the field of CR research, we would like to point\nout that the turbulent damping of Alfv ´en waves can be responsible for many fundamental astrophysical processes. For instance,\ndifferent processes of damping were discussed for heating of stellar corona by Alfv ´en waves, as well as for launching of stellar\nwinds (see Suzuki & Inutsuka 2005, Verdini et al. 2005, Evans et al. 2009, Vidotto & Jatenco-Pereira 2010, Verdini et al. 2010,\nSuzuki 2015). It is clear that the turbulent damping of Alfv ´en waves can be very important in these settings. More recently,\nlaunching galactic winds by turbulent damping of the Alfv ´en waves generated by galactic activity was considered in Suzuki &\nLazarian (2017). Accounting for the dependence of turbulent damping on MAis important for the quantitative modeling of the\nprocess. A similar process is important for launching winds from other types of active disk systems, e.g. circumstellar disks.15\n100101102ECR [GeV]1026102710281029D [cm2 s-1]MA = 0.7MA = 1.4 ECR1.1\nFigure 2. Diffusion coefficient vs. ECRof streaming CRs in super and subAlfv ´enic turbulence in the WIM. From Xu & Lazarian (2022).\nApart from launching galactic winds by turbulent damping of Alfv ´en waves generated by the galaxy, the turbulent damping\nof streaming instability also plays a very important role in coupling CRs and magnetized galactic matter. The pressure of CRs\nin galactic settings is significant and it can modify interstellar dynamics. Galactic winds driven by CRs present an important\nexample of this modification.\nIn general, the importance of galactic winds is easy to understand. For galaxies of the Milky Way luminosity, about 20 percent\nof baryons are accounted for when matching the observed luminosity to the halo mass function. Observing absorption lines in\nspectra of background quasars testifies for the efficient expulsion of galactic baryons from the galaxies. In fact there is evidence\nthat galaxies with significant star formation can drive mass outflows up to 10 times the rate of star formation (Brand-Hawthorn\net al. 2007).\nNumerical simulations have demonstrated that CRs indeed influence the generation of global outflows and the local structure of\nthe interstellar medium (ISM) (see Ruszkowski et al. 2017). The exact properties of the simulated outflows depend sensitively on\nhow CR transport is modeled. Recent simulations by Holguin et al. (2019) employed Lazarian (2016) model of turbulent damping\nand obtained the results that differ significantly from the earlier modeling in e.g., Ruszkowski et al. (2017). The difference\nstemmed from the fact that the earlier calculations employed the model by FG04, which is only applicable to transAlfv ´enic\nturbulence, i.e. MA= 1. However, the actual MAof gas can vary significantly in simulations.\nThe results of the numerical simulations in Holguin et al. (2019) are presented in Figure 3. Some of the implications include,\nfist of all, when turbulent damping of CR streaming instability is included, there is an increase of star formation rate, and the\nincrease is more significant at a higher level of turbulence. The reason is that the turbulent damping increases the average CR\nstreaming speed. This allows CRs to leave the dense mid-plane, reducing the pressure support from CRs to the gas. As a result,\nthe gas in the disk collapses and stars form more efficiently. Furthermore, the higher efficiency of star formation results in more\nCRs produced in the mid-plane. The increased streaming speed of CRs leads to a more extended CR distribution away from the\nmid-plane. It is also important that the escape of CRs from the dense regions allows them to interact with lower-density gas. This\nwidens the gas distribution in height and accelerates the gas to form CR-driven galactic winds.\nIn addition, the theory of Alfv ´en wave damping by turbulence suggests that Alfv ´en waves can propagate across longer distances\nin highly magnetized regions of solar atmosphere (small MA) compared to the regions with higher MA. This prediction can be\nobservationally tested. This effect should be accounted in both modelling of solar wind launching and modelling of plasma\nheating. For instance, it is likely that the turbulent damping can be important in order to explain the observed “unexpected”\ndamping of Alfv ´en waves in the regions above the Sun’s polar coronal holes (Hahn et al. 2012).\n8.SUMMARY\nAlfv ´en waves are damped in turbulent media and the damping depends on the Alfv ´en Mach number MAof the turbulence. At\nthe same wavelength, the wave damping depends on whether the waves are generated in the local reference system of magnetic\neddies by the CR streaming or they are injected at an angle relative to the large-scale mean magnetic field from an extended\nastrophysical source. The latter is, e.g., the case of the Alfv ´en waves arising from magnetic reconnection, or oscillations in16\nni[ cm-3]\nncr[ cm-3]5\n-505\n0\n-5-1-111-11-11\nFigure 3. Simulations of the galactic ISM evolution in the presence of star formation and CR driven outflows. The figure shows the gas ( ni)\nand CR (ncr) density slices \u00065kpc alongzdirection perpendicular to the midplane obtained in two simulations over time 200 Myr. The\nCR streaming is affected by turbulent damping of streaming instability with the turbulent velocity \u001b= 10 km/s. The results obtained in the\nabsence of turbulent damping on the left side of each pair of plots are clearly different from those with turbulent damping on the right side. The\ndistribution of both gas and CRs is more extended in the presence of turbulent damping. From Hoguin et al. (2019).\naccretion disks and stellar atmospheres. The difference in their damping rates arises from the difference between the local and\nglobal systems of reference where the Alfv ´en waves are generated.\nThe dependence of damping rate on the wavelength \u0015of the Alfv ´en waves in the local system of reference is \u0015\u00001=2, as opposed\nto a stronger dependence \u0015\u00002=3for the waves in the global reference system.\nThe turbulent damping also depends on whether Alfv ´en waves interact with weak or strong Alfv ´enic turbulence. For MA<1,\nthe turbulence from the injection scale Lto the scaleLM2\nAis weak and is strong at smaller scales. Weak turbulence can play an\nimportant role in turbulent damping of streaming instability driven by high-energy CRs at a small MA.\nIn a partially ionized gas, the turbulent damping still dominates the damping of streaming instability when the ionization\nfraction is sufficiently high, e.g., in the warm ionized medium (Xu & Lazarian 2022). In star burst galaxies, the ionization\nfraction is low and the ion-neutral collisional damping can be more important (e.g., Krumholz et al. 2020).\nThe turbulent damping of streaming instability has important implications on propagation of CRs in the Galaxy, star formation,\ncoupling between CRs and magnetized gas and thus driving galactic winds. In addition, the turbulent damping of Alfv ´en waves\nresults in heating of the medium and transfer of the momentum from Alfv ´enic flux to the medium. The latter is also important\nfor launching winds.17\nACKNOWLEDGMENTS\nThe research is supported by NASA TCAN 144AAG1967. 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A., et al. 2014,\nNature, 515, 85"    },    {        "title": "1509.01487v1.Damped_transverse_oscillations_of_interacting_coronal_loops.pdf",        "content": "arXiv:1509.01487v1  [astro-ph.SR]  4 Sep 2015Astronomy&Astrophysics manuscriptno.ms c/ci∇clecopy∇tESO2021\nJune27,2021\nDamped transverseoscillations ofinteracting coronal loo ps\nRoberto Soler1,2& Manuel Luna3,4\n1Departament de Física,Universitat de les IllesBalears, E- 07122 Palmade Mallorca, Spain.\n2Institut d’Aplicacions Computacionals de Codi Comunitari (IAC3), Universitat de les Illes Balears, E-07122 Palma de Mallor ca,\nSpain.\n3Institutode Astrofísicade Canarias,E-38200 La Laguna, Te nerife, Spain.\n4Departamento de Astrofísica,Universidad de La Laguna, E-3 8206 La Laguna, Tenerife,Spain.\ne-mail:roberto.soler@uib.es, mluna@iac.es\nReceived XXX;accepted XXX\nABSTRACT\nDamped transverse oscillations of magnetic loops are routi nely observed in the solar corona. This phenomenon is interp reted as\nstanding kink magnetohydrodynamic waves, which are damped by resonant absorption owing to plasma inhomogeneity acros s the\nmagnetic field. The periods and damping times of these oscill ations can be used to probe the physical conditions of the cor onal\nmedium. Some observations suggest that interaction betwee n neighboring oscillating loops in an active region may be im portant\nand can modify the properties of the oscillations compared t o those of an isolated loop. Here we theoretically investiga te resonantly\ndamped transverse oscillations of interacting non-unifor m coronal loops. We provide a semi-analytic method, based on the T-matrix\ntheory of scattering, to compute the frequencies and dampin g rates of collective oscillations of an arbitrary configura tion of parallel\ncylindrical loops. The e ffect of resonant damping is included in the T-matrix scheme in the thin boundary approximation. Analytic\nand numerical results inthe specific case of twointeracting loops are givenas anapplication.\nKey words. Magnetohydrodynamics (MHD) —Sun: atmosphere —Sun: corona —Sun: oscillations — Waves\n1. INTRODUCTION\nTransverse oscillations of magnetic loops in the solar coro na\nare under intense research since the first observational rep orts\nby theTransition Region And Coronal Explorer (TRACE) (see,\ne.g., Nakariakovetal. 1999; Aschwandenetal. 1999). Large -\namplitude coronal loop oscillations are usually excited af -\nter energetic events as, e.g., solar flares, coronal mass eje c-\ntions, or low coronal eruptions (see Zimovets&Nakariakov\n2015). Based on magnetohydrodynamic (MHD) wave theory\n(e.g., Nakariakov&Verwichte 2005), transverse loop oscil la-\ntions have been interpreted as standing kink MHD waves.\nKink MHD modes are nearly incompressible waves, mainly\ndriven by magnetic tension, and responsible for global tran s-\nverse motions of the flux tube (see, e.g., Edwin& Roberts\n1983; Goossensetal. 2009, 2012). A relevant feature of larg e-\namplitude loop oscillations is that they are strongly dampe d. It\nhas been shown that resonant absorption, caused by plasma in -\nhomogeneityinthedirectionperpendiculartothemagnetic field,\nisanefficientdampingmechanismofkinkMHDwavesincoro-\nnal loops (see, e.g., Ruderman&Roberts 2002; Goossensetal .\n2002). Due to resonant absorption, the energy from the globa l\nkink motion of the flux tube is transferred to small-scale, un re-\nsolved rotational motions around the nonuniform boundary o f\nthe tube (see, e.g., Terradasetal. 2006; Goossenset al. 201 4;\nSoler&Terradas 2015). As a result of this process, the globa l\nkink oscillation of the coronal loop is quickly damped in tim e.\nTheinterestedreaderisreferredtoGoossenset al.(2011), where\nthe theoryandapplicationsof resonantwavesin the solar at mo-\nspherearereviewed.\nObservations often show that neighboring oscillating loop s\ninanactiveregioninteractwitheachotherandexhibitcoll ectivebehaviour(e.g.,Schrijver&Brown2000;Verwichteetal.20 04;\nSchrijveret al. 2002; White et al. 2013). Interaction betwe en\nloops can modify the properties of their transverse oscilla tions\ncompared to those of the classic kink mode of an isolated loop .\nTherefore,advancedmodelsdescribingcoronallooposcill ations\nshould take into account interactions within loop systems. A\nnumber of works have studied collective transverse oscilla tions\nin Cartesian geometry (e.g., Díaz etal. 2005; Díaz &Roberts\n2006; Lunaetal. 2006; Arreguiet al. 2007, 2008). In cylindr i-\ncal geometry, Lunaetal. (2008) numerically investigated t rans-\nverse oscillations of two cylindrical loops, and Ofman (200 9)\nperformed numerical simulations in the case of four interac t-\ning loops. Concerning analytical works in cylindrical geom e-\ntry, Lunaet al. (2009, 2010) used the T-matrix theory of scat -\ntering (see, e.g., Twersky 1952; Waterman 1969; Bogdan 1987 ;\nKeppensetal.1994)toinvestigatetransverseoscillation softwo\nandthreeloops(Lunaetal. 2009)andofbundlesofmanyloops\n(Lunaetal. 2010) in the β=0 approximation,where βrefersto\ntheratioofthegaspressuretothemagneticpressure.Soler et al.\n(2009) later extended the method of Lunaet al. (2009, 2010)\nby incorporating gas pressure and longitudinal flows and stu d-\nied collective oscillationsof flowingprominencethreads. These\nworksshowedthatloopinteractiona ffectsthepropertiesoftheir\noscillations. Lunaet al. (2008, 2009) obtained that a syste m of\ntwo loops of arbitrary radii supports four kink-like collec tive\nmodes.Theshiftofthecollectivemodefrequencieswithres pect\ntotheindividualkinkfrequenciesoftheloopsissignifican twhen\nthe distance between loops is small (of the order of the loop r a-\ndius)andwhen loopshavesimilar densities. Conversely,th e os-\ncillatingloopsshowlittleinteractionwhentheyarefarfr omeach\notherandwhentheir densitiesaresubstantiallydi fferent.Onthe\nother hand, VanDoorsselaereetal. (2008) and Robertsonet a l.\nArticlenumber, page 1of 11A&Aproofs: manuscript no. ms\n(2010) used a different method based on bycilindrical coordi-\nnates to study transverse oscillations of two pressure-les s loops\ninthethintube(TT)approximation.Ofthefourkink-likeco llec-\ntivemodesobtainedbyLunaet al.(2009)intheT-matrixtheo ry,\nonly two different modes remain in the TT approximation con-\nsidered by VanDoorsselaereet al. (2008) and Robertsonetal .\n(2010).\nConcerning the damping of the oscillations, Arreguietal.\n(2007,2008)investigatedresonantlydampedoscillations oftwo\nslabs, while Terradaset al. (2008) numerically studied the res-\nonant damping of transverse oscillations of a multi-strand ed\nloop. Those works showed that the process of resonant damp-\ning is not compromised by the irregular geometry of a real-\nistic loop model and still produces the e fficient attenuation of\nglobal transverse oscillations. The damping of transverse oscil-\nlations of two cylindrical loops was analytically investig atedby\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014),whoconsideredtheTTapproximationandusedbycili n-\ndrical coordinates. Results obtained with bycilindrical c oordi-\nnates should be treated with caution when the distance betwe en\nloops is small. Geometrical e ffects intrinsically associated to\nthe bycilindrical coordinates may produce unphysical resu lts.\nOur purpose is to use the T-matrix method of Lunaet al. (2009,\n2010) to investigate resonantly damped oscillations of bun dles\nof loops. The present paper is partially based on unpublishe d\nresults included in Soler (2010)1. The effect of resonant ab-\nsortion in the Alfvén continuum is incorporated to the T-mat rix\nscheme by using the methodthat combinesthe jump conditions\nof the perturbationsat the resonance position with the so-c alled\nthinboundary(TB)approximation(see,e.g.,Sakuraietal. 1991;\nGoossenset al. 1992). A similar method has previously been\nused by Keppens (1995) to investigate absorption of acousti c\nwaves. We provide a general analytic theory, which is valid f or\nbundles of many transversely nonuniform parallel loops of a r-\nbitrary radii. Specific results in the case of two loops are ob -\ntained and compared to those given in Robertson&Ruderman\n(2011) and Gijsen &VanDoorsselaere (2014). Our results are\nalso compared to those of Arreguiet al. (2007, 2008) obtaine d\ninCartesian geometry.\nThispaperisorganizedasfollows.Section2containsthede -\nscription of the equilibrium configurationand the basic gov ern-\ning equations. The general analytic T-matrix theory of scat ter-\ning to compute the frequencies and damping rates of collecti ve\nlooposcillationsisgivenin Section3.Later,thespecificc ase of\ndamped oscillations of two loops is discussed both analytic ally\nandnumericallyin Section 4. Finally,someconcludingrema rks\naregiveninSection5.\n2. MODEL AND GOVERNINGEQUATIONS\nOurequilibriumconfigurationiscomposedof Nstraightandpar-\nallelmagneticcylindersoflength Lembeddedinauniformcoro-\nnalplasma.Theendsofthemagnetictubesarefixedattworigi d\nwalls representing the solar photosphere. We set the z-direction\nto be along the axes of the tubes. The magnetic field is straigh t\nalongthe z-direction,namely B=Bˆez,whereBisaconstantev-\nerywhere.Weusesubscripts‘i’and‘e’toreferto,ingenera l,the\ninternalregionofthetubesandtheexternalplasma,respec tively.\nThe subscript or superscript ‘j’ is used to refer to a particu lar\nloop.Wedenoteby Rjtheradiusofthejthtube.Thedistancebe-\ntweenthecentersofthejthandj’thloopsis djj′.Wedenotebyρj\n1The full text of Soler (2010) is available at\nhttp://www.uib.es/depart/dfs/Solar/thesis_robertoso ler.pdfFig. 1.Sketch of the equilibrium configuration in the specific case o f\ntwotransverselynon-uniform coronal loops.\ntheinternaldensityofthejth tube,while ρedenotestheexternal\ndensity,i.e.,thedensityofthecoronalenvironment.Inou rmodel\nρjandρeareconstants.Thereisa transverselynonuniformtran-\nsitional layer surroundingeach magnetictube in which the d en-\nsity continuously varies from the internal density, ρj, to the ex-\nternal density,ρe. The thicknessesof the non-uniformboundary\nlayer of the jth cylinter is lj. A sketch of the equilibriumconfig-\nuration in the case of two magnetic tubes ( N=2) is given in\nFigure1.\nWe adopt theβ=0 approximation,where βrefersto the ra-\ntio of the thermal pressure to the magnetic pressure. This is an\nappropriateapproximationtoinvestigatetransversewave sinthe\nsolar corona. In the β=0 approximation,the ideal MHD equa-\ntions governing linear perturbations superimposed on the s tatic\nequilibriumstate are\nρ∂2ξ\n∂t2=1\nµ(∇×b)×B, (1)\nb=∇×(ξ×B), (2)\nwhereξis the plasma Lagrangian displacement, bis the mag-\nnetic field Eulerian perturbation, ρis the density, and µis the\nmagneticpermittivity.\nWe assume the temporal dependence of perturbations as\nexp(−iωt), whereωis the oscillation frequency. In the case of\ntransverselynonuniformtubes,theglobaltransverseosci llations\narequasi-modeswhosefrequencyiscomplexowingtodamping\nby resonant absorption, i.e., ω=ωR+iωI, whereωRandωI\nare the real and imaginary parts of the frequency, respectiv ely.\nThereal partofωisrelatedtotheperiodandtheimaginarypart\ncorrespondstothedampingrateoftheoscillations.We cons ider\nthat the oscillating flux tubes are line-tied at the photosph ere,\nwhich acts as a perfectly reflecting wall in this model owing t o\nits largedensity comparedto the coronaldensity.Hence, we as-\nsume perturbations to be proportional to exp (ikzz), withkzthe\nlongitudinalwavenumber.Forstandingoscillations, kzgivenby\nkz=nπ\nL,withn=1,2,... (3)\nWeshallrestictourselvestothefundamentalmodeofoscill ation,\nsowe take n=1.\nArticlenumber, page 2of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nIn the regions with constant density, Equations (1) and (2)\ncanbereducedtofollowingequation,\n∇2\n⊥P′+k2\n⊥P′=0, (4)\nwhereP′=B·b/µis the total pressure Eulerian pertubation\nand the subscript⊥refers to the direction perpendicular to the\nmagnetic field. Thus, ∇2\n⊥denotes the perpendicular part of the\n∇2operator.Inturn,thequantity k⊥playstheroleoftheperpen-\ndicularwavenumberandisdefinedas\nk2\n⊥=ω2−ω2\nA\nv2\nA, (5)\nwhereω2\nA=k2\nzv2\nAis the square of the Alfvén frequency and\nv2\nA=B2/µρis the square of the Alfvén velocity. We stress that\nEquation(4)isonlyvalidintheregionswithconstantdensi ty,so\nitdoesnotapplywithinthenonuniformboundariesoftheloo ps.\n3. T-MATRIXTHEORY OF SCATTERING\nEquation (4) is the two-dimensional Helmholtz Equation. To\nsolve Equation (4) we use the scattering theory in its T-matr ix\nformalism (see, e.g., Waterman 1969). In the solar context, the\nT-matrixtheoryhaspreviouslybeenusedtoinvestigatethe scat-\ntering and absorption properties of bundles of magnetic flux\ntubes (e.g. Bogdan&Zweibel 1985; Bogdan&Cattaneo 1989;\nKeppenset al. 1994; Keppens 1995, among others). Lunaetal.\n(2009, 2010) used of this techniqueto compute the eigenmode s\nof systems of magnetic tubes. Because of the inhomogeneityo f\nthe tubes in the transverse direction, the modes with freque n-\ncies between the internal Alfvén frequencies of the loops an d\nthe external Alfvén frequency are resonant in the Alfvén con -\ntinuum.As a result, the oscillationsare dampedby resonant ab-\nsorption. The effect of resonant absorption was not considered\nbyLunaet al.(2009,2010).Herewe extendtheirtheoryto con -\nsiderresonantdamping.\n3.1. SolutionsintheInternaland ExternalPlasmas\nWe use local polar coordinates associated to the jth loop. We\ndenoteby rjandϕjtheradialandazimuthalcoordinates,respec-\ntively, of the coordinate system whose origin is located at t he\ncenter of the jth tube. We can define an equivalent coordinate\nsystem in each tube. In this local coordinate system, the sol u-\ntion to Equation (4) in the internal region of the jth tube can be\nexpressedas\nP′j\ni=∞/summationdisplay\nm=−∞Aj\nmJm/parenleftig\nk⊥jrj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n, (6)\nwheremis the azimuthal wavenumber, Jmis the usual Bessel\nfunctionofthefirstkindoforder m,andAj\nmareconstants.Unlike\nthecase ofisolatedtubes(see,e.g.,Edwin&Roberts1983), the\nsolutionisnotentirelydescribedbyasinglevalueof m.Because\nof interaction between tubes, the values of mare coupled. For\ntransverse, kink-like oscillations the dominant terms in t he ex-\npansion are those with m=±1, but the contribution from other\nm’s is not negligible unless the tubes are far from each other\n(Lunaet al.2009).\nThesolutionto Equation(4)in theexternalregioniswritte n\nusingtheprincipleofsuperposition,whichisapplicablet olinearwaves.Thetotalexternalsolutioniscomputedbyaddingthe net\ncontributionsofall fluxtubes,namely\nP′\ne=/summationdisplay\njP′j\ne, (7)\nwhereP′j\neis the net contribution of the jth tube to the external\nsolution.Thekeyideabehindthescatteringtheoryisthatt heso-\nlutionofEquation(4)intheexternalplasmacanbedecompos ed\ninto several fieldswith different physical meanings, namely the\ntotal,exciting,andscatteredfields .Herewegiveanoverviewof\nthe method. Interested readers are referred to Lunaetal. (2 009,\n2010)forextensiveexplanations.\nIn the external plasma, the total field associated to the jth\ncylindercanbeexpressedas\nP′j\ntotal=∞/summationdisplay\nm=−∞/bracketleftig\nαj\n1mH(1)\nm/parenleftig\nk⊥erj/parenrightig\n+αj\n2mH(2)\nm/parenleftig\nk⊥erj/parenrightig/bracketrightig\nexp/parenleftig\nimϕj/parenrightig\n,(8)\nwhereH(1)\nmandH(2)\nmare the usual Hankel functions of the first\nand second kind, respectively, and αj\n1mandαj\n2mconstants. The\nfirst term on the right-hand side of Equation (8) represents o ut-\ngoing waves from the jth tube, and the second term represents\nincomingwavestowardthejthtube.Importantly,we notetha t\nP′\ne/nequal/summationdisplay\njP′j\ntotal. (9)\nThe reason for this inequality is that the outgoing wave asso -\nciated to a particular tube contributes as an incoming wave f or\nall the other tubes. In other words,/summationtext\njP′j\ntotalis not the net exter-\nnalsolution.Toovercomethisproblem,the totalfield associated\nto the jth cylinder, P′j\ntotal, is decomposed into a scattered field ,\nP′j\nscat,andaexcitingfield ,P′j\nexcit.Conceptually,the scatteredfield\nrepresentsthe actual contributionof the varioustubes to t he net\nexternal solution, whereas the exciting field can be understood\nasthecross-talkmechanismresponsibleforinteractionbe tween\nfluxtubes(seedetailsin, e.g.,Bogdan&Cattaneo1989).\nThe full net external solution, P′\ne, is defined so that it corre-\nsponds to the sum of the scattered fields associated to all tubes,\nnamely\nP′\ne=/summationdisplay\njP′j\nscat. (10)\nConversely,the excitingfield associatedtothejthtubeisdefined\nas the difference between the full net contribution and the scat-\nteredfield ofthejthtube,namely\nP′j\nexcit=P′\ne−P′j\nscat=/summationdisplay\nn/nequaljP′n\nscat. (11)\nWaterman (1969) introduced the T-matrix operator, Tj, which\nlinearlyrelatesthe scatteredandexcitingfieldsas\nP′j\nscat=TjP′j\nexcit. (12)\nBogdan (1987) showed that for cylindrical scatterers the T-\nmatrixisdiagonal,andKeppenset al.(1994)gaveanexpress ion\nofits elements,namely\nTj\nmm=1\n21−αj\n1m\nαj\n2m, (13)\nArticlenumber, page 3of 11A&Aproofs: manuscript no. ms\nwhereαj\n1mandαj\n2mare the same constants the appear in Equa-\ntion(8).WecanuseEquation(13)toeliminate αj\n1mandwriteall\ntheexpressionsintermsof αj\n2malone.Withthehelpoftheselast\nformulae, and after some algebraic manipulations using wel l-\nknown properties of the Bessel functions, we can rewrite Equ a-\ntion(8) as\nP′j\ntotal=∞/summationdisplay\nm=−∞2αj\n2m/bracketleftig\nJm/parenleftig\nk⊥erj/parenrightig\n−Tj\nmmH(1)\nm/parenleftig\nk⊥erj/parenrightig/bracketrightig\nexp/parenleftig\nimϕj/parenrightig\n,\n(14)\nfrom where it is straightforward to identify both exciting a nd\nscatteredfields,namely\nP′j\nexcit=∞/summationdisplay\nm=−∞2αj\n2mJm/parenleftig\nk⊥erj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n, (15)\nP′j\nscat=−∞/summationdisplay\nm=−∞2αj\n2mTj\nmmH(1)\nm/parenleftig\nk⊥erj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n.(16)\nFinally,weusetheexpressionof P′j\nscatintoEquation(10)toarrive\nat thetotalnet solutionintheexternalplasma,namely\nP′\ne=−/summationdisplay\nj∞/summationdisplay\nm=−∞2αj\n2mTj\nmmH(1)\nm/parenleftig\nk⊥erj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n. (17)\nEquations (6) and (17) formally describe the total pressure\nperturbation in the interior and in the exterior of the tubes , re-\nspectively.However,we recall that these expressionsdo no t ap-\nply in the nonuniform boundary layers. The T-matrix element s,\nTj\nmm, contain the information about how the solutions are con-\nnectedacrossthenon-uniformboundariesofthe tubes.\n3.2. T-matrixElementsinthe ThinBoundaryApproximation\nAtthisstageweincorporatethee ffectofthenonuniformbound-\narylayers.Inthenonuniformlayerofthejthtubetheglobal wave\nmodesare resonantin the Alfvén continuumat the resonantpo -\nsition,rj=rA,j, where the global oscillation frequency matches\nthe local Alfvén frequency. The resonant position, rA,j, is de-\nfined through the resonant condition ω2=k2\nzv2\nA(rA,j). We use\nthe TB approximation and restrict ourselves to lj/Rj≪1. The\nTB approximation assumes that the jump of the perturbations\nacross the resonant layer is the same as their jump across the\nwhole nonuniform layer. Thus, the connection formulae of th e\nwave perturbations across the resonance are used as jump con -\nditions for the total pressure and the Lagrangian displacem ent\nat the boundariesof the tubes. This method and its applicati ons\nhave been reviewed by Goossenset al. (2011). The TB approx-\nimation within the formalism of the T-matrix theory has prev i-\nouslybeenusedbyKeppenset al.(1994)andKeppens(1995).\nGeneral expressions of the connection formulae for the\nperturbations across the resonant layer can be found in, e.g .,\nSakuraiet al.(1991).Inlocalcoordinates,theconnection formu-\nlae forthe total pressure Eulerianpertubation, P′, and the radial\ncomponentoftheLagrangiandisplacement, ξr,atrj=rA,jare\n/bracketleftbigP′/bracketrightbig=0,/bracketleftbigξr/bracketrightbig=−iπm2/r2\nA,j\n|ρ∆A|jP′, (18)\nwhere [X]=Xe−Xjdenotes the jump of the quantity Xacross\ntheresonantlayerand |ρ∆A|jisdefinedas\n|ρ∆A|j=ρ(rA,j)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndrj/parenleftig\nω2−k2\nzv2\nA/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglerA,j=ω2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ\ndrj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglerA,j, (19)wherewe have used the resonantcondition ω2=k2\nzv2\nA(rA,j). Af-\nterimposingthejumpconditionsgiveninEquation(18),weo b-\ntainboththeT-matrixelementsandthedispersionrelation ofthe\ncollectivemodes.\nOnthe onehand,theT-matrixelementsare\nTj\nmm=k⊥e\nρe(ω2−k2zv2\nAe)J′\nm(k⊥eRj)\nJm(k⊥eRj)−k⊥j\nρj/parenleftig\nω2−k2zv2\nAj/parenrightigJ′\nm(k⊥jRj)\nJm(k⊥jRj)+iπm2/r2\nAj\n|ρ∆A|\nk⊥e\nρe(ω2−k2zv2\nAe)H(1)\nm′(k⊥eRj)\nH(1)\nm(k⊥eRj)−k⊥j\nρj/parenleftig\nω2−k2zv2\nAj/parenrightigJ′m(k⊥jRj)\nJm(k⊥jRj)+iπm2/r2\nAj\n|ρ∆A|\n×Jm/parenleftig\nk⊥eRj/parenrightig\nH(1)\nm/parenleftig\nk⊥eRj/parenrightig, (20)\nwheretheprime′denotesthe derivativeoftheBessel orHankel\nfunctionwithrespecttoitsargument.Intheabsenceofreso nant\ndamping,Equation (20) consistently reverts to Equation (1 7) of\nLunaet al.(2009).\nOntheotherhand,theconstants αj\n2msatisfyanalgebraicsys-\ntemofequations,namely\nαj\n2m+/summationdisplay\nj′/nequalj∞/summationdisplay\nm′=−∞αj′\n2m′Tj′\nm′m′H(1)\nm′/parenleftig\nk⊥edjj′/parenrightig\nexp/parenleftig\ni(m′−m)ϕjj′/parenrightig\n=0,\n(21)\nfor−∞<m<∞, whereϕjj′is the angle formed by the vec-\ntor positions of the centers of the two tubes with respect to t he\nglobal reference frame. Equation (21) is system with an infin ite\nnumberofalgebraicequations.Theconditionthatthereisa non-\ntrivial solution, i.e., the determinant formed by the coe fficients\nset equal to zero, provides us with the dispersion relation ( see\ndetails in Lunaetal. 2009, 2010). The solution of the disper -\nsion relation is the complex frequencyof the damped collect ive\nquasi-mode. The imaginary part of the frequency is the reso-\nnant damping rate. In the case of Lunaet al. (2009, 2010), the\nsolution frequency was real because of the absence of resona nt\ndamping.\nThe dispersion relation is a complicated expression and has\nto be solved by numericalmethods.For practicalcomputatio nal\npurposes, in Equation (21) the indices mandm′must be trun-\ncated into a finite number. The truncation term must be large\nenough to avoid inaccuracy of the solutions. Then, the dispe r-\nsion relation can be solved using standardnumericalroutin esto\nfindthe rootsoftranscendentalequations.\n4. APPLICATION TODAMPEDOSCILLATIONS OF\nTWOLOOPS\nWerecallthattheT-matrixtheorydescribedintheprevious Sec-\ntionisvalidforanarbitrarynumberofloops.Hereweapplyt hat\nmethod to investigate damped transverse oscillations in a l oop\nsystem composed of two magnetic tubes alone. First we derive\napproximate expressions of the period and damping time in th e\ncase of two identical thin tubes. Later, we consider two tube s\nwith different properties and arbitrary radii and perform a nu-\nmericalstudy.\n4.1. Approximatesolutionsfortwo identicalthintubes\nWe consider the paradigmatic case of two tubes with identica l\nproperties. We denote by ρi,R, andlthe internal density, the\nradius, and the thickness of the boundary layer of both tubes ,\nArticlenumber, page 4of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nrespectively. The external density remains ρe. The distance be-\ntweenthecentersofthetwo tubesis d.\nTo obtain an approximate dispersion relation for kink-like\nmodes we assume that the contribution from the azimuthal\nwavenumbers m=±1 to the collectiveoscillation is muchmore\nimportant than the contributions from other values of m. Thus,\nwe takem,m′=±1 in Equation (21) and roughly neglect the\nother terms. As pointed out by Lunaet al. (2009), the couplin g\nbetweenthedifferentvaluesof mgetsstrongerasthe separation\nbetween the cylinders decreases. In terms of the parameters of\nthemodel,thismeansthatourapproximationappliestothec ase\noflargeseparations,i.e., R/d≪1.\nThe system defined in Equation (21) now becomes an al-\ngebraic system of four equations for the unknowns α1\n2,−1,α1\n2,1,\nα2\n2,−1, andα2\n2,1. We obtain the dispersion relation from the con-\nditionthatthereisanon-trivialsolutionofthesystem,na mely\n/parenleftig\nH(1)\n0(k⊥ed)2−H(1)\n2(k⊥ed)2/parenrightig2T4\n11\n−2/parenleftig\nH(1)\n0(k⊥ed)2+H(1)\n2(k⊥ed)2/parenrightig\nT2\n11+1=0, (22)\nwhere we used the property that T11=T−1−1according to\nthe symmetry relations of the Bessel and Hankel functions (s ee\nAbramowitz&Stegun 1972). Equation (22) is an equation for\nω, which is enclosed in the definitions of T11andk⊥e. To go\nfurther analytically, we perform a series expansion of the H an-\nkelfunctionsforsmall argumentsandretainthefirsttermal one.\nThis would be approximately valid for thin tubes. After some\nalgebraicmanipulations,Equation(22) canberecastas\nT11±iπ\n4(k⊥ed)2≈0. (23)\nNow, we look for a simplified expression of T11. We use the\nTT approximation,i.e., R/L≪1. In Equation (20) we perform\nan asymptoticexpansionofthe Bessel and Hankel functionsf or\nsmall arguments.Equation(20)becomes\nTmm≈ −1\nρe(ω2−k2zv2\nAe)−1\nρi(ω2−k2zv2\nAi)+iπm/R\nω2\nR|dρ/dr|R\n1\nρe(ω2−k2zv2\nAe)+1\nρi(ω2−k2zv2\nAi)−iπm/R\nω2\nR|dρ/dr|R\n×iπ\n22m(k⊥eR)2m\nm!(m−1)!, (24)\nwhere we also assumed rA≈Rfor simplicity. We take m=\n1, substitute Equation (24) into Equation (23), and arrive a t the\nfollowingexpression,\nρi/parenleftig\nω2−k2\nzv2\nAi/parenrightig\nδ+ρe/parenleftig\nω2−k2\nzv2\nAe/parenrightig\n−iπ\nRρiρe\n|dρ/dr|R/parenleftig\nω2−k2\nzv2\nAi/parenrightig/parenleftig\nω2−k2\nzv2\nAe/parenrightig\nω2=0, (25)\nwheretheparameter δisdefinedas\nδ=1∓/parenleftigR\nd/parenrightig2\n1±/parenleftigR\nd/parenrightig2. (26)\nEquation (25) is the dispersion relation for resonantly\ndampedkinkmodesoftwoidenticaltubesintheTTandTBap-\nproximationsandforlargeseparations.Inthelimitthatth etubes\nare far from each other, R/d→0 and soδ→1. Then, Equa-\ntion (25) consistently reverts to the dispersion relation o f reso-\nnant kink modes in an isolated thin tube (e.g., Goossensetal .\n1992).4.1.1. Frequencyof the Oscillations\nFirst we neglect the presence of the nonuniform boundary lay -\ners and study undamped oscillations. We drop the last term on\nthe left-hand side of Equation (25) and the dispersion relat ion\nsimplifiesto\nρi/parenleftig\nω2−k2\nzv2\nAi/parenrightig\nδ+ρe/parenleftig\nω2−k2\nzv2\nAe/parenrightig\n=0. (27)\nTheexactsolutionis\nω2=ω2\nk\n1∓ζ−1\nζ+1/parenleftigR\nd/parenrightig2, (28)\nwhereζ=ρi/ρeis the density contrast and ωkis the frequency\nof the kink mode in an individual thin tube (Edwin&Roberts\n1983),namely\nω2\nk=ρiv2\nAi+ρev2\nAe\nρi+ρek2\nz. (29)\nEquation (28) can be compared to Equation (51) of\nVanDoorsselaereetal.(2008).Bothexpressionsagreeifth epa-\nrameterEof VanDoorsselaereetal. (2008) is approximated as\nE=exp[−2arccosh(d\n2R)]≈/parenleftigR\nd/parenrightig2, which is valid for R/d≪1.\nIn agreementwith VanDoorsselaereetal. (2008), in the TT ap -\nproximationwe obtain two kink-likesolutionscorrespondi ngto\nthe+and−signs in Equation (28). As explained by Lunaet al.\n(2009), there are actually four kink-like modes beyond the T T\napproximation. For arbitrary radii, the low-frequency sol ution\n(that with the+sign in Equation (28)) splits in the SxandAy\nmodes of Lunaet al. (2009), while the high-frequency soluti on\n(that with the−sign in Equation(28)) becomestheir SyandAx\nsolutions.\nFrom Equation (28) we compute the period of the oscilla-\ntions,P=2π/ω,as\nP=Pk/radicaligg\n1∓ζ−1\nζ+1/parenleftbiggR\nd/parenrightbigg2\n, (30)\nwithPk=2π/ωkthe period of the kink mode of an isolated\ntube.Forlargeseparationsthetubesfeel little interacti on.Then,\nthe periods of the two solutions consistently tend to that of the\nkinkmodeofanisolatedtube.\n4.1.2. DampingRate\nHereweincorporatethee ffectofresonantdampingandconsider\nthefull expressionofthedispersionrelation(Equation(2 5)).To\nobtain an approximate expression of the damping rate, we sub -\nstituteω=ωR+iωIinEquation(25)andassumeweakdamping,\ni.e.,|ωI|≪|ωR|. Then,we are allowed to neglect terms of order\nω2\nIand higher orders. After some algebraic maniputations (not\ngivenhere for the sake of simplicity) we arrive at an express ion\nfortheratioωI/ωR, namely\nωI\nωR=π\n21\nRρiρe\nρiδ+ρe1\n|dρ/dr|R/parenleftig\nω2\nR−k2\nzv2\nAi/parenrightig/parenleftig\nω2\nR−k2\nzv2\nAe/parenrightig\nω2\nR.(31)\nTosimplifyEquation(31)weconsiderasmoothmonotonicpro -\nfile for the density in the nonuniformboundariesso that we ca n\nexpressthederivativeofthe densityprofileas\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ\ndr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleR=π2\n4Fρi−ρe\nl, (32)\nArticlenumber, page 5of 11A&Aproofs: manuscript no. ms\nwithFa factor that depends on the form of the profile. For in-\nstance,F=4/π2for a linear profile and F=2/πfor a sinu-\nsoidal profile. In addition, we approximate ωRby the value of\nthe frequencyin the undampedcase (Equation(28)) and subst i-\ntutethe expressionof δ(Equation(26))to arriveat\nωI\nωR≈−1\n2π1\nFl\nRζ−1\nζ+11±/parenleftigR\nd/parenrightig2\n1∓ζ−1\nζ+1/parenleftigR\nd/parenrightig2/bracketleftigg\n1−/parenleftbiggR\nd/parenrightbigg4/bracketrightigg\n. (33)\nFor a linear density profile, Equation (33) becomes Equa-\ntion(86)ofRobertson&Ruderman(2011)iftheapproximatio n\nexp[−2arccosh(d\n2R)]≈/parenleftigR\nd/parenrightig2valid for R/d≪1 is again per-\nformedintheirexpression.\nNow, we define the damping time as τD=1/|ωI|and use\nEquation (33) to obtain the expression for the ratio τD/P. By\nkeepingtermsupto( R/d)2,theexpressionforthedampingratio\nis\nτD\nP≈/parenleftbiggτD\nP/parenrightbigg\nk/bracketleftigg\n1∓2ζ\nζ+1/parenleftbiggR\nd/parenrightbigg2/bracketrightigg\n, (34)\nwhere\n/parenleftbiggτD\nP/parenrightbigg\nk=FR\nlζ+1\nζ−1, (35)\nis the damping ratio of the individual kink mode (see, e.g.,\nRuderman&Roberts 2002; Goossensetal. 2002). As for indi-\nvidual kink modes, the damping ratio of collective modes is i n-\nversely proportional to the thickness of the nonuniform lay er.\nEquation (34) shows that the solution with the +sign is more\nefficientlydampedbyresonantabsorptionthanthesolutionwit h\nthe−sign.Thisdifferenceinthedampingratesofthetwomodes\nqualitatively agrees with the results of Robertson&Ruderm an\n(2011).\n4.2. NumericalStudy\nHereweperformanumericalstudybeyondtheapproximatelim -\nitsstudiedbefore.Todoso,weconsiderthefulldispersion rela-\ntionobtainedfromEquation(21)byusingthegeneralexpres sion\nof the T-matrix elements (Equation (20)). The truncated dis per-\nsion relation is then numerically solved. Convergence test s of\ntheresultshavebeenperformedtomakesurethatthetruncat ion\nvalue of the azimuthal series is large enoughfor the errorin the\nsolutions to be negligible. In short, we found that for kink- like\nmodes the truncation value, namely mt, has no important e ffect\nunlessmt<5andthetwoloopsarenexttoeachother( d/R≈2).\nWhenmt>5and/ortheloopsarefarfromeachother,weessen-\ntiallyobtainthattheresultsareindependentof mt.Wehaveused\nmt=30 in all computationsgivenhere, which reducesthe error\nand assures the excellent converge of the solutions even whe n\nd/R≈2.\nThe numerical method follows a two-step procedure. First,\nwe solve the dispersion relation in the absence of nonunifor m\nboundary layers. In that case, the solution is a real frequen cy\nthat corresponds to an approximation to ωR. This approximate\nvalue ofωRis used to compute the resonance positions and the\nderivative of the density profile at the resonances. We assum e\nsinusoidal density profiles in the nonuniform boundary laye rs.\nThen, we use these parameters to solve the complete dispersi on\nrelation,whichnowincludesthee ffectofresonantdamping.The\nfrequency obtained from the second run is complex, so that it\nprovides us with a more accurate value of ωRand also gives us\nthevalueofωI.4.2.1. Identicalloops\nWe initially study the case of two identical tubes. We set the\nCartesiancoordinatessystemsothatthe xy-planeisperpendicu-\nlar to the axes of the loops. In that plane, the centers of the t wo\nloops are located on the x-axis. We use the notation introduced\nbyLunaetal. (2008)to denotethefourkink-likemodesprese nt\nin a two-loop configuration. The modes are labeled as Sx,Ax,\nSy, andAy, whereSandAdenotesymmetric or anti-symmetric\nmotions of the two loops, respectively, and the subscripts xand\nyindicate the main direction of polarization of the oscillat ions\nin the coordinates system defined above. The eigenfunctions of\nthesefourmodesinthecaseofloopswithoutnonuniformboun d-\narylayerscanbefoundinFigure2ofLunaet al.(2008).\nFigure 2 shows the dependenceof ωRand the ratio|ωI|/ωR\non the separation between loops, d/R, for a particular set of\nparameters given in the caption of the figure. We note that the\ncurves corresponding to the SxandAymodes, and those of the\nAxandSymodes,arealmostsuperimposedbecauseweareinthe\nTTregime(weused L/R=100).Concerningthebehaviorof ωR,\nthe frequencies of the four solutions tend to the kink freque ncy\nofanisolatedloopinthelimit d/R≫1.Conversely,thesmaller\nthe separation between loops, the more important the splitt ing\nof the collective frequencies with respect to the kink frequ ency\nof an isolated loop. Figure 2(a) can be compared to Figure 3 of\nLunaet al. (2008) and with Figure 4 of VanDoorsselaereet al.\n(2008).However,unlikeinthosepreviousworkswenotethat in\nourcasethefrequenciesofthehigh-frequencysolutions( Axand\nSy)donottendtotheexternalAlfvénfrequencywhen d/R→2.\nThe reason for this di fference is probably that the AxandSy\nmodes are strongly damped when d/R→2, and this fact has\nsomeimpactonthereal partofthefrequencyaswell.\nOntheotherhand,Figure2(b)showsthatthehigh-frequency\nmodes (AxandSy) are more efficiently damped by resonant\nabsorption than the low-frequency modes ( SxandAy). This\nresult agrees with that of Robertson&Ruderman (2011) for\nlarge separations. However, the behavior of |ωI|/ωRobtained\nhere for small separations is dramatically di fferent from that\nof Robertson& Ruderman (2011). They found that the oscil-\nlations become undamped in the limit d/R→2, while we\nfind that the modes remain damped. Although the analysis of\nRobertson&Ruderman (2011) is mathematically correct, we\nfind no physical reason for which the oscillations should be\nundamped when the loops are close to each other. Our com-\nputations show that the damping of the low-frequency modes\n(SxandAy) is roughly independent on d/R, whereas the damp-\ning of the high-frequency modes ( AxandSy) gets stronger as\nthe separation between loops is reduced. As pointed out by\nRobertson&Ruderman (2011) and Gijsen &VanDoorsselaere\n(2014),thephysicalsignificanceoftheresultsobtainedwi thby-\ncilindrical coordinates should be treated with caution whe n the\nseparationbetweenthetubesissmall.\nPanels(c)and(d)ofFigure2showthesameresultsaspanels\n(a) and (b), respectively, but now log10(l/R−2)is used in the\nhorizontal axes. These additional graphs are included to sh ow\nin more detail the behavior of the solutions obtained with th e\nT-matrixmethodforsmall separationsbetweentubes.\nWe haveoverplottedinFigure2theanalyticapproximations\nofωRand|ωI|/ωRgiven in Equations (28) and (33). These ap-\nproximationswere derivedin the limit d/R≫1 and reasonably\nagree with the numericalsolutions when d/R/greaterorsimilar3. As expected,\nthe approximationsdo not work well for small separations. T he\nanalytic approximationswere derived considering the cont ribu-\ntionsfrom m=±1 alone,but the contributionof high m’s to the\nArticlenumber, page 6of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nFig.2.Numerical resultsinthe case oftwoidentical coronal loops . (a)Dependence of ωR/ωA,eond/R,whereωA,e=kzvA,eistheexternal Alfvén\nfrequency. (b)Dependence of |ωI|/ωRond/R.The meaningof thevarious linesisindicatedwithinthepan els. Thedashed lines correspond tothe\nanalytic approximations in the limit d/R≫1 (Equations (28) and (33)). We have used ζ=5,L/R=100, andl/R=0.2. Panels (c) and (d) show\nthe same results as panels (a)and (b),respectively, but asf unction of log10(d/R−2).\nFig.3.Numericalresultsinthecase oftwoidentical coronal loops . (a)Dependence of ωR/ωA,eonL/R,whereωA,e=kzvA,eistheexternal Alfvén\nfrequency. (b)Dependence of |ωI|/ωRonL/R.Themeaning ofthevarious linesisindicatedwithinthefigu re.Wehave usedζ=5,l/R=0.2, and\nd/R=2.5. Wenote thatthe horizontal axes of both panels are inlogar ithmic scale.\nfullsolutionisimportantforsmallseparationbetweenloo ps(see\nLunaet al. 2009).\nFigure 3 displays the dependence of the solutions on L/R.\nThis figure is included to show that the almost degenerate cou -\nplesSx–AyandAx–Sysplit into four different solutions for\nsmall values of L/Rbeyondthe TT regime. We point out, how-ever, that the impact of the value of L/Ron the solutions is\nnot relevant when realistic values of this parameter are con sid-\nered. The TT limit used by Robertson&Ruderman (2011) and\nGijsen&VanDoorsselaere(2014) isthereforeadequate.\nNow we plot in Figure 4 the dependenceof ωRand|ωI|/ωR\non the nonuniform layer thickness, l/R. We consider a small\nArticlenumber, page 7of 11A&Aproofs: manuscript no. ms\nseparation, namely d/R=2.5, and the remaining parameters\nare the same as in Figure 2. Consistently, when l/R=0 the\nmodes are undamped. The real part of the frequency of the\nlow-frequency modes is almost independent of l/R, while their\n|ωI|/ωRis roughly linear with l/R. Conversely, the real part of\nthe frequency of the low-frequency modes decreases when l/R\nincreases,andtheir |ωI|/ωRisonlylinearwith l/Rforsmallval-\nues of this parameter. As discussed before, the behavior of t he\nlow-frequency modes ( SxandAy) is similar to that of the kink\nmode of an isolated loop. However, the high-frequency modes\n(AxandSy) seem to be more a ffected by the interaction be-\ntween loops and show a somewhat di fferent behavior when l/R\nincreases.We notethatbecauseoftheTBapproximationwear e\nrestrictedtoconsidersmall valuesof l/R.\nIt is useful to relate the present results with those of\nArreguietal. (2007, 2008), who studied the damping of trans -\nverse oscillations of two nonuniformslabs. They found that the\nratio|ωI|/ωRcorresponding to the symmetric kink mode of\nthe two slabs is weakly dependent of the separation between\nthe two slabs (see Arreguietal. 2007, their Figure 4). In tur n,\nArreguietal.(2008)foundthattheanti-symmetrickinkmod eof\nthetwoslabsismoree fficientlydampedthanthesymmetrickink\nmode (see their Figure 6). The symmetric and anti-symmetric\nkink modes of two slabs would be equivalent to the SxandAx\nmodesoftwocylinders.Thus,ourresultsincylindricalgeo metry\nareconsistentwithpreviousfindingsin Cartesiangeometry .\nIt is also convenient to consider the physical arguments of\nArreguietal. (2007, 2008) to explain why the high-frequenc y\nmodes damp more e fficiently than the low-frequency modes.\nArreguietal. (2007, 2008) related the e fficiency of the damp-\ning with the magnitude of the total pressure perturbationwi thin\nthe resonant layers. According to Andrieset al. (2000), the ef-\nficiency of the resonant coupling between the global transve rse\nmodeandtheAlfvéncontinuummodesisproportionaltotheto -\ntalpressureperturbationsquared.Soler(2010)plottedth esquare\nof the total pressure perturbation corresponding to the Sxand\nAxmodes (see his Figures 9.7 and 9.8) and found that, when\nthe quantities are normalized, the perturbation of the Axmode\nreaches a larger value in the resonant layers than that of the Sx\nsolution. This result qualitatively explains why resonant damp-\ning is more efficient for the Axmode than for the Sxmode.\nEquivalently,asimilarreasoninghelpusunderstandthedi fferent\nattenuationofthe SyandAymodes.Nevertheless,amorerobust\nstudy of the process of resonant absorption in two-dimensio nal\nconfigurationswouldbeneededforacompleteunderstanding of\nthedifferentdampingrates(see Russell &Wright2010).\n4.2.2. Non-identicalloops\nHere we consider two loops with di fferent properties and com-\npare our results to those of Gijsen&VanDoorsselaere (2014) .\nWe usesubscripts1and2torefertothetwodi fferentloops.For\nsimplicity,we take R1=R2=RandL/R=100in all following\ncomputations.\nFirst, we assume the same density contrast in the two loops,\nnamelyζ1=ζ2=5, and vary l2/Rwhilel1/Ris kept fixed\ntol1/R=0.2. These results are shown in Figure 5 and can\nbe compared to those already displayed in Figure 4 in the case\nofl1=l2. Importantly, we find that the collective oscillations\nremain damped when l2/R=0. Although this is a rather par-\nticular situation, the results have interestingimplicati ons. When\nl2/R=0 resonant absorption only occurs in the boundarylayer\nof loop #1. However, this is enough for the collective oscill a-\ntions of the two loops to be e fficiently damped. We note that inthe study by Gijsen &VanDoorsselaere (2014) the thicknesse s\nofthe nonuniformlayersare linkedtothe coordinatesystem .\nNow we consider the case of two loops with di fferent den-\nsity contrasts. We fix ζ2=3 and compute the solutions as func-\ntions ofζ1. These results are displayed in Figure 6, where the\nremaining parameters used in the computations are specified in\nthe caption. For consistency, we keep the same notation as be -\nfore to denote the various modes according to the ordering of\ntheirfrequencies,althoughtheydonotrepresenttrulycol lective\noscillations ifζ1/nequalζ2(Lunaet al. 2009). Figure 6 can be com-\nparedtoFigure6ofGijsen &VanDoorsselaere(2014).Tomake\na proper comparison, we note that Gijsen &VanDoorsselaere\n(2014) plotted the signed damping ratio, while here we plot t he\nabsolutevalue.\nConcerning the real part of the frequency (Figure 6(a)), we\nfind the same results as Lunaetal. (2009). When ζ1<ζ2, the\nhigh-frequencymodes are associated to loop #1 alone, where as\nthe low-frequency modes represent individual oscillation s of\nloop #2. The opposite happenswhen ζ1>ζ2. Conversely,when\nζ1≈ζ2the four modes approach and interact in the form of an\n‘avoided crossing’. Only in that case the modes represent tr uly\ncollectiveoscillations(Lunaetal. 2009).\nFigure 6(a) also shows that the high-frequency modes are\nalwayswithintheAlfvéncontinuaofthe twoloops,i.e.,the fre-\nquencies of the AxandSymodes are always larger than ωA,1\nandωA,2and smaller thanωA,e. This is true except in the limit\nζ1→1, where the frequencies of the AxandSymodes tend to\nωA,e. On the contrary, the low-frequency SxandAymodes are\nbelow the Alfvén continuumof loop #1 when ζ1/lessorsimilar2 and below\nthe Alfvén continuum of loop #2 when ζ1/greaterorsimilar5. This result may\nhaveimplicationsforthedampingbyresonantabsorption.\nFigure 6(b) displays the damping ratio of the modes as a\nfunction ofζ1. The result for the high-frequency modes can be\nunderstood as follows. When ζ1→1 the frequencies of the\nhigh-frequencymodestendto ωA,eand,asa consequence,these\nmodes become undamped in that limit. When 1 <ζ1<ζ2, the\ndampingratioincreaseswhen ζ1increases.When1<ζ1<ζ2the\nhigh-frequency modes represent individual oscillations o f loop\n#1. Then, the damping ratio reaches a maximum when ζ1≈ζ2.\nWhenζ1>ζ2the damping ratio saturates to a constant value\nbecause the high-frequencymodes representnow individual os-\ncillationsofloop#2,andthevalueof ζ2isfixedinthecomputa-\ntions.Thebehaviorofthedampingofthehigh-frequencymod es\nagreeswiththatplottedbyGijsen&VanDoorsselaere(2014) in\ntheirFigure6.\nThe overall behavior of the damping ratio of the low-\nfrequency modes displayed in Figure 6(b) also agrees with\nGijsen&VanDoorsselaere (2014). The damping ratio of the\nlow-frequencymodes is roughly constant when ζ1<ζ2and in-\ncreaseswhenζ1>ζ2.Thefactthatthelow-frequencymodesare\nbelow the Alfvén continuumof loop #1 when ζ1/lessorsimilar2 and below\ntheAlfvéncontinuumofloop#2when ζ1/greaterorsimilar5havenoimportant\nimpact on the damping. Again, these results can be understoo d\nby considering that the low-frequency modes are associated to\nloop#2whenζ1<ζ2,whiletheyareassociatedtoloop#1when\nζ1>ζ2.\nThe low-frequency modes computed here do not show the\npronounced minimum of the damping rate seen in the solu-\ntion plotted by Gijsen& VanDoorsselaere (2014) when ζ1≈\nζ2. There are several e ffects that may explain this di fference.\nThe most obvious one is the di fferent geometry considered\nin Gijsen &VanDoorsselaere (2014) and here. Another pos-\nsible explanation is that the density profile in the nonuni-\nform layers used by Gijsen& VanDoorsselaere (2014) is dif-\nArticlenumber, page 8of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nFig.4.Numerical results inthe case of two identical coronal loops . (a) Dependence of ωR/ωA,eonl/R, whereωA,e=kzvA,eis the external Alfvén\nfrequency. (b) Dependence of |ωI|/ωRonl/R. The meaning of the various lines is indicated within the figu re. We have usedζ=5,L/R=100,\nandd/R=2.5.\nFig.5.Numericalresultsinthecaseoftwonon-identicalcoronall oopswith R1=R2=R.(a)Dependence of ωR/ωA,eonl2/R,whereωA,e=kzvA,e\nistheexternalAlfvénfrequency.(b)Dependence of |ωI|/ωRonl2/R.Themeaningofthevariouslinesisindicatedwithinthefigu re.Wehaveused\nζ1=ζ2=5,L/R=100,l1/R=0.2andd/R=2.5.\nFig. 6.Numerical results in the case of two non-identical coronal l oops with R1=R2=R.(a) Dependence of ωR/ωA,eonζ1, whereωA,e=kzvA,e\nistheexternalAlfvénfrequency.(b)Dependence of |ωI|/ωRonζ1.Thedottedlinescorrespond totheAlfvénfrequenciesofth etwoloops,andthe\nmeaning of the remaining lines isindicated withinthe figure . Wehave usedζ2=3,L/R=100,l1/R=l2/R=0.2andd/R=3.\nferent from that used here. A linear density profile is used in\nGijsen &VanDoorsselaere(2014),sothatthederivativeofd en-sity at the resonancepositionis independentof thefrequen cyof\nthemode.Hereweuseasinusoidalprofileandtakeintoaccoun t\nArticlenumber, page 9of 11A&Aproofs: manuscript no. ms\nthat the positionof theresonanceandthe valueof the deriva tive\nof density at the resonance position are functions of the mod e\nfrequency.\nInFigure6(b),thedampingrateofthelow-frequencymodes\nshowsasmallbumparound ζ1≈2,whentherealpartofthefre-\nquency approximately crosses the internal Alfvén frequenc y of\nloop #1. The reason for this bump is that the the low-frequenc y\nmodes intersect with and ‘avoid cross’ the fluting modes that\ncluster toward the internal Alfvén frequency. This bump is a b-\nsent from Figure 6 of Gijsen& VanDoorsselaere (2014) prob-\nably because coupling between kink and fluting modes is not\ndescribedinthe TTapproximation.\n5. CONCLUDING REMARKS\nIn this paper we have extended the analytic T-matrix theory\nof scattering of Lunaet al. (2009, 2010) to investigate reso -\nnantly damped oscillations of an arbitrary configuration of par-\nallel cylindrical coronal loops. After presenting the gene ral the-\nory,we have performeda specific applicationin the case of tw o\nloops. This work is partially based on unpublished results i n-\ncluded in Soler (2010), where collective damped oscillatio ns of\nprominencethreadswerestudied.\nWe have compared our results to those of the papers by\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014). They investigated the damping of collective oscill a-\ntions of two loops in the TT approximation and used a method\nbased on bicylindrical coordinates. In general, the result s of\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014) are in good agreementwith the present results, speci ally\nwhen the separation between loops is large. However, when\nthe separation between the loops is small, i.e., for separat ions\nof few radii, the results of those previous works show impor-\ntant differences compared to the present findings. For instance,\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014) obtained that by decreasing the distance between loo ps,\nthe efficiency of resonant damping is reduced. In their compu-\ntations,bothlow-andhigh-frequencymodesbecomeundampe d\nwhen the loops are in contact. However, this result lacks of a\nphysical explanationand contradictspreviousfindingsin C arte-\nsian geometry (Arreguiet al. 2007, 2008). In our computatio ns,\nwe find that the damping of the high-frequency modes gets\nstronger by decreasing the separation between loops, while the\ndampingof the low-frequencymodesis roughlyindependento f\nthe separation. Our solutions do not become undamped when\nthe two tubes are in contact. Thus, the results obtained here are\nconsistentwithpreviousresultsbyArreguietal.(2007,20 08)of\ncollectiveoscillationsoftwo slabs.\nAlthough the mathematical analysis of\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014) is flawless, their results by may be a ffected by un-\navoidable geometrical problems related to the bicylindric al\ncoordinates when the loops are close to each other. In bicyli n-\ndricalcoordinatestheshapesofnonuniformboundarylayer sare\nnot symmetric and change when the separation between tubes\ndecreases.Thenonuniformlayersgetthickerintheouterpa rtsof\nthetubesandthinnerintheinnerparts.Asalreadymentione dby\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014), these geometrical limitations may lead to unphysi-\ncal results for small separations. The T-matrix method used\nhere is not constrained by the geometrical problems of the\nbicylindrical coordinates. Therefore, we may conclude tha t the\nresults given here are more generally applicable than those ofRobertson&Ruderman (2011) and Gijsen &VanDoorsselaere\n(2014) whentheloopsareclose toeachother.\nBecauseoftheTBapproximationwe wererestrictedtocon-\nsider small values of l/R, i.e.,l/R≪1. The effect of thick\nnonuniform layers could be included into the T-matrix forma l-\nism with the method of Frobeniusused by Soleretal. (2013) in\nthecaseofanisolatedloop.Thiswouldsubstantiallyincre asethe\nmathematicalcomplexityof the problembut, on the otherhan d,\nit would provide a more accurate description of the damping\nof largely nonuniform loops. It has been shown by Soleret al.\n(2014) that the error in the damping rate associated to the us e\nof the TB approximation can be important when the loops are\nlargely non-uniform. Apart from a numerical factor, the dam p-\ning rate in the TB approximation is independent of the specifi c\ndensity profile considered within the nonuniform boundary, but\nthe density profile can have a more important impact when the\nnon-uniformlayersarethick.Inaddition,therealpartoft hefre-\nquency depends on l/Rbeyond the limit l/R≪1. The effect\nof thick nonuniform boundaries on collective loop oscillat ions\ncouldbeexploredinthefuture.\nThe method given here to compute resonantly damped col-\nlective oscillations can have multiple applications in the future.\nFor instance, damped oscillations of a coronal arcade could be\nstudied by modeling the arcade as a long line of parallel loop s.\nAnother interesting application is the investigation of os cilla-\ntionsofloopsformedbymanystrands.AsshownbyLunaet al.\n(2010), the global oscillation of the whole loop would be de-\ntermined by the interaction of the oscillations of the indiv idual\nstrands.Theresonantabsorptionprocessworkingintheind ivid-\nual strands would a ffect the damping on the global loop motion\n(see Terradaset al. 2008). In principle, the presence of mul tiple\nresonances in the system may cause the transverse oscillati ons\nof a multi-strandedloop to damp more quickly than the oscill a-\ntions of an equivalent monolithic loop. This idea could be co n-\nfirmedusingtheT-matrixmethod.Also,asshownbySoleret al .\n(2009),theeffectsofgaspressureandmassflowalongtheloops\ncaneasilybeincludedintheT-matrixformalism,thusexten ding\ntheapplicabilityofthemethod.\nAcknowledgements. We thank Ramón Oliver for useful comments on a draft\nof this paper. R.S. acknowledges support from MINECO throug h a ‘Juan de\nla Cierva’ grant and through projects AYA2011-22846 and AYA 2014-54485-P,\nfromMECDthrough projectCEF11-0012,fromthe‘Vicerector at d’Investigació\ni Postgrau’ of the UIB, and from FEDER funds. M.L. acknowledg es support\nby MINECO through projects AYA2011-24808 and AYA2014-5507 8-P. M.L.is\nalso grateful to ERC-2011-StG 277829-SPIA.\nReferences\nAbramowitz, M.& Stegun, I. 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V.&Nakariakov, V.M. 2015, A&A,577, A4\nArticlenumber, page 11of 11"    },    {        "title": "1903.02802v1.Investigating_optically_excited_THz_standing_spin_waves_using_noncollinear_magnetic_bilayers.pdf",        "content": "Investigating optically-excited THz standing spin waves using noncollinear magnetic\nbilayers\nM.L.M. Lalieu,1,\u0003R. Lavrijsen,1R.A. Duine,2, 1and B. Koopmans1\n1Department of Applied Physics, Institute for Photonic Integration,\nEindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands\n2Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584 CC Utrecht, The Netherlands\n(Dated: November 4, 2021)\nWe investigate optically excited THz standing spin waves in noncollinear magnetic bilayers. Us-\ning femtosecond laser-pulse excitation, a spin current is generated in the \frst ferromagnetic (FM)\nlayer, and \rows through a conductive spacer layer to be injected into the second (transverse) FM\nlayer, where it exerts a spin-transfer torque on the magnetization and excites higher-order stand-\ning spin waves. We show that the noncollinear magnetic bilayer is a convenient tool that allows\neasy excitation of THz spin waves, and can be used to investigate the dispersion and thereby the\nspin wave sti\u000bness parameter in the thin-\flm regime. This is experimentally demonstrated using\nwedge-shaped Co and CoB (absorption) layers. Furthermore, the damping of these THz spin waves\nis investigated, showing a strong increase of the damping with decreasing absorption layer thickness,\nmuch stronger than expected from interface spin pumping e\u000bects. Additionally, a previously un-\nseen sudden decrease in the damping for the thinnest \flms is observed. A model for the additional\ndamping contribution incorporating both these observations is proposed.\nI. INTRODUCTION\nAbout a decade ago, it was discovered that spin cur-\nrents are generated upon femtosecond (fs) laser-pulse ex-\ncitation of a ferromagnetic (FM) thin \flm. This was\n\frst discovered in a collinear magnetic bilayer, in which\nthe laser-induced transfer of angular momentum between\nthe two FM layers was demonstrated by their in\ruence\non the demagnetization dynamics in both layers1. In\nthe years that followed, several experiments have demon-\nstrated the direct measurement of the optically excited\nspin current in a FM/NM (non-magnetic metal) bilayer.\nIn these experiments, the spin current is generated by\nlaser-pulse excitation of the FM, and is detected at the\nouter NM surface2{5. One of the motivations for the re-\nsearch into the laser-pulse-excited spin current is its po-\ntential use in the \feld of spintronics, in which (electri-\ncal) spin currents are already heavily used to manipulate\nmagnetic information in future magnetic data storage\ndevices6,7. The manipulation of the magnetization can\nbe pushed to the ultrafast time scale by using the op-\ntically generated spin currents. This was demonstrated\nin recent years using noncollinear magnetic bilayers, in\nwhich the laser-induced spin current excited in one FM\nlayer was used to exert a spin-transfer torque (STT) on a\nsecond, transversely magnetized, FM layer3,8{10. More-\nover, it has been demonstrated that the optically excited\nspin current is absorbed very locally near the injection\ninterface11, which allowed the excitation of THz stand-\ning spin waves11{13. This shows that in addition to its\ngeneral importance in the \feld of spintronics, the opti-\ncally excited spin currents could also be of high potential\nfor future THz magnonics.\n\u0003Corresponding author: m.l.m.lalieu@tue.nlIn this paper, it is experimentally demonstrated that\nthe noncollinear magnetic bilayer is a convenient tool\nto generate and investigate optically-excited THz spin\nwaves. Using a wedge-shaped absorption layer (Co or\nCoB), it is shown that the dispersion and thereby the spin\nwave sti\u000bness parameter is easily accessible for magnetic\nlayer thicknesses down to a few nanometers. Addition-\nally, the structure allows the investigation of the damp-\ning of the THz spin waves and its dependence on the\n\flm thickness. The measured damping behavior shows\na strong increase of the damping as the layer thickness\ndecreases down to \u001910 nm, which is attributed to the\ninhomogeneous nature of the spin waves. Moreover, a\npreviously unseen reduction of the damping is seen upon\nfurther decrease of the layer thickness. A model describ-\ning the observed damping behavior is proposed. For the\nanalysis of the THz standing spin waves, the e\u000bective\nmagnetization and Gilbert damping parameter (bulk and\ninterface spin pumping contributions) are needed as a\nfunction of the thickness of the absorption layer. These\nproperties are determined using the homogeneous (fun-\ndamental) precession, of which the analysis will be dis-\ncussed \frst.\nII. SAMPLE STRUCTURE AND\nCHARACTERIZATION\nThe basic structure of the noncollinear mag-\nnetic bilayers used in this work is given by\nSi:B(substrate)/Ta(4)/Pt(4)/[Co(0.2)/Ni(0.6)] 4/Co(0.2)\n/Cu(5)/FM IP/Pt(1) (thickness in nm), in which two\ndi\u000berent wedge-shaped in-plane (IP) magnetized (top)\nFM layers are used; a Co wedge ranging from 0 to 20\nnm, and a Co 77B23wedge ranging from 0 to 15 nm.\nThese samples are referred to as the Co and the CoB\nsample in the following. The bottom FM layer is anarXiv:1903.02802v1  [cond-mat.mes-hall]  7 Mar 20192\nout-of-plane (OOP) magnetized Co/Ni multilayer. The\ntwo FM layers are separated by a 5nm-thick Cu spacer\nlayer which allows for the transfer of spin currents and\ndecouples both FM layers. All samples are fabricated\nusing dc magnetron sputtering at room temperature.\nThe measurements are performed using a standard\ntime-resolved magneto-optic Kerr e\u000bect setup in the\npolar con\fguration. The probe and pump pulses have\na spot size of\u001910\u0016m and a pulse length of \u0019150 fs.\nThe pulses are produced by a Ti:sapphire laser with a\nwavelength of 790 nm and a repetition rate of 80 MHz.\nDuring the experiments, the pump pulse excites the\nspin dynamics, and the probe pulse measures the OOP\nmagnetization component of both FM layers. In case of\nthe homogeneous precession measurements, an external\nmagnetic \feld is applied parallel to the sample surface.\nThe e\u000bective magnetization Me\u000bof the IP (absorption)\nlayer at a certain thickness is determined by measuring\nthe frequency fIPof the homogeneous (fundamental) pre-\ncession as a function of the applied magnetic \feld Bapp.\nThe value of Me\u000bis obtained by \ftting the \feld depen-\ndent frequency using the standard Kittel equation for IP\nmagnetized layers,\nfIP=\r\n2\u0019q\nBapp(Bapp+\u00160Me\u000b): (1)\nIn this equation, \rcorresponds to the gyromagnetic ratio.\nThe excitation mechanism of the homogeneous preces-\nsion is the same ultrafast STT mechanism as used for\nthe standing spin wave excitation presented later, and\nis illustrated in the inset of Fig. 1(a). In this mecha-\nnism, a fs laser pulse is used to excite a spin current\nin the OOP (generation) layer. This spin current \rows\nthrough the Cu spacer layer and is injected into the top\nFM layer, exerting a STT on the IP magnetization. As\na result, the IP magnetization is canted slightly OOP,\nwhereafter it starts a damped precession around the IP\napplied magnetic \feld. A more detailed characterization\nand validation of the excitation mechanism can be found\nin Refs.8,11.\nA measurement of the homogeneous precessions in the\nCo sample at a thickness of tCo= 3 nm, and for six dif-\nferent IP magnetic \feld amplitudes, is shown in Fig. 1(a).\nThe background (remagnetization) signal is subtracted,\nand an o\u000bset is added to the signal for clarity. A clear\nincrease in the precession frequency with the applied \feld\namplitude is observed, as is expected from the Kittel re-\nlation. The precessions are \ftted using a damped sine,\nfrom which the precession frequency fIPand the charac-\nteristic damping time \u001care obtained. Using Eq. (1), the\ne\u000bective magnetization at each measured Co thickness is\ndetermined by \ftting the \feld dependent precession fre-\nquency. Figure 1(b) shows the e\u000bective magnetization\nas a function of tCo. The observed thickness dependence\nofMe\u000bresults from an out-of-plane surface anisotropy,\nwhich decreases Me\u000b, and of which the contribution falls\no\u000b ast\u00001\nCo. The obtained thickness dependent Me\u000bis later\nused in the analysis of the THz standing spin waves. The\nKittel \fts also allow the determination of the gfactor us-\ning the \ftted value of \r. For the Co sample, a gfactor\n0 100 200 300 400 500 600-10123456\nAppliedfield\n47.2mT\n65.1mT\n84.0mT\n101.2mT\n118.9mT\n134.4mTMOKE signal(arb. units)\nDelay (ps)\n0 5 10 15 200.00.20.40.60.81.01.2Meff(MA/m)\nCothickne ss(nm)024681012141618200.000.010.020.030.040.050.06Damping\nCothickness(nm)(a)\n(b)FIG. 1. (a) Time-resolved MOKE measurement on the Co\nsample at a Co thickness of 3 nm. The \fgure shows the homo-\ngeneous precession for six di\u000berent magnetic \feld amplitudes.\nThe background (remagnetization) signal is subtracted, and\nan o\u000bset is added for clarity. The inset shows an illustration\nof the precession excitation mechanism, based on the ultra-\nfast laser-induced STT. (b) E\u000bective magnetization Me\u000bas\na function of the Co thickness. The inset shows the Gilbert\ndamping parameter as a function of the Co thickness, in which\nthe damping determined with the di\u000berent magnetic \feld am-\nplitudes are averaged. The red curve represents a \ft to the\ndata using Eq. 3.\nof 2:30\u00060:06 was found, which is similar as found in\nliterature14.\nThe damped sine \fts of the precession data also pro-\nvide the characteristic damping time \u001c. Together with\nthe previously determined value of Me\u000b, the Gilbert\ndamping constant \u000bat each thickness and \feld can be\ndetermined using\n\u000b=\u0014\n\r\u001c\u0012\nBapp+\u00160Me\u000b\n2\u0013\u0015\u00001\n: (2)\nThe damping as a function of the Co thickness is shown in\nthe inset of Fig. 1(b), in which the damping determined\nwith the di\u000berent magnetic \feld amplitudes are averaged.3\nThe damping shows a clear t\u00001\nCobehavior. This thickness\ndependence is known to be the result of spin pumping\ninto neighboring layers15, in this case at the Cu/Co and\nCo/Pt interfaces. The interface spin pumping enhances\nthe damping, and since it is an interface e\u000bect it falls o\u000b\nast\u00001\nCo. The damping as a function of thickness is \ftted\nusing\n\u000b=\u000bbulk+\u000bpump =\u000bbulk+Apump\nt; (3)\nin which\u000bbulkis the (intrinsic) bulk damping, and \u000bpump\nis the interface spin pumping contribution to the damp-\ning. The interface spin pumping amplitude Apump in-\ncludes the contribution of both interfaces. The \ft-\nted values are equal to \u000bbulk= (4:5\u00060:4)\u000110\u00003and\nApump = (1:29\u00060:06)\u000110\u000010m. The value for \u000bbulk\nagrees well with literature values16. The value of Apump\ncan be used to calculate the e\u000bective spin-mixing conduc-\ntance of the interfaces17, but due to the complex nature\nof the used multilayers, this is out of the scope of the\npresented work. Both values are used later when evalu-\nating the damping of the THz standing spin waves. A\nsimilar analysis of the homogeneous precessions for the\nCoB sample is presented in Supplementary Note 1.\nIII. RESULTS\nWith the e\u000bective magnetization and the damping\nof the homogeneous precession mode characterized, the\nTHz standing spin waves can be investigated using the\nsame noncollinear magnetic bilayers. The higher-order\nstanding spin waves are excited using the same time-\nresolved polar MOKE measurement as before. Di\u000berent\nfrom the previous measurements is that there is no ex-\nternal magnetic \feld applied, which is not needed since\nthe standing spin waves are driven by the exchange in-\nteraction. Furthermore, to achieve a better sensitivity of\nthe MOKE signal to the THz spin waves, a quarter-wave\nplate was added to the probe beam11,18.\nAn illustration of the excitation mechanism of the\nstanding spin waves is shown in the inset of Fig. 2(a).\nAs discussed earlier [inset Fig. 1(a)], a short and intense\ntransverse (OOP) spin current is injected into the top IP\nmagnetized layer after the fs laser-pulse excitation. The\nspin current is absorbed very locally near the injection\ninterface11, creating a strong gradient in the OOP mag-\nnetization component in the top layer, as illustrated in\nthe \fgure ( t= 0). This highly non-equilibrium magneti-\nzation state relaxes by the excitation of (damped) higher-\norder standing spin waves, as illustrated for n= 0;1;2\nand 3. In the following, only the \frst-order ( n= 1)\nstanding spin wave is investigated. It is noted, however,\nthat up to the third-order standing spin waves have been\nobserved using a 20 nm thick CoB absorption layer.\nA typical measurement of the \frst-order standing spin\nwave is presented in Fig. 2(a), in which the measure-\nment at a Co thickness of 13 nm is shown. The ob-\nserved dynamics is a superposition of two damped os-\ncillations, which can be separated using a \ft including\n(a)\n(b)0 10 20 30 40 50 600.000.050.100.150.200.250.30MOKE signal(arb. unit.)\nDelay (ps)0.15 THz\n0.08 THz 12 0 3 n = t = 0\n2 4 6 8 101214161820220.00.20.40.60.81.01.21.4CoIPlayer\nCoBIPlayerFrequency(THz)\nFMlayerthickne ss(nm)FIG. 2. (a) Typical precession measurement of \frst-order\nstanding spin wave, measured in the Co sample at a Co thick-\nness of 13 nm. The observed dynamics is a superposition of\ntwo damped oscillations, which are illustrated by the blue and\nblack solid lines in the \fgure. The inset shows an illustration\nof the excitation mechanism of the standing spin waves. (b)\nStanding spin wave frequency as a function of the FM layer\nthickness, for both the Co (black dots) and CoB (blue dots)\nsamples. The red solid lines are \fts to the data using Eq. 4.\ntwo damped sines and a double exponential background\n(red solid line). The two \ftted precessions are illustrated\nby the black (0 :15 THz) and blue (0 :08 THz) solid lines\nin the \fgure. Although the presence of two precessions\ncould be explained by two di\u000berent standing spin wave or-\nders, it turns out that the slower precessions (blue curve)\ncorresponds to an acoustic strain wave traveling along\nthe depth of the multilayer. The acoustic strain wave is\npresent in the polar MOKE measurement due to a lattice-\ndeformation-induced change in the magneto-optical sig-\nnal from the Co/Ni multilayer when the acoustic wave\npasses through it. A more detailed analysis of the acous-\ntic strain wave can be found in Supplementary Note 2.\nThe faster precession, indicated by the black curve,\nbelongs to the (\frst-order) ferromagnetic standing spin\nwave. Measuring this precession at di\u000berent positions4\nalong the Co wedge allows to extract the standing spin\nwave frequency fswas a function of the Co thickness, of\nwhich the result is presented in Fig. 2(b) (black dots). In\nthis \fgure, also the result of the same measurement on\nthe CoB sample is presented (blue dots). The dispersion\nrelation for the standing spin waves is given by11(using\nMe\u000b)\nfsw=\r\n2\u0019\u0014\u0012\nBapp+Dsw\n\r~k2\u0013\n\u0002\u0012\nBapp+\u00160Me\u000b+Dsw\n\r~k2\u0013\u00151=2\n;(4)\nin whichDswcorresponds to the spin wave sti\u000bness, and\nthe wave number kof thenthorder standing spin wave\nis given by\nk=\u0019n\nt: (5)\nThe red solid lines in Fig. 2(b) are \fts to the data using\nEq. (4). The \fts are done using the earlier obtained g\nfactor and thickness dependent Me\u000b. Furthermore, with\nBapp= 0 andn= 1, this leaves Dswas the only \ftting\nparameter. As can be seen, the measured standing spin\nwave frequencies are well described by the dispersion rela-\ntion. In case of the Co sample, however, a deviation from\nthe dispersion curve can be seen around a Co thickness of\ntCo\u00197\u000010 nm. The exact reason for this is not known.\nIt is noted, however, that a change in crystallographic\nstructure has been reported from the fcc structure for\ntCo<6 nm to the hcp structure for bulk16. In case of\nthe CoB top layer, which is known to be amorphous, such\na crystallographic change should not be present. Looking\nat the measured dispersion for CoB (blue dots), it can be\nseen that there is no such deviation from the \ftted dis-\npersion curve. This suggests that the deviation seen for\nthe Co top layer could indeed be related to the change\nin crystallographic structure. A more elaborate investi-\ngation should be performed (e.g., using XRD) to con\frm\nthis hypothesis.\nThe \ftted values of the spin wave sti\u000bness for the\nCo and CoB samples are Dsw= 882\u00068 meV \u0017A2and\nDsw= 582\u00067 meV \u0017A2, respectively. In the case of Co,\nexperimental values for thin \flms ( <140 nm) range be-\ntween 250\u0000520 meV \u0017A219. Surprisingly, the measured\nvalue for the present Co sample is much higher. In the\ncase of the (amorphous) CoB sample, the measured Dsw\nis also high compared to a (bulk) literature value of \u0019170\nmeV\u0017A220. This suggests that the enhanced value of Dsw\nis not related to the crystalline structure. Moreover, the\nratio of the measured values for Co and CoB is compa-\nrable to the ratio of the literature values. This indicates\nthat the origin of the enhanced spin wave sti\u000bness is the\nsame for both used absorption layers.\nThe large value of Dswmight be related to strain or\nintermixing at the absorption layer boundaries. In the\ncase of Co, this can lead to lattice deformations. Since\nthe spin wave sti\u000bness is highly dependent on the lat-\ntice constant aof the material ( Dsw= 2JSa2withJtheexchange constant and Sthe atomic spin), such lattice\ndeformations are expected to have a signi\fcant in\ruence\non the spin wave sti\u000bness. In case of the amorphous CoB,\nthis e\u000bect would be present in the pair distribution func-\ntion. It is also noted, without going into details, that the\namplitude of the standing spin waves in the measured\nsignal depends on the depth pro\fle of the polar MOKE\nsensitivity within the absorption layer, which is known\nto be in\ruenced by (amongst others) the attenuation of\nthe laser and interface e\u000bects. If, for instance, the MOKE\nwould be more sensitive to both interface regions and less\nto the bulk of the absorption layer, the (net) signal of the\nodd-order spin waves would be suppressed [see inset Fig.\n2(a)]. In that case, the measured spin waves in Fig. 2(b)\nare the second-order standing spin waves ( n= 2), and\nthe resulting spin wave sti\u000bness for the Co layer would\nbeDsw\u0019(882=4 =)220 meV \u0017A2. Although this value\nseems to be more in line with the literature values, the\nvalidity of such a MOKE-sensitivity-pro\fle related sup-\npression of the odd-order spin waves should be tested\n(especially for the thicker absorption layer thicknesses).\nClearly, more research is needed in order to fully com-\nprehend the enhanced value of the spin wave sti\u000bness,\nfor which the presented noncollinear bilayers could be of\ngreat value.\nNext to the precession frequency, the damped sine \fts\nof the standing spin waves [Fig. 2(a)] also provide the\ncharacteristic damping time \u001csw, which can be used to\ndetermine the Gilbert damping parameter \u000bswof the THz\nspin waves. The damping is calculated using\n\u000bsw=\u0014\n\r\u001csw\u0012\nBapp+\u00160Me\u000b\n2+Dsw\n\r~k2\u0013\u0015\u00001\n;(6)\nwhich is similar as the equation used for the homogeneous\nprecession [Eq. (2)], with an additional term resulting\nfrom the exchange interaction.\nThe measured damping as a function of the Co layer\nthickness is presented in Fig. 3 (black dots). Similar\nas for the homogeneous precessions, both the intrinsic\ndamping (\u000bbulk) and interface spin pumping ( \u000bpump) are\ncontributing to the damping. In case of the (inhomoge-\nneous) standing spin waves, the damping due to interface\nspin pumping is twice as large as for the homogeneous\nprecession15. The\u000bbulkand 2\u000bpump contributions to the\ntotal damping are illustrated by the black and blue solid\ncurves in the \fgure. Note that the values of \u000bbulkand\n\u000bpump are the ones determined from the homogeneous\nprecessions [inset Fig. 1(b)].\nThe \fgure clearly shows that there is an additional\ncontribution to the damping \u000badd, which has a surprising\nthickness dependence, and enhances the damping up to\nabout an order of magnitude compared to the damping of\nthe homogeneous precession. The thickness dependence\nof\u000baddcan be divided into two regions. For tCo\u001510\nnm, a strong increase in \u000baddis seen when decreasing the\nCo thickness. For tCo<10 nm, the additional damping\nvanishes upon further reduction of the Co thickness. The\nsame behavior was found in the CoB sample, which is\nshown in Supplementary Note 3.5\n0 2 4 6 8 1012141618200.000.050.100.150.200.250.300.35Damp ing\nCothickne ss(nm)\nFIG. 3. Gilbert damping for the higher-order standing spin\nwaves as a function of the Co thickness. The \u000bbulkand 2\u000bpump\ncontributions to the total damping are illustrated by the black\nand blue solid curves. The red solid line represents the \ft to\nthe data using Eq. (7).\nThe additional source of damping might be the re-\nsult of the inhomogeneity of the standing spin waves,\nfor which an additional contribution to the damping\nwas modelled in Ref.21. In this model, the additional\ndamping originates from spin pumping between regions\nin the magnetic material that are precessing at a di\u000ber-\nent phase. This damping term was calculated to scale\nwithk2, which is proportional to t\u00002\nCoin the present case.\nThe damping based on this model (including \u000bbulkand\n2\u000bpump) is illustrated in the \fgure by the red dashed line.\nAt \frst sight, the behavior of this additional source of\ndamping does not seem to agree with the measurement.\nThet\u00002\nCodependence does not include the reduction in\ndamping for tCo<10 nm, and a thickness dependence\nmuch stronger than t\u00002\nCois observed for tCo\u001510 nm.\nThe derivation of the additional damping in Ref.21was\ndone for low frequencies, i.e., for slow dynamics, thereby\nneglecting frequency dependent terms in the transport\nequation for the spin current. Adding these terms in\nthe derivation results in the following equation for the\nadditional damping (see Supplementary Note 4 for the\nderivation)\n\u000badd=ARe\"\n\u001c?(1 +i\u001c?2\u0019fsw)\n(\u001c?\u0001xc=~)2\u0000(\u0000i+\u001c?2\u0019fsw)2#\nk2:(7)\nIn this equation, Ais a constant prefactor (discussed\nlater),\u001c?is the transverse spin scattering time, \u0001 xcthe\nexchange energy and fswthe precession frequency, given\nby the \ft in Fig. 2(b). A \ft to the data using this equa-\ntion is shown by the red solid line in Fig. 3. From a\nqualitative point of view, it can be seen that the data\nis well described by Eq. (7), which is also the case for\nthe CoB sample (Supplementary Note 3). As can be\nseen in the \fgure, the extended model correctly describes\nthe strong thickness dependence for tCo\u001510 nm, wherethe increased dependence on tCowith respect to the ini-\ntial model (dashed red curve) results from the thickness\ndependence of fsw. Moreover, the extended model re-\nproduces the reduction of the damping for tCo<10\nnm, reducing \u000badddown to zero when tCo!0, i.e., for\nfsw!1 . In this high frequency limit, where fsw\u001d\u001c\u00001\n?,\nthe angular momentum dissipation ( /\u001c\u00001\n?) becomes too\nslow, and its damping e\u000bect on the spin wave precession\nbecomes negligible.\nA more quantitative analysis of the \ft can be done by\nlooking at the \ftted values of A,\u001c?and \u0001 xc. ForA,\na value of (3 :1\u00060:2)\u000110\u00006m2s\u00001is obtained. This\nprefactor is equal to21\nA=ne~2\n4m\u0003S; (8)\nwithnethe electron number density, ~the reduced\nPlanck constant, m\u0003the e\u000bective electron mass, and S\nthe spin density. Using the Drude conductivity \u001bD, which\nis given by\u001bD= (nee2\u001cD)=m\u0003, the spin wave density can\nbe calculated using\nS=~2\u001bD\n4\u001cDe2A: (9)\nIn this equation eand\u001cDare the charge and mean free\ntime of the electron, respectively. The spin density can\nin turn be used to calculate the amount of spins per Co\natom. With a mean free path of \u001910 nm and a Fermi\nvelocity of 2 :55\u0001105m s\u00001in Co22, the mean free time\nis equal to \u001cD\u001939 fs. Together with a conductivity of\n\u001bD= 1:79\u0001107S m\u00001in Co23, and the assumption of an\nfcc lattice with a lattice constant of a0= 3:54\u0017A24, a spin\ndensity of 1 :69~per Co atom is calculated. This value\nis close to the known value of 1 :72 for Co, and thereby\nsupports the validity of the \ft.\nThe transverse spin scattering time was found to be\n\u001c?= 1:5\u00060:2 ps. This scattering time is related to the\ndisorder scattering time \u001cdisand electron-electron scat-\ntering time \u001ceevia21,\u001c\u00001\n?=\u001c\u00001\ndis+\u001c\u00001\nee. Unfortunately,\nno corresponding values for Co were found in the lit-\nerature. Lastly, the \ftted exchange energy is equal to\n\u0001xc= 0:93\u00060:04 meV. This is much lower than the\nexchange energy known for the delectrons in Co, which\nis in the order of 0 :1 eV. However, the \ftted exchange\ninteraction might need to be compared to the exchange\nenergy for the selectrons at the Fermi surface, which is\nexpected to be much smaller.\nIV. CONCLUSION\nIn conclusion, it has been demonstrated that the non-\ncollinear magnetic bilayer is a convenient tool to ex-\ncite and investigate THz standing spin waves, thereby\nshowing high potential for future THz magnonics. Us-\ning wedge-shaped absorption layers, the spin wave dis-\npersion in Co and CoB was measured. Analysis of the\ndispersion resulted in a surprisingly high spin wave sti\u000b-\nness for both materials, for which further investigation6\nis needed in order to clarify the enhanced values. Addi-\ntionally, the noncollinear magnetic bilayers were used to\ninvestigate the damping of the THz standing spin waves,\ndemonstrating a large damping contribution, additional\nto the bulk damping and damping resulting from inter-\nface spin pumping. The additional damping displayed a\nstrong increase of the damping with decreasing absorp-\ntion layer thickness, and a previously unseen sudden de-\ncrease in the damping for the thinnest \flms. A model\nfor the additional damping contribution was proposed.\nThe observed decrease in the (additional) damping for\nthe highest spin wave frequencies might be of great rele-vance for future magnonics, in which high frequency spin\nwaves with low damping are desired.\nACKNOWLEDGMENTS\nWe thank Y. Tserkovnyak for valuable discussions.\nThis work is part of the Gravitation program 'Research\nCentre for Integrated Nanophotonics', which is \fnanced\nby the Netherlands Organisation for Scienti\fc Research\n(NWO).\n1G. Malinowski, F. 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Cam-\nbridge University Press, 2010.7\nSupplementary Note 1: Analysis\nhomogeneous precession for the CoB\nsample\nFor the analysis of the THz standing spin waves, the\ne\u000bective magnetization and Gilbert damping parame-\nter (bulk and interface spin pumping contributions) are\nneeded as a function of the thickness of the absorption\nlayer. These properties are determined using the homo-\ngeneous precession, of which the analysis for the non-\ncollinear magnetic bilayer with a Co absorption layer is\nshown in the main paper. In this section, the results of\nsimilar measurements performed on the sample with the\nCoB absorption layer are presented.\nAs was mentioned in the main paper, the e\u000bective mag-\nnetizationMe\u000bof the absorption layer at a certain thick-\nness is determined by measuring the frequency fIPof\nthe homogeneous precession as a function of the applied\nmagnetic \feld Bapp, and using the Kittel relation [Eq.\n(1) of the main paper] to \ft the value of Me\u000b(and\r).\nFigure 4 presents the resulting e\u000bective magnetization\nas a function of the CoB thickness tCoB. The observed\nthickness dependence of Me\u000bresults from an out-of-plane\nsurface anisotropy, which decreases Me\u000b, and of which\nthe contribution falls o\u000b as t\u00001\nCoB. The obtained thickness\ndependentMe\u000bis later used in the analysis of the THz\nstanding spin waves (Fig. 2(b) of the main paper, and\nSupplementary Note 3). The Kittel \fts also allows the\ndetermination of the gfactor using the \ftted value of \r.\nFor the CoB sample, a gfactor of 2:31\u00060:08 was found,\nwhich is similar as the one found for the Co absorption\nlayer.\nUsing the characteristic damping time \u001cof the ho-\nmogeneous precessions, together with the previously de-\ntermined value of Me\u000b, the Gilbert damping constant\n\u000bat each CoB thickness can be determined, using Eq.\n(2) of the main paper. The measured damping as a\nfunction of the CoB thickness is shown in the inset\nof Fig. 4, in which the damping determined with the\ndi\u000berent magnetic \feld amplitudes are averaged. The\n\ftted values for the bulk damping and interface spin\npumping amplitude are \u000bbulk= (5:5\u00060:2)\u000110\u00003and\nApump = (0:94\u00060:02)\u000110\u000010m, respectively. Both val-\nues are used later when evaluating the damping of the\nTHz standing spin waves (Supplementary Note 3).\nSupplementary Note 2: Acoustic strain\nwaves\nThe precession measurement of the \frst-order standing\nspin wave in the Co sample (at a thickness of tCo= 13\nnm) presented in Fig. 2(a) of the main paper showed\na superposition of two precessions. The second slower\nprecession of 0 :08 THz was attributed to a longitudinal\nacoustic strain wave traveling along the depth of the mul-\ntilayer. In this section a more elaborate analysis of the\nslow precessions is presented, which demonstrates that it\nindeed belongs to a laser-induced acoustic strain wave.\n02468101214160.000.010.020.030.040.05Damping\nCoBthickness(nm)\n0 2 4 6 8 10 12 14 160.00.20.40.60.81.01.2Meff(MA/m)\nCoB thickne ss(nm)FIG. 4. E\u000bective magnetization Me\u000bas a function of the CoB\nthickness. The inset shows the Gilbert damping parameter\nas a function of the CoB thickness, in which the damping\ndetermined with the di\u000berent magnetic \feld amplitudes are\naveraged. The red curve represents a \ft to the data using Eq.\n(3) of the main paper.\nThe frequency of the precession as a function of Co\nthickness is displayed in the inset of Fig. 5. At \frst\nsight, due to the Co thickness dependency, it appears\nthat the precession exists in the Co layer. However, (in\na di\u000berent measurement) it turned out that the preces-\nsion was observed along the Co wedge down to a thick-\nness oftCo= 0, i.e., without the Co layer. Moreover,\nthe decrease in frequency with increasing Co thickness\nappears to be close to linear (dotted line, guide to the\neye), which neither \fts the Co thickness dependence of\nthe fundamental precession nor the higher-order stand-\ning spin wave dispersion. The observed behavior does \ft\nwith a (laser-induced) longitudinal acoustic strain wave\nthat travels through the full multilayer. For such a wave,\nthe periodpof one round trip is given by\np=2t\nvl; (10)\nin whichtis the thickness of the multilayer, and vlis the\nlongitudinal sound velocity.\nTo check that the measured precession indeed belongs\nto the acoustic strain wave, the frequency data in the\ninset of Fig. 5 is converted to the precession period as a\nfunction of the Co thickness, which is presented in the\nmain \fgure and \ftted using a linear \ft. Looking at Eq.\n10, it can be seen that the \ftted slope is equal to 2 =vl;Co,\nwithvl;Cothe longitudinal sound velocity in Co. The re-\nsulting sound velocity is vl;Co\u00196:6 km s\u00001. This value is\nclose to the literature value of 5 :7 km\u00001(Ref.25). More-\nover, measurements on a similar noncollinear magnetic\nbilayer with a wedge-shaped Co absorption layer resulted\nin a sound velocity of vl;Co\u00195:5 km s\u00001. Lastly, using\na wedge-shaped Pt layer instead of the Co absorption\nlayer, a sound velocity in Pt of vl;Pt\u00193:6 km s\u00001was\nfound, again close to the literature value of 4 :08 km s\u000018\n0 2 4 6 8 10 12 14 16 1802468101214\n0 2 4 6 810121416180.000.020.040.060.080.100.12Frequency(THz)\nCothickness(nm)Perio d(ps)\nCothickn ess(nm)\nFIG. 5. Precession period as a function of the Co thickness\nfor the second (0 :08 THz) precession measured in Fig. 2(a) of\nthe main paper (blue curve). The solid red curve is a linear\n\ft to the data. The inset shows the precession frequency as a\nfunction of the Co thickness, where the black dotted line is a\nguide to the eye.\n(Ref.25).\nIn conclusion, the analysis presented in this section\nshow that the second (0 :08 THz) precession observed in\nFig. 2(a) of the main paper correspond to the a laser-\ninduced longitudinal acoustic strain wave. Although such\nwaves have been measured before using time-resolved\nmeasurements of the re\rectivity, here, they are measured\nin the magneto-optical signal. It is believed that this\nis due to a lattice-deformation-induced change in the\nmagneto-optical signal coming from the Co/Ni multi-\nlayer (bottom OOP magnetic layer in the noncollinear\nmagnetic bilayer) when the acoustic strain wave passes\nthrough it. One of the reasons for this conclusion is that\nthe sign of the precession was seen to invert when the\nmagnetization in the Co/Ni multilayer was reversed.\nSupplementary Note 3: Damping THz\nstanding spin waves in the CoB\nabsorption layer\nIn the main paper, the damping of the THz standing\nspin waves in the Co absorption layer was investigated\n(Fig. 3 of the main paper), which demonstrated an ad-\nditional contribution to the damping \u000badd(on top of the\nbulk damping \u000bbulkand the damping resulting from in-\nterface spin pumping 2 \u000bpump). Moreover, the additional\ndamping displayed an unexpected thickness dependence,\nshowing a strong increase in \u000baddwith decreasing Co\nthickness for tCo\u001510 nm, while the additional damping\nvanished upon further reduction of the Co thickness for\ntCo<10 nm. In this section, it is demonstrated that the\nsame additional damping is present in case of the CoB\nabsorption layer, with the same thickness dependence.\nThe measured damping as a function of the CoB layer\n0 2 4 6 8 10 12 14 160.000.020.040.060.080.100.120.140.160.18Damping\nCoB thickne ss(nm)FIG. 6. Gilbert damping for the higher-order standing spin\nwaves as a function of the CoB thickness. The \u000bbulkand\n2\u000bpump contributions to the total damping are illustrated by\nthe black and blue solid curves. The red solid line represents\nthe \ft to the data using Eq. (7) of the main paper.\nthickness is presented in Fig. 6 (black dots). The \u000bbulk\nand 2\u000bpump contributions to the total damping are illus-\ntrated by the black and blue solid curves in the \fgure.\nNote that the values of \u000bbulkand\u000bpump are the ones de-\ntermined from the homogeneous precessions in Supple-\nmentary Note 1. The red solid line represents the \ft to\nthe data using Eq. (7) of the main paper. As can be seen,\nthe same dependence of the additional damping on the\nmagnetic layer thickness is found as for the Co sample in\nFig. 3 of the main paper, including a strong increase in\ndamping for tCoB\u00158 nm, and a decrease in damping for\ntCoB<8 nm.\nSupplementary Note 4: Derivation THz\nadditional damping equation\nIn this section, the equation for the additional damp-\ning\u000baddas given in Eq. (7) of the main paper is derived,\nwhich is an extension to the more elaborate derivation\ngiven in Ref.21. For simplicity, only the exchange interac-\ntion is taken into account in the e\u000bective \feld, i.e., there\nis no applied \feld and no anisotropy contributions, and\nthe canting of the magnetization away from its equilib-\nrium direction (+ y) is assumed to be small. As a result,\nthe magnetization can be described by a precession in\nthex;zplane. With the inclusion of a kzphase term to\nallow standing spin waves in the zdirection, the normal-\nized magnetization is described by\n~ m=2\n64mx\nmy\nmz3\n75=2\n64m0ei(kz+!t)\n1\nim0ei(kz+!t)3\n75; (11)9\nwhere the real parts represent the physical components\nof the magnetization. The e\u000bective \feld ~Hexis equal to\n~He\u000b=2Aex\n\u00160Msr2~ m; (12)\nin whichMsandAexare the saturation magnetization\nand exchange sti\u000bness of the material, respectively.\nFollowing Ref.21, and omitting the Gilbert damping\nterm, the LLG equation including the additional spin-\nwave damping term is given by\nd~ m\ndt=\u0000\r\u00160\u0010\n~ m\u0002~He\u000b\u0011\n\u0000\u0011(!)\nS\u0012\n~ m\u0002r2d~ m\ndt\u0013\n;(13)\nin whichSis the spin density and \u0011(!) a phenomenolog-\nical parameter characterizing spin-wave damping, which\ncan be dependent on the precession frequency !. Evalu-\nating either the xorzcomponent, the equation can be\nsolved for the precession frequency\n!=2\rAex\nMsk2+\u0011(!)\nSk2i!: (14)\nThe \frst term on the right-hand side can be recognized\nas the standard (exchange) spin-wave frequency given by\n!sw=Dsw=~k2, in which the relation between Aexand\nthe spin wave sti\u000bness Dsw, as given by26\nAex=MsDsw\n2\r~; (15)\nis used. Looking at the second term, it can be seen\nthat the imaginary part of \u0011(!) alters the precessions fre-\nquency, while the real part contributes to the damping.\nUsing a \frst order approximation, in which the contri-\nbution of\u0011(!) to the precession frequency (Im [ \u0011(!sw)])\nis considered small, and therefore can be neglected, the\nprecession frequency can be rewritten to\n!'!sw+Re [\u0011(!sw)]\nSk2i!sw: (16)The imaginary term on the right-hand side corresponds\nto the spin-wave damping, in which the additional spin-\nwave damping parameter is given by\n\u000badd=Re [\u0011(!sw)]\nSk2: (17)\nFrom Ref.21[Eqs. (22) and (28)], the (frequency inde-\npendent) phenomenological spin-wave damping parame-\nter is given by\n\u0011=ne~2\n4m\u0003\u001c?\n1 + (\u001c?\u0001xc=~)2; (18)\nwithnethe electron number density, ~the reduced\nPlanck constant, m\u0003the e\u000bective electron mass, \u001c?the\ntransverse spin scattering time and \u0001 xcthe exchange en-\nergy. This derivation was performed for low frequency,\nthereby neglecting the frequency dependent term in the\ntransport equation for the spin current [Eq. (24)]. The\nfrequency dependence can be included by using the sub-\nstitution\n1\n\u001c?!1\n\u001c?+i!sw: (19)\nFinally, combining this substitution with Eqs. (17) and\n(18), and using !sw= 2\u0019fsw, the \ft equation for the\nadditional damping as shown in Eq. (7) of the main paper\nis obtained,\n\u000badd=ARe\"\n\u001c?(1 +i\u001c?2\u0019fsw)\n(\u001c?\u0001xc=~)2\u0000(\u0000i+\u001c?2\u0019fsw)2#\nk2;\n(20)\nin which\nA=ne~2\n4m\u0003S: (21)"    },    {        "title": "0808.0694v2.Radiation_damping__noncommutativity_and_duality.pdf",        "content": "arXiv:0808.0694v2  [hep-th]  15 Oct 2008Radiation damping, noncommutativity and duality\nE. M. C. Abreua∗, A. C. R. Mendesb†, C. Nevesc‡and W. Oliveirad§\naGrupo de F´ ısica Te´ orica e Matem´ atica F´ ısica,\nDepartamento de F´ ısica,\nUniversidade Federal Rural do Rio de Janeiro\nBR 465-07, 23890-000, Serop´ edica, Rio de Janeiro, Brazil\nbCampus Rio Parana´ ıba,\nUniversidade Federal de Vi¸ cosa,\nBR 354 - km 310, 38810-000,\nRio Parana´ ıba, Minas Gerais, Brazil\ncDepartamento de Matem´ atica e Computa¸ c˜ ao,\nUniversidade do Estado do Rio de Janeiro\nRodovia Presidente Dutra, km 298,\n27537-000, Resende, Rio de Janeiro, Brazil\ndDepartamento de F´ ısica,ICE,\nUniversidade Federal de Juiz de Fora,\n36036-330, Juiz de Fora, MG, Brazil\nOctober 30, 2018\nDedicated to the memory of Prof. Emerson da Silva Guerra\nIn this work, our main objective is to construct a N= 2 supersymmetric extension of the nonrel-\nativistic (2+1)-dimensional model describing the radiati on damping on the noncommutative plane\nwith scalar (electric) and vector (magnetic) interactions by theN= 2 superfield technique. We also\nintroduce a dual equivalent action to the radiation damping one using the Noether procedure.\nPACS numbers: 11.15.-q; 11.10.Ef; 11.10.-z; 41.60.-m\nI. INTRODUCTION\nThe study of dissipative systems in quantum theory is\nof strong interest and relevance either for fundamental\nreasons [1] and for its practical applications [2, 3]. The\nexplicit time dependence of the Lagrangianand Hamilto-\nnian operators introduces a major difficulty to this study\nsince the canonical commutation relations are not pre-\nservedby time evolution. Thendifferent approacheshave\nbeen used in order to apply the canonical quantization\nscheme to dissipative systems [4, 5].\nOne of these approaches is to focus on an isolated sys-\ntem composed by the original dissipative system plus a\nreservoir. One start from the beginning with a Hamil-\ntonian which describes the system, the bath and the\nsystem-bath interaction. Subsequently, one eliminates\nthe bath variables which give rise to both damping and\nfluctuations, thus obtaining the reduced density matrix\n[2, 5, 6, 7, 8].\nAnother way to handle the problem of quantum dissi-\npative systems is to double the phase-space dimensions,\nso as to deal with an effective isolated system composed\nby the original system plus its time-reversed copy [9, 10].\nThe new degrees of freedom thus introduced may repre-\n∗E-mail: evertonabreu@ufrrj.br\n†E-mail: albert@ufv.br\n‡E-mail: clifford.neves@gmail.com\n§E-mail: wilson@fisica.ufjf.brsented by a single equivalent (collective) degree of free-\ndom for the bath, which absorbs the energy dissipated\nby the system.\nThe study of the quantum dynamics of an accelerated\ncharge is appropriated to use the indirect representation\nsince it loses the energy, the linear momentum, and the\nangular momentum carried by the radiation field [11].\nTheeffect oftheselossestothemotionofchargeisknown\nas radiation damping (RD) [11].\nThe reaction of a classical point chargeto its own radi-\nation was first discussed by Lorentz and Abraham more\nthan one hundred years ago, and never stopped being\na source of controversy and fascination [12, 13]. The\nmost disputable aspects of the Abraham-Lorentz the-\nory are the self-acceleration and preacceleration. Self-\nacceleration refers to classical solutions where the charge\nis under acceleration even in the absence of an external\nfield. Preacceleration means that the charge begins to\naccelerate before the force is actually applied.\nTheprocessofradiationdampingisimportantinmany\nareas of electron accelerator operation [14], as in recent\nexperiments with intense-laser relativistic-electron scat-\ntering at laser frequencies and field strengths where radi-\nation reaction forces begin to become significant [15, 16].\nIn this paper we introduce a N= 2 supersymmet-\nric extension of the radiation damping model complet-\ning theN= 1 supersymmetric version introduced in\n[17, 18]. Also a new action dual equivalent to the RD\none is obtained using the Noether dualization procedure.\nUsing the variables introduced in [19] we obtain a new\nnonvanishing phase space of Poisson brackets. The La-2\ngrangian is so divided in two parts describing “external”\nand “internal” degrees of freedom in a noncommutative\nphase space. In this work, our objective is to describe\naN= 2 supersymmetric extension of the nonrelativis-\ntic (2+1)-dimensional model describing the radiation-\ndamping (represented by the equation (3) below), on the\nnoncommutative plane introducing an interaction to the\nfree model by the N= 2 superfield technique. The in-\ntroduction of a scalar superpotential for the interaction\nterm permit us to construct N= 2 supersymmetric La-\ngrangians.\nThis paper is organized in the following distribution:\nin section 2 we present briefly the D= 2+1 model and\nobtain a dual equivalent model through the Noether du-\nalizationprocedureinsection3. Insection4weintroduce\na symplectic structure in the model in order to introduce\nthe noncommutativity through the variables used in [19].\nIn section 5 we promote the supersymmetric extension\nof the model. A supersymmetric version through the\nHamiltonian formalism is depicted in section 6. Finally,\nas usual the conclusions and perspectives are described\nin the last section.\nII. THE MODEL\nIn [20] it was introduced a nonrelativistic classical me-\nchanicsfreeparticlemodelwithaChern-Simonsliketerm\nL=1\n2m˙xi˙xj+λεijxi˙xj, i,j= 1,2,(1)\nwhereλhas dimension of mass/time, and we can realize\nthe secondterm asa particularelectromagneticcoupling.\nTo make (1) quasi-invariant under D= 2 Galilei symme-\ntry the second term in (1) was modified and we have\nthat\nL=1\n2m˙xi˙xj−κεij˙xi¨xj, i,j= 1,2,(2)\nwhereκhas dimensions of mass ×time. It can be shown\n[21] the quasi-invariance of this Lagrangian just above.\nThe authors in [20] demonstrated that the model de-\nscribes the superposition of a free motion in noncommu-\ntativeD= 2 spaces. A N= 2 supersymmetric extension\nof(2) wasaccomplishedin[22]describingparticlesonthe\nnoncommutative space with electric and magnetic inter-\nactions.\nIn [18, 23] a new approach in the study of radiation\ndamping [11] was presented, introducing a Lagrangian\nformalism to the model in D= 2+1 dimensions given by\nL=1\n2mgij˙xi˙xj−γ\n2εij˙xi¨xj, i,j= 1,2,(3)\nwhereεijis the Levi-Civita antisymmetric metric, gijis\nthe pseudo-Euclidean metric given by\ng=/parenleftbigg\n1 0\n0−1/parenrightbigg\n, (4)and where, as will be the case throughout the paper, the\nEinstein convention on the summation of repeated in-\ndices is employed. The Lagrangian (3) describes, in the\nhyperbolicplane, the dissipativesystemofachargeinter-\nacting with its own radiation, where the 2-system repre-\nsents the reservoir or heat bath coupled to the 1-system\n[17, 18]. The model (3) was shown to have the (2+1)-\nGalilean symmetry and the dynamical group structure\nassociated with that system is the SU(1,1) [23]. Note\nthat this Lagrangian is similar to the one discussed in\n[20] (that is a special nonrelativistic limit of the parti-\ncle with torsion investigated in [24]), but in this case we\nhave a pseudo-Euclidean metric and the RD constant, γ,\nis the coupling constant of a Chern-Simons-like term. It\nis important to note that, despite the results obtained\nin this paper being very closely related with the ones\nfrom [22], the difference between them is not just the\npseudo-Euclidean metric. The physical systems studied\nare different, where the constant γis not a simple cou-\npling constant, but depends on the physical properties of\nthe charged particle, like its charge eand massm, be-\ning related to the term in its equation of motion which\ndescribes an interaction of the charge with its own radi-\nation field. The radiation-damping constant γmake the\nrole of the “exotic” parameter κin [20, 22].\nA supersymmetrized version of the model (3) was pre-\nsented by us in [25] to N= 1, where we employ a super-\nsymmetric enlargement of the Galileo algebra obtained\nin [23] and shown that the supersymmetric action can be\nsplit into dynamically independent external and internal\nsectors.\nIII. DUALITY THROUGH DUALIZATION\nThe bosonization technique that express a theory of\ninteracting fermions in terms of free bosons provides a\npowerful non-perturbative tool for investigations in dif-\nferent areas of theoretical physics with practical applica-\ntions [26]. In two-dimensions these ideas have been ex-\ntended in an interpolating representation of bosons and\nfermions which clearly reveals the dual equivalence char-\nacter of these representation [27]. In spite of some diffi-\nculties, the bosonization program has been extended to\nhigher dimensions [28, 29].\nThis new technique to perform duality mappings in\nany dimensions that is alternative to the master action\napproach has been used in the literature [30]. It is based\non the traditional idea of a local lifting of a global sym-\nmetry and may be realized by an iterative embedding of\nNoether counter terms. This technique was originally ex-\nplored in the context of the soldering formalism [31, 32]\nand is explored [33, 34, 35] since it seems to be the most\nappropriate technique for non-Abelian generalization of\nthe dual mapping concept.\nThere has been a number of papers examining the ex-\nistence of gauge invariance in systems with second class\nconstraints [36]. Basically this involves disclosing using3\nthe language of constraints, hidden gauge symmetries in\nsuch systems. This situation may be of usefulness since\none can consider the non-invariant model as the gauge\nfixed version of a gauge theory. By doing so it has some-\ntimes been possible to obtain a deeper and more illu-\nminating interpretation of these systems. Such hidden\nsymmetries may be revealed by a direct construction of\na gauge invariant theory out of a non-invariant one [37].\nThe former reverts to the latter under certain gauge fix-\ning conditions. The advantage in having a gauge theory\nlies in the fact that the underlying gauge invariant the-\nory allows us to establish a chain of equivalence among\ndifferent models by choosing different gauge fixing con-\nditions.\nAs said above, we will use the iterative Noether\ngauging technique (also called dualization procedure) to\nachieve this objective. The important point to stress in\nthis application of the Noether technique is the ability to\nimplementspecificsymmetriesleadingtodistinctmodels.\nIt is an alternative route to establish dual equivalences\nbetweengaugeandnon-gaugetheories. Inanutshell, it is\nbasedonthe locallifting ofthe globalsymmetriespresent\nin the non-gauge action. This is done by iteratively in-\ncorporating counter-terms into the action along with a\nset of auxiliary fields. Clearly, the resulting embedded\ntheory is dynamically equivalent to the original one [34].\nThis alternative approach to the dual transformation is\ndimensionally independent and sufficiently general to en-\ncompass both Abelian and non-Abelian theories.\nAs the first step, let us rewrite our RD model equation\n(3),\nL0=1\n2mgij˙xi˙xj−γ\n2εij˙xi¨xj, i,j= 1,2,(5)\nhence the variation of this action is\nδL0=Ji\n1˙ηi+Ji\n2¨ηi (6)\nwhereδxi=ηiand the Noether currents are\nJ1i=mgij˙xj−1\n2γεij¨xj (7)\nJ2i=1\n2γεij˙xj. (8)\nThe second step in the iterative method is to construct\nthe action,\nL1=L0−Di\n1J1i−Di\n2J2i (9)\nwhereDi\n1andDi\n2are auxiliary fields which will be elim-\ninated in the process.\nThe variation of L1will give us,\nδL1=−mgijDi\n1δDj\n1+1\n2γεijδ(Di\n1Dj\n2) (10)\nand the final gauge invariant model is\nL2=L1+1\n2mgijDi\n1Dj\n1−1\n2γεijDi\n1Dj\n2(11)that automatically compensates for equation (10), mak-\ningL2gauge invariant and ending the iterative chain.\nWe have therefore succeed in transforming the global\nRD theory into a locally invariant gauge theory. The\nnext step would be to take advantage of the Gaussian\ncharacter of the auxiliary fields D1andD2to rewrite\n(11) as an effective action depending only on the original\nvariablexi[33].\nIV. NONCOMMUTATIVITY\nIntroducing a Lagrangian multiplier which equates ˙ x\ntoz, and substituting all differentiated x-variables in the\nLagrangian (3) by z-variables, one has a first- order La-\ngrangian:\nL(0)=pi(˙xi−zi)+m\n2gijzizj−γ\n2εijzi˙zj,(12)\nwhich equations of motion can be written, by employing\nthe symplectic structure [38], as\nωij˙ξj=∂H(ξj)\n∂ξi(13)\nwhere the symplectic two form is\n(ω) =/parenleftBigg0−120\n120 0\n0 0 −γε/parenrightBigg\n(14)\nwith\n12=/parenleftbigg\n1 0\n0 1/parenrightbigg\n, ε=/parenleftbigg\n0 1\n−1 0/parenrightbigg\n,(15)\nand0denotes the 2 ×2 null matrix. H(ξl) is the Hamil-\ntonian and ξiare the symplectic variables.\nNow, using the variables introduced in [19] modified as\nQi=γ(mgijzj−pi), Xi=xi+εijQij, Pi=pi(16)\none obtains that\nL(0)=L(0)\next+L(0)\nint (17)\nwhere\nL(0)\next=Pi˙Xi+γ\n2εijPi˙Pj−1\n2mgijPiPj,(18)\nand\nL(0)\nint=1\n2γεijQi˙Qj+1\n2mγ2gijQiQj,(19)\nwith the following nonvanishing Poisson brackets:\n{Xi,Xj}=γεij,{Xi,Pj}=δij,\n{Qi,Qj}=−γεij. (20)4\nWe can see that our Lagrangian is now separated into\ntwodisconnectedpartsdescribingthe“external”and“in-\nternal” degrees of freedom in a noncommutative phase\nspace, parametrized by the variables ( Xi,Pi)(external\nstructure) and Qi(internal structure) [19].\nNowweshallintroduceaninteractiontothe“external”\nsector, equation (18)(which do not modify the internal\nsector), represented by a potential energy term U(X)\ninvolving noncommutative variables, as follows\nLext=Pi˙Xi+γ\n2εijPi˙Pj−1\n2mgijPiPj−U(X).(21)\nThis leads to a deformation of the constraint algebra,\nsince the constraint now involves a derivative of the po-\ntential [23].\nV. THE SUPERSYMMETRIC MODEL IN N= 2\nTo obtain the supersymmetric extension of the model\ndescribed by the Lagrangian (21), for each space com-\nmuting coordinate, representing the degrees of freedom\nof the system, we associate one anticommuting vari-\nable, which are the well known Grassmannian variables.\nWe are considering only the N= 2 SUSY for a non-\nrelativisticparticle, whichisdescribedbytheintoduction\nof two real Grassmannian variables Θ and ¯Θ (the Hermi-\ntian conjugate of Θ) in the configuration space, but all\nthe dynamics are represented by the time t[39, 40].\nFurthermore, intoducing the Taylor expansion for the\nreal scalar supercoordinate as\nXi→ Xi(t,Θ,¯Θ) =Xi(t)+iψi(t)Θ+i¯Θ¯ψi(t)+¯ΘΘFi(t)\n(22)\nand their canonical supermomenta\nPi(t)→ Pi(t,Θ,¯Θ) =iηi(t)−iΘ(Pi(t)+ifi(t))−¯ΘΘ˙ηi(t),\n(23)\nwhich under the infinitesimal supersymmetry transfor-\nmation law\nδt= (ǫQ+¯ǫ¯Q)t,\nδΘ = (ǫQ)Θ, (24)\nδ¯Θ = (¯ǫ¯Q)¯Θ,\nfurnish\nδXi= (ǫ¯Q+¯ǫQ)Xi (25)\nδPi= (ǫ¯Q+¯ǫQ)Pi, (26)\nwhereQand¯Qare the two SUSY generators\nQ=∂\n∂¯Θ+iΘ∂\n∂t,¯Q=∂\n∂Θ+i¯Θ∂\n∂t.(27)\nIn terms of ( Xi(t),Pi(t),Fi,fi) bosonic (even) com-\nponents and ( ψi(t),¯ψi(t),ηi(t)) fermionic (odd) compo-\nnents, we get the following transformations,\nδXi=i(¯ǫ¯ψi−ǫψi)δψi=−¯ǫ(˙Xi−iFi)\nδ¯ψi=−ǫ(˙Xi+iFi) (28)\nδFi=ǫ˙ψi+¯ǫ˙¯ψi,\nand\nδηi=−ǫ(Pi+ifi)\nδPi= 0\nδfi= 2¯ǫ˙ηi (29)\nδ˙ηi=−ǫ(˙Pi+i˙fi).\nNotice that the supersymmetry mixes the even and odd\ncoordinates. Carrying out a variation in the even com-\nponents we obtain the odd components and vice-versa.\nThe super-Lagrangian for the superpoint particle with\nN= 2, invariant under the transformations (28) and\n(29), can be written as the following integral (we use for\nsimplicity that m= 1):\n¯Lext=1\n2/integraldisplay\ndΘd¯Θ/bracketleftBig/parenleftbig¯DXi¯Pi+PiDXi/parenrightbig\n+γ\n2εij/parenleftBig\nPi˙¯Pj+˙Pj¯Pi/parenrightBig\n−1\n2gij/parenleftbig\nPi¯Pj+Pj¯Pi/parenrightbig/bracketrightbigg\n−/integraldisplay\ndΘd¯ΘU[X(t,Θ,¯Θ)] (30)\nwhereDis the covariant derivative ( D=∂Θ−i¯Θ∂t) and\n¯Dis its Hermitian conjugate. The U[X] is a polynomial\nfunction of the supercoordinate\nExpanding the superpotential U[X] in Taylor series\nand maintaining Θ ¯Θ (because only these terms remain\nafter integrations on Grassmannian variables Θ and ¯Θ),\nwe have that\nU[X] =Xi∂U[X(t)]\n∂Xi+XiX∗\nj\n2∂2U[X(t)]\n∂Xi∂Xj+...(31)\n=Fi¯ΘΘ∂iU[X(t)]+¯ΘΘψi¯ψj∂i∂jU[X(t)]+...\nwhere the derivatives ∂i=∂\n∂Xiare such that Θ = 0 = ¯Θ,\nwhich are functions only of the X(t) even coordinate.\nSubstituting euqation (31) in equation (30), we obtain\nafter integrations\n¯Lext=L(0)\next−1\n2gijfifj−Fifi+γ\n2εijfi˙fj\n−i/parenleftbig¯ψi˙¯ηi−˙ηiψi/parenrightbig\n−igij˙ηi¯ηj+iγεij˙ηi˙¯ηj\n−Fi∂iU[X(t)]−ψi¯ψj∂i∂jU[X(t)],(32)\nwhich is the complete Lagrangian for N= 2.\nThe bosonic component Fiis not a dynamic variable.\nIn this case, using the Euler-Lagrange equations for the\nauxiliary variables fiandFi, we obtain:\nfi(t) =∂iU[X(t)], (33)\nFi(t) =gijfj−γεij˙fj\n=gij∂jU[X(t)]−γεij∂j∂kU[X]˙Xk(t),(34)5\nwhere we have to eliminate the variable fias well as its\nderivative in Fi. Now, substituting the (33) and (34) in\n(32) the auxiliaryvariablescanbe completelyeliminated,\nso that\n¯L(N=2)ext=L(0)\next−1\n2gij∂iU∂jU+γ\n2εij∂iU∂j∂kU˙Xk\n−i/parenleftbig¯ψi˙¯ηi−˙ηiψi/parenrightbig\n−igij˙ηi¯ηj+iγεij˙ηi˙¯ηj\n−ψi¯ψj∂i∂jU, (35)\nNote that, as in [22], we can rewrite equation (35) as\n¯L(N=2)ext=L(0)\next+Ak(X,t)˙Xk+A0(X,t)+\n−i/parenleftbig¯ψi˙¯ηi−˙ηiψi/parenrightbig\n−igij˙ηi¯ηj+iγεij˙ηi˙¯ηj\n−ψi¯ψj∂i∂jU, (36)\nthat is invariant under standard gauge transformations\nAµ→A′\nµ=Aµ+∂µΛ, where\nA0(X,t) =−1\n2gij∂iU∂jU (37)\nand\nAk(X,t) =γ\n2εij∂iU∂j∂kU. (38)\nwere identifided in [22] with the scalar potential A0(that\nin this case have a pseudo-Euclidean metric) and the vec-\ntor potential Ak. The vector potential introduce a mag-\nnetic fieldB=εij∂iAjgiven by\nB(X) =γ\n2εikεlj(∂i∂lU)(∂j∂kU) (39)\nwhere we see that the noncommutativity introduced by\nthe parameter γgenerates a constant magnetic field [22].\nThe Euler-Lagrange equations, in this case, are\nm∗˙Xi=gijPj−meγεijEj+mγεijψl¯ψk∂l∂k∂jU,\n˙Pi=eBεij˙Xj+eEi−ψl¯ψj∂l∂j∂iU, (40)\nwhereEiandBare the electric and magnetic field, re-\nspectively, and m∗=m(1−eγB) is an effective mass.\nBut, such a way of introducing electromagnetic inter-\naction modifies the symplectic structure of the system\nwhich determines the noncommutative phase-space ge-\nometry, for the bosonic sector, equation (20),\n{Xi,Xj}=m\nm∗γεij,{Xi,Pj}=m\nm∗δij,\n{Pi,Pj}=m\nm∗eBεij, (41)\nwhich implies an analysis of the value eγB/ne}ationslash= 1 in order\nto avoid a singularity [41, 42]. To the fermionic sector,\nthe Euler-Lagrange equations are\niγεij¨¯ηj−igij˙¯ηj+i˙ψ= 0,\n−iγεij¨ηj−igij˙ηj−i˙¯ψ= 0, (42)for the fermionic varialbles ( η,¯η). For the fermionic vari-\nables (ψi,¯ψi) the Euler-Lagrange equations are\ni˙ηi+¯ψj∂i∂jU= 0\ni˙¯ηi−ψl∂i∂jU= 0. (43)\nwhere the fermionic variables ( ψi,¯ψi) do not have dy-\nnamics.\nThe canonical Hamiltonian for the N= 2 SUSY is\ngiven by\n¯H=˙Xi∂¯L\n∂˙Xi+∂¯L\n∂˙ψi˙ψi+∂¯L\n∂˙¯ψi˙¯ψi+∂¯L\n∂˙ηi˙ηi+∂¯L\n∂˙¯ηi˙¯ηi−¯L(N=2)\n=1\n2mgijPiPj−A0+ψi¯ψj∂i∂jU(X), (44)\nwhich provides a mixed potential term, with a dynam-\nical variable of the particle A0and the Grassmannian\nvariables (ψi,¯ψi).\nThere is one other way to introduce the minimal elec-\ntromagneticinteraction. It isthroughthe transformation\nPi→ Pi=Pi+eAi(Xi,t) in the Hamiltonian, that pre-\nserve the symplectic structure of equation (20). In [22]\nthis transformation has been considered and it leads to\nthe same expression for the magnetic field Eq.(39).\nVI. REMARKS AND CONCLUSIONS\nA fundamental property of all charged particles is that\nthe electromagnetic energy is radiated whenever they are\naccelerated. The recoil momentum of the photons emit-\nted during this process is equivalent to a reaction force\ncorresponding to the self-interaction of the particle with\nits own electromagnetic field, which originates radiation\ndamping.\nThe process of RD is important in many areas of elec-\ntron accelerator operation, like in recent experiments\nwith intense laser relativistic electron scattering at laser\nfrequencies and field strengths where radiation reaction\nforces begin to become significant.\nIn [23] some of us introduced an alternative approach\nto canonical quantization of the RD based on doubling\nthe degrees of freedom. A Lagrangian model for the\nsystem with a Chern-Simons-like term with high order\nderivative was obtained. In [25] it was introduced the\nN= 1 supersymmetric version of the RD in the Grass-\nman superspace and the N= 2 version was constructed\nin [22].\nHere the supersymmetric model was split into “exter-\nnal” and “internal” degrees of freedom of the supersym-\nmetric model in terms of new variables, where the RD\nconstant introduced noncommutativity in the coordinate\nsector. We presented a way to introduce an electromag-\nnetic coupling.\nWe carried out the supersymmetric N= 2 extension\nofthe RD model and realized that the noncommutativity\nintroduced by the parameter generates a constant mag-\nnetic field.6\nAlso in this work, we used an alternative way to con-\nstruct a dual equivalent action to the RD one, a dual-\nization procedure. It used the Noether technique which\nis independent of dimensions and imposes a gauge sym-\nmetry which is believed to be hidden in the theory. The\nmain ingredient is an auxiliary field which is eliminated\nthrough the equations of motion and the final action is\nan effective one depending only on this original variables.VII. ACKNOWLEDGMENTS\nThis work is partially supported by FAPERJ,\nFAPEMIG and CNPq, Brazilian Research Agencies.\n[1] W. G. Unruh and W. H. Zurek, Phys. Rev. D 40 (1961)\n37.\n[2] A. O. Caldeira and A. J. Legget, Phys. Rev. Lett. 46\n(1981) 211.\n[3] W. S. Warren, S. L. Hammes and J. L. Bates, J. Chem.\nPhys. 91 (1989) 5895.\n[4] A. C. R. Mendes and F. I. Takakura, Phys. Rev. E 64\n(2001) 056501.\n[5] P. M. V. B. Barone and A. O. Caldeira, Phys. Rev. A 43\n(1991) 57.\n[6] R. P. Feynman and F. L. Vernon Jr., Ann. Phys. 24\n(1963) 118.\n[7] A. O. Caldeira and A. J. Legget, Ann. Phys. 121 (1983)\n587.\n[8] A. O. Caldeira and A. J. Legget, Physica A 149 (1983)\n374.\n[9] R. Banerjee and P. Mukherjee, quant-ph/0205143; ibid,\nquant-ph/0108055.\n[10] H. Feshbach and Y. Tikochinsky, Trans. N. Y. Acad. 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B 595 (2004) 547.\n[20] J. Lukierski, P. C. Stichel and W. J. Zakrzewski, Ann.\nPhys. 260 (1997) 224.\n[21] J. Lopuszanski and P. C. Stichel, Fortschr. Phys. 45\n(1997) 79.\n[22] J. Lukierski, P. C. Stichel and W. J. Zakrzewski, Phys.Lett. B 602 (2004) 249.\n[23] A. C. R. Mendes, F. I. Takakura, C. Neves and W.\nOliveira, Eur. Phys. J. C 45 (2006) 257.\n[24] M. S. Plyushchay, Nucl. Phys B 362 (1991) 54; Nucl.\nPhys. B 262 (1991) 71.\n[25] A. C. R. Mendes, F. I. Takakura, C. Neves and W.\nOliveira, J. Phys. A 38 (2005) 9387.\n[26] S. Coleman, Phys. Rev. D 11(1975) 2088; S. Mandelstan,\nPhys. Rev. D 11 (1975) 3026.\n[27] P. H. Damgaard, H. B. Nielsen and R. Scollacher, Nucl.\nPhys. B 385 (1992) 227; Nucl. Phys. B 414 (1994) 541;\nPhys. Lett. B 296 (1992) 132; Phys. Lett. B 332 (1994)\n131; P. H. Damgaard and H. B. Nielsen, Nucl. Phys. B\n433 (1995) 671.\n[28] M. L¨ uscher, Nucl. Phys. B 326 (1989) 557.\n[29] E. C. Marino, Phys. Lett. B 263 (1991) 63.\n[30] A. Ilha and C. Wotzasek, Nucl. 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Galv˜ ao and C. Teitelboim, J. Math. Phys. 21\n(1980) 1863.\n[40] G. Junker and S. Mattiensen, J. Phys. A 27(1995) L751.\n[41] P. A. Horv´ athy and M. S. Plyushchay, Nucl. Phys. B 714\n(2005) 269, hep-th/0502040.\n[42] C. Duvaland P. A.Horv´ athy, J. Phys. A34 (2001) 10097.\n[43] N. Seiberg and E. Witten, JHEP 9909 (1999) 032."    },    {        "title": "2209.14179v1.Unidirectional_magnetic_coupling.pdf",        "content": "Unidirectional magnetic coupling\nH. Y. Yuan,1R. Lavrijsen,2and R. A. Duine1, 2\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584 CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: September 29, 2022)\nWe show that interlayer Dzyaloshinskii-Moriya interaction in combination with non-local Gilbert\ndamping gives rise to unidirectional magnetic coupling. That is, the coupling between two magnetic\nlayers | say the left and right layer | is such that dynamics of the left layer leads to dynamics of\nthe right layer, but not vice versa. We discuss the implications of this result for the magnetic sus-\nceptibility of a magnetic bilayer, electrically-actuated spin-current transmission, and unidirectional\nspin-wave packet generation and propagation. Our results may enable a route towards spin-current\nand spin-wave diodes and further pave the way to design spintronic devices via reservoir engineering.\nIntroduction. | Non-reciprocal transmission of elec-\ntrical signals lies at the heart of modern communication\ntechnologies. While semi-conductor diodes, as an exam-\nple of an electronic component that underpins such non-\nreciprocity, have been a mature technology for several\ndecades, new solutions are being actively pursued [1, 2].\nSuch research is spurred on by the emergence of quan-\ntum technologies that need to be read out electrically\nbut should not receive unwanted back-action from their\nelectronic environment.\nComplementary to these developments, spintronics has\nsought to control electronic spin currents and, more\nrecently, spin currents carried by spin waves | i.e.,\nmagnons | in magnetic insulators [3]. Devices that im-\nplement non-reciprocal spin-wave spin currents have been\nproposed [4{7]. Most of these proposals rely on dipolar\ninteractions [8{11] or Dzyaloshinskii-Moriya interactions\n(DMI) [12{16]. Other proposals involve the coupling of\nthe spin waves to additional excitations such that the spin\nwaves are endowed with non-reciprocity. Examples are\nthe coupling of the spin waves to magnetoelastic, optical,\nand microwave excitations [17{22].\nMost of these proposals have in common that they con-\nsider spin-wave dispersions that are asymmetric in wave\nvector. For example, due to the DMI spin waves at one\nparticular frequency have di\u000berent wave numbers and ve-\nlocities for the two di\u000berent directions. There are there-\nfore spin waves travelling in both directions. This may\nbe detrimental for some applications. For example, one\nwould like to shield quantum-magnonic technologies from\nspin-current noise [23], and completely quench the spin-\ncurrent transmission in one of the two directions along a\nwire.\nHere, we propose a set-up that realizes unidirectional\nmagnetic coupling between two magnetic layers or be-\ntween two magnetic moments. The ingredients are DMI\nand dissipative coupling between the two layers or mo-\nments. The dissipative coupling takes the form of a non-\nlocal Gilbert damping and may arise, for example, from\nthe combined action of spin pumping and spin transfer.Then, one magnet emits spin current when it precesses,\nwhich is absorbed by the other. The resulting dissipa-\ntive coupling turns out to, for certain parameters, pre-\ncisely cancel the DMI in one direction. As a result, an\nexcitation of one of the magnets leads to magnetization\ndynamics of the other, but not vice versa. This yields\nspin-wave propagation that is truly uni-directional: for\nspeci\fc direction and magnitude of the external \feld, all\nspin waves travel in one direction only.\nMinimal model. | Let us start with the minimal set-\nup that demonstrates the unidirectional coupling. We\n\frst consider two identical homogeneous magnetic lay-\ners that are coupled only by an interlayer DMI with\nDzyaloshinskii vector Dand by interlayer spin pumping\n(see Fig. 1). The magnetization direction in the layers\nis denoted by mi, wherei2f1;2glabels the two lay-\ners. We also include an external \feld H. The magnetic\nenergy is given by\nE[m1;m2] =D\u0001(m1\u0002m2)\u0000\u00160MsH\u0001(m1+m2);(1)\nwhereMsis the saturation magnetization of both layers\nand\u00160is the vacuum susceptibility. The magnetization\ndynamics of layer 1 is determined by the Landau-Lifshitz-\nGilbert (LLG) equation\n@m1\n@t=\r\nMsm1\u0002\u000eE\n\u000em1+\u000bnlm1\u0002@m2\n@t; (2)\nwhere\ris the gyromagnetic ratio and \u000bnlcharacterizes\nthe strength of the non-local damping that in this set-up\nresults from the combination of spin pumping and spin\ntransfer torques, as described in the introduction. The\nequation of motion for the magnetization dynamics of the\nsecond layer is found by interchanging the labels 1 and\n2 in the above equation. Working out the e\u000bective \feldsarXiv:2209.14179v1  [cond-mat.mes-hall]  28 Sep 20222\nFM\nFM\nm2\nm1xy\nz, H, D\nFIG. 1. Schematic of two magnetic moments coupled by an\ninterlayer DMI and by interlayer spin pumping. The dynam-\nics of m1induces the motion of m2, but not vice versa for\nappropriate parameters.\n\u000eE=\u000emiyields\n@m1\n@t=\r\nMsm1\u0002(m2\u0002D\u0000\u00160MsH) +\u000bnlm1\u0002@m2\n@t;\n(3a)\n@m2\n@t=\r\nMsm2\u0002(D\u0002m1\u0000\u00160MsH) +\u000bnlm2\u0002@m1\n@t;\n(3b)\nwhere the sign di\u000berence in e\u000bective-\feld contribution\nfrom the DMI stems from the asymmetric nature of the\nDMI. We show now that depending on the magnitude and\ndirection of the e\u000bective \feld, this sign di\u000berence leads\nfor one of the layers to cancellation of the torques due\nto interlayer DMI and non-local damping. As the can-\ncellation does not occur for the other layer, and because\nthe DMI and non-local damping are the mechanisms that\ncouple the layers in the model under consideration, this\nleads to uni-directional magnetic coupling.\nTaking the external \feld to be much larger than the\ninterlayer DMI, i.e., \u00160jHj\u001djDj=Ms, and taking \u000bnl\u001c\n1, we may replace @mi=@tby\u0000\r\u00160mi\u0002Hon the right-\nhand side of Eqs. (3) because the external \feld then is\nthe dominant contribution to the precession frequency.\nFor the \feld H=D=\u000bnl\u00160Ms, one then \fnds that\n@m1\n@t=\u0000\r\n\u000bnlMsm1\u0002D; (4a)\n@m2\n@t=2\r\nMsm2\u0002(D\u0002m1)\u0000\r\n\u000bnlMsm2\u0002D:(4b)\nHence, the coupling between the two magnetic layers is\nunidirectional at the \feld H=D=\u000bnl\u00160Ms: the magne-\ntization dynamics of layer 1 leads to dynamics of layer\n2 as evidenced by Eq. (4b), but not vice versa as im-\nplied by Eq. (4a). This one-way coupling is reversed by\nchanging the direction of the \feld to \u0000Hor the sign of\nthe non-local coupling \u000bnl.\nMagnetic susceptibility. | Let us now take into ac-\ncount the Gilbert damping within the layers, exchange,\nand anisotropies and discuss the in\ruence of the unidi-\nrectional coupling on the magnetic susceptibility. Theenergy now reads\nE[m1;m2] =\u0000Jm1\u0001m2+D\u0001(m1\u0002m2)\n\u0000\u00160MsH\u0001(m1+m2)\u0000K\n2\u0000\nm2\n1;z+m2\n2;z\u0001\n;(5)\nwith the constant Kcharacterizing the strength of the\nanisotropy and Jthe exchange. We shall focus on the\nferromagnetic coupling ( J >0) without loss of generality.\nThe LLG equation now becomes\n@m1\n@t=\r\nMsm1\u0002@E\n@m1+\u000bm1\u0002@m1\n@t\n+\u000bnlm1\u0002@m2\n@t; (6)\nwith\u000bthe Gilbert damping constant of each layer, and\nwhere the equation for the second layer is obtained from\nthe above by interchanging the labels 1 and 2. We\ntake the external \feld in the same direction as the\nDzyaloshinskii vector and D=D^z,H=H^z, while\n\u00160MsH;K\u001dD, so that the magnetic layers are aligned\nin the ^z-direction. Linearizing the LLG equation around\nthis direction we write mi= (mi;x;mi;y;1)Tand keep\nterms linear in mi;xandmi;y. Writing\u001ei=mi;x\u0000imi;y,\nwe \fnd, after Fourier transforming to frequency space,\nthat\n\u001f\u00001(!)\u0012\u001e1(!)\n\u001e2(!)\u0013\n= 0: (7)\nTo avoid lengthy formulas, we give explicit results below\nfor the case that J= 0, while plotting the results for\nJ6= 0 in Fig. 2. The susceptibility tensor \u001fij, or magnon\nGreen's function, is given by\n\u001f(!) =1\n((1 +i\u000b)!\u0000!0)2\u0000(\rD=Ms)2\u0000\u000b2\nnl!2)\n\u0002\u0012(1 +i\u000b)!\u0000!0i(\rD=Ms\u0000\u000bnl!)\n\u0000i(\rD=Ms+\u000bnl!) (1 +i\u000b)!\u0000!0\u0013\n;(8)\nwith!0=\r(\u00160H+K=Ms) the ferromagnetic-resonance\n(FMR) frequency of an individual layer. The poles of\nthe susceptibility determine the FMR frequencies of the\ncoupled layers and are, for the typical case that \u000b;\u000b nl\u001c\n1, given by\n!\u0006=!r;\u0006\u0000i\u000b!r;\u0006; (9)\nwith resonance frequency\n!r;\u0006=\r(\u00160H+K=Ms\u0006D=Ms): (10)\nWhen\r\u00160H= (1\u0007\u000bnl)D=(\u000bnlMs)\u0000K=Ms\u0019\nD=(\u000bnlMs)\u0000K=Mswe have for J= 0 that\u001f12(!r;\u0006) = 0\nwhile\u001f21(!r;\u0006)6= 0, signalling the non-reciprocal cou-\npling. That is, the excitation of layer 1 by FMR leads\nto response of magnetic layer 2, while layer 1 does not\nrespond to the excitation of layer 2. For opposite direc-\ntion of \feld the coupling reverses: the excitation of layer3\n|χ21(J=0)|\n|χ21(J=0.5D)|\n|χ21(J=15D)|\n|χ12(J=0)|\n|χ12(J=0.5D)|\n|χ12(J=15D)|\n0.96 0.98 1.00 1.02 1.040100200300400\nω/ωH\nFIG. 2. Magnetic susceptibilities of two magnetic layers as a\nfunction of frequency at di\u000berent exchange couplings. !H\u0011\n\r(\u00160H+K=M s). The resonance frequencies are located at\nthe peak positions. The parameters are D=! H= 0:001;\u000bnl=\n0:001;\u000b= 0:002.\n2 by FMR leads in that case to response of magnetic\nlayer 1, while layer 2 does not respond to the excitation\nof layer 1. As is observed from Fig. 2, for \fnite but\nsmallJ\u001cD, the coupling is not purely unidirectional\nanymore but there is still a large non-reciprocity. For\nJ\u001dD, this non-reciprocity is washed out.\nElectrically-actuated spin-current transmission. | In\npractice, it may be challenging to excite the individual\nlayers independently with magnetic \felds, which would\nbe required to probe the susceptibility that is determined\nabove. The two layers may be more easily probed inde-\npendently by spin-current injection/extraction from ad-\njacent contacts. Therefore, we consider the situation that\nthe two coupled magnetic layers are sandwiched between\nheavy-metal contacts (see Fig. 3(a)). In this set-up, spin\ncurrent may be transmitted between the two contacts\nthrough the magnetic layers.\nFollowing the Green's function formalism developed by\nZheng et al. [24], the spin-current from the left (right)\nlead to its adjacent magnetic layer is determined by the\ntransmission function of the hybrid system T12(T21)\ngiven by\nTij(!) = Trh\n\u0000i(!)G(+)(!)\u0000j(!)G(\u0000)(!)i\n: (11)\nHere,G(+)(!) is the retarded Green's function for\nmagnons in contact with the metallic leads that is de-\ntermined by Dyson's equation\u0002\nG(+)\u0003\u00001(!) =\u001f\u00001(!)\u0000\n\u0006(+)\n1(!)\u0000\u0006(+)\n2(!), where the retarded self energy\n~\u0006(+)\ni(!) accounts for the contact with the metallic lead\ni. These self energies are given by\n~\u0006(+)\n1(!) =\u0000i~\u000b0\n1\u0012\n!0\n0 0\u0013\n; (12)and\n~\u0006(+)\n2(!) =\u0000i~\u000b0\n2\u00120 0\n0!\u0013\n: (13)\nThe rates for spin-current transmission from the heavy\nmetal adjacent to the magnet iinto it, are given by\n\u0000i(!) =\u00002Imh\n\u0006(+)\ni(!)i\n=~. The couplings \u000b0\ni=\n\rRe[g\"#\ni]=4\u0019Msdiare proportional to the real part of the\nspin-mixing conductance per area g\"#\nibetween the heavy\nmetal and the magnetic layer i, and further depend on\nthe thickness diof the magnetic layers. Finally, the ad-\nvanced Green's function is G(\u0000)(!) =\u0002\nG(+)\u0003y.\nIn the analytical results below, we again restrict our-\nselves to the case that J= 0 for brevity, leaving the\ncaseJ6= 0 to plots. Using the above ingredients,\nEq. (11) is evaluated. Taking identical contacts so that\n\u000b0\n1=\u000b0\n2\u0011\u000b0, we \fnd that\nT12=4(\u000b0)2!2(\rD=Ms+\u000bnl!)2\njC(!)j2; (14)\nwhile\nT21=4(\u000b0)2!2(\rD=Ms\u0000\u000bnl!)2\njC(!)j2; (15)\nwith\nC(!) = [!H\u0000(1 +i(\u000b\u0000\u000bnl+\u000b0))!]\u0001\n[!H\u0000(1 +i(\u000b+\u000bnl+\u000b0))!]\u0000(\rD=Ms)2:(16)\nFrom the expression for C(!) it is clear that, since\n\u000b;\u000b nl;\u000b0\u001c1, the transmission predominantly occurs\nfor frequencies equal to the resonance frequencies !r;\u0006\nfrom Eq. (9). Similar to the discussion of the suscepti-\nbilities, we have for \felds \r\u00160H=D=\u000b nl\u0000K=Msthat\nthe transmission T12(!=D=\u000b nl)6= 0, while T21(!=\nD=\u000b nl) = 0. As a result, the spin-current transmis-\nsion is unidirectional at these \felds. For the linear spin-\nconductances Gij, given byGij=R\n~!(\u0000N0(~!))Tij(!),\nwe also have that G126= 0, while G21= 0. Here,\nN(~!) = [e~!=kBT\u00001]\u00001is the Bose-Einstein distri-\nbution function at thermal energy kBT. For the oppo-\nsite direction of external \feld we have G12= 0, while\nG216= 0. Like in the case of the susceptibility discussed\nin the previous section, a \fnite but small exchange cou-\npling makes the spin current transport no longer purely\nunidirectional, while maintaining a large non-reciprocity\n(see Fig. 3(b)).\nSpin-wave propagation. | Besides the unidirectional\ncoupling of two magnetic layers, the above results may\nbe generalized to a magnetic multilayer, or, equivalently,\nan array of coupled magnetic moments that are labeled\nby the index isuch that the magnetization direction of\nthei-th layer is mi. This extension allows us to engi-\nneer unidirectional spin-wave propagation as we shall see4\nm1\nm2\nLead Lead FM FM(a)\n(b)\nT21(J=0)\nT21(J=0.5D)\nT21(J=15D)\nT12(J=0)\nT12(J=0.5D)\nT12(J=15D)\n0.96 0.98 1.00 1.02 1.040.0000.0020.0040.0060.008\nω/ωH\nFIG. 3. (a) Schematic of the system that the two coupled\nmagnetic layers are sandwiched between two heavy-metal con-\ntacts. (b) Transmission of the hybrid system as a function of\nfrequency.\nbelow. We consider the magnetic energy\nE[m] =X\nk[D\u0001(mk\u0002mk+1)\u0000\u00160MsH\u0001mk];(17)\nand \fnd | within the same approximations as for our\ntoy model above | for the magnetization dynamics that\n@mk\n@t=2\r\nMsmk\u0002(D\u0002mk\u00001)\u0000\r\n\u000bnlMsmk\u0002D;(18)\nfor the \feld H=D=\u000bnl\u00160Ms. This shows that for these\n\felds the magnetic excitations travel to the right | cor-\nresponding to increasing index k| only. The direction\nof this one-way propagation is reversed by changing the\nmagnetic \feld to \u0000Hor by changing the sign of the non-\nlocal damping.\nTo study how spin waves propagate in an array of cou-\npled magnetic moments described by the Hamiltonian in\nEq. (17). We start from the ground state mk= (0;0;1)T\nand perturb the left-most spin ( k= 0) to excite spin\nwaves. Since the dynamics of this spin is not in\ruenced\nby the other spins for the \feld H=D=\u000b nl\u00160Ms, its\nsmall-amplitude oscillation can be immediately solved\nas\u001e0(t) =\u001e0(t= 0) exp(\u0000i!0t\u0000\u000b!0t) with\u001ek=\nmk;x\u0000imk;yas used previously. The dynamics of the\nspins to the right of this left-most spin is derived by solv-\ning the LLG equation (18) iteratively, which yields\n\u001ek(t) =\u001e0(t= 0)e\u0000i!0te\u0000\u000b!0t\nk!(\u00002\u000bnl!0t)k;(19)wherek= 0;1;2;:::N\u00001.\nTo guarantee the stability of the magnetization dynam-\nics, the dissipation matrix of the N-spin system should\nbe negative-de\fnite, which imposes a constraint on the\nrelative strength of Gilbert damping and non-local damp-\ning, i.e.,\u000b > 2\u000bnlcos\u0019\nN+1. For an in\fnitely-long chain\nN!1 , we have\u000b>2\u000bnl. Physically, this means that\nthe local dissipation of a spin has to be strong enough to\ndissipate the spin current pumped by its two neighbors.\nFor a spin chain with \fnite number of spins, \u000b= 2j\u000bnljis\nalways su\u000ecient to guarantee the stability of the system.\nTaking this strength of dissipation simpli\fes Eq. (19) to\n\u001ek(t) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)e\u0000t=\u001c\nk!(\u0000t=\u001c)k; (20)\nwhere\u001c\u00001=\u000b!0is the inverse lifetime of the FMR\nmode. This spatial-temporal pro\fle of spins is the same\nas a Poisson distribution with both mean and variance\nequal to\u001b=t=\u001cexcept for a phase modulation, and it\ncan be further approximated as a Gaussian wavepacket\non the time scale t\u001d\u001c, i.e.\n\u001e(x) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)\np\n2\u0019\u001be\u0000(x\u0000\u001b)2\n2\u001b: (21)\nSuch similarity suggests that any local excitation of the\nleft-most spin will generate a Gaussian wavepacket prop-\nagating along the spin chain. The group velocity of the\nmoving wavepacket is v=a=\u001c, whereais the distance\nbetween the two neighboring magnetic moments. The\nwidth of the wavepacket spreads with time as ap\nt=\u001c,\nwhich resembles the behavior of a di\u000busive particle. Af-\nter su\u000eciently long time, the wavepacket will collapse.\nOn the other hand, the excitation is localized and can-\nnot propagate when the right-most spin ( k=N\u00001) is\nexcited, because its left neighbor, being in the ground\nstate, has zero in\ruence on its evolution. These results\ndemonstrate the unidirectional properties of spin-wave\ntransport in our magnetic array.\nDiscussion, conclusion, and outlook. | We have\nshown that the ingredients for unidirectional coupling be-\ntween magnetic layers or moments are that they are cou-\npled only by DMI and non-local Gilbert damping. While\nin practice it may be hard to eliminate other couplings,\nthe DMI and non-local coupling need to be su\u000eciently\nlarger than the other couplings to observe unidirectional\ncoupling.\nThere are several systems that may realize the unidi-\nrectional coupling we propose. A \frst example is that of\ntwo magnetic layers that are coupled by a metallic spacer.\nSuch a spacer would accommodate non-local coupling via\nspin pumping and spin transfer. For a spacer that is\nmuch thinner than the spin relaxation length, we \fnd,\nfollowing Refs. [25{27], that \u000bnl=\r~Re[~g\"#]=4\u0019dMs,\nwith ~g\"#the spin-mixing conductance of the interface\nbetween the magnetic layers and the spacer, dthe thick-\nness of the magnetic layers. For simplicity, we took the5\nmagnetic layers to have equal properties. The two mag-\nnetic layers may be coupled by the recently-discovered\ninterlayer DMI [28, 29], tuning to a point (as a function\nof thickness of the spacer) where the ordinary RKKY ex-\nchange coupling is small. We estimate \u000bnl= 4:5\u000210\u00003\nford= 20 nm, Re[~g\"#] = 4:56\u00021014\n\u00001m\u00002and\nMs= 1:92\u0002105A=m (YIGjPt). The required mag-\nnetic \feld for unidirectional magnetic coupling is then\naround 4.5 T for D= 1 mT. Another possible platform\nfor realizing the unidirectional coupling is the system of\nFe atoms on top of a Pt substrate that was demonstrated\nrecently [30]. Here, the relative strength of the DMI and\nexchange is tuned by the interatomic distance between\nthe Fe atoms. Though not demonstrated in this experi-\nment, the Pt will mediate non-local coupling between the\natoms as well. Hence, this system may demonstrate the\nunidirectional coupling that we proposed.\nThe non-local damping is expected to be generically\npresent in any magnetic material and does not require\nspecial tuning, though it may be hard to determine its\nstrength experimentally. Hence, an attractive implemen-\ntation of the unidirectional coupling would be a magnetic\nmaterial with spins that are coupled only via DMI, with-\nout exchange interactions. While such a material has\nto the best of our knowledge not been discovered yet,\nit is realized transiently in experiments with ultrafast\nlaser pulses [31]. Moreover, it has been predicted that\nhigh-frequency laser \felds may be used to manipulate\nDMI and exchange, even to the point that the former is\nnonzero while the latter is zero [32, 33].\nPossible applications of our results are spin-wave and\nspin-current diodes and magnetic sensors, where a weak\n\feld signal can be ampli\fed and transported through\nthe unidirectional coupling to the remote site to be read\nout without unwanted back-action. Finally, we remark\nthat the unidirectional magnetic coupling that we pro-\npose here may be thought of as reservoir engineering, cf.\nRef. [34]. In our proposal, the reservoir is made up by the\ndegrees of freedom that give rise to the non-local damp-\ning, usually the electrons. We hope that this perspective\nmay pave the way for further reservoir-engineered mag-\nnetic systems\nAcknowledgements. | It is a great pleasure to\nthank Mathias Kl aui and Thomas Kools for discus-\nsions. 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X 5,021025 (2015)."    },    {        "title": "1002.4958v1.Correlation_Effects_in_the_Stochastic_Landau_Lifshitz_Gilbert_Equation.pdf",        "content": "arXiv:1002.4958v1  [cond-mat.mes-hall]  26 Feb 2010Correlation Effects in Stochastic Ferromagnetic Systems\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗\n(Dated: June 16, 2018)\nAbstract\nWe analyze the Landau-Lifshitz-Gilbert equation when the p recession motion of the magnetic\nmoments is additionally subjected to an uniaxial anisotrop y and is driven by a multiplicative cou-\npled stochastic field with a finite correlation time τ. The mean value for the spin wave components\noffers that the spin-wave dispersion relation and its damping is strongly influenced by the deter-\nministic Gilbert damping parameter α, the strength of the stochastic forces Dand its temporal\nrangeτ. The spin-spin-correlation function can be calculated in t he low correlation time limit by\nderiving an evolution equation for the joint probability fu nction. The stability analysis enables us\nto find the phase diagram within the α−Dplane for different values of τwhere damped spin wave\nsolutions are stable. Even for zero deterministic Gilbert d amping the magnons offer a finite life-\ntime. We detect a parameter range wherethe deterministic an d the stochastic damping mechanism\nare able to compensate each other leading to undamped spin-w aves. The onset is characterized by\na critical value of the correlation time. An enhancement of τleads to an increase of the oscillations\nof the correlation function.\nPACS numbers: 75.10.Hk, 05.40.-a, 75.30.Ds,72.70.+m,76.60.Es\n∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1I. INTRODUCTION\nMagnetism can be generally characterized and analyzed on different length and time scales.\nThe description of fluctuations of the magnetization, the occurre nce of damped spin waves\nand the influence of additional stochastic forces are successfully performed on a mesoscopic\nscale where the spin variables are represented by a continuous spa tio-temporal variable [1].\nIn this case a well established approach isbased uponthe Landau-L ifshitz equation [2] which\ndescribes the precession motion of the magnetization in an effective magnetic field. This\nfield consists of a superposition of an external field and internal fie lds, produced by the in-\nteracting magnetic moments. The latter one is strongly influenced b y the isotropic exchange\ninteraction and the magnetocrystalline anisotropy, for a recent r eview see [3]. The studies\nusing this frame are concentrated on different dynamical aspects as the switching behav-\nior of magnetic nanoparticles which can be controlled by external tim e-dependent magnetic\nfields [4] and spin-polarized electric currents [5, 6]. Such a current- induced spin transfer\nallows the manipulation of magnetic nanodevices. Recently, it has bee n demonstrated that\nan electric current, flowing through a magnetic bilayer, can induce a coupling between the\nlayers [7]. Likewise, such a current can also cause the motion of magn etic domain walls in\na nanowire [8]. Another aspect is the dynamical response of ferrom agnetic nanoparticles\nas probed by ferromagnetic resonance, studied in [9]. In describing all this more complex\nbehavior of magnetic systems, the Landau-Lifshitz equation has t o be extended by the in-\nclusion of dissipative processes. A damping term is introduced pheno menologically in such a\nmanner, that the magnitude of the magnetization /vectorSis preserved at any time. Furthermore,\nthe magnetization should align with the effective field in the long time limit. A realization\nis given by [2]\n∂S\n∂t=−γ[S×Beff]−ε[S×(S×Beff)]. (1)\nThe quantities γandεare the gyromagnetic ratio and the damping parameter, respectiv ely.\nAn alternative equation for the magnetization dynamics had been pr oposed by Gilbert [10].\nThe Gilbert equation yields an implicit form of the evolution of the magne tization. A com-\nbination of both equations, called Landau-Lifshitz-Gilbert equation (LLG) will be used as\nthe basic relation for our studies, see Eq. (2). The origin of the dam ping term as a non-\nrelativistic expansion of the Dirac equation has been discussed in [11 ] and a generalization\nof the LLG for conducting ferromagnetics is offered in [12]. The form of the damping seems\n2to be quite general as it has been demonstrated in [13] using symmet ry arguments for fer-\nroelectric systems.\nAs a new aspect let us focus on the influence of stochastic fields. Th e interplay between\ncurrent and magnetic fluctuations and dissipation has been studied recently in [14]. Via the\nspin-transfer torque, spin-current noise causes a significant en hancement of the magnetiza-\ntion fluctuations. Such a spin polarized current may transfer mome ntum to a magnet which\nleads to a spin-torque phenomenon. The shot noise associated with the current gives rise to\na stochastic force [15]. In our paper we discuss the interplay betwe en different dissipation\nmechanism, namely the inherent deterministic damping in Eq. (1) and t he stochastic mag-\nnetic field originated for instance by defect configurations giving ris e to a different coupling\nstrength between the magnetic moments. Assuming further, tha t the stochastic magnetic\nfield is characterized by a finite correlation time, the system offers m emory effects which\nmight lead to a decoherent spin precession. To that aim we analyze a f erromagnet in the\nclassical limit, i.e., the magnetic order is referred to single magnetic at oms which occupy\nequivalent crystal positions, and the mean values of their spins exh ibit a parallel orientation.\nThe last one is caused by the isotropic exchange interaction which will be here supplemented\nby a magneto-crystalline anisotropy that defines the direction of t he preferred orientation.\nEspecially, we discuss the influence of an uniaxial anisotropy. The co upling between differ-\nent dissipation mechanisms, mentioned above, leads to pronounced correlations, which are\ndiscussed below. Due to the multiplicative coupling of the stochastic fi eld and the finite\ncorrelation time the calculation of the spin-spin correlation function is more complicated.\nTo that aim we have to derive an equivalent evolution equation for the joint probability\ndistribution function. Within the small correlation time limit this approa ch can be fulfilled\nin an analytical manner. Our analysis is related to a recent paper [16] in which likewise the\nstochastic dynamics of the magnetization in ferromagnetic nanopa rticles has been studied.\nFurther, we refer also to a recent paper [17] where the mean first passage time and the\nrelaxation of magnetic moments has been analyzed. Different to tho se papers our approach\nis concentrated on the correlation effects in stochastic system wit h colored noise.\nOur paper is organized as follows: In Sec.II we discuss the LLG and ch aracterize the ad-\nditional stochastic field. The equations for the single and the two pa rticle joint probability\ndistribution are derived in Sec.III. Using these functions we obtain t he mean value of the\nspin wave variable and the spin-spin correlation function. The phase diagram, based on the\n3stability analysis, is presented in Sec.IV. In Sec.V we finish with some co nclusions.\nII. MODEL\nIn order to develop a stochastic model for the spin dynamics in ferr omagnetic systems let\nus first consider the deterministic part of the equation of motion. W e focus on a description\nbased upon the level of Landau-Lifshitz phenomenology [2], for a r ecent review see [3]. To\nfollow this line we consider a high spin systems in a ferromagnet sufficien tly below the\nCurie temperature. In that regime the dynamics of the magnet are dominated by transverse\nfluctuationsofthespatio-temporalvaryinglocalmagnetization. Theweakexcitations, called\nspin waves or magnons, are determined by a dispersion relation, the wavelength of which\nshould be large compared to the lattice constant a, i.e., the relation q·a≪1 is presumed to\nbe satisfied, where qis the wavenumber. In this limit the direction of the spin varies slowly\nwhile its magnitude |S|=msremains constant in time. A proper description for such a\nsituation is achieved by applying the Landau-Lifshitz-Gilbert equatio n (LLG) [4, 10, 18].\nThe spin variable is represented by S=msˆ n, whereˆ n(r,t) is a continuous variable which\ncharacterizes the local orientation of the magnetic moment. The e volution equation for that\nlocal orientation reads\n∂ˆ n\n∂t=−γ\n1+α2ˆ n×[Beff+α[ˆ n×Beff]]. (2)\nThe quantities γandαare the gyromagnetic ratio and the dimensionless Gilbert damping\nparameter, respectively, where αis related to εintroduced in Eq. (1). Beffis the effective\nmagnetic field that drives the motion of the spin density. Generally, it consists of an internal\npart originated by the interaction of the spins and an external field . This effective field is\nrelated to the Hamiltonian of the system by functional variation with respect to ˆ n\nBeff=−m−1\nsδH\nδˆ n. (3)\nIn absence of an external field the Hamiltonian can be expressed as [19, 20]\nH=/integraldisplay\nd3r{wex+wan},with\nwex=1\n2msκ(∇ˆ n)2andwan=1\n2msΓ sin2θ.(4)\nThereby, the constants κand Γ denote the exchange energy density and the magneto-\ncrystalline anisotropy energy density. To be more precise, κ∝Ja2,Jbeing the coupling\n4strength that measures theinteraction between nearest neighb ors inthe isotropic Heisenberg\nmodel [21]. Once again ais thelattice constant. Notice that the formof theexchange ener gy\nin the Hamiltonian (4) arises from the Heisenberg model in the classica l limit. The quantity\nθrepresents the angle between ˆ nand the anisotropy axis ˆν= (0,0,1), where ˆνpoints\nin the direction of the easy axis in the ground state in the case of zer o applied external\nfield. Thus, the constant Γ >0 characterizes anisotropy as a consequence of relativistic\ninteractions (spin-orbital and dipole-dipole ones [20]). In deriving Eq . (4) we have used\nˆ n2= 1. Although it is more conventional to introduce the angular coord inates (θ,Φ) [2, 4],\nwe find it more appropriate to use Cartesian coordinates. To proce ed, we divide the vector\nˆ ninto a static and a dynamic part designated by µandϕ, respectively. In the linearized\nspin wave approach let us make the ansatz\nˆ n(r,t) =µ(r)+ϕ(r,t) =µˆν+ϕ, µ= const., (5)\nwhereˆ n2= 1 is still valid. The effective field can now be obtained from Eqs. (3) an d (4).\nThis yields\nBeff=κ∇2ϕ−Γϕ′;ϕ′= (ϕ1,ϕ2,0). (6)\nEq. (2) together with Eqs. (3) and (4) represent the determinist ic model for a classical\nferromagnet. In order to extent the model let us supplement the effective magnetic field in\nEq. (6) by a stochastic component yielding an effective random field Beff=Beff+η(t). The\nstochastic process η(t) is assumed to be Gaussian distributed with zero mean and obeying\na colored correlation function\n˜χij(t,t′) =∝an}b∇acketle{tηi(t)ηj(t′)∝an}b∇acket∇i}ht=˜Dij\n˜τijexp/bracketleftbigg\n−|t−t′|\n˜τij/bracketrightbigg\n. (7)\nHere,˜Dijand ˜τijare the noise strength and the finite correlation time of the noise η.\nDue to the coupling of the effective field to the spin orientation ˆ nthe stochastic process\nis a multiplicative one. Microscopically, such a random process might be originated by\na fluctuating coupling strength for instance. The situation associa ted with our model is\nillustrated in Fig. 1 and can be understood as follows: The stochastic vector fieldη(t) is able\nto change the orientation of the localized moment at different times. Therefore, fixed phase\nrelations between adjacent spins might be destroyed. Moreover, theη(tk) are interrelated\ndue to the finite correlation time τ. The anisotropy axis defines the preferred orientation of\nthe mean value of magnetization. Due to the inclusion of η(t) the deterministic Eq. (2) is\n5xyz\nanisotropy axis ˆν\nexchange ∝Jaη(t1)η(t2)η(t3)random field at\ndifferent times ti\nFIG. 1. Part of a ferromagnetic domain influenced by stochast ic forces for the example of cubic\nsymmetry with lattice constant a. The black spin in the center only interacts with its nearest\nneighbors (green), where Jis a measure for the exchange integral.\ntransformed into the stochastic LLG. Using Eq. (5) it follows\n∂ϕ\n∂t=−γ\n1+α2(µ+ϕ)×[Beff+α[(µ+ϕ)×Beff]]. (8)\nThe random magnetic field is defined by\nBeff=κ∇2ϕ−Γϕ′+η(t), (9)\nwhereϕ′is given in Eq. (6). With regard to the following procedure we suppose the random\nfield to be solely generated dynamically, i.e., ˆ n×η(t) =ϕ×η(t). So far, the dynamics\nof our model (Eqs. (8) and (9)) are reflected by a nonlinear, stoc hastic partial differential\nequation (PDE). Using Fourier transformation, i.e., ψ(q,t) =F{ϕ(r,t)}and introducing\nthe following dimensionless quantities\nβ= (l0q)2+1, l2\n0=κ\nΓ, ω=γΓ,¯t=ωt ,λ(t) =η(t)\nΓ,(10)\nthe components ψi(q,t) fulfill the equation\nd\ndtψi(q,t) = Ωi(ψ(q,t))+Λ ij(ψ(q,t))λj(t). (11)\n6The quantity l0is the characteristic magnetic length [22]. The vector Ωand the matrix Λ\nare given by\nΩ=ξµβ\n−(αµψ1+ψ2)\nψ1−αµψ2\n0\n, ξ=1\n1+α2, (12)\nand\nΛ =ξ\nαµψ3ψ3−(ψ2+αµψ1)\n−ψ3αµψ3ψ1−αµψ2\nψ2−ψ1 0\n. (13)\nFor convenience we have substituted ¯t→tagain. The statistical properties of λ(t) are\nexpressed as ∝an}b∇acketle{tλ(t)∝an}b∇acket∇i}ht= 0 and\nχkl(t,t′) =∝an}b∇acketle{tλk(t)λl(t′)∝an}b∇acket∇i}ht=Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\nτkl→0− −− →2Dklδklδ(t−t′).(14)\nIncidentally, in the limit τ→0 the usual white noise properties are recovered. We empha-\nsize that although we regard the long-wavelength limit ( a·q≪1), wave vectors for which\nl0·q≫1 (in Eq. (10)) can also occur [22]. But this case is not discussed in the present\npaper and will be the content of future work. Whereas, in what follo ws we restrict our\nconsiderations to the case q→0 so that, actually, l0·q≪1 is fulfilled. Hence, we can set\nβ= 1 approximately in Eq. (10). Due to the anisotropy the spin wave dis persion relation\noffers a gap at q= 0. Owing to this fact ψis studied at zero wave vector. For this situation\nthe assumption of a space-independent stochastic force ηi(t), compare Eq. (7), is reasonable.\nFor non-zero wave vector the noise field should be a spatiotempora l fieldηi((r,t). Because\nour model is based on a short range interaction we expect that the corresponding noise\ncorrelation function is δ-correlated, i.e. instead of (14) we have\nχkl(r,t;r′,t′) =Dkl\nτklδklexp/bracketleftbigg\n−|t−t′|\nτkl/bracketrightbigg\n2Mδ(r−r′),\nwhereMis the strength of the spatial correlation. Using this relation we are able to study\nalso the case of small qwhich satisfies l0·q≪1. In the present paper we concentrate on\nthe case of zero wave vector q= 0.\n7III. CORRELATION FUNCTIONS\nIn the present section let us discuss the statistical behavior of th e basic Eqs. (11)-(14). They\ndescribe a non-stationary, non-Markovian process attributed t o the finite correlation time.\nDue to their common origin both characteristics can not be analyzed separately. In the\nlimitτ→0, Eq. (11) defines a Markovian process which provides also station arity by an\nappropriate choice of initial conditions [23]. However, the present s tudy is focused on the\neffectofnonzerocorrelationtimes. Tothatpurposeweneedapro perprobabilitydistribution\nfunction which reflects the stochastic process defined by Eqs. (1 1)-(14). In deriving the\nrelevant joint probability distribution function we follow the line given in [24], where the\ndetailedcalculationshadbeencarriedout, seealsothereferences citedtherein. Inparticular,\nit has been underlined in those papers that in order to calculate corr elation functions of type\n∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hta single probability distribution function P(ψ,t) is not sufficient. Instead of\nthat one needs a joint probability distribution of the form P(ψ,t;ψ′,t′). Before proceeding\nlet us shortly summarize the main steps to get the joint probability dis tribution function.\nTo simplify the calculation we assume τkl=τδklandDkl=Dδkl. Notice that our system\nhas no ergodic properties what would directly allow us to relate the st ochastic interferences\nwith temperature fluctuations by means of a fluctuation-dissipatio n theorem. Based on\nEq. (11) the appropriate joint probability distribution is defined by [2 4, 25], for a more\ngeneral discussion compare also [26]:\nP(ψ,t;ψ′,t′) =∝an}b∇acketle{tδ(ψ(t)−ψ)δ(ψ(t′)−ψ′)∝an}b∇acket∇i}ht. (15)\nHere the average is performed over all realizations of the stochas tic process. In defining the\njoint probability distribution function we follow the convention to indic ate the stochastic\nprocess by the function ψ(t) whereas the quantity without arguments ψstands for the\nspecial values of the stochastic variable. These values are even re lalized with the probaility\nP(ψ,t;ψ′,t′). The equation of motion for this probability distribution reads acco rding to\n8[24]\n∂\n∂tP(ψ,t;ψ′,t′)\n=−∂\n∂ψit/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t)\nδλk(t1)/bracketrightbigg\nψ(t)=ψ·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1\n−∂\n∂ψ′it′/integraldisplay\n0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t′)\nδλk(t1)/bracketrightbigg\nψ(t′)=ψ′·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg\ndt1,(16)\nwhere Novikov’s theorem [27] has been applied. Expressions for the response functions\nδψi(t)/δλk(t1) andδψi(t′)/δλk(t1) can be found by formal integration of Eq. (11) and\niterating the formal solution. After a tedious but straightforwar d calculation including the\ncomputation of the response functions to lowest order in ( t−t1) and (t′−t1) and the\nevaluation of several correlation integrals referring to χklfrom Eq. (14), Eq. (16) can be\nrewritten in the limit of small correlation time τas\n∂\n∂tPs(ψ,t;ψ′,t′) =/braceleftbig\nL0(ψ,τ)\n+exp[−(t−t′)/τ]D∂\n∂ψiΛik(ψ)∂\n∂ψ′\nnΛnk(ψ′)/bracerightbigg\nPs(ψ,t;ψ′,t′).(17)\nThereby, transient terms and terms of the form ∝τexp[−(t−t′)/τ] (these terms would lead\nto terms of order τ2in Eq. (22)) have been neglected. The result is valid in the stationary\ncase characterized by t→ ∞andt′→ ∞but finites=t−t′. In Eq. (17) L0is the operator\nappearing in the equation for the single probability density. Following [2 4, 28] the operator\nreads\nL0(ψ,τ) =−∂\n∂ψiΩi(ψ)+∂\n∂ψiΛik(ψ)∂\n∂ψn/braceleftigg\nD/bracketleftbig\nΛnk(ψ)−τMnk(ψ)/bracketrightbig\n+D2τ/bracketleftbigg\nKnkm(ψ)∂\n∂ψlΛlm(ψ)+1\n2Λnm(ψ)∂\n∂ψlKlkm(ψ)/bracketrightbigg/bracerightigg\n,(18)\nwith\nMnk= Ωr∂Λnk\n∂ψr−Λrk∂Ωn\n∂ψr\nKnlk= Λrk∂Λnl\n∂ψr−∂Λnk\n∂ψrΛrl.(19)\nThe equation of motion for the expectation value ∝an}b∇acketle{tψi∝an}b∇acket∇i}htscan be evaluated from the single\nprobability distribution in the stationary state\n∂\n∂tPs(ψ,t) =L0Ps(ψ,t). (20)\n9One finds\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi∝an}b∇acket∇i}hts+D/angbracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/angbracketrightbigg\ns−D2τ/braceleftigg/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/angbracketrightbigg\ns/bracerightigg\n.(21)\nThe knowledge of the evolution equation of the joint probability distr ibutionP(ψ,t;ψ′,t′)\ndue to Eqs. (17) and (18) allows us to get the corresponding equat ion for the correlation\nfunctions. Following again [24], it results\nd\ndt∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi(ψ(t))ψj(t′)∝an}b∇acket∇i}hts+D/angbracketleftbigg/bracketleftbigg∂Λik\n∂ψn/parenleftbig\nΛnk−τMnk/parenrightbig/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n−D2τ/braceleftigg/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnKnkm/parenrightbigg\nΛrm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns\n+1\n2/angbracketleftbigg/bracketleftbigg∂\n∂ψr/parenleftbigg∂Λik\n∂ψnΛnm/parenrightbigg\nKrkm/bracketrightbigg\ntψj(t′)/angbracketrightbigg\ns/bracerightigg\n+Dexp/bracketleftbigg\n−t−t′\nτ/bracketrightbigg\n∝an}b∇acketle{tΛik(ψ(t))Λjk(ψ(t′))∝an}b∇acket∇i}hts,(22)\nwhere the symbol [ ...]tdenotes the quantity [ ...] at timet. As mentioned above the result\nis valid for t, t′→ ∞whiles=t−t′>0 remains finite. The quantities MnkandKklmare\ndefined in Eq. (19). The components Ω iand Λ ijare given in Eqs. (12) and (13). Performing\nthe summation over double-indices according to Eqs. (21) and (22) we obtain the evolution\nequations for the mean value and the correlation function\nd\ndt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t)∝an}b∇acket∇i}hts, (23)\nand\nd\ndsCij(s) =d\nds∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t′+s)ψj(t′)∝an}b∇acket∇i}hts\n+Dexp/bracketleftig\n−s\nτ/bracketrightig\n∝an}b∇acketle{tΛik(ψ(t′+s))Λjk(ψ(t′))∝an}b∇acket∇i}hts.(24)\nNotice, that in the steady state one gets Cij(t,t′) =Cij(s) withs=t−t′. The matrix\ncomponents of Gikare given by\nGik=\n−A1A20\n−A2−A10\n0 0 −A3\n, (25)\n10where\nA1=−D2τ(6µ2α2−1)ξ4+2µ2αDτξ3−D(µ2α2−2)ξ2+µ2αξ\nA2=1\n2µαD2τ/parenleftbig\n11−3µ2α2/parenrightbig\nξ4+µDτ/parenleftbig\nµ2α2−1/parenrightbig\nξ3+3µDαξ2−µξ\nA3= +D2τ/parenleftbig\n3µ2α2+1/parenrightbig\nξ4−4µ2αDτξ3+2Dξ2,(26)\nandξis defined in Eq. (12). At this point let us stress that in the case t′= 0 the term\n∝exp[−(t−t′)/τ] on the rhs. in Eqs. (22) and (24), respectively, would vanish in the steady\nstate, i.e.\n∝an}b∇acketle{tψi(t′+s)ψj(t′)∝an}b∇acket∇i}hts∝ne}ationslash=∝an}b∇acketle{tψi(s)ψj(0)∝an}b∇acket∇i}hts.\nTheoccurrenceofsuchatermisastrongindicationforthenon-st ationarityofourmodel. An\nexplicit calculation shows, that in general this inequality holds for non -stationary processes\n[23].\nIV. RESULTS\nThe solution of Eq. (23) can be found by standard Greens function methods and Laplace\ntransformation. As the result we find\n∝an}b∇acketle{tψ(t)∝an}b∇acket∇i}hts=\ne−A1tcos(A2t)e−A1tsin(A2t) 0\n−e−A1tsin(A2t)e−A1tcos(A2t) 0\n0 0 e−A3t\n·∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts, (27)\nwhere∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts=∝an}b∇acketle{tψ(t= 0)∝an}b∇acket∇i}htsare the initial conditions. The parameters A1,A3andA2defined\nin Eqs. (26) play the roles of the magnon lifetime and the frequency o f the spin wave at\nzero wave vector, respectively. As can be seen in Eq. (26) all of th ese three parameters are\naffected by the correlation time τand the strength Dof the random force. Moreover, the\nGilbert damping parameter αinfluences the system as well. The solution of Eq. (24) for\nthe correlation function in case of t′= 0 is formal identical to that of Eq. (27). The more\ngeneral situation t′∝ne}ationslash= 0 allows no simple analytic solution and hence the behavior of the\ncorrelation function C(s) is studied numerically. In order to analyze the mean values and\nthe correlation function let us first examine the parameter range w here physical accessible\nsolutions exist. In the following we assume ∝an}b∇acketle{tψ1(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ2(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ0∝an}b∇acket∇i}htand∝an}b∇acketle{tψ3(0)∝an}b∇acket∇i}ht= 0, since\nthe solutions for ψ1(t) andψ2(t) on the one hand and ψ3(t) on the other hand are decoupled\n11in Eq. (27). Therefore, spin wave solutions only exists for non-zer o averages ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htand\n∝an}b∇acketle{tψ2(t)∝an}b∇acket∇i}ht. The existence of such non-trivial solutions are determined in depe ndence on the\nnoise parameters Dandτand the deterministic damping parameter α. Notice, that the\ndimensionless quantity D=˜D/Γ, i.e.,Dis the ratio between the strength of the correlation\nfunction (Eq. (7)) and the anisotropy field in the original units. The stability of spin wave\nsolutions is guaranteed for positive parameters A1andA3. According to Eqs. (26) the\nphase diagrams are depicted in Fig. 2 within the α−Dplane for different values of the\ncorrelation time τ. The separatrix between stable and unstable regions is determined by\nthe condition A1= 0. The second condition A3= 0 is irrelevant due to the imposed initial\nconditions. As the result of the stability analysis the phase space dia gram is subdivided into\nfour regions where region IV does not exist in case of τ= 0, see Fig. 2(a). For generality,\nwe take into account both positive and negative values of Dindicating correlations and\nanti-correlations of the stochastic field. Damped spin waves are ob served in the areas I and\nIV, whereas the sectors II and III reveal non-accessible solutio ns. In those regions the spin\nwave amplitude, proportional to exp[ −A1t], tends to infinity which should not be realized,\ncompare Figs. 2(b)-2(d). Actually, a reasonable behavior is obser ved in regions I and IV. As\nvisible from Fig. 2 damped spin waves will always emerge for D>0 even in the limit of zero\ndamping parameter αand vanishing correlation time τ. This behavior is shown in Fig. 3,\nwhere theevolution of ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htis depicted for different valuesof α. As canbeseen in Fig.2(a)\nthesolutionfor D<0isunlimitedandconsequently, itshouldbeexcludedfurther. Contr ary\nto this situation, additional solutions will be developed in region IV in ca se ofτ >0 and\nsimultaneously α= 0, see Figs. 2(b)-2(d). Thereby the size of area IV grows with inc reasing\nτ. Likewise, the extent of region I decreases for an enhanced τ. However, in the limit of\nD= 0 and consequently for τ= 0, too, only damped spin waves are observed. Immediately\non the separations line undamped periodic solutions will evolve, compa re the sub-figures in\nFig. 2. This remarkable effect can be traced back to the interplay be tween the deterministic\ndamping and the stochastic forces. Both damping mechanism are co mpensated mutually\nwhich reminds of a kind of resonance phenomenon. The difference to conventional resonance\nbehavior consists of the compensation of the inherent determinist ic Gilbert damping and the\nstochastic one originated from the random field. This statement is e mphasized by the fact\nthat undamped periodic solutions do not develop in the absence of st ochastic interferences,\ni.e.,D= 0. The situation might be interpreted physically as follows: the requ ired energy\n12(a)τ= 0 (b)τ= 0.1\n(c)τ= 1 (d)τ= 10\nFIG. 2.α−Dplane for fixed magnetization µ= 0.9 and different values of τ.\nthat enables the system to sustain the deterministic damping mecha nisms is delivered by\nthe stochastic influences due to the interaction with the environme nt. To be more precise, in\ngeneral, the Gilbert damping enforces the coherent alignment of th e spin density along the\nprecession axis. Contrary, the random field supports the dephas ing of the orientation of the\nclassical spins. Surprisingly, the model predicts the existence of a critical value τ=τc≥0\n13FIG. 3. Evolution of the mean value ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}ht, withµ= 0.9,D= 0.1 andτ= 0.αvaries from 0\n(dash-dotted line), 0 .05 (solid line), 0 .5 (dotted line) and 1 (dashed line).\ndepending on αandDwhich determines the onset of undamped periodic solutions. Notice,\nthat negative values of τcare excluded. The critical value is\nτc=−[µ2(α3−Dα2+α)+2D](1+α2)2\n2Dµ2(α3−3Dα2+α)+D2. (28)\nHence, this result could imply the possibility of the cancellation of both damping processes.\nExamples according to the damped and the periodic case are displaye d in Fig. 4. An increas-\ningτfavors the damping process as it is visible in Fig. 4(a). Based on estima tions obtained\nfor ferromagnetic materials [29] and references therein, the Gilbe rt damping parameter can\nrange between 0 .04<α<0.22 in thin magnetic films, whereas the bulk value for Co takes\nαb≈0.005. The phase space diagram in Fig. 2 offers periodic solutions only fo r values of\nαlarger than those known from experiments. Therefore such perio dic solutions seem to be\nhard to see experimentally. We proceed further by analyzing the be havior of the correlation\nfunction by numerical computation of the solution of Eq. (24) with E qs. (25) and (26). As\ninitial values we choose Cik(t=t′,t′) =Cik(s= 0) =C0for every combination i,k={1,2,3}.\nThe results are depicted in Figs. 5 and 6. Inspecting Figs. 5(a)-5(c ) one recognizes that an\nenhancement of the correlation time τleads to an increase of the oscillations within the\ncorrelation functions C1k,k={1,2,3}. Moreover, Fig. 5(d) reveals that the oscillatory\n14(a) (b)\nFIG. 4. Evolution of the mean values ∝an}b∇acketle{tψ1,2(t)∝an}b∇acket∇i}ht, withµ= 0.9. (a):D= 0.1,α= 0.005 andτvaries\nfrom 10 (solid line), 1 (dotted line) and 0 (dash-dotted line ). (b):D= 2,α= 1 andτ=τc≈1.79\n(Eq. (28)). The solid line represents ∝an}b∇acketle{tψ1∝an}b∇acket∇i}htand the dash-dotted line is ∝an}b∇acketle{tψ2∝an}b∇acket∇i}ht.\nbehavior of C31seems to be suppressed. Obviously, the decay of the correlation f unction is\nenhanced if τgrowths up. The pure periodic case for τ=τc, corresponding to Fig. 4(b),\nis depicted in Fig. 6. Exemplary, C12andC31are illustrated. The behavior of the latter is\nsimilar to the damped case, displayed in Fig. 5(d), unless slight oscillatio ns occur. However,\nif one compares the form of C12in Fig. 5(b) and Fig. 6 the differences are obvious. The am-\nplitude of the correlation function for the undamped case grows to the fourfold magnitude\nin comparison with C0, whereas the damped correlation function approaches zero. Fur ther,\na periodic behavior is shown in Fig. 6, and therefore the correlation w ill oscillate about zero\nbut never vanish for all s=t−t′>0.\nV. CONCLUSIONS\nIn this paper we have analyzed the dynamics of a classical spin model with uniaxial\nanisotropy. Aside from the deterministic damping due to the Landau -Lifshitz-Gilbert\nequation the system is subjected to an additional dissipation proce ss by the inclusion of a\nstochastic field with colored noise. Both dissipation processes are a ble to compete leading to\n15(a) (b)\n(c) (d)\nFIG. 5. Correlation functions Cik(s) forµ= 0.9,D= 0.1 andα= 0.005.τtakes 0 (dotted line),\n1 (solid line) and 10 (dash-dotted line).\na more complex behavior. To study this one we derive an equation for the joint probability\ndistribution which allows us to find the corresponding spin-spin-corr elation function. This\nprogram can be fulfilled analytically and numerically in the spin wave appr oach and the\nsmall correlation time limit. Based on the mean value for the spin wave c omponent and\n16FIG. 6. Correlation functions Cik(s) forτ=τc≈1.79 (Eq. (28)), µ= 0.9,D= 2 andα= 1. The\ndotted line represents C12and the solid line is C31.\nthe correlation function we discuss the stability of the system in ter ms of the stochastic\nparameters, namely the strength of the correlated noise Dand the finite correlation time\nτ, as well as the deterministic Gilbert damping parameter α. The phase diagram in the\nα−Dplane offers that the system develops stable and unstable spin wave solutions due to\nthe interplay between the stochastic and the deterministic damping mechanism. So stable\nsolutions evolve for arbitrary positive Dand moderate values of the Gilbert damping α.\nFurther, we find that also the finite correlation time of the stochas tic field influences the\nevolution of the spin waves. In particular, the model reveals for fix edDandαa critical\nvalueτcwhich characterizes the occurrence of undamped spin waves. The different situa-\ntions are depicted in Fig. 2. Moreover, the correlation time τaffects the damped spin wave\nwhich can be observed in regions I and IV in the phase diagram. If the parameters Dand\nαchanges within these regions, an increasing τleads to an enhancement of the spin wave\ndamping, cf. Fig. 4(a). The influence of τon the correlation functions is similar as shown\nin Figs. 5(a)-5(c). The study could be extended by the inclusion of fi nite wave vectors and\nusing an approach beyond the spin wave approximation.\n17ACKNOWLEDGMENTS\nOne of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which is\nsupported by the Saxony-Anhalt State, Germany.\n18[1] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of continuous media (Pergamon\nPress, Oxford, 1989).\n[2] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935).\n[3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halpe rin, Rev. Mod. Phys. 77, 1375\n(2005).\n[4] A. Sukhov and J. Berakdar, J. Phys. - Cond. 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Daniel and M. Lakshmanan, Physica A 120, 125 (1983).\n[19] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984).\n[20] V. G. Bar’Yakhtar, M. V. Chetkin, B. A. Ivanov, andS. N. G adetskii, Dynamics of Topological\nMagnetic Solitons: Experiment and Theory (Springer Tracts i n Modern Physics) (Springer,\n1994).\n[21] M. Lakshmanan, T. W. Ruijgrok, and C. J. Thompson, Physi ca A84, 577 (1976).\n[22] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep .194, 117 (1990).\n[23] A. Hernandez-Machado and M. San Miguel, J. Math. Phys. 25, 1066 (1984).\n[24] A. Hernandez-Machado, J. M. Sancho, M. San Miguel, and L . Pesquera, Zeitschr. f. Phys. B\n52, 335 (1983).\n19[25] N. G. van Kampen, Braz. J. Phys. 28, 90 (1998).\n[26] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amster-\ndam, 1981).\n[27] E. A. Novikov, Sov. Phys. JETP 20, 1290 (1965).\n[28] H. Dekker, Phys. Lett. A 90, 26 (1982).\n[29] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev . Lett.88, 117601 (2002).\n20"    },    {        "title": "1104.4215v2.Spin_Damping_Monopole.pdf",        "content": "arXiv:1104.4215v2  [cond-mat.mes-hall]  27 Feb 2012Typeset with jpsj2.cls <ver.1.2.2> Letter\nSpin Damping Monopole\nAkihitoTakeuchi∗and Gen Tatara\nDepartment of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan\nWe present theoretical evidence that a magnetic monopole emerge s in dynamic magnetic\nsystems in the presence of the spin-orbit interaction. The monopo le field is expressed in\nterms of spin damping associated with magnetization dynamics. We de monstrate that the\nobservation of this spin damping monopole is accomplished electrically u sing Amp` ere’s law\nfor monopole current. Our discovery suggests the integration of monopoles into electronics,\nnamely, monopolotronics.\nKEYWORDS: magnetic monopole, spin damping, inverse spin Ha ll effect, Maxwell’s equations\nThe magnetic monopole predicted by Dirac in 19311is a unique particle that arises from\nsingularity.2In high-energy physics, a monopole emerges if the electroma gnetic interaction\nin the world, described by a U(1) algebra, is a result of the sy mmetry breaking of a unified\nforce having a higher symmetry of SU(5).3,4Intensive effort has been exerted to find a grand\nunified theory (GUT) monopole by waiting for a monopole creat ed in the early universe to\ngo through superconducting detectors5and by detecting its ionization;6however, no evidence\nhas been found thus far. The energy needed to create a GUT mono pole is about 1017GeV,\nand so creating one in an accelerator on earth is impossible.\nSince a monopole is a consequence of symmetry breaking, it ex ists in solids too. Symmetry\nbreaking in solids occurs at an energy much lower than the GUT energy, typically below 1 eV,\nand therefore experiments on this are feasible. The most wel l-known monopole in solids is the\nhedgehog monopole (HHM), which arises in magnetic material s;7it is the SU(2) counterpart\nof the GUT monopole. The key interaction for the HHM is the cou pling of the conduction\nelectron to the magnetization represented by the vector M(r,t), which depends on the space\ncoordinate rand the time t. The electronic spin, represented by a vector σ, is polarized by\nMowing to the coupling\nHsd=−JM·σ, (1)\nwhereJis the coupling constant. Since Mis an external field for electrons, the SU(2) symme-\ntry of the electronic spin is broken. By diagonalizing Hsdand choosing the spin quantization\ndirection to be along the z-axis, the electrons are described as spin-polarized and in teracting\nwith a SU(2) gauge field,7Aa\nµ(µ=t,x,y,zanda=x,y,zare the indices in the coordi-\n∗E-mail: atake@phys.se.tmu.ac.jp\n1/9J. Phys. Soc. Jpn. Letter\nnate space and spin space, respectively). In the adiabatic l imit, i.e., when Jis large, only the\nzcomponent of the gauge field, Az\nµ, survives and it acts as a U(1) gauge field. The HHM\noriginates from the deviation from the perfect U(1) symmetr y, i.e., from the perpendicular\ncomponents Ax\nµandAy\nµ. In fact, the electrons feel the SU(2) gauge fields whose stre ngth is\nFa\nµν=∂µAa\nν−∂νAa\nµ−(2e//planckover2pi1)ǫabcAb\nµAc\nν, whereeis the electric charge, /planckover2pi1is the Planck constant\ndivided by 2 π, andǫabcrepresents the antisymmetric tensor. When this field is proj ected onto\nthe U(1) space, we obtain the electromagnetic field tensor as Fz\nµν=∂µAz\nν−∂νAz\nµ+ Φµν,\nwhere Φ µν≡ −(2e//planckover2pi1)(Ax\nµAy\nν−Ay\nµAx\nν). The anomalous field strength Φ µν, which in terms\nofn≡M/|M|reads Φ µν=−(/planckover2pi1/2e)n·(∂µn×∂νn), represents the HHM. In fact, the\nfield strength satisfies (1 /2)ǫµνσρ∂νFz\nσρ= 0, whose components read ∇×E+˙B=−jh, and\n∇·B=ρh,wheretheelectric andmagneticfieldsare Ei≡ −∇iAz\nt−˙Az\ni=−(/planckover2pi1/2e)n·(˙n×∇in)\nandBi≡ǫijk∇jAz\nk= (/planckover2pi1/4e)ǫijkn·(∇jn× ∇kn), respectively. The monopole current ( jh)\nand its density ( ρh) for the HHM are given as jh,i=−(3/planckover2pi1/4e)ǫijk˙n·(∇jn× ∇kn) and\nρh= (/planckover2pi1/4e)ǫijk∇in·(∇jn× ∇kn), respectively. Although the HHM is mathematically al-\nlowed, experimental realization has not been achieved thus far. In fact, as seen from the\nexpression for jhandρh, the HHM disappears in common magnets where the length of the\nlocal magnetization, n, is constant. In addition, the boundary condition at infinit y for the\nHHM would not be easy to realize experimentally.\nIn this paper, we search for a different monopole in magnets, wh ich exists in conventional\nferromagnets where the local magnetization length is const ant. Such a monopole current\ncreates the rotational electric field via Amp` ere’s law. Thi s means that the monopole current is\nananomalousangularmomentumsourcethatinducestherotat ional motionofelectriccharges.\nTo realize such a monopole in magnetic systems, a coupling be tween spin and electron orbital\nmotion is, therefore, essential. Such a coupling is known to emerge from a relativistic effect,\nnamely, the spin-orbit interaction. The spin-orbit intera ction exists in all elements including\nmagnetic ones and is particularly strong in heavy elements s uch as platinum and gold.\nOur aim in this study is to prove the existence of the above mon opole theoretically.\nSince the spin-orbit interaction explicitly breaks the SU( 2) invariance, we cannot follow the\nderivationoftheHHMshownabove.Instead,wewilldirectly calculatetheeffectiveelectricand\nmagnetic fields for the electron spin based on a nonequilibri um Green’s function formalism,\nand derive Maxwell’s equations they satisfy.\nWe consider two types of spin-orbit interaction. The first is that from a uniform field, ER,\nnamely the Rashbainteraction.8Such a field is realized at interfaces and surfaces.9Thesecond\nis that from a random potential, vi, induced by heavy impurities. The spin-orbit interaction\nthus reads\nHso=−1\n/planckover2pi1(λRER−λi∇vi)·(p×σ), (2)\nwherepis the electron momentum and λis the coupling constant (the subscripts R and i\n2/9J. Phys. Soc. Jpn. Letter\ncharacterize Rashba and impurity-induced spin-orbit inte reactions, respectively). The inter-\naction with the magnetization is described by Hsd. The Hamiltonian of the present system is,\ntherefore, given as H= (p2/2m)+vi+Hsd+Hso, wheremis the electron mass. The magneti-\nzation we consider in Hsdis dynamic. Dynamic magnetization, when coupled to the spin -orbit\ninteraction, generates an electric charge flow.10–12The pumped electric current jis calculated\nby evaluating a quantum field theoretical expectation value of the electron velocity operator\nˆv=−(i/planckover2pi1/m)∇+(1//planckover2pi1)(λRER−λi∇vi)×σ. By using field operators for electrons, c†andc,\ntheelectric current thus reads j=−etr/an}b∇acketle{tc†ˆvc/an}b∇acket∇i}ht, wheretr denotes the trace over spin indices and\nthe bracket represents the expectation value. It is written in terms of the lesser component of\nthe nonequilibrium Green’s function,13defined as G<\nss′(r,t;r′,t′)≡(i//planckover2pi1)/an}b∇acketle{tc†\ns′(r′,t′)cs(r,t)/an}b∇acket∇i}ht(s\nands′are spin indices), as\nj(r,t) =etr({/planckover2pi12\n2m(∇r−∇r′)+i[λRER\n−λi∇vi(r)]×σ}G<(r,t;r′,t))r′=r. (3)\nThis quantum field theoretical expectation value is evaluat ed by solving the Dyson’s equation\nfor the nonequilibrium Green’s function defined on the Keldy sh contour (C),\nGss′(r,t;r′,t′) =δs,s′gs(r,t;r′,t′)\n+/integraldisplay\nd3r′′/integraldisplay\nCdt′′gs(r,t;r′′,t′′)\n×(δs,s′′vi(r′′)−JM(r′′,t′′)·σss′′\n+i{[λRER−λi∇vi(r′′)]×∇r′′}·σss′′)\n×Gs′′s′(r′′,t′′;r′,t′), (4)\nwhereGss′(r,t;r′,t′)≡ −(i//planckover2pi1)/an}b∇acketle{tTC[cs(r,t)c†\ns′(r′,t′)]/an}b∇acket∇i}ht(TCis the path-ordering operator) and\ngdenotes the free Green’s function.\nIn the calculation, the impurities are approximated as rand om point scatterers, and aver-\naging is carried out as /an}b∇acketle{tvi(r)vi(r′)/an}b∇acket∇i}hti=niu2\niδ3(r−r′) (nianduiare the impurity concentration\nand the strength of scattering, respectively).14The impurities give rise to an elastic lifetime\nfor the electron, τ, which is calculated in metals as τ=/planckover2pi1/2πniu2\niν(νis the density of states\nper volume).15The Dyson’s equation is solved by treating λandJperturbatively to the first\nand second orders, respectively. We consider a sufficiently s low dynamics of magnetization,\nnamely, Ω τ≪1 (Ω is the frequency of magnetization dynamics), and assume that the mag-\nnetization structure varies smoothly in the space compared with the electron mean free path\nℓ, i.e.,qℓ≪1 (qis the wave number of magnetization profile). The leading con tribution in\n3/9J. Phys. Soc. Jpn. Letter\nthis case turns out to be16,17\nj(r,t) =eJ2\nV/summationdisplay\nk,k′,q1,q2/summationdisplay\nω,Ω1,Ω2e−i(q1+q2)·r+i(Ω1+Ω2)tΩ1\n×dfω\ndω(Mq1,Ω1×Mq2,Ω2)×[iλRτ\n/planckover2pi1ER|gr\nk,ω|2\n+4/planckover2pi12λi\n3πντ2(q1+q2)εk|gr\nk,ω|2|gr\nk′,ω|4]\n−D∇ρ(r,t), (5)\nwhereVis the system volume, D≡2εFτ/3mis the electron diffusion constant ( εFrepresents\nthe Fermi energy), fωis the Fermi distribution function, gris the retarded Green’s function\ndefined as gr\nk,ω= [/planckover2pi1ω−εk+ (i/planckover2pi1/2τ)]−1, andεk=/planckover2pi12k2/2m. The last term is the diffusive\ncontribution arising from vertex corrections, where the el ectric charge density ρis19\nρ(r,t) =4eνλRJ2τ3\n/planckover2pi12V∇·/integraldisplay\nd3r′/integraldisplay\ndt′\n×/summationdisplay\nq/summationdisplay\nΩe−iq·(r−r′)+iΩ(t−t′)\nDq2τ+iΩτ\n×{ER×[M(r′,t′)×˙M(r′,t′)]}. (6)\nSumming over the wave vectors and frequencies in eq. (5), the electric current is obtained as\nj=−16eνλiJ2εFτ2\n3/planckover2pi12∇×(M×˙M)\n−4eνλRJ2τ2\n/planckover2pi12ER×(M×˙M)−D∇ρ. (7)\nThisresultis rewritten usingtheeffective electric andmagn etic fields, EsandBs, respectively,\nas (µandσcare the magnetic permeability and electric conductivity, r espectively)\nj=1\nµ∇×Bs+σcEs−D∇ρ, (8)\nwhere the effective fields are defined as20,21\nEs≡ −αRER×N,\nBs≡ −βiN. (9)\nHere,N≡M×˙Mis a vector representing the spin damping torque (inset in Fi g. 1).22The\ncoefficients αRandβiareαR≡4eνλRJ2τ2/σc/planckover2pi12andβi≡16eνµλiJ2εFτ2/3/planckover2pi12, respectively.\nThe effective fields calculated here are those acting on the ele ctronic spin in the same manner\nas the effective fields from the HHM. Clearly, the fields [eq. (9) ] do not satisfy Faraday’s law\nor Gauss’s law of conventional electromagnetism, but those with monopole contribution:\n∇×Es+˙Bs=−jm,\n∇·Bs=ρm, (10)\n4/9J. Phys. Soc. Jpn. Letter\nx\nxyzz\njjmM\n+++---\nρmd\nM˙ M \nN\nN θERjm\nFig. 1. (Color online). Schematic illustration of monopole pumping and d etection in thin ferromag-\nnetic film attached to a nonmagnetic layer. Magnetization ( M) precession is induced by applying\nan oscillating magnetic field. The Rashba field ERexists at the interface and creates the monopole\ncurrentjmnear the interface. The width of the monopole current distribution ,d, is comparable to\nthe decay length of the magnetization at the interface. The monop ole current induces the electric\ncurrentjvia Amp` ere’s law at the interface. The impurity spin-orbit interactio n directly induces\npositive (+) and negative ( −) monopole chargedistributions, ρm, at the two edges. This monopole\ndistribution generates an electric current again perpendicular to t he monopole current, as is seen\nfrom eqs. (8) - (10). Inset: depiction of the spin damping vector N≡M×˙Marising from the\nprecession of magnetization. The component of the spin damping ve ctor perpendicular to the\nprecession axis vanishes when time-averaged, leaving Nalong the axis as the DC component.\nwhere the monopole current and monopole density, respectiv ely read\njm=αR∇×(ER×N)+βi˙N, (11)\nand\nρm=−βi∇·N. (12)\nEquations (10) - (12) are the central results of this paper. W e have thus proved that\na monopole exists when spin damping occurs, namely, a spin da mping monopole. The spin\ndamping monopole is a composite object made from a magnetiza tion configuration in the\nsame manner as the HHM. It satisfies the conservation law ˙ ρm+∇·jm= 0.\nOur result obtained in a disordered system is a general one an d can be extended to a\nclean system. In fact, the same monopole exists in a clean cas e, but only the coefficients αR\nandβiappearing in the monopole density and current are changed. T he same applies to the\nHHM; when we take into account the third-order contribution inJ, our analysis correctly\nreproduces the HHM, which was discussed in a clean limit only . Our approach is, therefore, a\nnovel method of identifying monopoles.\nThe spin damping monopole is unique since it does not require a particular non-coplanar\nspin structure like a hedgehog, and so it exists quite genera lly in magnetic systems. The\nsimplest candidate for creating the monopole would be a thin ferromagnetic film put on a\n5/9J. Phys. Soc. Jpn. Letter\nnonmagnetic material, as shown in Fig. 1. We choose the z-axis perpendicular to the film.\nA Rashba-type spin-orbit field would then arise at the interf ace along the z-direction.9We\nexcite the precession of the uniform magnetization by apply ing the alternating magnetic\nfield in the yz-plane in the presence of a static field along the x-axis (ferromagnetic reso-\nnance22). The precession results in a spin damping vector with a finit e time average, N,\nalongx-direction (inset in Fig. 1). In the present case with unifor m magnetization, spatial\nderivatives in eqs. (11) and (12) arise at the interface and e dges, where the magnetization\nvanishes. The Rashba interaction contributes to the DC mono pole current at the interface\nasjRm,x=−αRER(∂N/∂z)≃ −(αR/d)ERN, wheredis the spatial scale of the magnetiza-\ntion decay at the interface. The monopole current driven by r andom spin-orbit impurities,\non the other hand, vanishes when time-averaged. The total DC monopole current thus reads\njm=−ex(αR/d)ERN(exrepresents the unit vector along the x-direction). This monopole\ncurrent at the interface generates an electromotive force a long the y-direction via Amp` ere’s\nlaw for the monopole. The monopole density induced by the ran dom spin-orbit interaction\narises at the edge of the ferromagnetic film since ∇·N≃∂Nx/∂xis finite there. The induced\nmonopole density at the two edges is ρm=∓(βi/d)N, where the sign is positive on one side\nof the edge and negative on the other side. The monopole then p roduces a magnetic field\nalong the x-direction as Bs=−exβiN. This field creates an electric current in the y-direction\nvia the conventional Amp` ere’s law. The total electric curr ent density generated by the spin\ndamping monopole [eq. (8)] thus reduces to j=−ey[σcαRER+(βi/µd)]N.\nWhen spin damping arises from the magnetization precession with a frequency\nΩ and an angle θ, the monopole-induced current density is estimated as |¯j|=\n(ek2\nFΩ/π2)(Jτ//planckover2pi1)2sinθ[(∆R/εF)+(4/3kFd)(∆i/vi)], where ∆ Rand ∆ iare the energy of the\nRashba and impurity spin-orbit couplings, respectively ( kFis the Fermi wave vector). In dis-\nordered ferromagnets, J/εF∼0.1,εFτ//planckover2pi1∼10 andk−1\nF∼2˚A. The Rashba interaction\ncan be enhanced on surfaces and at interfaces, resulting in ∆ R/εF∼0.1 (∆i/εFis gener-\nally smaller).23Whenθ= 30◦and Ω = 1 GHz, the electric current density is thus 2 ×107\nA/m2which is sufficiently large for experimental detection. In ad dition to DC, there is an AC\ncomponent in eq. (8), which would be accessible by time-reso lved measurement.\nWe note that the electric current estimated here is an initia l current that arises when\nthe pumping of monopoles starts. When the monopole current i s pumped steadily, monopole\naccumulation grows at the edges of the system, inducing a diffu sive current. The steady\nmonopole distribution is then determined by the balance of t his backward diffusion and the\npumped monopole current.\nDirect evidence of the spin damping monopole is given by dete cting the electric current\ndiscussed above. Surprisingly, the signal from the spin dam ping monopole might have already\nbeen detected. In fact, the electric voltage due to magnetiz ation precession has been observed\n6/9J. Phys. Soc. Jpn. Letter\nin a junction of a ferromagnet on a Pt film in a pioneering work b y Saitoh et al..24The\nmechanism of voltage generation has been explained by the in verse spin Hall effect. According\ntotheinversespinHall scenario, magnetization precessio ngenerates aspincurrentviathespin\npumpingeffect,25and the spin current jsis converted into an electric current by the spin-orbit\ninteraction (the inverse of the spin Hall effect). This explan ation assumes that the conversion\nmechanism of ji=ǫijkjk\ns,jwherekis the index representing the spin polarization of the spin\ncurrent.26A recent theoretical study has revealed, however, that the c onversion formula is\nnot universal; it does not apply to the slowly varying magnet ization configuration or in the\npresence of disorder.11Rather, the conversion formula is an approximated one conne cting a\nphysical observable (electric current) to aspincurrent wh osedefinition dependson the specific\nsystem considered. Our result obtained in the present paper suggests a different scenario that\nis universal owing to the symmetry of Maxwell’s equations.\nFor the experimental confirmation of the spin damping monopo le, of crucial importance is\ntheseparation ofthemonopolesignalfromtheinversespinH allsignal drivenbyaspincurrent.\nThis is accomplished by applying an electric field ( Es) perpendicular to the junction of Fig. 1.\nThe monopole contribution then leads to a transverse electr ic current as a result of the Hall\neffect of the monopole,27while the contribution of the spin current is not affected. In a nother\nexperiment, the strongest evidence of the spin damping mono pole is given by observing a\nmagnetic field ( Bs) produced by monopoles via Gauss’s law by electron holograp hy.\nWe have shown analytically that a magnetic monopole is a comm on object in dynamic\nmagnetic systems with damping. The control of spin damping m onopoles is as feasible as that\nof electrons, for both are governed symmetrically by Maxwel l’s equations of electromagnetism.\nWe here propose the monopolotronics, i.e., the control of mo nopoles, as a novel concept of\nrealizing spintronic devices.\nAcknowledgements\nThe authors are grateful to N. Kitazawa, R. Matsumoto, S. Mur akami, and N. Nagaosa\nfor discussion. This work was supported by a Grant-in-Aid fo r Scientific Research in Priority\nAreas (Grant No. 1948027) from the Ministry of Education, Cu lture, Sports, Science and\nTechnology, a Grant-in-Aid for Scientific Research (B) (Gra nt No. 22340104) from Japan\nSociety for the Promotion of Science. One of the authors (A.T .) is financially supported by\nthe Japan Society for the Promotion of Science for Young Scie ntists.\n7/9J. Phys. Soc. Jpn. Letter\nReferences\n1) P. A. M. Dirac: Proc. R. Soc. London 133(1931) 60.\n2) J. D. Jackson: Classical Electrodynamics (Wiley, New York, 1998).\n3) G. ’t Hooft: Nucl. Phys. B79(1974) 276.\n4) A. M. Polyakov: JETP Lett. 20(1974) 194.\n5) B. Cabrera: Phys. Rev. Lett. 48(1982) 1378.\n6) M. Ambrosio et al.: Eur. Phys. J. C 25(2002) 511.\n7) G. E. Volovik: J. Phys. C 20(1987) L83.\n8) E. I. Rashba: Sov. Phys. Solid State 2(1960) 1109.\n9) L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch¨ on, and K. Ensslin : Nat. Phys. 3(2007) 650.\n10) K. Hosono, A. Takeuchi, and G. Tatara: J. Phys. Soc. Jpn. 79(2010) 014708.\n11) A. Takeuchi, K. Hosono, and G. Tatara: Phys. Rev. B 81(2010) 144405.\n12) K. M. D. Hals, A. Brataas, and Y. Tserkovnyak: Eur. Phys. Le tt.90(2010) 47002.\n13) H.HaugandA.-P.Jauho: Quantum Kinetics in Transport and Optics of Semiconductors (Springer,\nNew York, 2007).\n14) J. Rammer: Quantum Transport Theory (Westview Press, Boulder, CO, 2004).\n15) Small corrections on the order of /planckover2pi1/εFτare neglected.14This corresponds to the Born approxi-\nmation.\n16) Equation (5) is obtained by evaluating three and six distinct self- energy types of Feynman dia-\ngrams for the the Rashba and impurity-induced spin-orbit interact ions, respectively, and vertex\ncorrections including infinite summation of the impurity scattering.14\n17) The contribution linear in λiarisesfrom the averagingof the correlationof the impurity potent ials\ninHsoand in the bare scattering term vi, as in the calculation of the anomalous Hall effect.18\n18) A. Cr´ epieux and P. Bruno: Phys. Rev. B 64(2001) 014416.\n19) The factor of ( Dq2τ+iΩτ)−1represents the effect of electron diffusion.14\n20) In eq. (8), electron diffusion ( D∇ρ) induces the effective electric and magnetic polarizations Ps\nandMs, respectively. By using the relation ( −D∇2+∂t)ρ=−σc∇·Es, the electric current\n[eq. (8)] is described as the rotation of the magnetic field and the tim e derivative of the electric\nfield,j=∇×Hs−˙Ds, whereHsandDsare the fields defined as Hs≡(1/µ)Bs−Msand\nDs≡εEs+Ps(ε=−σcτbeing the permittivity), respectively. Thus, EsandBsare identified\nwith the electric and magnetic fields, respectively.\n21) Equations (8) and (9) apparently contain an arbitrariness, Es→Es+ (1/σc)∇×CandBs→\nBs−µC, whereCis an arbitrary vector field. However, such a transform in Maxwell’s e quations\nwith monopoles is generally not allowed because of the gauge invarianc e in the original space\nwith a higher symmetry. In fact, in the case of HHM, the SU(2) gaug e invariance forbids this\narbitrariness.7We expect the same argument applies to the present spin damping mo nopole.\n22) S. Chikazumi: Physics of Ferromagnetism (Oxford University Press, New York, 1997).\n23) C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pac il´ e, P. Bruno, K. Kern, and M.\nGrioni: Phys. Rev. Lett. 98(2007) 186807.\n24) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara: Appl. Phys. Let t.88(2006) 182509.\n25) Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer: Phys. Rev. Lett.88(2002) 117601.\n26) J. E. Hirsch: Phys. Rev. Lett. 83(1999) 1834.\n8/9J. Phys. Soc. Jpn. Letter\n27) The Lorentz force from an electric field acts on a magnetic mono pole (see §6.8 in ref. 2).\n9/9"    },    {        "title": "2103.09011v1.Efficient_field_free_perpendicular_magnetization_switching_by_a_magnetic_spin_Hall_effect.pdf",        "content": "1 \n Efficient field -free perpendicular magnetization  switching  \nby a magnetic spin Hall effect   \n \nShuai Hu1,†, Ding- Fu Shao2,†, Huanglin Yang1,†, Meng Tang1, Yumeng Yang3,*, Weijia Fan1, \nShiming Zhou1, Evgeny Y. Tsymbal2,* and Xuepeng Qiu1,* \n \n1Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology and \nSchool of Physics Science and Engineering, Tongji University, Shanghai 200092, China \n2Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, \nUniversity of Nebraska, Lincoln, Nebraska 68588- 0299, USA  \n3School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, \nChina \n \n†These authors contributed equally to this work. \n*Email: yangym1@shanghaitech.edu.cn, tsymbal@unl.edu & xpqiu@tongji.edu.cn \nCurrent induced s pin-orbit torque s driven by the conventional spin Hall effect are  \nwidely  used to manipulate the magnetization . This approach, however, is non deterministic \nand inefficient for the switching of magnet s with  perpendicular magnetic anisotropy that are \ndemanded by the high -density magnetic  storage and memor y devices . Here, we demonstrate \nthat this limitation  can be overcome by exploiting a magnetic spin Hall effect  in noncollinear \nantiferromagnet s, such as  Mn 3Sn. The magnetic group  symmetry  of Mn 3Sn allows  \ngeneration of the out-of-plane spin current carrying spin polarization  induced by an in -plane \ncharge current . This spin current drives  an out -of-plane anti -damping torque providing  \ndeterministic switching of perpendicular magnetization  of an adjacent Ni/Co  multilayer . \nCompared to the  conventional spin- orbit torqu e devices, the observed switching does not \nneed any external magnetic field  and requires much lower current  density . Our results  \ndemonstrate great prospects of  exploit ing the magnetic spin Hall effect  in noncollinear \nantiferromagnet s for low -power spintronics.   2 \n A spin-orbit torque (SOT) provides an effective approach to manipulate the magnetization \nin spintronic  devices1-5. In a typical  SOT  device, an in -plane charge current 𝑱𝑱 generates a spin \npolarization 𝒑𝒑 via spin- orbit coupling and  exerts a torque on the magnetizatio n 𝒎𝒎 of a neighboring \nferromagnetic layer . The most efficient SOT driven magnetization switching would require 𝒑𝒑 to \nbe parallel to the easy axis of 𝒎𝒎, so that its  associated  anti-damping torque ~𝒎𝒎×(𝒎𝒎×𝒑𝒑) can \ndirectly change the effective damping of a ferromagnet  (FM)6. In the conventional spin Hall effect \n(SHE)3,4 or the Rashba -Edelstein effect (RE E)1,2, the induced spin polarization 𝒑𝒑, either  in a heavy \nmetal (HM) or at a nonmagnetic metal (NM)/FM interface, is always  aligned the in -plane direction \ndetermined by 𝑱𝑱×𝒛𝒛 (where 𝒛𝒛 is a normal to the film plane )7,8. The associated  in-plane anti -\ndamping torque  is thus favorable  to switch  a magnet  with in -plane magnetization , while  its use for \nthe switching of a magnet with perpendicular magnetization is nondeterministic6. Although this \ndeficiency could be alleviated by applying a n external assisting magnetic field and a high current \ndensity (Fig. 1a) , these additional requirements  would eventually hinder the a pplication  of the SOT \ntechnique  in high- density and low -power spintronic devices .  \nA more efficient  approach  is to exploit an  out-of-plane an ti-damping torque ~𝒎𝒎×(𝒎𝒎×𝒛𝒛) \nthat directly counteract s the magnetization  damping in a perpendicular magnet  (Fig. 1b)9-12. \nRecently, extensive studies  have been aimed at creating  z-polarized spins  that are capable to \nproduce this kind of SOTs13. In particular, i t has been shown that using a spin source with low  \ncrystal symmetry14,15 and/or an additional magnetic order15-18 can give rise to  the z-polarized spins  \nthat are not allowed to appear in the conventional HM  spin source s. Alternatively,  a proper \ninterface engineering has been reported as a possible mechanism to polarize spins along the  z-\ndirection  through combined actions of spin- orbit filtering, spin precession, and scattering19-24. In \nterms of practical device application, it would be more straightforward and reliable to  optimize  the \nbulk property of the spin- source  layer  rather than to modify the interface between different layers . \nSo far, h owever, switching of a perpendicular magnet  has only been experimentally  realized  in \nFM/NM/FM  trilayer s, where the out-of-plane anti-damping torque  was induced by the FM/NM  \ninterface24.  \nHere, we demonstrate  that an out-of-plane anti-damping torque can be directly generated in \na SOT device based on the   noncollinear antiferromagnet Mn 3Sn due to the  magnetic spin Hall \neffect25,26, thus providing an efficient mechanism for perpendicular magnetization switching. We 3 \n show that this switching is deterministic even in the absence of an applied magnetic field and \nrequires a much lower critical current density than  that based on the  conventional  SHE .  \nBulk Mn 3Sn is a hexagonal compound with the Ni 3Sn-type crystal structure27. As depicted \nin Fig ure 1c, Mn atoms form Kagome -type lattice planes stacked along the c  axis, where as Sn \natoms are located at the center of the Mn-hexagons. The frustration from the triangular geometry \nof the Kagome lattice results in the chiral alignment of the Mn mome nts within each plane, leading \nto a noncollinear antiferromagneti c order  of Mn 3Sn with the Néel temperature ( TN) of ~420 K28-30. \nThe noncollinear antiferromagneti sm in Mn 3Sn results in many interesting  properties, such as the \nlarge room temperature anomalous Hall effect27, the Weyl semimetal phase31, and the  magnetic \nspin Hall effect (MSHE)32 relevant to our studies. Different from  the conventional SHE being  even \nunder time reversal symmetry, the MSHE is odd with respect to this symmetry and hence \nreversable by flipping the magnetic moments . Being a consequence of symmetry breaking caused \nby the  noncollinear magnetic structure , the MSHE  can be intuitively regarded  as a “magnetic” \nversion of the SHE25,33,34. The MSHE gives rise to the unconventional spin currents with distinct \nsymmetr ies providing useful implications for SOT devices. For example , an out -of-plane spin \ncurrent with a finite z -spin component  can be induced by the in-plane charge current (along  the x \nor y direction) in the Mn 3Sn (0001) film (Fig. 1c). Th is spin current is related to the magnetic spin \nHall conductivity (MSHC) 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 or 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 (in the form of 𝜎𝜎𝑗𝑗𝑗𝑗𝑖𝑖, where 𝑖𝑖, 𝑗𝑗 and 𝑘𝑘 are the spin \npolarization, spin- current and charge- current directi ons, respectively), whose spin polarization 𝒑𝒑 \ncan be reversed by flipping the magnetic moments in Mn 3Sn. There are two types of the inverse \ntriangular alignment of the magnetic moments in Mn 3Sn, denoted as AFM1 and AFM2 in Figure  \n1c. The magnetic group symmetry of these alignments supports a  small but nonvanishing in- plane \nnet magnetizations along  the x ([011�0]) direction for AFM1 and along the y ([21�1�0]) direction \nfor AFM228,29. Th e presence of this magnetization allows, in particular, the control of \nantiferromagnetic domains by a  small in -plane magnetic field28. This is reflected by our first-\nprinciples calculation ( Fig. 1d) , showing that the 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 and 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 are finite and reversable for AFM1 \nand AFM2 of Mn 3Sn, respectively. These sizable MSHC s allow using Mn 3Sn as a spin source \nmaterial to generate the out -of-plane anti-damping SOT  and switch the magnetization of an \nadjacent ferromagnet .  4 \n On this basis , we design a MSHE SOT  device consist ing of a  crystalline Mn 3Sn (0001) film  \nto generate the MSHE , a [Ni/Co] 3 multilayer with perpendicular magnetic anisotropy to control its \nmagnetization , and a thin Cu spacer layer to magnetically decouple Mn 3Sn and [Ni/Co] 3. Figure \n2a schematically shows  a stack of Mn 3Sn(7)/Cu (1)/[Ni(0.4) /Co(0.2) ]3 (number  in the parentheses \nindicate thickness in nanometer s) is epitaxially prepared by dc magnetron sputtering on cubic MgO \n(111) substrate . Hall bar devices are subsequently fabricated by standard lithography and etching \nprocess for electrical  measurements . Through combined characterization techniques of  X-ray \ndiffraction, high- resolution transmission electron microscopy , and magneto -transport \nmeasurement s (Supplementary Information S2, S3), the Mn 3Sn film is confirmed to  have well-\ndefined (0001) orientation, and its quality is comparable to the previous ly fabricated bulk Mn 3Sn \ncrystal31 and sputtered epitaxial  films35-37. Strong perpendicular anisotropy of the [Ni/Co] 3 \nmultilayer is confirmed by anomalous  Hall effect (AHE) measurement s (Supplementary \nInformation  S4).  \nWe first verify t he existence of the z -polarized  component in the spin current  by measur ing \nthe hysteresis loop of the anomalous Hall resistance R AHE versus the  out-of-plane magnetic field \nHz in the presence of  a bias dc current  I along the x ([011�0]) direction. If there was a z -polarized  \nspin component , the associated  out-of-plane anti -damping torque  would cause  an abrupt shift of \nthe RAHE-Hz hysteresis loop when  the torque is sufficiently strong to overcome the intrinsic \ndamping of the Ni/Co multilayer24. Indeed, we find no shift of the  loop when  the amplitude of I is \nat 4 mA  (Fig. 2 b), while  a sizable positive  or negative  shift occurs  when I = +16 mA or −16 mA, \nrespectively  (Fig. 2c ). Figure 2d, shows that the threshold current to produce the RAHE-Hz hysteresis \nloop shift is around  ~10 mA , above which the shift increases almost linearly with the increase of \nI. A similar  current induced shift and threshold effect  are also observed when I is applied along \nthe y  ([21�1�0]) direction  (Supplementary Information S5).  \nThe emergence of the z -polarized spin Hall current in Mn 3Sn allows the realization of the \ndeterministic field -free magnetization switching. Fig ure 3a shows that the measured AHE \nresistance as a function of a pulse  current I applied along the [011�0] direction  exhibits a hysteretic \nbehavior and a sign change in the absence of applied magnetic field . This behavior reflects \nmagnetization reversal in the Ni/Co  multilayer . We note that in our SOT device, the exchange \ncoupling between the Mn 3Sn and Ni/Co  layer s is eliminated by the Cu spacer , and hence it does 5 \n not contribute to the magnetization switching. Instead , the field -free switching is entirely caused \nby the out -of-plane anti-damping torque . Furthermore, w e find that about 60% of  the magnetic \nNi/Co  layer volume switches as estimated from the measured  RAHE-I loop (Fig. 3a ) with respect to \nthe saturation AHE resistance. This partial switching  behavior is due to  the multi- domain structure \nof the AFM  Mn 3Sn film. Due to small magnetic anisotropy and nearly identical energies of AFM1 \nand AFM2 magnetic  orders , as-grown Mn 3Sn films exhibit multiple antiferromagnetic  domains  \nwhich  have a tend ency  to align randomly in zero field . This gives rise to variation in magnitude \nand sign of MSHC 𝜎𝜎 𝑧𝑧𝑧𝑧𝑧𝑧 among different  domains , and eventually leads to  a reduced effective 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧. \nTherefore, the switching ratio is expected to increase if we align the AFM domains . This can be \ndone  by applying a finite magnetic field  due to a small in-plane magnetization in Mn 3Sn28. Figure \n3b shows  the measured RAHE-I loops  for different magnetic field s Hx along the [011�0] direction , \nand Figure 3c displays the corresponding switching ratio  as a function of H x. As seen, by aligning \nmore domains into one configuration, the switching ratio increases as the magnitude of the field \nincreases, with a maximum  value  of ~80% at a field as small as ~30 Oe.  Importantly , the switching \npolarity is reversed upon reversing the field to opposite direction. All these observations  \ncorroborate with the MSHE scenario showing that the MSHC 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 is controlled by the reorientation \nof Mn 3Sn domains35, 38. It is not able that flipping the switching polarity, accompanied by a \nvanished switching ratio, does not appear at zero field but at a small negative field of about −8 Oe  \n(Fig. 3c) . It is likely that some preferred AFM domain orientation  is induced by structur al \nreconstructions at interfaces or near defects  during the deposition process , which as if there were \na small bias field (~ −8 Oe) on Mn 3Sn. Similar behavior is found when the current and magnetic \nfield are both along the y  ([21�1�0]) direction  (Supplementary I nformation S 5). \nFinally , we compare the SOT switching efficiency of the Mn 3Sn based MSHE device with a \nconventional β-Ta based  SHE device. The two devices have the same geometry  and thickness of  \nthe corresponding layers (Figs. 4a and 4b) . Here, a thicker 2 nm Cu layer is used in both devices \nas this is the minimum thickness require d to achieve good perpendicular anisotropy in a [Ni/Co ]3 \nmultilayer for the β -Ta based  device. As a widely explored spin source, β -Ta has a large charge-\nspin conversion efficiency to generate the  spin current with a conventional in -plane spin \npolarization4. This source, however, require s an assistive magnetic field applied parallel to the \ncharge current for deterministic switching. We find that only a small switching ratio  of ~17% is  6 \n realized  in this device at the sacrifice of a large current density of 1.3×107 A cm-2 and a moderate \nfield of 300 Oe  (Fig. 4a). On the contrary , a much larger switching ratio of ~ 60% is achieved in  \nthe Mn 3Sn based  MSHE  device even  in the absence of external field and  under application of a  \nmuch lower current density of 4.6× 106 A cm-2 (Fig. 4b). T his evidence unambiguously proves the \nsuperior efficiency of the MSHE induced out -of-plane anti -damping torque for the deterministic \nswitching of perpendicular magnetization. \nIn conclusion, we have demonstrate d the efficient current -induced  field-free switching of  a \nferromagnet with  perpendicular magnetization . The magnetization switching is driven by an out -\nof-plane anti -damping SOT generated by non- collinear antiferromagnet Mn 3Sn resulting from the \nMSHE. Due to the magnetic spin Hall current  being collinear to its spin polarization , this observed \nmechanism of switching requires much lower critical current  density compared to the conventional \nSHE based SOT  devices  and allows control through the reorientation of magnetic domains in \nMn 3Sn. Our findings pinpoint the enormous  potential of the  MSHE as the spin- torque source to \nengineer novel energy- efficient  spintronic devices.  \nMethods  \nFirst-principles density functional theory (DFT) calculations were performed using a plane -wave \npseudopotential method with the fully -relativistic ultrasoft pseudopotentials39 implemented in \nQuantum -ESPRESSO40. The exchange and correlation effects were treated within the generalized \ngradient approximation (GGA)41. In the calculations, we used the plane -wave cut -off energy of 35 \nRy and a 16 × 16 × 16 k- point mesh in the irreducible Brillouin zone. The experimental lattice \nparameters a = 5.689Å and c = 4.522Å measured in this work were used in the calculation. Al l \nthe atomic co -ordinates were relaxed until the force on each atom was less than 0.001 eV/Å. In the \ncalculations, we firstly set the initial magnetic configurations according to the experimentally \nobserved noncollinear antiferromagnetic states, and then perform ed full relaxations of the \nmagnetic structure, and the electronic structure without any constraint.  \nThe tight-binding Hamiltonians  were obtained from the maximally localized Wannier functions42 \nwithin the Wannier90 code43 to calculate the magnetic spin Hall conductivity (MSHC)25 \n𝜎𝜎𝑖𝑖𝑗𝑗𝑗𝑗=−𝑒𝑒ℏ\n𝜋𝜋�𝑑𝑑3𝑘𝑘�⃗\n(2𝜋𝜋)3�Γ2Re��𝑛𝑛𝑘𝑘�⃗�𝐽𝐽𝑖𝑖𝑗𝑗�𝑚𝑚𝑘𝑘�⃗��𝑚𝑚𝑘𝑘�⃗�𝑣𝑣𝑗𝑗�𝑛𝑛𝑘𝑘�⃗��\n��𝐸𝐸𝐹𝐹−𝐸𝐸𝑛𝑛𝑗𝑗�⃗�2+Γ2���𝐸𝐸𝐹𝐹−𝐸𝐸𝑚𝑚𝑗𝑗�⃗�2+Γ2�\n  𝑛𝑛,𝑚𝑚, (1) 7 \n where 𝐽𝐽𝑖𝑖𝑗𝑗=1\n2{𝑣𝑣𝑖𝑖,𝑠𝑠𝑗𝑗} is the spin -current operator, 𝑣𝑣𝑖𝑖 and 𝑠𝑠𝑗𝑗 are velocity and spin operators, \nrespectively, and 𝑖𝑖,𝑗𝑗,𝑘𝑘 = 𝑥𝑥,𝑦𝑦,𝑧𝑧. Γ=50 meV25 and a 200 × 200 × 200 k-point mesh were used \nto evaluate the integral of Eq. (1).  \nThe symmetry determined geometries of conductivity tensors were obtained using the linear \nresponse symmetry code44. \n \nSample preparation.  All samples were deposited on MgO (111)  substrates using DC ma gnetron \nsputtering with a base pressure of 5×10-8 Torr. Mn 3Sn films were deposited with a stoichiometric  \ntarget  at the Ar pressure of  2×10-3 Torr, the deposition was  performed at 100 °C, followed by \nannealing  at 400 °C  for 1h . After the Mn 3Sn film cooled down to the room temperature, Cu and \nNi/Co  multilayers  were then deposited on the  Mn 3Sn film with a Cu , Ni, and Co target, respectively. \nStandard photolithography and Ar  ion etching  were used to fabricate the 8 μm wide and 35 μm \nlong Hall bar .  \n \nSample characterization. Structural properties of the samples were characterized using a Bruker \nD8 Discover X -ray diffraction (XRD) system with Cu Kα radiation . The HR -TEM were performed \nwith an electron microscope operated at 200 kV (FEI Titan Themis 200).  For current induced \nmagnetization switching measurements, current pulses with a constant duration of 800 μs but \nvarying amplitude were  applied to the Hall bar. In -between two adjacent writing pulses, the \nmagnetization state of the Ni/Co multilayer was read by current pulses with the same duration as \nthat of the writing pulse but at a much smaller amplitude of 1 mA.   \n \n  8 \n Data availability  \nThe data that support the findings of this study are available from the  corresponding author upon \nreasonable request.  \n \nAcknowledgements  \nThis work was supported by the National Key R&D Program of China Grand No. 2017YFA0303202 and 2017YFA0305300, the National Natural Science Foundation of China \nGrant Nos. 52022069, 11974260, 11674246, 51501131, 51671147, 11874283, 51801152, and \n11774064, Natural Science Foundation of Shanghai Grant No. 19ZR1478700, and the \nFundamental Research Funds for the Central Universities.  \n Author contributions  \nS.H. and X.Q. conceived and designed the experiment . S.H. and H.Y. fabricated the samples and \nperformed the measurements . S.H., D.S., H.Y., M.T., Y.Y., W.F., S.Z. and X.Q. analyzed and \ndiscussed the experiment results. F .S., E.Y.T. analy zed the data and performed the numerical \ncalculation. S. H., Y.Y.  wrote the manuscript with contributions  from all the authors . All authors \ndiscussed the results and commented on the manuscript. \n \nCompeting interests  \nThe authors declare no competing interests. 9 \n  \nFigure  1: Switching of perpendicular magnetization by damping -like SOTs . a A schematic of a \nconventional bilayer SOT device. An in-plane charge current passes along x direction in the bottom spin \nsource layer, generates an out -of-plane spin current with y- polarized  spin through SHE. This spin current \nexerts an in-plane  damping -like torque  ~𝒎𝒎×𝒎𝒎×𝒚𝒚 on the perpendicular magnetization in the top \nferromagnetic layer. In this case, a sizable external magnetic field is required for a deterministic switching, \nand the charge current required is large. b A schematic of a bilayer SOT  device supporting the out -of-plane \nanti-damping torque , where the in- plane charge current generate s an out -of-plane spin current with z-\npolarized spin. This spin current exerts an out -of-plane anti -damping torque ~𝒎𝒎×𝒎𝒎×𝒛𝒛 on the  \nperpendicular magnetization in the top layer to realize a field- free switching , which does not require a lar ge \ncharge current. c The structure  of the noncollinear antiferromagnetic Mn 3Sn. The left panel is the side view \nof the unit cell. The right panel shows the top view of the triangular magnetic alignments  of Mn moments  \nwithin each Mn -Sn Kagome plane . There are two types of the magnetic alignments observed in Mn 3Sn, \ndenoted as AFM1 and AFM2. d The calculated magnetic spin Hall  conductivity 𝜎𝜎 𝑧𝑧𝑧𝑧𝑧𝑧 and 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 in Mn 3Sn. \nLeft panel show s the 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 and 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 as a function of energy for AFM1 and AFM2. Right panel shows the sign \nchange of 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 and 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 at EF when the magnetic moments in AFM1 and AFM2 are reversed by  in-plane \nmagnetic fields. The finite 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 and 𝜎𝜎𝑧𝑧𝑧𝑧𝑧𝑧 indicate Mn 3Sn can be a spin source for the device shown in b to \nsupport an out- of-plane anti -damping torque. \n10 \n  \nFigure 2: Z-polarized  spin current generated by Mn 3Sn thin film . a The schematic of  Mn 3Sn (7) /Cu \n(1)/FM (1.8) stack (left) and optical image of the device using for electrical transport measurements (right). \nThe spins with both ± y and ± z polarizations generated by bottom Mn 3Sn thin film will act on the \nferromagnetic layer and induce spin orbit torques simultaneous ly. b, c RAHE vs. Hz curve when the bias \ncurrents are ±4 mA and ± 16 mA.  d A summary of the shift (Δ Hz) at different bias currents ( I). The threshold \nI to cause a shift in AHE curve is about 10 mA. + I will shift the AHE curve to the + x while -I leads to the \nopposite shift.  \n  \nb c d\nMn3SnJ\nI+ I-\nV+V-\nxy\nzCu (1 nm )FM (1.8 nm )11 \n  \nFigure 3: External magnetic field tuning polarity of current induced magnetization switching . a \nCurrent induced magnetization switching with clockwise polarity in  the absence of an external magnetic \nfield. b The switching  curve under different external magnetic fields from negative to positive. c The \nevolution of the switching polarity and switching ratio under different magnetic fields. H ere the direction \nof magnetic field is in -plane and parallel to the current. T he two opposite Mn 3Sn domains contribute \nopposite z spins and thus induce clockwise and anticlockwise switching polarity as indicated as solid and \nhollow dots. \n  \na b\n c12 \n  \nFigure 4: High MSHE based SOT efficiency with the assistance of z  spin polarization. a The \nconventional SHE based SOT device with a structure of β-Ta (7)/Cu (2)/FM (1.8) and  its maximum current \ninduced magnetization switching curve at the external magnetic field of 300 Oe. b . The novel MSHE based \nSOT device with a structure of Mn 3Sn (7)/Cu (2)/FM (1.8)  and its current induced magnetization switching \ncurve of Mn 3Sn based device with the absent of external magnetic field. Note that the current density is \ncalculated by considering the shunting effect of Cu and FM layer in both devices. \n  \nMn3Sn (7 nm)Cu (2 nm)FM (1.8 nm )\nβ-Ta (7 nm)Cu (2 nm)FM (1.8 nm )\nHx= 0 Oe Hx= 300 Oeb a13 \n Reference  \n1. Miron, I. 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Wang, X., Feng, Z., Qin, P., Yan, H., Zhou, X., Guo, H., Leng, Z., Chen, W., Jia, Q., Hu, Z., Wu, H., Zhang, X., Jiang, C. & Liu, Z. Integration of the noncollinear antiferromagnetic metal Mn3Sn onto ferroelectric oxides for electric -field control. Acta Materialia  181, 537 \n(2019).  \n38. Bai, H., Zhu, W., You, Y., Chen, X., Zhou, X., Pan, F. & Song, C. Size -dependent \nanomalous Hall effect in noncollinear antiferromagnetic Mn3Sn films. Applied Physics Letters  117, 052404 (2020). \n39. Vanderbilt, D. Soft self -consistent pseudopotentials in a generalized eigenvalue formalism. \nPhysical Review B  41, 7892 (1990).  \n40. Espresso, Q. a modular and open- source software project for quantum simulations of \nmaterials/P. Giannozzi [et al.]. Journal of Physics: Condensed Matter  21, 395502 (2009).  \n41. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made  \nsimple. Physical Review Letters  77, 3865 (1996). \n42. Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: Theory and applications. Reviews of Modern Physics  84, 1419 (2012). \n43. Mostofi, A. A., Yates , J. R., Pizzi, G., Lee, Y.- S., Souza, I., Vanderbilt, D. & Marzari, N. \nAn updated version of wannier90: A tool for obtaining maximally- localised Wannier \nfunctions. Computer Physics Communications  185, 2309 (2014).  \n44. Železný, J. https://bitbucket.org/zeleznyj/linear\n  \n "    },    {        "title": "0705.1990v1.Identification_of_the_dominant_precession_damping_mechanism_in_Fe__Co__and_Ni_by_first_principles_calculations.pdf",        "content": "arXiv:0705.1990v1  [cond-mat.str-el]  14 May 2007Identification of the dominant precession damping mechanis m in Fe, Co, and Ni by\nfirst-principles calculations\nK. Gilmore1,2, Y.U. Idzerda2, and M.D. Stiles1\n1National Institute of Standards and Technology, Gaithersb urg, MD 20899-8412\n2Physics Department, Montana State University, Bozeman, MT 59717\n(Dated: October 23, 2018)\nThe Landau-Lifshitz equation reliably describes magnetiz ation dynamics using a phenomenolog-\nical treatment of damping. This paper presents first-princi ples calculations of the damping param-\neters for Fe, Co, and Ni that quantitatively agree with exist ing ferromagnetic resonance measure-\nments. This agreement establishes the dominant damping mec hanism for these systems and takes\na significant step toward predicting and tailoring the dampi ng constants of new materials.\nMagnetic damping determines the performance of\nmagnetic devices including hard drives, magnetic ran-\ndom access memories, magnetic logic devices, and mag-\nnetic field sensors. The behavior of these devices can be\nmodeled using the Landau-Lifshitz (LL) equation [1]\n˙m=−|γ|m×Heff−λ\nm2m×(m×Heff),(1)\nor the essentially equivalent Gilbert (LLG) form [2, 3].\nThe first term describes precession of the magnetization\nmabouttheeffectivefield Heffwhereγ=gµ0µB/¯histhe\ngyromagnetic ratio. The second term is a phenomeno-\nlogical treatment of damping with the adjustable rate λ.\nTheLL(G) equationadequatelydescribesdynamicsmea-\nsured by techniques as varied as ferromagnetic resonance\n(FMR) [4], magneto-optical Kerr effect [5], x-ray absorp-\ntion spectroscopy [6], and spin-current driven rotation\nwith the addition of a spin-torque term [7, 8].\nAccess to a range of damping rates in metallic mate-\nrials is desirable when constructing devices for different\napplications. Ideally, one would like the ability to de-\nsign materials with any desired damping rate. Empiri-\ncally, dopingNiFe alloyswith transitionmetals[9]orrare\nearths [10] has produced compounds with damping rates\nin the range of α= 0.01 to 0.8. A recent investigation\nof adding vanadium to iron resulted in an alloy with a\ndamping rate slightly lower than that for pure iron [11],\nthe systemwiththe lowestpreviouslyknownvalue. How-\never, the damping rate of a new material cannot be pre-\ndicted because there has not yet been a first-principles\ncalculation of damping that quantitatively agrees with\nexperiment. The challenging pursuit of new materials\nwith specific or lowered damping rates is further com-\nplicated by the expectation that, as device size contin-\nues to be scaled down, material parameters, such as λ,\nshould change [12]. A detailed understanding of the im-\nportant damping mechanisms in metallic ferromagnets\nand the ability to predictively calculate damping rates\nwould greatly facilitate the design of new materials ap-\npropriate for a variety of applications.\nThe temperature dependence of damping in the tran-\nsition metals has been carefully characterized through\nmeasurement of small angle dynamics by FMR [13].While one might na¨ ıvely expect damping to increase\nmonotonically with temperature, as it does for Fe, both\nCo and Ni also exhibit a dramatic rise in damping at low\ntemperature as the temperature decreases. These ob-\nservations indicate that two primary mechanisms are in-\nvolved. Subsequent experiments [14, 15] partition these\nnon-monotonic damping curves into a conductivity-like\nterm that decreases with temperature and a resistivity-\nliketermthatincreaseswithtemperature. Thetwoterms\nwere found to give nearly equal weight to the damping\ncurve of Ni and have temperature dependencies similar\nto those of the conductivity and resistivity, suggesting\ntwo distinct roles for electron-lattice scattering.\nThe torque-correlation model of Kambersky [16] ap-\npears to qualitatively match the data. However, like\nmost of the various models presented by Kambersky\n[16, 17, 18, 19] and others [20], it has not been quan-\ntitatively evaluated in a rigorous fashion. This has left\nthe community to speculate, based on rough estimates or\nless, astowhichdampingmechanismsareimportant. We\nresolve this matter in the present work by reporting first-\nprinciples calculations of the Landau-Lifshitz damping\nconstant according to Kambersky’s torque-correlation\nexpression. Quantitative comparison of the present cal-\nculations to the measured FMR values [13] positively\nidentifies this damping pathway as the dominant effect\nin the transition metal systems. In addition to present-\ning these primary conclusions, we also describe the re-\nlationship between the torque-correlation model and the\nmore widely understood breathing Fermi surface model\n[18, 21], showing that the results of both models agree\nquantitatively in the low scattering rate limit.\nThe breathing Fermi surface model of Kambersky pre-\ndicts\nλ=g2µ2\nB\n¯h/summationdisplay\nn/integraldisplaydk3\n(2π)3η(ǫn,k)/parenleftbigg∂ǫn,k\n∂θ/parenrightbigg2τ\n¯h.(2)\nThis model offers a qualitative explanation for the low\ntemperature conductivity-like contribution to the mea-\nsureddamping. The modeldescribesdamping ofuniform\nprecession as due to variations ∂ǫn,k/∂θin the energies\nǫn,kof the single-particle states with respect to the spin2\ndirection θ. The states are labeled with a wavevector\nkand band index n. As the magnetization precesses,\nthespin-orbitinteractionchangestheenergyofelectronic\nstatespushingsomeoccupiedstatesabovetheFermilevel\nand some unoccupied states below the Fermi level. Thus,\nelectron-hole pairs are generated near the Fermi level\neven in the absence of changes in the electronic popula-\ntions. The ηfunction in Eq. (2) is the negative derivative\nof the Fermi function and picks out only states near the\nFermi level to contribute to the damping. gis the Land´ e\ng-factor and µBis the Bohr magneton. The electron-hole\npairs created by the precession exist for some lifetime τ\nbeforerelaxingthroughlattice scattering. The amountof\nenergy and angular momentum dissipated to the lattice\ndepends on how far from equilibrium the system gets,\nthus damping by this mechanism increases linearly with\nthe electron lifetime as seen in Eq.2. Since the electron\nlifetime is expected to decrease as the temperature in-\ncreases, this model predicts that damping diminishes as\nthe temperature is raised.\nBecause the predicted damping rate is linear in the\nscattering time the damping rate cannot be calculated\nmore accurately than the scattering time is known. For\nthis reason it is not possible to make quantitative com-\nparisonsbetween calculationsof the breathing Fermi sur-\nface and measurements. Further, while the breathing\nFermi surface model can explain the dramatic temper-\nature dependence observed in the conductivity-like por-\ntion of the data it fails to capture the physics driving the\nresistivity-like term. This is a significant limitation from\na practical perspective because the resistivity-like term\ndominates damping at room temperature and above and\nis the only contribution observed in iron [13] and NiFe\nalloys [22]. For these reasons it is necessary to turn to\nmore complete models of damping.\nKambersky’s torque-correlation model predicts\nλ=g2µ2\nB\n¯h/summationdisplay\nn,m/integraldisplaydk3\n(2π)3/vextendsingle/vextendsingleΓ−\nnm(k)/vextendsingle/vextendsingle2Wnm(k) (3)\nand we will show that it both incorporates the physics of\nthe breathing Fermi surface model and also accounts for\ntheresistivity-like terms. The matrix elements Γ−\nnm(k) =\n/angbracketleftn,k|[σ−, Hso]|m,k/angbracketrightmeasure transitions between states\nin bands nandminduced by the spin-orbit torque.\nThese transitionsconservewavevector kbecausethey de-\nscribe the annihilation of a uniform precession magnon,\nwhich carries no linear momentum. The nature of these\nscattering events, which are weighted by the spectral\noverlapWnm(k) = (1/π)/integraltext\ndω1η(ω1)Ank(ω1)Amk(ω1),\nwill be discussedin moredetail below. The electronspec-\ntral functions Ankare Lorentzians centered around the\nband energies ǫnkand broadenedbyinteractionswith the\nlattice. The width ofthe spectralfunction ¯ h/τprovidesa\nphenomenological account for the role of electron-lattice\nscattering in the damping process. The ηfunction is theh/τ(eV)\n108109λ (1/s)\n0.001α0.001 0.01 0.1 1\nFe\n1081091010λ (1/s)\n101310141015\n 1/τ (1/s)0.0010.010.1αNi108109λ (1/s)\n0.0010.01αCo\nFIG. 1: Calculated Landau-Lifshitz damping constant for Fe ,\nCo, and Ni. Thick solid curves give the total damping param-\neter while dotted curves give the intraband and dashed lines\nthe interband contributions. The top axis is the full-width -\nhalf-maximum of the electron spectral functions.\nsame as in Eq. (2) and enforces the requirement of spec-\ntral overlap at the Fermi level.\nEquation (3) captures two different types of scatter-\ning events: scattering within a single band, m=n, for\nwhich the initial and final states are the same, and scat-\ntering between two different bands, m/negationslash=n. As explained\nin [16] the overlap of the spectral functions is propor-\ntional (inverse) to the electron scattering time for intra-\nband (interband) scattering. From this observation the\nqualitative conclusion is made that the intraband contri-\nbutions matchthe conductivity-like terms while the inter-\nband contributions give the resistivity-like terms. While\nthis seems promising, evaluation of Eq. (3) is more com-\nputationally intensive than that of the breathing Fermi\nsurface model and until now only a few estimates for Ni\nand Fe have been made [19].3\nTABLE I: Calculated and measured [13] damping parameters. V alues for λ, the Landau-Lifshitz form, are reported in 109s−1,\nvalues of α, the Gilbert form, are dimensionless. The last two columns l ist calculated damping due to the intraband contribution\nfrom Eq. (3) and from the breathing Fermi surface model [12], respectively. Values for λ/τare given in 1022s−2. Published\nnumbers from [13] and [12] have been multiplied by 4 πto convert from the cgs unit system to SI.\nαcalcλcalcλmeasλcalc/λmeas(λ/τ)intra(λ/τ)BFS\nbcc Fe/angbracketleft001/angbracketright0.0013 0.54 0.88 0.61 1.01 0.968\nbcc Fe/angbracketleft111/angbracketright0.0013 0.54 – – 1.35 1.29\nhcp Co/angbracketleft0001/angbracketright0.0011 0.37 0.9 0.41 0.786 0.704\nfcc Ni/angbracketleft111/angbracketright0.017 2.1 2.9 0.72 6.67 6.66\nfcc Ni/angbracketleft001/angbracketright0.018 2.2 – – 8.61 8.42\nWe have performed first-principles calculations of the\ntorque-correlation model Eq. (3) with realistic band\nstructures for Fe, Co, and Ni. Prior to evaluating Eq. (3)\nthe eigenstates and energies of each metal were found us-\ning the linear augmented plane wave method [23] in the\nlocal spin density approximation (LSDA) [24, 25, 26].\nDetails of the calculations for these materials are de-\nscribed in [27]. The exchange field was fixed in the cho-\nsen equilibrium magnetization direction. Calculations of\nEq. (3) presented in this paper are converged to within\na standard deviation of 3 %, which required sampling\n(160)3k-points for Fe, (120)3for Ni, and (100)2k-points\ninthe basalplaneby57alongthe c-axisforCo. Electron-\nlattice interactions were treated phenomenologically as a\nbroadeningofthe spectralfunctions. The Fermi distribu-\ntion was smeared with an artificial temperature. Results\ndid not vary significantly with reasonable choices of this\ntemperature since the broadening of the Fermi distribu-\ntion was considerably less than that of the bands. The\ndamping rate was calculated for a range of scattering\nrates (spectral widths) just as damping has been mea-\nsured over a range of temperatures.\nThe results ofthese calculations arepresented in Fig. 1\nand are decomposed into the intraband and interband\nterms. The downward sloping line in Fig. 1 represents\nthe intraband contribution to damping. Damping con-\nstants were recently calculated using the breathing Fermi\nsurface model [12, 21] by evaluating the derivative of\nthe electronic energy with respect to the spin direc-\ntion according to Eq. (2). The results of the breathing\nFermi surface prediction are indistinguishable from the\nintraband terms of the present calculation even though\nthe computational approaches differed significantly; the\nagreement is quantified in Table I.\nThe breathing Fermi surface model could not be quan-\ntitatively compared to the experimental results because\nthe temperature dependence of the scattering rate has\nnot been determined sufficiently accurately. While the\npresent calculations also require knowledge of the scat-\ntering rate to determine the damping rate the non-monotonic dependence of damping on the scattering rate\nproduces a unique minimum damping rate. In the same\nmanner that the calculated curves of Fig. 1 have a mini-\nmumwithrespecttoscatteringrate,themeasureddamp-\ning curves exhibit minima with respect to temperature.\nWhatever the relation between temperature and scatter-\ning rate, the calculated minima may be compared di-\nrectly and quantitatively to the measured minima. Ta-\nble I makes this comparison. The agreement between\nmeasured and calculated values shows that the torque-\ncorrelationmodelaccountsforthedominantcontribution\nto damping in these systems.\nOur calculated values are smaller than the measured\nvalues. Using measured gvalues instead of setting g=\n2 would increase our results by a factor of ( g/2)2, or\nabout 10 % for Fe and 20 % for Co and Ni. Other pos-\nsible reasons for the difference include a simplified treat-\nment ofelectron-latticescatteringin which the scattering\nrates for all states were assumed equal, the mean-field\napproximation for the exchange interaction, errors asso-\nciatedwith thelocalspindensityapproximation(LSDA),\nand numerical convergence (discussed below). Other\ndamping mechanisms may also make small contributions\n[28, 29, 30].\nSince the manipulations involved with the equation of\nmotion techniques employed in deriving Eq. (3) obscure\nthe underlying physics we now discuss the two scatter-\ning processes and connect the intraband terms to the\nbreathing Fermi surface model. The intraband terms in\nEq. (3) describe scattering from one state to itself by\nthe torque operator, which is similar to a spin-flip oper-\nator. A spin-flip operation between some state and itself\nis only non-zero because the spin-orbit interaction mixes\nsmall amounts of the opposite spin direction into each\nstate. Since the initial and final states are the same, the\noperation is naturally spin conserving. The matrix ele-\nments do not describe a real transition, but rather pro-\nvide a measure of the energy of the electron-hole pairs\nthat are generated as the spin direction changes. The\nelectron-hole pairs are subsequently annihilated by a real4\nelectron-lattice scattering event.\nTo connect the derivatives ∂ǫ/∂θin Eq. (2) and the\ntorque matrix elements in Eq. (3) we imagine first point-\ning the magnetization in some direction ˆ z. The only\nenergy that changes with the magnetization direction is\nthe spin-orbit energy Hso. As the spin of a single parti-\ncle state |/angbracketrightrotates along ˆθabout ˆxits spin-orbit energy\nis given by ǫ(θ) =/angbracketleft|eiσxθHsoe−iσxθ|/angbracketright. The derivative\nwith respect to θis∂ǫ(θ)/∂θ=i/angbracketleft|eiσxθ[σx, Hso]e−iσxθ|/angbracketright.\nEvaluating this derivative at the pole ( θ= 0) gives\n∂ǫ/∂θ=i/angbracketleft|[σx, Hso]|/angbracketright. Similarly, rotating the spin along\nˆθabout ˆyleads to ∂ǫ/∂θ=i/angbracketleft|[σy, Hso]|/angbracketright. The torque\nmatrix elements in Eq. (3) are Γ−=/angbracketleft|[σ−, Hso]|/angbracketright=\n/angbracketleft|[σx, Hso]|/angbracketright−i/angbracketleft|[σy, Hso]|/angbracketright. Using the relations between\nthe commutators and derivatives just found the torque is\nΓ−=−i(∂ǫ/∂θ)x−(∂ǫ/∂θ)ywhere the subscripts in-\ndicate the rotation axis. Squaring the torque matrix\nelements gives |Γ−|2= (∂ǫ/∂θ)2\nx+ (∂ǫ/∂θ)2\ny. For high\nsymmetry directions ( ∂ǫ/∂θ)x= (∂ǫ/∂θ)yand we de-\nduce|Γ−|2= 2(∂ǫ/∂θ)2demonstrating that the intra-\nband terms of the torque-correlation model describe the\nsame physics as the breathing Fermi surface.\nThe monotonically increasing curves in Fig. 1 indi-\ncate the interband contribution to damping. Uniform\nmode magnons, which have negligible energy, may in-\nduce quasi-elastic transitions between states with differ-\nent energies. This occurs when lattice scattering broad-\nens bands sufficiently so that they overlap at the Fermi\nlevel. Thesewavevectorconservingtransitions, whichare\ndriven by the precessing exchange field, occur primarily\nbetween states with significantly different spin character.\nThe process may roughly be thought of as the decay of\na uniform precession magnon into a single electron spin-\nflip excitation. These events occur more frequently as\nthe band overlaps increase. For this reason the interband\nterms, which qualitatively match the resistivity-like con-\ntributions in the experimental data, dominate damping\nat room temperature and above.\nWe have calculated the Landau-Lifshitz damping pa-\nrameterfortheitinerantferromagnetsFe, Co, andNiasa\nfunction ofthe electron-latticescatteringrate. Theintra-\nband and interband components match qualitatively to\nconductivity- andresistivity-like terms observed in FMR\nmeasurements. A quantitative comparison was made be-\ntweentheminimaldampingratescalculatedasafunction\nof scattering rate and measured with respect to temper-\nature. This comparison demonstrates that our calcula-\ntions account for the dominant contribution to damping\nin these systemsand identify the primarydamping mech-\nanism. At room temperature and above damping occurs\noverwhelmingly through the interband transitions. The\ncontribution of these terms depends in part on the band\ngap spectrum around the Fermi level, which could be\nadjusted through doping.\nK.G. and Y.U.I. acknowledge the support of the Officeof Naval Research through grant N00014-03-1-0692 and\nthroughgrantN00014-06-1-1016. We wouldlike tothank\nR.D. McMichael and T.J. Silva for valuable discussions.\n[1] L. LandauandE. Lifshitz, Phys. Z.Sowjet. 8, 153(1935).\n[2] T. L. Gilbert, Armour research foundation project No.\nA059, supplementary report, unpublished (1956).\n[3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[4] D. Twisselmann and R. McMichael, J. Appl. Phys. 93,\n6903 (2003).\n[5] T. Gerrits, J. Hohlfeld, O. Gielkens, K. Veenstra, K. Bal ,\nT. Rasing, and H. van den Berg, J. Appl. Phys. 89, 7648\n(2001).\n[6] W. Bailey, L. Cheng, D. Keavney, C. Kao, E. Vescovo,\nand D. Arena, Phys. Rev. B 70, 172403 (2004).\n[7] I. Krivorotov, D. Berkov, N. Gorn, N. Emley, J. Sankey,\nD. Ralph, and R. Buhrman, Phys. Rev. B (2007).\n[8] M. Stiles and J. Miltat, Spin dynamics in confined mag-\nnetic structures III (Springer, Berlin, 2006).\n[9] J. Rantschler, R. McMichael, A. Castiello, A. Shapiro,\nJ. W.F. Egelhoff, B. Maranville, D.Pulugurtha, A. Chen,\nand L. Conners, J. Appl. Phys. 101, 033911 (2007).\n[10] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE\nTrans. Mag. 37, 1749 (2001).\n[11] C. Scheck,L.Cheng, I.Barsukov, Z.Frait, andW.Bailey ,\nPhys. Rev. Lett. 98, 117601 (2007).\n[12] D. Steiauf and M. Faehnle, Phys. Rev. B 72, 064450\n(2005).\n[13] S. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n[14] B. Heinrich, D. Meredith, and J. Cochran, J. Appl. Phys.\n50, 7726 (1979).\n[15] J.F.Cochran and B. Heinrich, IEEE Trans. Magn. 16,\n660 (1980).\n[16] V. Kambersky, Czech. J. Phys. B 26, 1366 (1976).\n[17] B. Heinrich, D. Fraitova, and V. Kambersky,\nPhys. Stat. Sol. 23, 501 (1967).\n[18] V. Kambersky, Can. J. Phys. 48, 2906 (1970).\n[19] V. Kambersky, Czech. J. Phys. B 34, 1111 (1984).\n[20] V. Korenman and R. Prange, Phys. Rev. B 6, 2769\n(1972).\n[21] J. Kunes and V. Kambersky, Phys. Rev. B 65, 212411\n(2002).\n[22] S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczews ki,\nP. Trouilloud, and R. Koch, Phys. Rev. B 66, 214416\n(2002).\n[23] L. Mattheiss and D. Hamann, Phys. Rev. B 33, 823\n(1986).\n[24] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864\n(1964).\n[25] W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965).\n[26] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).\n[27] M. Stiles, S. Halilov, R. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[28] R. McMichael and A. Kunz, J. Appl. Phys. 91, 8650\n(2002).\n[29] E. Rossi, O. G. Heinonen, and A. H. MacDonald,\nPhys. Rev. B 72, 174412 (2005).\n[30] Y. Tserkovnyak, G. Fiete, and B. Halperin,\nAppl. Phys. Lett. 84, 5234 (2004)."    },    {        "title": "2301.12797v1.Investigation_of_Ultrafast_Demagnetization_and_Gilbert_Damping_and_their_Correlation_in_Different_Ferromagnetic_Thin_Films_Grown_Under_Identical_Conditions.pdf",        "content": "1  Investigation  of Ultrafast  Demagnetization  and Gilbert  Damping  and their  Correlation  in Different  \nFerromagnetic  Thin  Films  Grown  Under  Identical  Conditions1 \nSuchetana Mukhopadhyay, Sudip Majumder, Surya Narayan Panda, Anjan Barman*  \nDepartment  of Condensed  Matter  and Materials  Physics,  S.N. Bose  National  Center  for Basic  \nSciences,  Block -JD, Sector  III, Salt Lake,  Kolkata  700106,  India  \nEmail:  abarman@bose.res.in  \n \n \nAbstract  \nFollowing the demonstration of laser -induc ed ultrafast demagnetization in ferromagnetic nickel,  several  \ntheoretical  and phenomenological  propositions  have sought  to uncover  its underlying  physics.  In this work \nwe revisit the three temperature model (3TM) and the microscopic three  temperature  model  (M3TM)  to \nperform  a comparative  analysis  of ultrafast  demagnetization  in 20-nm-thick  cobalt,  nickel  and permalloy  \nthin films  measured  using  an all-optical  pump - probe  technique.  In addition  to the ultrafast  dynamics  at \nthe femtosecond  timescales,  the nano second  magnetization  precession  and damping  are recorded  at various  \npump  excitation  fluences  revealing  a fluence -dependent  enhancement  in both the demagnetization  times  \nand the damping  factors.  We confirm  that the Curie  temperature  to magnetic  moment  ratio of a given \nsystem acts as a figure of merit for the demagnetization time, while the demagnetization  times  and damping  \nfactors  show  an apparent  sensitivity  to the density  of states  at the Fermi  level for a given system.  \nFurther, from numerical simulations of the ultrafast demagnetization  based on both the 3TM and the M3TM, \nwe extract the reservoir coupling parameters that best  reproduce  the experimental  data and estimate  the \nvalue  of the spin flip scattering  probability  for each system. We discuss how the f luence -dependence of \ninter-reservoir coupling parameters  so extracted  may reflect  a role played  by nonthermal  electrons  in the \nmagnetization  dynamics  at low laser  fluences.  \n \n1. Introduction  \nOptical  excitation  of a magnetic  material  with a short  and intense  laser pulse  sets in motion  a chain  \nof microscopic  events  that result  in a macroscopic,  measurable  quenching  of the net magnetization.  \nSince  the processing  speed  of magnetic  storage  devices  is limited  by the maximum  speeds  at which  the \nmagnetization  can be manipulated,  the possibility  of controlling  magnetization  at sub-picosecond  timescales  \nby employing  ultrashort  laser pulses  holds  tremendous  potential  for applications  in spin-based  memory  and \nstorage  devices  with ultrafast  processing  speeds  including  laser -induced  opto-magnetism  and all-optical  \nswitching  [1–4] as well as THz spintronic  devices  [5–7]. Meanwhile,  the relaxation  timescales  of the laser -\ninduced  magnetization  precession  in ferromagnetic  thin films  associated  with the magnetic  damping  set \nfundamen tal limits  on magnetization  switching  and data transfer  rates  in spintronic  devices.     \nPicosecond  laser -induced  magnetization  quenching  in ferromagnetic  gadolinium  was reported  by \nVaterlaus  et al. in 1991,  who also reported  for the first time a character istic timescale  of 100±80  ps \n[8] associated  with the magnetization  loss. In this early  work,  the timescale  was identified  with the \ntimescale  of the spin-lattice  interactions  mediating  the disruption of magnetic ordering due to laser \nheating of the lattice,  though later works  contradicted  this interpretation  [9, 10]. In fact, by the mid-\n1990s,  it had begun  to be recognized that finer time resolution was necessary to probe this phenomenon \nto uncover a  fuller  picture  of the associated  relaxation  processes  occu rring  at sub-picosecond  timescales.  \nIn 1996,  Beaurepaire  et al. demonstrated  in a seminal  work  that faster  demagnetization  occurring at sub -\npicosecond timescales could be triggered by femtosecond laser pulses in a  nickel thin film [11].  The \nultrafast demag netization phenomenon went on to be demonstrated  in a wide  array  of ferromagnetic  \nsystems  [12–19], triggering  a flurry  of research  that continues  till date. A few years after the pioneering \nexperiments, Koopmans et al. characterized for the  first time the full magneto -optical response to \nfemtosecond laser pulses in a ferromagnet by  time-resolved measurements of Kerr ellipticity and \nrotation [13].  However, the observation of  nonmagnetic contributions to the Kerr rotation signal \nnaturally led some to question  whether  the ultrafast quenching in response to the laser excitation was \nindeed a magnetic phenomenon  [13, 14]. In 2003, Rhie et al.  used time -resolved photoemission  \n \n1First  submitted  in March  2022  2  spectroscopy to probe the collapse of the 3d exchange s plitting in nickel as an unambiguous signature of a \nphotoinduced demagnetization occurring over 300±70 fs [15]. This was soon followed by the first reports \nof an estimate of the characteristic timescale of femtosecond laser -induced demagnetization derived from \nquantum -mechanical principles [16].  \nOne of the major reasons behind this sustained interest is the need to achieve a complete understanding of \nthe microscopic mechanisms that underlie the ultrafast demagnetization phenomenon that have so far \nremained elusive. Much inquiry has been focused on how the laser -induced loss of magnetic order is \ncompensated for via the transfer of angular momentum of the spin system at the associated timescales. \nOver the years, several mechanisms have been proposed for explai ning the angular momentum \nconservation associated with the ultrafast demagnetization, which may be broadly categorized in two \ndistinct classes. One of these argues in favor of a dominant contribution to the demagnetization arising \nfrom nonlocal transport p rocesses driven by the laser pulse, such as superdiffusive transport of spin -\npolarized hot electrons [20, 21] or heat currents [22]. The other narrative relies on local spin -flip scattering \nprocesses occurring by the collision of excited electrons with imp urities, phonons, and magnons [17, 23, \n24]. In this category, electron -lattice and electron -spin relaxation mechanisms are accepted as the major \ndemagnetization channels in the picoseconds following the laser excitation [25], with several works \nattributing  a pivotal role to the material -specific spin -orbit interaction [23, 25 –27]. On the other hand, it is \npossible to model the demagnetization by adopting a thermal description, considering the laser excitation \nas a “heat” source that excites electrons close to the Fermi level instantaneously to very high energies, \nwhose eventual relaxation processes drive the magnetization loss. The laser excitation energy controlled \nby the applied laser pump fluence has a direct influence on the maximum electron temperatures  reached \nand thus plays a pivotal role in the ultrafast magnetization dynamics that follow as a result.  \nIn this work, employing an all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) technique, \nwe investigate the ultrafast spin dynamics along w ith the nanosecond magnetization precession and \ndamping at different laser pump fluences in three ferromagnetic thin films: 20 -nm-thick cobalt, nickel and \npermalloy grown under uniform deposition conditions. Cobalt and nickel are elementary 3d ferromagneti c \ntransition metals well studied in the literature in both their elementary forms and as constituting elements \nin multilayered structures where their strong spin -orbit coupling mediates interfacial effects such as spin \npumping. Permalloy, an alloy comprise d of approximately 80% nickel and 20% iron, is a prototype \nmaterial for spintronic applications due to several desirable qualities such as low coercivity,   large  \npermeability  and  in  particular,   low  Gilbert  damping,   which  aids  in minimizing the  maximum power \nconsumption of devices and allows spin wave propagation on length scales of the order of device size. The \nsimultaneous investigation of the fluence -dependent modulation of ultrafast demagnetization and damping \nin all the above ferromagnetic thin films is motivated by the primary objectives of exploring the dominant \nmicroscopic processes underlying the demagnetization in these systems, correlating the observed \nmagnetization dynamics to the material properties of each system, and in the process  exploring their \ntunability with laser pump fluence. We set out to achieve this in two ways: (a) by modelling the ultrafast \ndemagnetization at various pump fluences using two theoretical models sharing a common link and (b) by \ncorrelating the laser -induced  changes in ultrafast demagnetization to the changes in the Gilbert damping \nfactor which characterizes the magnetization precession. TR -MOKE is a well established local  and  non -\ninvasive all -optical measurement technique tunable to an extremely fine time resolution limited only by \nthe laser pulse width and is thus well suited to our purpose.  \nFor the detailed analysis of ultrafast demagnetization in our samples, we extract the values of \ndemagnetization time and fast relaxation time for each sample using a p henomenological fitting function \nand record its systematic variation with the pump fluence. We subsequently model the demagnetization \ndata using two well known theoretical models: the phenomenological three -temperature model (3TM) and \nthe microscopic three  temperature model (M3TM), and thereby extract values for the microscopic \nparameters relevant for the demagnetization process and calculate the temporal evolution of the simulated \ntemperatures of the electron, spin and lattice systems within the first few picoseconds of the laser \nexcitation. Both these models assume a thermal picture to explain the initial electron temperature rise and \nsubsequent relaxation but differ significantly in their approach with regards to the treatment of the \nmagnetization. A syst ematic study implementing both models to analyze TR -MOKE data recorded under \nuniform experimental conditions on identically prepared samples is absent in the literature and will prove \ninstructive. Further, a thorough investigation and characterization of t he ultrafast demagnetization, \nmagnetic damping and their intercorrelation in three different systems deposited under identical deposition \nconditions has not been carried out before. For the purposes of a comparative analysis, it is  vital to study \nsamples g rown under the same deposition conditions.  The conductivity of the  substrate  can also directly  \ninfluence  demagnetization  time by promoting  or hindering  ultrafast spin transport processes [20, 28] while \nthe deposition technique determines the structural  properties of the samples that may indirectly affect the 3  demagnetization time.  Moreover,  since the experimentally obtained demagnetization timescale has been \nshown to be quite  sensitive to extrinsic factors such as the laser pulse duration and the spectral ba ndwidth \n[29],  it is useful to measure the demagnetization times for various thin films from experiments  performed  \nin the same  experimental  setup  under  near-identical  experimental  conditions.  \n \n2. Materials  and Methods  \n2.1 Sample  fabrication  \n20 nm -thick cobalt,  nickel and permalloy thin films were deposited by electron beam  evaporation \nunder an average base pressure of 1 ×10−7 Torr and at a very low deposition  rate of 0.2 Å/s chosen to \nachieve uniform deposition.  Each film was deposited on an  insulating  8 mm × 8 mm silicon  substrate  \ncoated  with  285-nm-thick  SiO 2. Subsequently, the films were capped in-situ with a 5 -nm-thick \nprotective gold layer (base  pressure  ~1 × 10−7 Torr,  deposition  rate 0.1 Å/s) to prevent  surface  \noxidation  of the ferromagnetic layer and protect it from possible degradation due to high power laser  \nexposure during the optical pump -probe measurements [ 30, 31]. The thickness of the capping  layer  was \nkept more  than three  times  smaller  than the optical  penetration  depth  of 400 nm light in gold.  After  \nthe deposition  of the capping  layer,  the surface  topography  of the samples  was investigated  by atomic  \nforce  microscopy  (AFM)  using  a Tap190Al -G tapping  mode  AFM  probe as shown in Figure 1 ( a). The \naverage surface roughness  Ra of the cobalt,  nickel and  permalloy  films  were  obtained  as 0.91 nm, 0.36 nm \nand 0.41 nm respectively.  Thus,  reasonably  low surface  roughn ess in the sub-nm range  was obtained  \nwhich  is comparable  for all the samples.  In addition,  the static  magneto -optical  Kerr effect  was used to \nstudy  the magnetic  hysteresis  of the deposited  samples  to confirm  their ferromagnetic  nature  (Figure  \n1(b)). \n \n 2.2 All-optical  measurement  of ultrafast  spin dynamics  \nMeasurement of laser -induced magnetization dynamics is carried out using a TR -MOKE  technique \nin two -color optical pump -probe arrangement using a 400 nm pump beam and 800  nm probe beam \nhaving a pump -probe cr oss-correlation width of ~100 fs [30] shown  schematically in Figure 1( c). A \ntypical TR -MOKE trace is shown in Figure 1( d), comprising  ultrafast  demagnetization  (Regime  I), fast \nremagnetization  (Regime  II) and damped  magnetization precession (Regime III).Th is technique enables \nthe direct observation of the  spin dynamics in the femto - and picosecond time domain. Femtosecond \npulses are generated  by an amplified laser system (Libra, Coherent Inc.)  employing a chirped -pulse \nregenerative  amplifier and a Ti:sapphi re laser oscillator (Coherent Inc.)  pumped by a neodymium -doped  \nyttrium lithium fluoride (Nd -YLF) laser.  The second harmonic of the amplified laser output  \n(wavelength = 400 nm, repetition rate = 1 kHz, pulse width >40 fs), generated through a  lithium  \ntribo rate (LBO)  nonlinear  crystal,  is used  for laser  excitation  of the ferromagnetic  thin films.  The \ntime-delayed fundamental beam (wavelength = 800 nm, repetition rate = 1 kHz, pulse width ~40 fs) is \nused to probe the ensuing magnetization dynamics.  In our  setup, different wavelengths are employed \nfor the pump and the probe pulse to eliminate the  possibility of state blocking effects arising from the \nuse of identical wavelengths for pumping  and probing [32]. A computer -controlled variable delay \ngenerator offers  precise control of the  delay time between pump and probe. Before commencing \nmeasurements on any sample, the  zero delay was carefully estimated by maximizing the transient \nreflectivity signal of a bare  silicon  substrate  placed  adjacent  to the sample  on the same  sample  holder.  \nTR-MOKE  experiments are performed with a non -collinear pump -probe geometry.  The pump beam,  \nfocused to a spot size of ~300 µm, is incident obliquely on the sample while the probe  beam, with a \nspot size of ~100 µm, is incident normal to the sample surface and aligned  to the center of the pump \nspot. The pump -probe spatial overlap on the sample was carefully  maintained. The choice of a \nrelatively smaller spot size of the probe beam as compared to the  pump beam facilitates optical \nalignment and ensures that the probe beam detects the local  magnetization changes from a part of the \nsample uniformly irradiated by the pump.  Before  reflection on the samples, the probe beam is polarized \northogonally to the linearly polarized  pump beam. After reflec tion, the Kerr -rotated probe beam is split \nand one part is fed directly  into a Si -photodiode to measure the time -resolved reflectivity signal.  The \nother part is fed  into an identical photodiode after passing through a Glan -Thompson polarizer adjusted \nto a small  angle  from  extinction  to minimize  optical  artifacts  in the Kerr  rotation  signal.  In this way,  \nsimultaneous  measurement  of the time-resolved  total reflectivity  and Kerr  rotation  signals is possible.  \nAn optical chopper operating at 373 Hz placed in th e path of the pump  beam provides a reference signal \nfor the digital signal processing lock -in amplifiers (Stanford  Research  Systems,  SR830)  which  record  \nthe modulated  signal  in a phase  sensitive  manner.  All experiments  were  carried  out under  ambient  \ncondit ions of temperature  and pressure.  4  3. Results  and Discussion  \n 3.1 Theoretical  models  for ultrafast  demagnetization  \nThe phenomenon  of optically  induced  ultrafast  demagnetization  starts  with the irradiation  of the \nmagnetic  sample  with  a brief  and intense  optical  laser  pulse,  exciting  electrons  momentarily a few \nelectron volts above the Fermi level.  Though the exact sequence of events  following  the initial  excitation  \nis difficult  to trace  due to the highly  nonequilibrium  conditions  created  by it, a qualitative  overview  of \nthe complete  demagnetization  process  is fairly  well   established.  The laser  excitation  generates  a \nnonthermal  pool of excited  electrons  which  thermalize  rapidly  within  several  femtoseconds  via \nelectron -electron  interactions.  Spin -dependent  scatter ing events  taking  place  during  this transient  regime  \nlead to a sharp  drop  in magnetization  observable  around  a few hundred  femtoseconds  in the \nexperimental  Kerr  rotation  signal.  Subsequently,  the thermalized  electrons  may release  their  excess  \nenergy  via a variety  of relaxation  channels,  such  as by excitation  of phonons  or magnons.  This results  in a \npartial  recovery  of the magnetization  beyond  which  heat dissipation  into the environment  promotes  further  \nrecovery  on a longer  timescale.  The 3TM  posits  that the thermodynamics  of the demagnetization  \nphenomenon  can be described  simply  by considering  energy  exchange  between  three  thermal  reservoirs  \n[33],  each of which  is assigned  a temperature:  the electrons  at temperature  Te, the lattice  at temperature  \nTl and the electronic  spin reservoir  at temperature  Ts. Since  the reservoirs  are in thermal  contact  and \nthe overall  process  is adiabatic,  equilibration  of the excited  electrons  with the spin and lattice  reservoirs  \nvia energy  transfer  may be described  by coupled  rate equations  in the following  manner:   \n \n \n \n   (1) \n \n \n \nwhere,  Ce, Cl and Cs denote the specific heats of the electron,  lattice and spin reservoirs  respectively,  \nwhile  Gel, Ges, and Gsl denote  the inter -reservoir  coupling  parameters.  The term  P (t) describes  the action  \nof the laser  pulse  as a source  term driving  the excitation  of the electron  reservoir  to high temperatures.  The \nthermal  diffusion  term  \n  describes  heat dissipation  occurring  via thermal  conduction  \nalong  the sample  thicknes s. Under  this description,  the observed  demagnetization  is attributed  to a \nrise in the spin temperature  Ts occurring  shortly  after  the electron  temperature  rise. The coupling  \nstrengths  between  the electron -spin and electron -lattice  subsystems  qualitatively  determine  the efficiency  \nof energy  transfer  between  them  and hence  influence  the timescales  associated  with the demagnetization \nand fast relaxation.  However, the 3TM is purely phenomenological and does  not explicitly consider any \nmicroscopic mechanisms un derlying the phenomenon it describes.  On the other  hand,  the M3TM  \nproposed  by Koopmans  et al. [23] provides  a “spin -projected”  perspective  [27] to explain  the ultrafast  \ntransfer  of angular  momentum  highlighting  the role of Elliot -Yafet  type ultrafast  spin-flip scattering  in \nthe demagnetization  process.  The initial  excitation  by the laser  pulse  disturbs  the electronic  subsystem  \nfrom  equilibrium  which  leads  to an imbalance  in spin-up and spin-down  scattering  rates,  resulting  in the \nobserved  loss of magnetic  order.  The process  is mediated  by spin-orbit  interactions  leading  to the \nformation  of hot spots  in the band  structure  where  spin-up and spin-down  channels  are intermixed.  An \nelectron  scattered  into these  hot spots  via a phonon - or impurity -mediated  scatteri ng will flip its spin with  \na finite  probability.  The individual  scattering  events  are characterized  by a parameter asf across  the  \nsample,  identified  with the probability  of spin-flip due to  electron -phonon  scattering.  The magnitude  \nof this parameter  will directly  depend  on the extent of  spin-orbit  coupling and  hence  is expected  to be \ncomparable  in materials with  similar  spin-orbit  coupling  strengths.  The M3TM  retains  the coupled  rate \nequations  for the electron  and lattice  temperatures,  similar  to the thermal description  provided  by the \n3TM.  However  the fundamental  difference  from  the 3TM  is that in the framework  of the M3TM,  the \nspin bath is formed  by a collection  of two-level  systems  obeying  Boltzmann  statistics.  Instead  of \nassigning  a temperature  to the spin bath,  the normalized  magnetization  is directly  calculated  from  the \nassociated  exchange  splitting.  The rate of change  of the magnetization,  derived  analytically \nconsidering an Elliot -Yafet scattering -driven demagnetization, is parametrized by  asf and c oupled to the \nelectron and the lattice subsystem temperatures. The assignment of a  characteristic temperature to the spin \nsubsystem is replaced in the M3TM by an evolution  equation  for the magnetization:   \n 5  \n        (2) \n \nThe quantity  R is a mate rial-specific  factor  which  influences  the demagnetization  rate and is \nproportional  to asfTc2/μat where  TC is the material Curie temperature and µat is the atomic magnetic \nmoment. Though the two models  differ in their approaches, one can immediately discern certain \nsimilarities in their domains of  validity.  Both models are appropriate only when th e nonlocal \nmechanisms driving ultrafast  demagnetization such as superdiffusive spin transport can be neglected. \nWe note here that it  has been  reported  that spin transport  is not a major  contributor  to the ultrafast  \ndemagnetization in transition metals [34] . We nevertheless use an insulating SiO 2-coated Si  substrate \nfor our samples to minimize spin transport effects such that analysis with the local  models described \nabove should suffice in our case. Any additional contributions arising from  the gold capping  layer  \nwould  be uniform  across  all samples  investigated  and therefore  unlikely  to impact the main results of our \ncomparative study.  Moreover, since the thickness of the  capping layer is much smaller than the \npenetration depth of both 400 nm and 800 nm light  in gold [35], the pump excitation fully penetrates \ndown to the magnetic layer ensuring that the  effect of direct laser excitation of the ferromagnet is \nprobed in our case.  Thus, we set up  numerical calculations based on the models described above in \norder  to extract microscopic  information  from  the experimentally  obtained  demagnetization  traces.  \n 3.2 Ultrafast  demagnetization  in cobalt,  nickel,  and permalloy  thin films  \nWe proceed by performing time -resolved measurements of the polar magneto -optical Kerr  effect in \nthe cobalt, nickel and permalloy thin film samples as a function of the laser fluence.  Measurements are \ncarried out under a strong enough external magnetic field kept constant at  around 2 kOe tilted at a small \nangle from the sample plane to saturate  the magnetization of  the samples. The pump fluence is varied \nbetween 0.8 -8.7 mJ/cm2 by varying the power of the  pump pulse.   The results are presented in Figure 2.   \nTo ascertain that the measured Kerr  signal  reflects  the true magnetization  dynamics  without any \nspurious  contribution  from  optical effects triggered in the initial stages of laser excitation, we also \nexamine the transient  reflectivity signal for each sample. The fluence dependent variation in the \nreflectivity can be  found in Figure S1 of the  Supplementary Materials, demonstrating that at any given \nfluence  the amplitude  of the reflectivity  signal  is negligibly  small  compared  to that of the  Kerr  rotation.  \nWe nevertheless restrict ourselves to the low fluence regime to avoid nonlinear  effects a nd sample \ndamage.  For all our experiments, the probe fluence is kept constant at a  value about half that of the \nlowest pump fluence used to prevent additional contribution to  the spin dynamics by probe excitation.  \nAs seen in Figure 2, the ultrafast demagne tization  completes within 1 ps for all three samples \nconsidered which is followed by a fast recovery of  the magnetization,  all observed  within  the \nexperimental  time window  of 4 ps. These  experimental traces clearly exhibit the “Type -I” or “one -step” \ndemagn etization expected for  transition metal thin films at room temperature and under low -to-\nmoderate pump fluence  [23]. The amplitude of the maximum quenching of the Kerr rotation signal \nincreases with the  laser  fluence,  allowing  us to rule out nonlinear  effec ts [36].  Closer  inspection  of the \ntraces  also reveals an increase of the time taken to demagnetize the samples with increasing fluence for  \nall three  samples.  To quantify  this increase,  we fit our demagnetization  traces  to a phenomenological  \nexpression  based on the 3TM  and valid  in the low laser  fluence  regime  [37]:   \n \n(3) \n \nwhere  Θ(t) is the Heaviside  step function,  δ(t) is the Dirac  delta  function  and Γ(t) is the \nGaussian laser pulse.  The constant A1 represents the value of the normalized  magnetization after \nremagnetization has completed and equilibrium between the electron,  spin and lattice  reservoirs  has \nbeen  re-established.  A2 is proportional  to the initial  rise in electron  temperature  and hence  to the \nmaximum  magnetization  quenching.  A3 represents  the magnitude of state -filling effects present during \nthe onset of the demagnetiz ation response,  which is negligible in our case.  τM and τE are the \ndemagnetization time and fast relaxation  time,  respectively.  Prior  to the fitting,  all the experimental  \ntraces  were  normalized  by hysteresis measurements of the Kerr rotation signal under the saturating \nmagnetic field in the  absence of la ser excitation.  We find that within the range of fluence values \nconsidered,  permalloy  exhibits  the largest  magnetization  quenching  of 54.6%,  followed  by a 23.7%  \nquenching achieved in nickel, while the magnetization of cobalt the least, only about 8% for  the largest  \napplied  fluence.  The demagnetization  occurs  at a characteristic  timescale  of 230-280 fs for cobalt, 160 -6  210 fs for nickel, and 220 -250 fs for permalloy, increasing with the  laser fluence.  This effect can be \nattributed to enhanced spin -fluctuation s at elevated spin  temperatures  for  higher  fluences  [38].      \nAt  a  fluence  of  4.8  mJ/cm2,   the  extracted  demagnetization  times  are 276.6  ± 3.41 fs for cobalt,  \n187.3  ± 2.89 fs for nickel  and 236.8  ± 2.45 fs for permalloy.  The timescale  for the magnetization  \nrecovery  τE also increases  with increasing pump fluence. The variation of these characteristic \ntimescales with laser fluence is  shown  in Figure  3. These  fluence -dependent  trends  in τM and τE hint at a \nspin-flip process -dominated ultrafast demagnetization in our stu died systems [23, 39, 40]. The values  of \nτM extracted  from  our experiments  lie within  the typical  range  of 100-300 fs consistent  with \nprevious reports of the ultrafast demagnetization times in these metals [17, 23], and are  too large  to \nrepresent  a superdi ffusive  transport -driven  demagnetization  [41].  \n For the 3TM and M3TM simulations, we choose a laser pump term given by \nproportional to  the pump fluence F and following a Gaussian temporal profile.  \nThe maximum rise of the  electron temperature a nd thus also the extent of demagnetization depends \nsensitively on this  term which is hence adjusted to reproduce the maximum quenching observed \nexperimentally.  We use a pulse width τp = 100 fs determined by the pump -probe cross -correlation in all  \ncalculations.  Intrinsic  to both the models  we consider  is the assumption  that electron  thermalization occurs \nextremely fast.  The thermal  diffusion term  can be neglected in our case since th e thicknesses of the films \nwe study are kept slightly greater than the optical penetration depth of 400 nm pump beam in  those films.  \nThis ensures uniform heating of the films in the vertical direction while also  avoiding laser penetration \ninto the substrat e in which case heat dissipation into the substrate  would have to be taken into account. \nBesides, the timescales associated with heat dissipation  are generally  tens to hundreds  of picoseconds,  \nmuch  longer  than the demagnetization  and fast relaxation  times,  and hence  unlikely  to significantly  \ninfluence  our observations  at these  timescales.  Since both models we consider are thermal in their \napproach, choosing correct  values for the reservoir specific heats is vital for a proper simulation of the \ndemagnetizati on. For the electronic  specific  heat Ce, we assume  a linear  dependence  on the electronic  \ntemperature Ce(Te) = γTe  derived from the Sommerfeld free -electron approximation where γ is determined \nby the electronic density of states at the Fermi level [42].  The value of γ for permalloy is approximated as \na weighted average of the individual γ values of nickel and iron  in permalloy. The lattice specific heat Cl is \ncalculated at each value of the lattice temperature  according to the following relation derived from Debye \ntheory:  \n(4) \n \nwhere NA is the  Avogadro’s  number,  kB is the Boltzmann’s  constant  and θD  is  the  Debye  temperature.  \nFinally,  we fix the spin specific  heat Cs to its value  at room  temperature  for the 3TM  calculations,  \nobtained  by subtracting  the electronic  and lattice  contributions  from  the experimental values of the total \nspecific heat found in the literature [43].  Considering a  spin-temperature -dependent form of Cs was not \nfound to significantly affect our conclusions  as described in Section IV of the Supplementary Materials.  \nThe fixed parameter set used in  our calculations have been listed in Table 1. To relate the experimental \ndemagnetization to  the temperature  of the spin subsystem  under  the 3TM  framework,  the spin \ntemperature  Ts is  mapped to the magnetization of the system via the Weiss’ mean field theory [44], \nwhich is  then fitted with the experimental magnetization traces to obtain the empirical inter -reservoir  \ncoupling parameters Gel and Ges consistent with the observed dynamics.  We neglect the  spin-lattice \ncoupling parameter Gsl for the 3TM simulations since in ferromagnetic transitio n metals the energy \ntransfer between electrons and lattice is far greater than that between  lattice  and spins  [45].  \n \n Table 1 . Fixed parameter set used in the calculations. Literature values have been used for all parameters listed   \n[30, 31, 34, 35].  \n \n For the 3TM simulations, we proceeded to extract Gel and Ges by first fitting the  demagnetization \ndata at the lowe st fluence to the model. However, fitting the higher fluence  data using identical values \nof the coupling parameters as extracted at the lowest fluence did  not result in a good match to our \nexperimental results.  The coupling parameters extracted  from the lo w fluence data led to an \noverestimation of the demagnetization time at the higher  fluences. It was seen that a 5 -10% increase in Sampl e Tc (K) θD (K) γ (Jm-3K-2) Cs (Jm-3K-1) at (B) \nNi 627  450  1065  3.07 × 105 0.62  \nCo 1388  445  714  1.59 × 105 1.72  \nPy 860  454  992  2.67 × 105 1.00  7  Ges from its value at its adjacent lower fluence  value rectifies the overestimation of the demagnetization \ntime.  On the other h and, the remagnetization dynamics is most sensitive to the Gel parameter so that the \noverall dynamics  is best reproduced  only  by adjusting  both  Gel and Ges. As shown  in Figure  4, the \nresulting  fit shows excellent agreement with the experimental data.  This exercise reveals the crucial role  \nplayed by the electron -spin relaxation channels in determining the timescale associated with  the initial \ndemagnetization while the magnetization recovery is primarily mediated by the  electron -lattice \ninteraction.  We also f ind that the mismatch between model and experiment  can be resolved by \nconsidering an increasing trend of Gel and Ges with pump fluence arising  from a faster demagnetization \nprocess for the same percentage quenching as compared to the  model predictions with in the studied \nfluence range. The values of the microscopic parameters  extracted from the least -squares fits with their \ncorresponding error bounds can be found in  Supplementary Tables S1 -S3. Since the exact values of the \ncoupling parameters extracted  from  the fits naturally  depend  on the values  chosen  for the fixed  \nparameters,  the interpretation  of the results  from  these  fits is best limited  to a comparative  one. \nFor the M3TM  simulations,  the demagnetization  traces  are fitted  directly  to Equation  2 yielding  \nGel and asf as fit parameters.  In this case,  asf plays  a role  in  determining  the  maximum extent and the \nassociated timescale of the demagnetization via the scaling factor R while Gel continues to influence \nmainly the magnetization recovery process.  How ever, the  demagnetization  time is less sensitive  to \nchanges  in asf than it is to Ges in the 3TM  case.  This results in somewhat higher values of Gel and a \nsharper rise with pump fluence than those  extracted  from  the 3TM  simulations,  in order  to compensate  \nfor the overestimation  of demagnetization time that results from the model if Gel  at the lowest fluence is \nused for all  the fits.  We have obtained an asf of ~0.02 for cobalt,  ~0.05-0.06 for nickel, and  ~0.03-0.06 \nfor permalloy.  The value of asf we have ext racted for nickel is an order of  magnitude lower than the value \nasf = 0.185 first reported by Koopmans et al. [23] but quite  close to the value of 0.08 reported by Roth et \nal. [39].  This discrepancy is expected, as the  artificially high value of 0.185 aros e due to an \noverestimation of the electronic specific heat in  the original  work,  avoided  here by considering  \nexperimentally  determined  γ values  reported  in the literature.  The observation  that asf [Co] <asf [Ni] is \nconsistent  with previous  reports  [23, 40]. \nWith  regard  to thermal  treatments  of ultrafast  demagnetization  like the 3TM  and M3TM,  there is \nsome contention centered on the role o f the nonthermal electrons generated within the  first 50-100 \nfemtoseconds  after laser excitation  [9, 48, 49]. Within  this time frame,  the excited  hot electrons  have not \nyet thermalized  and their energies  cannot  be described  by a Fermi -Dirac  distribution.  However,  both \nthe 3TM  and the M3TM  completely  neglect  the finite  electron  thermalization  time and consider  only the \nenergy  exchange  between  the thermalized  electron  population  with the lattice  and/or  the spin subsystem  as the \ninitial  nonequilibrium  distribu tion is assumed  to last for much  shorter  durations  (≳ 10 fs) than the \ntimescale  over which  demagnetization  typically  occurs  (≳ 100 fs). However,  strictly  speaking,  this \nassumption  is valid  only when  the thermalization  proceeds  very rapidly  such as can be ensured with  high \nfluence  excitations  [50].  In the low  fluence  regime,  electron thermalization  is slower,  hence  the influence  \nof nonthermal  electrons  on the measured  dynamics  may become  more  important  with decreasing  pump  \nfluence.  The increasing  of Gel with fluence  deduced  from our calculations may be an indication of a \ndecrease in the efficiency of electron -lattice  interactions  at low laser  fluences  when  nonthermal  electrons  \nprevail.  However,  the specific  values of Gel and Ges values we have obtained for th e different systems \nremain comparable to  values  previously  reported  for these  metals  [11, 23, 37]. \nAfter completion of the fitting routine,  we set up simulations using the optimized values of  the free \nparameters to obtain the temporal evolution of Te, Tl and Ts as a function of the  pump -probe delay.   The \ncalculated profiles of Te, Tl and Ts for four different values of the  pump fluence are presented in Figure \n5 for nickel, showing good agreement between the  results predicted by the 3TM and the M3TM. Simil ar \nprofiles for cobalt and permalloy are  given in Figs.  S2 and S3 of the Supplementary Materials.  As the \npump fluence is increased,  laser excitation deposits a greater amount of energy in the electronic \nreservoir which leads to  a proportionate increase in the initial electron temperature rise.  However, the \nmaximum  electron temperature reached for the highest applied pump fluence remains well below the  \nFermi  temperature  for the respective  material,  ensuring  the assumption  of the linear  temperature  \ndependence  of the electronic  specific  heat remains  valid  [42].  Given  the fundamentally different origins \nof the 3TM and M3TM, the close agreement between the  temperature profiles calculated using the two \nmodels and among the extracted values of the  electron -lattice coupling parameter Gel is remarkable. \nThis observation reflects that, given the  strength of the laser excitation determining the initial \ntemperature excursion of the electron  system,  the Gel  parameter common to both the 3TM and the \nM3TM framework plays a  pivotal  role in determining  the profile  of the ensuing  magnetization  changes.  8  This  also  explains why Gel is seen to follow a similar fluence -dependent trend for a given system in \nboth models.  While the electron temperature is coupled to the spin temperat ure via the Ges parameter  in \nthe 3TM,  it enters  into the evolution  equation  for the magnetization  via  Equation 2 in the M3TM case. \nFrom Figure 5, a qualitative difference in the temporal profiles  of the electron,  spin and lattice \ntemperatures is also appa rent.  Referring to the figures  corresponding  to the 3TM,  while  Te exhibits  a \nsharp  peak  within  the first few hundred  femtoseconds, Ts exhibits a shorter and broader peak at slightly \nlonger time delays.  The delayed peak in Ts occurs when excited electrons t ransfer energy to the spin \ndegrees of  freedom  [51] through  scattering  events  mediated  by the coupling  parameter  Ges, which  \nqualitatively determines the maximum spin temperature value [11].  These scattering events  can result in \nthe generation of magnons, St oner excitations and spin fluctuations which may  play a part in the \nobserved demagnetization [52].  Lastly, the lattice temperature gradually  increases over the first few \npicoseconds after excitation, reaching an equilibrium value at  which  it stays  nearly  constant  over \nseveral  picoseconds.  \nIn Figure  6 we present  a comparison  between  the experimental  demagnetization  curves  and calculated \nelectron temperature profiles for each of our three samples. Under the same applied  pump  fluence,  different  \namplitudes  of magnetization  quenching  and electron  temperature  rise are observed. It is clearly seen that \nnickel quenches more strongly than cobalt, an observation  which  can be explained  by band  structure  \neffects  and a more  efficient  laser -induced  electronic  excitation i n nickel owing to its much lower Curie \ntemperature [34, 40, 53], reflected in the  M3TM as a higher extracted value of the spin -flip probability  \nasf  in nickel.   On the other  hand, a lower electronic specific heat capacity in cobalt as compared to \nnickel [42] predictably  leads  to a much  greater  rise in its electron  temperature  for the same  pump  fluence.  \nIn the case of permalloy,  alloying  with iron can reduce  the electronic  density  of states  at the Fermi  level  \n as compared to nickel, decreasing γ. Because o f the larger electron temperatures reached as a  result, a \nvalue of asf comparable to that of nickel is sufficient to drive a much larger reduction  in the \nmagnetization.  Permalloy also shows a somewhat larger demagnetization time at a  given pump fluence \nas compared to nickel.  Since a standard TR -MOKE optical pump -probe  measurement  cannot  be used  to \nprobe  the demagnetization  response  in alloys  in an element -specific manner, interpreting this finding is \ndifficult. It has been reported previously  using element -specific measurements in a nickel -iron alloy that \nthe demagnetization of nickel  proceeds  faster  than that of iron [54].  Our observation  of a slower  \ndemagnetization  in permalloy may be reflective of this result.  Finally, the demagnetization times for the \nthree systems  at any given  pump  fluence  is seen to sensitively  depend  on the ratio  of TC/µat, \nconfirming  its role as a ‘figure  of merit’  (FOM)  for the demagnetization  time [23].  As shown  in \nFigure 8 ( a), the demagnetization proceeds slower for a lower value  of the FOM. Thus at a  given \nincident laser pump fluence, the demagnetization times are found to correlate with the  values  of this \nquantifying  parameter,  rather  than the material  Curie  temperature  directly.  \n \n 3.3 Precessional  dynamics  and correlation  of Gilbert damping  parameter  with ultrafast  demagnetization  \ntime \nTo gain further  insight  into the microscopic  mechanism  underlying  the ultrafast  \ndemagnetization,  we study also the precessional magnetization dynamics in our thin film  systems. A \ntypical time -resol ved Kerr rotation data showing the different temporal regimes of  laser -induced \nmagnetization dynamics is given in Figure 1( d). Regime I and II represent the  ultrafast  demagnetization  \nand magnetization  recovery  processes  respectively,  while  the oscillatory  signal  of regime  III represents  \nthe magnetization  precession  with the Gilbert  damping  characteristic  of the ferromagnetic  material.  To \nextract  the Gilbert  damping  factor  α, the background -subtracted oscillatory Kerr signal is first fitted \nwith a damped sinusoidal  function  given  by:  \n \n  \n (5) \n   \nwhere  f is the precessional frequency, τ is the relaxation time, and φ is the initial phase of oscillation. \nThe depende nce of f on applied bias magnetic field H is fitted to the Kittel formula relevant for \nferromagnetic systems:  \n \n \n (6) \n \nwhere  γ0 = gµB/ℏ is  the  gyromagnetic  ratio  and  Meff  is the effective  saturation  magnetization  \nof the probed  volume.  From the Kittel  fit, the values  of g and Meff can be extracted  as fit \nparameters.  While the value  of g is found  to be 2 ± 0.1 for all the samples,  the extracted  values  9  of Meff are 1280.2  ± 1.69 emu/cm3 for cobalt,  400.5  ± 11.46  emu/cm3 for nickel  and 737.1  ± 5.32 \nemu/cm3 for permalloy  (Section  V of the Supplementary  Materials).  \nSubsequently,  the effective  damping  factor  α is calculated  depending  on the extracted  value  of τ and Meff \nas:  \n \n(7) \n \nAlthough  the timescale  of precessional  dynamics  and magnetic  damping  in ferromagnetic  thin films differ  \nby many orders of magnitude,  similar underlying  phenomena may oft en determine  the characteristics of \neither process.  Through several independent investigations relating these  two distinct processes, the \ncorrelation between the demagnetization time τM and the Gilbert  damping  factor  α emerged  as a critical  \nindicator  of the dominant  microscopic  contribution  to the ultrafast  demagnetization.  Koopmans  et al. in \n2005  analytically  derived  an inversely  proportional  relationship  between  the Gilbert  damping  factor  and \nultrafast  demagnetization  time based  on quantum  mechanical  considerations  [16].  However,  the validity  of \nthe model  could  not be confirmed  experimentally  when  τM and α were  tuned  by transition  metal  and rare \nearth  doping  [18].  Later  it was proposed  that the Gilbert  damping  parameter  and ultrafast  \ndemagnetization  time can be either  directly  or inversely  proportional  to each other  depending on the \npredominant micr oscopic mechanism contributing to the magnetic damping  [55].  A proportional  dependence  \nwould  suggest  a dominating  conductivity -like or “breathing  Fermi  surface”  contribution  to the damping  \ndue to the scattering  of intraband  electrons  and holes,  while  an inverse  dependence  may arise in materials  \nwith a dominant  resistivity -like damping  arising  from  inter-band  scattering  processes.  Zhang  et al. later \nsuggested  that an inverse  correlation  between  Gilbert  damping  factor  and ultrafast  demagnetization  time \nmay arise in the presence  of spin transport,  which  provides  an additional  relaxation  channel  resulting  in an \nenhancement  of the magnetic  damping  along  with an acceleration  of the demagnetization  process  at the \nfemtosecond  timescale  [56].  In cases  where  local  spin-flip scattering  processes  are expected  to dominate  the \ndemagnetization  process,  the breathing  Fermi  surface  model  is valid  predicting  a proportional  relationship  \nbetween  τM and α. \nTo study  the correlation  between  ultrafast  demagnetization  and damping  in a system,  both τM and α \nhave to be systematically  varied.  For a given  material,  the exciting  pump  fluence  can be used to \nmodulate the demagnetization time.  As discussed in the previous section, we  observe  an increment  of the \ndemagnetization  time with the pump  fluence  for the systems  under  investigation.  An enhancement  in α \nis also observed  as the pump  fluence  is increased  from  0.8 mJ/cm2 to 8.7 mJ/cm2 for each sample,  \nassoci ated with a corresponding  decrease  in the precessional  relaxation  time.  Within  the experimental  \nfluence  range,  α is found  to be enhanced from 0.0089 to 0.0099 for cobalt, from 0.0346 to 0.0379 for nickel \nand from 0.0106 to  0.0119  for permalloy.  This enhanc ement  of the damping  can be attributed  to the \ntransient  rise in the system temperature as the laser pump fluence is increased [57].  As a laser -induced  \nincrease in electron temperature also drives the demagnetization process at the femtosecond  timescale,  one \ncan qualitatively  correlate  the demagnetization  time with the magnetic  damping at various excitation \nenergies even for a single sample.  Clearly,  a direct correlation  between demagnetization time and the Gilbert \ndamping factor is obtained in our samples b y measuring the magnetization dynamics at various pump \nfluences, as shown in Figure 7 for the  nickel thin film.  Thus, we infer that similar relaxation processes likely \ndrive the laser induced  transient  magnetization  dynamics  at the two different  timescales .   At a low \nfluence  value  of 2.1 mJ/cm2, the extracted  value  of α is the highest  for the nickel  film,  and relatively  \nlow for cobalt  and permalloy.  The observed  trend  in Figure  8 (b) can be directly  correlated  with the \nvalues  of the Sommerfeld  coefficient  γ given  in Table  1. This observation  is in accordance  with \nKamber sky’s  spin flip scattering  theory  [58],  wherein  α is directly  proportional  to the density  of states  at \nthe Fermi  level,  which  in turn governs  the value  of γ. \n \n \n4. Summary  \nTo summarize,  we have investigated  fluence -dependent  ultrafast  demagnetization  in identi cally -grown  \ncobalt,  nickel  and permalloy  thin films  using  an all-optical  time-resolved  magneto -optical Kerr effect \ntechnique.  We study both the femtosecond laser -induced ultrafast  demagnetization  and the magnetization  \nprecession  and damping  in each of our samples  for various  pump  excitation  fluences.  Using  a \nphenomenological  expression,  we extract  the values  of demagnetization  time and fast relaxation  time 10  for each of our samples  to show  that they exhibit  an increasing  trend  with applied  laser fluence  consi stent  \nwith a spin-flip scattering -dominated  demagnetization  in our samples.  Two distinct  ultrafast  demagnetization  \nmodels  can each reproduce  our experimental  demagnetization  traces  remarkably  well.  By directly  fitting  \nour experimental  data to numerical  solutions  of these  models,  we have extracted  values  for the electron -\nlattice  and electron -spin coupling  parameters  and an empirical  estimate  of the spin flip scattering  \nprobability  for each of our samples  and demonstrated  the fluence -dependent  variation  in calculated  \nelectron,  spin,  and lattice  temperatures  as a function  of the pump -probe  delay.  We conjecture  that the \nincreasing  trend  of the extracted  electron -lattice  coupling  parameter  with fluence  may indicate  a decrease  \nin the efficiency  of electron -lattice  interactions  in the low laser fluences  when  nonthermal  electrons may \nplay a role in influencing the magnetization dynamics.  We have compared the  experimental  and theoretical  \nresults  obtained  from  each of the systems  under  investigation  and confirmed  that the ratio of the Curie  \ntemperature  to the atomic  magnetic  moment  acts as a figure  of merit  sensitive  to the demagnetization  time \nfor these  systems.  Our study  demonstrates the mutual agreement between the phenomenological three \ntemperature model  and the microscopic  three  temperature  model  for the systems  we study  and highlights  the \npossibility  of a fluence -dependent  enhancement  in the inter-reservoir  interactions  in the low laser  fluence  \nregime.  Finally,  we have reported  from  our experiments  a direct  correla tion between  the ultrafast  \ndemagnetization  time and the enhancement  of the magnetic  damping  that transiently appears as a result of \nthe laser excitation, reflecting that both phenomena are  sensitive  to similar  underlying  microscopic  \nprocesses.  \n \n5. Acknowledge ment  \nAB gratefully acknowledges the financial support from the S. N. Bose National Centre for  Basic \nSciences (SNBNCBS) under Project No.  SNB/AB/18 -19/211 and Department of Science  and \nTechnology  (DST),  Govt.  of  India  under  Grant  No.  DST/NM/TUE/QM -3/2019-1C-SNB.  SM \nacknowledges  DST,  India  for financial  support  from  INSPIRE  fellowship,  while  SM and SNP  \nacknowledge  SNBNCBS  for senior  research  fellowship.  \n \nReferences  \n[1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, T. Rasing,  All-\noptical  magnetic  recording  with  circularly  polarized  light,  Phys.  Rev.  Lett.  99 (2007)  047601.  \n[2] T. A. Ostler,  J. Barker,  R. F. L. Evans,  R. W. Chantrell,  U. Atxitia,  O. Chubykalo - \nFesenko,  S. El Moussaoui,  L. Le Guyader,  E. Mengotti,  L. J. Heyderman,  F. Nolting,  \nA. Tsukamoto,  A. Itoh,  D. Afanasiev,  B. A. Ivanov,  A. M. Kalashnikova,  K. Vahaplar,  \nJ. Mentink,  A. Kirilyuk,  T. 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The downward arrows represent \nincreased pump fluence.  The symbols  represent  experimental  data points  and the solid  lines  represent  \nfit to Equation  3. 16  \n \n \nFigure 3:  Variation of ( a) demagnetization time τM and ( b) fast relaxation time τE with the pump fluence \nfor 20-nm-thick cobalt, nickel and permalloy films. Both timescales show an increasing trend with \nincreasing pump  fluence.  \n \n \n \n \nFigure 4:  Sensitivity of the demagnetization traces to the various adju stable fit parameters.  \nSymbols  represent  data points  and solid  lines  represent  fits to the 3TM.  Data  shown  is for the \ncobalt  thin film.  For F =0.8  mJ/cm2, Gel=8.2  × 1017 Js−1m−3K−1 and Ges=6.9  × 1016 Js−1m−3K−1; \nfor F =2.1  mJ/cm2, Gel=1.1  × 1018 Js−1m−3K−1 and Ges=8 × 1017 Js−1m−3K−1. \n17   \n \n \nFigure  5: Temporal  evolution  of the reservoir  temperatures  of nickel  Te (blue),  Tl (red),  and Ts (green)  \ncalculated  using  the (a) phenomenological  three  temperature  model  and (b) the microscopic  three  \ntemperature  model.  \n \n \n \n \n \n \n \n \nFigure  6: Comparison  between  (a) the experimental  demagnetization  and (b) the simulated  electron  \ntemperature  profiles  for cobalt,  nickel  and permalloy  at a pump  fluence  of 4.8 mJ/cm2. \n18   \n \n \nFigure  7: (a) Fluence -dependent  variation  of precessional  dynamics  for the nickel  thin film.  \nSymbols  are experimental data points and the solid lines are fits to Equation 5.  (b) Variation of f \nwith pump fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  between  τM and \nα. \n \n \n \n \n \n \n \n \nFigure 8:  (a) Inverse variation of the demagnetization time with the ratio of Curie temperature to \natomic  magnetic moment and ( b) inverse correlation of the demagnetization time with ma gnetic \ndamping across three  different  materials.  \n19   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary  Materials  S1  I. Transient  reflectivity  measurements  \nTime -resolved  transient  reflectivity  signal  is recorded  by TR-MOKE  measurements  \nsimultaneously with the Kerr rotation signal  quantifying the magnetic response.At any  given \nfluence, the maximum amplitude of the transient reflectivity change is about a  factor of 100 \nsmaller as compared to that of the Kerr rotation.  This validates that the  Kerr rotation signal \nreflects the genuine  magnetic response with negligible nonmagnetic  contribution.  \n \n \nFigure S1:  Time -resolved transient reflectivity measured by TR -MOKE for cobalt, nickel, and \npermalloy.  The upward  arrows  represent  increased  pump  fluence.  \n \n \n \nII. Calculation of electron, spin, an d lattice temperature profiles  \nThe calculated  temporal  evolution  of the reservoir  temperatures  for the 20-nm-thick  cobalt  and \npermalloy  thin films  are represented  in Fig. S2 and Fig. S3, respectively.  The electron  \ntemperature  shows  a rapid  and dramatic  increase  within  one picosecond  of the laser excitation,  \nbeyond  which  it decays  to an equilibrium  value  over a few picoseconds.  Over  this period,  the lattice  \nsubsystem  is gradually  heated  above  the equilibrium  temperature  while  the heating  of the spin \nsubsyste m directly  leads  to the demagnetization.  The maximum  electron,  spin and lattice  temperatures  \nincrease  with increasing  laser  fluence.  \nS2  \n \n \nFigure  S2: Electron,  lattice,  and spin temperature  profiles  for the cobalt  thin film calculated  \nusing  (a) the three tem perature model and (b) the microscopic three temperature model for \nvarious laser fluences  (blue:  Te, dashed,  red: Tl and dotted,  green:  Ts). \n \n \nFigure S3: Electron, lattice, and spin temperature profiles for the permalloy thin film calculated \nusing (a)  the three  temperature  model  and (b) the microscopic  three  temperature  model  for various  \nlaser  fluences).  \n \nIII. Extraction  of inter -reservoir  coupling  constants  \nThe least -squares fitting routines set up based on the three temperature model and the  microscopic  \nthree  temperature  model  return  the values  of the inter-reservoir  coupling  \nS3  constants  that best reproduce  the experimental  demagnetization  traces.  The microscopic  three  \ntemperature  model  additionally  returns  the value  of spin-flip probability asf associated with  the \nspin-flip scattering events which directly lead to the  magnetization quenching.  The extracted \nvalues of these microscopic parameters are  given in Tables S1 -S3. Fig. S4 shows a comparison of \nthe values of the electron -lattice  coupling parameter Gel as extracted from the fits to the three \ntemperature model and  the microscopic three temperature model.  For reasons discussed in the \nmain text, the  spin-lattice coupling parameter Gsl is taken to be zero for the three temperature \nmodel  simulations.  Fig. S5 shows  that,  when  Gsl  is  set  equal  to  3  ×  1016 Jm−1s−1m−3K−1 as \nconsidered in Phys.  Rev.  Lett.  76, 4250 (1996), the simulated trace  and calculated electron \ntemperature profile are nearly indistinguishable from the case in  which  Gsl is neglected.  \n \nFigure  S4: Values  of the electron -lattice  coupling  parameter  Gel extracted  from  fitting  to the three  \ntemperature  model  and microscopic  three  temperature  model.  \n \n \n \n \nFigure  S5:  (a)  Results  of three  temperature  modelling  and (b) calculated  reservoir  temperature  \nprofiles  for two values  of Gsl. \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 0.82±0.081  0.85±0.097  0.69±0.037  0.19±0.023  \n2.1 1.08±0.063  1.18±0.105  0.8±0.029  0.16±0.014  \n3.5 1.26±0.057  1.48±0.106  0.91±0.028  0.16±0.009  \n4.8 1.2±0.037  1.57±0.096  0.97±0.015  0.17±0.008  \n6.0 1.21±0.042  1.49±0.072  1.02±0.022  0.17±0.008  \n7.3 1.56±0.076  1.98±0.087  1.15±0.041  0.16±0.004  \n8.7 1.65±0.045  2.18±0.066  1.28±0.036  0.17±0.003  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM s imulations for cobalt.  \n \n \n \nS4   \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8  1.29±0.154  1.41±0.172  2.1±0.182  0.53±0.03  \n2.1 1.52±0.061  1.74±0.077  2.84±0.087  0.55±0.012  \n3.5 1.53±0.0 51 1.77±0.069  2.93±0.076  0.56±0.011  \n4.8 1.65±0.047  1.9±0.06  2.86±0.064  0.51±0.008  \n6.0 1.49±0.036  1.73±0.039  2.74±0.048  0.57±0.007  \n7.3 1.58±0.028  1.88±0.033  2.87±0.038  0.59±0.006  \n8.7 1.58±0.048  1.77±0.039  2.77±0.062  0.62±0.008  \nTabl e S2:  Microscopic pa rameters extracted from the 3TM and M3TM simulations for nickel . \n \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 1.62±0.144  1.86±0.184  2.16±0.145  0.31±0.016  \n2.1 1.79±0.074  2.14± 0.084  2.34±0.073  0.34±0.007  \n3.5 2.05±0.081  2.46±0.065  2.65±0.084  0.39±0.005  \n4.8 2.08±0.068  2.55±0.038  2.81±0.071  0.45±0.006  \n6.0 2.02±0.053  2.5±0.065  2.72±0.058  0.49±0.007  \n7.3 2.07±0.083  2.67±0.052  2.87±0.089  0.56±0.009  \n8.7 2.11±0.084  2.71±0.069  2.87±0 .086 0.64±0.01  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM simulations for permalloy . \n \nIV. Temperature dependence of spin specific heat  \nConsidering the theory of Manchon et al. ( Phys.  Rev.  B 85, 064408 (2012)) as adopted  in Sci. \nRep. 10:6355 (2020), the specific heat Cs of the spin subsystem in the 3TM  framework  can be \ncalculated  as a function  of the spin temperature  Ts as: \n \n      (S1) \n \nFig. S6 below  shows  the trends  in the inter-reservoir  coupling  parameters  for two scenar ios: \nconsidering the  room temperature  value  of Cs(T0) or an explicit temperature  (and hence  time)  \ndependence  of Cs. \n \n \nFigure  S6: Fluence -dependent  variation  of (a) Gel and (b) Ges considering  two different  forms  \nfor the spin specific  heat Cs. \n \n \n \nS5  ∼ V. Bias fiel d dependence of precessional frequency and Gilbert damping  \nThe bias  magnetic  field-dependent variation of precessional dynamics in cobalt,  nickel and permalloy  \nis shown in Fig.  S7 - Fig. S9 for a pump fluence of 4.8 mJ/cm2. The oscillatory Kerr  signal is \nfitted with Eq. 5 of the main text to extract the precessional frequency f and relaxation time τ.   \nThe bias field dependence of the extracted frequency,  shown in Fig.  S7(b) for cobalt, is fitted to the \nKittel formula (Eq. 6) to extract the effective saturation  magnetization  Meff . Once  Meff have  been  \nderived,  the Gilbert  damping  factor  α is calcula ted using Eq. 7 of the main text. The extracted α \ndoes not show any systematic  variation  with  bias field.  \n \n \n \nFigure  S7: (a) Bias field -dependent  variation  of precessional  dynamics  for the cobalt  thin film.  \nSymbols  are experimental  data points  and the solid lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \nVI. Pump fluence dependence of precessional frequency and Gilbert \ndamping  \nThe variation of precessional dynamics in cobalt, nickel and permalloy with the laser  pump \nfluence is shown in Fig. S10 for the cobalt and in Fig. S11 for the permalloy thin  film at a \nsaturating  field  of   2 kOe.  The oscilla tory Kerr  signal  is fitted  with Eq.  5 of the main  text to \nextract  the precessional  frequency  f and relaxation  time τ. With  increasing  pump  fluence,  the \nprecessional  frequency  decreases  and the magnetic  damping  is enhanced.  A direct  correlation  \nbetween  the demagnetization  time τM and α is observed.  S6  \n \n \nFigure  S8: (a) Bias field -dependent  variation  of precessional  dynami cs for the nickel  thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \n \nFigure  S9: (a) Bias field -dependent  variation  of precessional  dynamics  for the  permalloy  thin  \nfilm.  Symbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. \n(b) Variation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ  \nwith H.  (d) Variation  of α with H. The solid  line is a guide  to the eye. \nS7  \n \n \nFigure S10: (a) Fluence -dependent variation of precessional dynamics for the cobalt thin film. \nSymbo ls are experimental data points and the solid lines are fits to Eq.  5 of the main text.  (b) \nVariation of f with pump  fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n \n \n \nFigure S11:  (a) Fluence -dependent variation of precessional dynamics for the permalloy thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with pump  fluence.  (c) Variation  of α with  pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n"    },    {        "title": "2303.11128v1.Nonlinear_Damping_and_Field_aligned_Flows_of_Propagating_Shear_Alfvén_Waves_with_Braginskii_Viscosity.pdf",        "content": "Draft version March 21, 2023\nTypeset using L ATEX default style in AASTeX631\nNonlinear Damping and Field-Aligned Flows of Propagating Shear Alfv\u0013 en Waves with Braginskii\nViscosity\nAlexander J. B. Russell1\n1School of Science and Engineering,\nUniversity of Dundee,\nDundee, DD1 4HN, Scotland, UK\nABSTRACT\nBraginskii MHD provides a more accurate description of many plasma environments than classical\nMHD since it actively treats the stress tensor using a closure derived from physical principles. Stress\ntensor e\u000bects nonetheless remain relatively unexplored for solar MHD phenomena, especially in nonlin-\near regimes. This paper analytically examines nonlinear damping and longitudinal \rows of propagating\nshear Alfv\u0013 en waves. Most previous studies of MHD waves in Braginskii MHD considered the strict\nlinear limit of vanishing wave perturbations. We show that those former linear results only apply to\nAlfv\u0013 en wave amplitudes in the corona that are so small as to be of little interest, typically a wave\nenergy less than 10\u000011times the energy of the background magnetic \feld. For observed wave ampli-\ntudes, the Braginskii viscous dissipation of coronal Alfv\u0013 en waves is nonlinear and a factor around 109\nstronger than predicted by the linear theory. Furthermore, the dominant damping occurs through the\nparallel viscosity coe\u000ecient \u00110, rather than the perpendicular viscosity coe\u000ecient \u00112in the linearized\nsolution. This paper develops the nonlinear theory, showing that the wave energy density decays with\nan envelope (1+ z=Ld)\u00001. The damping length Ldexhibits an optimal damping solution, beyond which\ngreater viscosity leads to lower dissipation as the viscous forces self-organise the longitudinal \row to\nsuppress damping. Although the nonlinear damping greatly exceeds the linear damping, it remains\nnegligible for many coronal applications.\nKeywords: Alfv\u0013 en waves (23), Solar corona (1483), Solar coronal heating (1989), Solar coronal holes\n(1484), Solar wind (1534), Magnetohydrodynamics (1964), Space plasmas (1544), Plasma\nastrophysics (1261), Plasma physics (2089)\n1.INTRODUCTION\nAlfv\u0013 enic waves are a ubiquitous feature of natural plasmas, including the solar corona (Tomczyk et al. 2007; De\nPontieu et al. 2007; Lin et al. 2007; Okamoto et al. 2007) and solar wind (Coleman 1967; Belcher & Davis 1971).\nIn solar physics, these waves contain su\u000ecient energy to heat the open corona and accelerate the fast solar wind\n(McIntosh et al. 2011), and they damp signi\fcantly within a solar radius above the surface (Bemporad & Abbo 2012;\nHahn et al. 2012; Hahn & Savin 2013; Hahn et al. 2022). How these Alfv\u0013 enic waves damp in astrophysical and space\nplasmas is an important question that has remained open for almost a century (see early papers by Alfv\u0013 en 1947 and\nOsterbrock 1961; modern reviews by De Moortel & Browning 2015, Arregui 2015 and Van Doorsselaere et al. 2020;\nand historical perspectives by Russell 2018 and De Moortel et al. 2020).\nMost theoretical knowledge about solar Alfv\u0013 enic waves is based on \\classical\" magnetohydrodynamics (MHD), a\nmathematical framework that originated from intuitive coupling of Maxwell's equations and Euler equations of inviscid\nhydrodynamics (Hartmann 1937; Alfv\u0013 en 1942, 1943, 1950; Batchelor 1950) and became widely adopted in large part\ndue to its success providing insight into diverse natural phenomena (see e.g. Priest 2014). However, classical MHD is\nCorresponding author: Alexander J. B. Russell\na.u.russell@dundee.ac.ukarXiv:2303.11128v1  [astro-ph.SR]  20 Mar 20232 Russell\nStart: General MHD with Braginskii  πSet geometryLinearise in  , includes setting    in  .b/B0h=̂zπ  removed. Dissipation by   only.η0η2Full calculation of QRatio of heating terms:  Q0/Q2Linearise in  , setting  . (ωiτi)−2η1,η2≪η0Dissipation by  . Nonlinear in  .η0b/B0\nVanishing waves (bB0)2≪(ωiτi)−2Coronal regime (bB0)2≫(ωiτi)−2\nFigure 1. Schematic paths of reasoning. The vertical branch gives priority to smallness of the wave amplitude and concludes\nthat damping is a linear process governed by the perpendicular viscosity coe\u000ecient \u00112, e.g. §8 of Braginskii (1965). The\nhorizontal branch gives priority to the smallness of \u00112=\u00110, leading to nonlinear damping via \u00110. This paper follows the diagonal\nbranch, which includes deriving the validity condition for the two outcomes. Nonlinear damping via \u00110is appropriate for most\ncoronal applications.\none member of a larger family of plasma descriptions, some of which o\u000ber a more complete description of the plasma.\nThis paper analytically examines Alfv\u0013 en wave damping in the more general framework of Braginskii MHD, which\nunlike classical MHD, retains the anisotropic viscous stress tensor.\nA number of authors, including §8 of Braginskii (1965), have previously investigated viscous damping of Alfv\u0013 en waves\nin the linear limit of vanishingly small wave amplitude. When priority is given to smallness of the wave amplitude,\nthe problem becomes framed as a matter of how anisotropic viscosity a\u000bects velocities that are perpendicular to the\nmagnetic \feld (the direction of which is treated as unchanging). With this approximation, damping is determined by\nthe \\perpendicular\" viscosity coe\u000ecient \u00112, which is extremely small in the corona. It was thus originally concluded\nthat viscous damping is very weak for coronal Alfv\u0013 en waves unless they have very short wavelengths. This path of\nreasoning is shown as the vertical branch in Fig. 1.\nThere is, however, another way to view the problem. Viscous damping of Alfv\u0013 en waves can alternatively be considered\nwith priority given to the largeness of parallel viscosity coe\u000ecient \u00110. Given that \u00112=\u00110&10\u000011is typical in the corona\n(Hollweg 1985), even a very small component of vparallel to the total magnetic \feld Bwould be expected to produce\nmajor departures from linear theory. This path of reasoning is shown as the horizontal branch in Fig. 1.\nThe second viewpoint of the problem takes impetus from the observation that (unless wave amplitudes vanish\nentirely) Alfv\u0013 en waves do have a non-zero velocity component parallel to the total magnetic \feld. Two e\u000bects contribute\nto this, which are separated if one expands V\u0001B=V\u0001(b+B0), where B0is the equilibrium magnetic \feld and bis\nthe magnetic perturbation. First, V\u0001bis non-zero for an Alfv\u0013 en wave, since the velocity perturbation perpendicular\ntoB0is aligned with the magnetic \feld perturbation b. In other words, de\rection of the magnetic \feld from its\nequilibrium direction implies there is a non-zero velocity component parallel to the total magnetic \feld. Second, in\ncompressible plasma, the magnetic pressure of the magnetic \feld perturbation drives a nonlinear ponderomotive \row\nparallel to the equilibrium magnetic \feld (e.g. Hollweg 1971). The ponderomotive \row makes V\u0001B0nonzero as well.Nonlinear Alfv \u0013en Waves with Viscosity Tensor 3\nBoth of these e\u000bects allow for the possibility of nonlinear viscous damping via the large parallel viscosity coe\u000ecient\n\u00110.\nWith the bene\ft of modern observations (e.g. McIntosh et al. 2011; Morton et al. 2015), it is known that normalised\nwave amplitudes b=B 0\u0018V=vA\u00180:1 are typical for the base of an open coronal \feld region, for example. The\n\\smallness\" of the square of this ratio is very modest in comparison to the extreme largeness of \u00110=\u00112. Thus, it is\nlikely from the outset that viscous damping of Alfv\u0013 en waves will be a nonlinear process governed by \u00110and the wave\namplitude. This paper provides mathematical evidence that this heuristic analysis holds true, along with detailed\nexamination of the consequences.\nVarious previous studies have explored e\u000bects of Braginskii viscosity on MHD waves since Braginskii (1965). In\nsolar physics, the e\u000bect of linearized Braginskii viscosity was revisited from the 1980s to the mid-1990s through the\nlens of phase mixing and resonant absorption, with the aim of determining how including the viscosity tensor modi\fes\nthese scale-shortening processes and their heating properties. At the time, it was common practice in solar MHD\nwave theory to work with linearized equations. Thus, due to linearization, Steinolfson et al. (1986), Hollweg (1987),\nRuderman (1991), Ofman et al. (1994) and Erdelyi & Goossens (1995) obtained analytical and numerical results that\nstrictly apply to Alfv\u0013 en waves of vanishing amplitude.\nIn adjacent \felds, the e\u000bect of anisotropic viscosity on MHD waves has also been investigated with an eye on\nMHD turbulence and the solar wind. Of particular note, Montgomery (1992) advocated that Braginskii viscosity is\nimportant in hot tenuous plasmas, that in many circumstances it should be treated using parallel ion viscosity, and\nthat plasma motions may self-organise to suppress damping. He further applied these ideas to anisotropy in MHD\nturbulence, on the basis that a quasi-steady turbulence is composed of the undamped modes. Quantitative elaboration\nin Montgomery (1992) was based on a linear normal mode analysis, which captures linear damping of magnetoacoustic\nwaves by parallel viscosity, but excludes nonlinear viscous damping of Alfv\u0013 en waves. The conclusion that a linearized\nstress tensor damps Alfv\u0013 en waves only negligibly, while damping magnetoacoustic waves signi\fcantly, was further\nreinforced by related work by Oughton (1996, 1997).\nSimilar ideas to ours regarding the importance of nonlinearity were advocated by Nocera et al. (1986), who modelled\nAlfv\u0013 en waves subject to the \u00110part of the Braginskii viscous stress tensor, retaining the leading-order nonlinear terms\nin the wave perturbations. Consistent with the argument above, their calculations found that coronal Alfv\u0013 en waves\ndamp nonlinearly by parallel viscosity. The current paper complements and extends the previous analysis by Nocera\net al. (1986), with the goal of producing a comprehensive understanding of the nonlinear damping and \feld-aligned\n\rows of propagating shear Alfv\u0013 en waves with Braginskii viscosity.\nA limitation of the mathematical techniques used in this paper is that they exclude certain other nonlinear e\u000bects\nthat may be important in plasmas, such as nonlinear interactions between waves. Numerical investigations will be\nrequired in future to verify the analytical theory presented here, compare the relative importance of viscous damping\nand other nonlinear e\u000bects such as parametric decay instability, and consider interactions between nonlinear processes\nin Braginskii MHD.\nThis paper is organised as follows. §2 provides scienti\fc background on single-\ruid Braginskii MHD and its\nrelationship to other single-\ruid plasma models. §3 quantitatively examines Alfv\u0013 en wave heating by the full Braginskii\nviscous stress tensor, demonstrating the importance of nonlinear \u00110terms and compressibility, and obtaining the wave\ndecay properties for the weakly viscous limit using energy principles. In §4, we argue that in highly viscous limit,\nviscous heating is suppressed by self-organisation of the ponderomotive \row, which implies that viscosity strongly\nalters the \feld-aligned \row associated with Alfv\u0013 en waves in this regime. §5 further strengthens the analysis, using\nmultiple scale analysis to obtain the decay properties without restrictions on the Alfv\u0013 enic Reynolds number, assuming\nthe framework of Braginskii MHD. The paper \fnishes with discussion in §6 and summary of main conclusions in §7.\n2.BRAGINSKII MHD\nBraginskii MHD is an important plasma description that treats anisotropic viscosity and thermal conduction using\nrigorous closure from physical principles. This section provides a short primer on single-\ruid Braginksii MHD, its\nconnection with pressure (or temperature) anisotropy, and its relation to classical MHD and the CGL double-adiabatic\nequations.\nAs is described in various plasma textbooks, \ruid variables can be rigorously and robustly de\fned as velocity\nmoments of the underlying particle distribution functions. Transport equations for each particle species are then4 Russell\nderived by taking moments of the kinetic Boltzmann equation, and combined to obtain the single \ruid equations.\nRecommended presentations can be found in Schunk & Nagy (2009) Chapter 7 and the Appendix of Spitzer (1962).\nAssuming quasi-neutrality, and conservation of mass, momentum and energy, this process yields the mass continuity\nequation,\n@\u001a\n@t+r\u0001(\u001aV) = 0; (1)\nmomentum equation\n\u001aDV\nDt=\u0000r\u0001P+\u001aG+j\u0002B; (2)\nenergy equation,\nD\nDt\u00123\n2p\u0013\n+5\n2pr\u0001V=\u0000\u0019:rV\u0000r\u0001 q+j\u0001(E+V\u0002B); (3)\nhigher-order transport equations if required, and the generalized Ohm's law.\nThe pressure tensor Pthat appears in Eq. (2) is the most fundamental representation of the internal forces associated\nwith thermal motions of particles. It is symmetric, so it represents six degree of freedom. The momentum equation\ncan also be reformulated by introducing the scalar pressure and stress tensor as\np=1\n3Trace ( P) =1\n3P\u000b\u000b; \u0019 \u000b\f=P\u000b\f\u0000p\u000e\u000b\f; (4)\nwhere\u000e\u000b\fis the Kronecker delta. So de\fned, the stress tensor \u0019is symmetric and traceless. These de\fnitions gives\nthe replacement\u0000r\u0001P=\u0000rp\u0000r\u0001\u0019.\nDeriving transport equations by moment taking meets with a fundamental closure problem: the transport equation\nfor each \ruid variable depends on a higher-order variable, producing an in\fnite regress unless the system can be\nclosed by other considerations. The method of closure is therefore a major distinguishing feature between di\u000berent\n\ruid models for plasmas. It is also a major source of validity caveats. Various di\u000berent methods of closure produce\ngoverning equations that conserve mass, momentum and energy, since these properties are already built into Eqs. (1){\n(3). However, the di\u000berent models discussed below disagree on the internal forces and heating, and can therefore\nproduce di\u000berent behaviors.\nClassical MHD (Hartmann 1937; Alfv\u0013 en 1942, 1943; Batchelor 1950) corresponds to a closure treatment in which the\nstress tensor and the heat \row vector are dropped from Eqs. (2) and (3). Dropping the stress tensor can be justi\fed\nwhen particle collisions or other forms of particle scattering such as wave-particle interactions are frequent enough\nthat the pressure tensor remains very close to isotropic. The resulting MHD equations are valid for many situations,\nfor instance modelling static equilibria, or dynamic situations in which the divergence of the stress tensor remains\nsmall compared to the Lorentz force. It is nonetheless a truncation since higher order variables are set to zero rather\nthan approximated. Furthermore, collisionality in environments such as the solar corona is low enough that the stress\ntensor can become signi\fcant for various dynamic phenomena, including MHD waves.\nBraginskii MHD uses a less restrictive method of closure. As is detailed by Braginskii (1965), when the collisional\nmean free path is signi\fcantly shorter than length scales over which \ruid quantities vary, the heat \row vector takes\nthe form of an anisotropic thermal conduction, and the stress tensor takes the form of an anisotropic viscosity. Closure\ncan therefore be achieved by expressing qand\u0019in terms of lower-order \ruid variables, which are traditionally derived\nusing methods similar to Chapman & Cowling (1939) or Grad (1949).\nThe anisotropy inherent in qand\u0019can be appreciated heuristically, by considering the helical motion of charged\nparticles in magnetized plasmas. The mean free path parallel to the magnetic \feld is the same as for unmagnetized\nplasmas, implying that transport parallel to the magnetic \feld is the same as for unmagnetized plasmas. Meanwhile,\nthe mean free path perpendicular to the magnetic \feld is the gyroradius, which is typically much less than the mean\nfree path parallel to the magnetic \feld, which supresses perpendicular transport. Hence both thermal conduction and\nviscous stresses are anisotropic with respect to the magnetic \feld direction, often extremely so.\nThe full Braginskii stress tensor, used in §3, involves \fve viscosity coe\u000ecients. A useful simpli\fcation, used in §5,\nis that for strong magnetizations, \n i\u001ci\u001d1, the parallel \u00110coe\u000ecient greatly exceeds the other viscosity coe\u000ecients.\nHence, one can often simplify by neglecting the smaller coe\u000ecients (although, as shown in §3 it can be necessary to\nretain other viscosity coe\u000ecients if length scales are highly anisotropic). In this simpli\fcation, one has the followingNonlinear Alfv \u0013en Waves with Viscosity Tensor 5\ncovariant expressions for parallel viscosity (Lifshitz & Pitaevskii 1981; Hollweg 1986):\n\u0019\u000b\f=\u00003\u00110\u0012\nh\u000bh\f\u0000\u000e\u000b\f\n3\u0013\u0012\nh\u0016h\u0017\u0000\u000e\u0016\u0017\n3\u0013\n@\u0016V\u0017;\nQvisc= 3\u00110\u0012\u0012\nh\u000bh\f\u0000\u000e\u000b\f\n3\u0013\n@\u000bV\f\u00132\n;(5)\nwhere h=B=jBjis the unit vector in the direction of the magnetic \feld. These expressions are di\u000berent to the\nisotropic viscosity that appears in the Navier-Stokes equations, owing to the anisotropy introduced by the magnetic\n\feld.\nParallel viscosity is closely related to pressure anisotropy. As pointed out by Chew et al. (1956), when \n i\u001ci\u001d1 the\nparticle Lorentz force makes the pressure tensor gyrotropic, giving it the form\nP\u000b\f=p?\u000e\u000b\f+ (pjj\u0000p?)h\u000bh\f: (6)\nThis is a signi\fcant simpli\fcation, since the six degrees of freedom of a general pressure tensor have been replaced\nwith two variables, pjjandp?. The de\fnitions in Eqs. (4) then yield p= (pjj+ 2p?)=3 and\n\u0019\u000b\f= (pjj\u0000p?)\u0012\nh\u000bh\f\u0000\u000e\u000b\f\n3\u0013\n: (7)\nEquation (7) shows that pressure anisotropy has an equivalent stress tensor, which is proportional to pjj\u0000p?.\nFurthermore, Eqs. (5) and (7) both have the form \u0019\u000b\f\u0018(h\u000bh\f\u0000\u000e\u000b\f=3), so equivalence of the stress tensors reduces\nto equivalence of the scalar factors in the two equations. An illuminating analysis of the conditions under which\nthey converge has been written by Hollweg (1985, 1986), the most important condition being that collisions (or other\nprocesses such wave-particle interactions) relax the pressure anisotropy driven by velocity gradients to an extent that\nthe pressure is only weakly anisotropic. Classical MHD, for comparison, assumes that pressure anisotropy can be\nneglected altogether.\nFor low collisionality, the quasistatic approximation in Braginskii MHD ceases to be valid and strong pressure\nanisotropy may develop. Under these conditions, separate evolution equations can be derived for pjjandp?(Chew\net al. 1956; Hollweg 1986). However, the closure problem rears its head again, because those equations depend on the\nheat \row vector. A simple approach to obtaining a closed system is to ignore the heat \row vector, thus obtaining\nthe CGL double adiabatic equations (Chew et al. 1956), which are commonly used for collisionless plasma. More\nsophisticated approaches also exist that solve for the evolution of the pressure anisotropy or the evolution of the stress\ntensor, retaining the heat \row vector and closing by other means. The works by Balescu (1988); Schunk & Nagy\n(2009); Zank (2014); Hunana et al. (2019a,b, 2022) provide further reading on this topic.\nSummarising, there exists a family of adjacent (sometimes overlapping) single-\ruid models for plasmas. The most\nappropriate choice for a particular problem and/or context depends on the collisionality. When MHD timescales are\ngreater than the ion collision time, Braginskii MHD provides rigorous closure and treats the internal forces and heat\n\row more accurately than classical MHD.\n3.ALFV \u0013EN WAVE HEATING BY BRAGINSKII VISCOSITY\n3.1. Model\nWe quantitatively examine the viscous dissipation for an Alfv\u0013 en wave, which is a transverse wave polarized so that the\nmagnetic perturbation is perpendicular to the equilibrium magnetic \feld and the wavevector. Setting the equilibrium\nmagnetic \feld in the z-direction, the magnetic perturbation in the x-direction and the wavevector in the yz-plane, we\nconsider a total magnetic \feld of the form\nB=b(y;z;t )ex+B0ez: (8)\nThis ansatz automatically satis\fes r\u0001B= 0. For the velocity \feld we assume the form\nV=Vx(y;z;t )ex+Vz(y;z;t )ez: (9)6 Russell\nTheVxis the dominant velocity component. In linearized theory it would be the only component of V. Additionally,\nwe have explicitly included a higher-order Vzterm that represents the nonlinear ponderomotive \row parallel to the\nequilibrium magnetic \feld, which is driven by gradients of the magnetic pressure perturbation b2=2\u00160associated with\na \fnite-amplitude Alfv\u0013 en wave (e.g. Hollweg 1971). The Vzterm can be dropped when the plasma is incompressible\n(see §3.3), however it is required for a nonlinear treatment of compressible plasma and a\u000bects the wave heating via the\nparallel viscosity coe\u000ecient \u00110(as remarked in §1). The expression for Vzin classical MHD is given later in Eq. (29).\nIn a full solution, derivatives of b2=2\u00160with respect to ygive rise to an additional nonlinear y-component of V,\nwhich in turn produces a nonlinear y-component of B. These terms are not shown explicitly in Eqs. (8) and (9). Such\nterms were included by Nocera et al. (1986) and appear not to a\u000bect our main conclusions, provided the perpendicular\nwavelength of the Alfv\u0013 en wave is su\u000eciently large.\nThe viscous force is determined from the viscous stress tensor \u0019\u000b\fby\nFvisc;\u000b =\u0000@\u0019\u000b\f\n@x\f; (10)\nand the viscous heating rate is determined using\nQvisc=\u0000\u0019\u000b\f@V\u000b\n@x\f; (11)\nwhere\u000b2fx;y;zg,\f2fx;y;zg, thex\fare components of the position vector, V\u000bare components of Vand repeated\nindices imply summation in the Einstein convention.\nA vital point is that the viscous stress tensor depends on the direction of the magnetic \feld given by the unit vector\nh=B=jBj, which for our Alfv\u0013 en wave model in Eq. (8) has\nhx=bp\nB2\n0+b2; hy= 0; hz=B0p\nB2\n0+b2; (12)\nwithh2\nx+h2\nz= 1. Our analysis di\u000bers from many past works by considering hx6= 0 and identifying the dominant\nheating contribution at the end, as opposed to setting hx= 0 before evaluating the damping e\u000bect on Alfv\u0013 en waves.\nApplying formulas from §4 of Braginskii (1965) (equivalent matrix expressions are given by Hogan 1984), the stress\ntensor is related to \fve viscosity coe\u000ecients by\n\u0019\u000b\f=\u00002X\ni=0\u0011iWi\u000b\f+4X\ni=3\u0011iWi\u000b\f: (13)\nThe gyroviscous \u00113and\u00114terms do not contribute to heating, so evaluating the heating rate Qviscrequires\nW0\u000b\f=3\n2\u0012\nh\u000bh\f\u00001\n3\u000e\u000b\f\u0013\u0012\nh\u0016h\u0017\u00001\n3\u000e\u0016\u0017\u0013\nW\u0016\u0017;\nW1\u000b\f=\u0012\n\u000e?\n\u000b\u0016\u000e?\n\f\u0017+1\n2\u000e?\n\u000b\fh\u0016h\u0017\u0013\nW\u0016\u0017;\nW2\u000b\f=\u0000\n\u000e?\n\u000b\u0016h\fh\u0017+\u000e?\n\f\u0017h\u000bh\u0016\u0001\nW\u0016\u0017;(14)\nwhere\u000e\u000b\fis the Kronecker delta,\n\u000e?\n\u000b\f=\u000e\u000b\f\u0000h\u000bh\f; (15)\nand the rate of strain tensor is\nW\u000b\f=@V\u000b\n@x\f+@V\f\n@x\u000b\u00002\n3\u000e\u000b\fr\u0001V: (16)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 7\nFor the shear Alfv\u0013 en wave geometry described by Eq. (9), the Witensors become\nW0=\u0000\nhxhz@zVx+\u00002\n3\u0000h2\nx\u0001\n@zVz\u00010\nB@3h2\nx\u00001 0 3hxhz\n0\u00001 0\n3hxhz0 3h2\nz\u000011\nCA; (17)\nW1=hx(hz@zVx\u0000hx@zVz)0\nB@\u0000h2\nz0hxhz\n0 1 0\nhxhz0\u0000h2\nx1\nCA+ (hx@yVx\u0000hz@yVz)0\nB@0hz0\nhz0\u0000hx\n0\u0000hx01\nCA; (18)\nW2=\u0000\u0000\n1\u00002h2\nx\u0001\n@zVx\u00002hxhz@zVz\u00010\nB@2hxhz0 1\u00002h2\nx\n0 0 0\n1\u00002h2\nx0\u00002hxhz1\nCA+ (hx@yVx+hz@yVz)0\nB@0hx0\nhx0hz\n0hz01\nCA; (19)\nThe viscous heating rate with hx6= 0 retained is thus\nQvisc=\u00110\n3\u0000\n3hxhz@zVx+ (2\u00003h2\nx)@zVz\u00012+\u00111h2\nx(hz@zVx\u0000hx@zVz)2+\u00111(hz@yVx\u0000hx@yVz)2\n+\u00112\u0000\u0000\n1\u00002h2\nx\u0001\n@zVx\u00002hxhz@zVz\u00012+\u00112(hx@yVx+hz@yVz)2:(20)\n3.2. Two small parameters\nAs anticipated in §1 (e.g. Fig. 1), two parameters determine the relative importance of individual terms in Eq. (20).\nThe \frst small parameter is ( b=B 0)2, the ratio of the wave's magnetic energy density to the energy density of the\nbackground magnetic \feld, which enters through hxandhz. In the modern era, extensive observations of coronal\nMHD waves (Nakariakov & Verwichte 2005; De Moortel & Nakariakov 2012) allow ( b=B 0)2to be quanti\fed with good\ncertainty, directly from resolved wave observations or indirectly from spectral line widths. For example, Morton et al.\n(2015) studied waves at the base of a coronal open \feld region using both approaches and reported a wave speed vA=\n400 km s\u00001and wave motions at v= 35 km s\u00001. Both measurements are consistent with earlier \fndings for coronal\nholes and the quiet Sun (e.g. McIntosh et al. 2011). From observations like these, h2\nx\u0018(b=B 0)2\u0018(v=vA)2\u001810\u00002.\nThe second small parameter is (\n i\u001ci)\u00002, which sets the viscosity coe\u000ecients \u00111and\u00112relative to\u00110. The value of\n\ni\u001cican vary signi\fcantly in the corona, but if magnetic null points are excluded one obtains values similar to the\nestimates made by Hollweg (1985), who found 3 :4\u0002105for a solar active region and 7 :2\u0002105near the base of a\ncoronal hole. We therefore expect (\n i\u001ci)\u00002.10\u000011under common conditions, and \u00111and\u00112simplify to\n\u00112=6408\n5125(\ni\u001ci)\u00002\u00110; \u0011 1=1\n4\u00112: (21)\nThe numerical coe\u000ecients in Eq. (21) are obtained in the limit (\n i\u001ci)\u00002!0, e.g. from Eq. (73) of Hunana et al.\n(2022). They are approximate for \fnite (\n i\u001ci)\u00002but have a high degree of accuracy because the corrections to the\ncoe\u000ecients are of the order of (\n i\u001ci)\u00002.10\u000011. Inspecting Eq. (21) and considering (\n i\u001ci)\u00002.10\u000011, the\u00112and\u00111\ncoe\u000ecients are both vastly smaller than \u00110.\nThe smallness of ( b=B 0)2and the smallness of (\n i\u001ci)\u00002compete to make di\u000berent terms dominate the viscous heating.\nIf one tries to simplify Eq. (20) by setting ( b=B 0)2to zero, then hx= 0 andVz= 0 givesQvisc=\u00112(@zVx)2+\u00111(@yVx)2,\nas obtained by Braginskii (1965). On the other hand, if one tries to simplify by \frst taking (\n i\u001ci)\u00002to zero then only\n\u00110terms remain, suggesting a di\u000berent conclusion. Thus, the quantitative results recover the two branches shown in\nFig. 1. To correctly determine the damping under coronal conditions, one must carefully compare terms in the full\nEq. (20), bearing in mind that there are two small parameters, which we do now (diagonal branch in Fig. 1).\n3.3. Heating rate for incompressible plasma\nThe analysis for incompressible plasma is relatively straightforward, which makes it a natural starting point for\ndiscussion. The assumption of incompressibility is appropriate for liquid metals or high-beta plasmas, but not, we\nnote, for the corona. The use of coronal wave amplitudes and magnetizations in this section is therefore intended to\nbe instructive only, with the compressible \fnite-beta treatment that follows later in this paper being required to treat\nthe corona.8 Russell\nIn the incompressible case, r\u0001V= 0 applied to our Alfv\u0013 en wave geometry implies Vz= 0. Thus Eq. (20) with\n\u00111=1\n4\u00112simpli\fes to\nQvisc=\u001a\n3\u00110h2\nxh2\nz+\u00112\u0012\u0000\n1\u00002h2\nx\u00012+1\n4h2\nxh2\nz\u0013\u001b\n(@zVx)2+\u00112\u00121\n4h2\nz+h2\nx\u0013\n(@yVx)2: (22)\nThe terms involving @zVxset the viscous dissipation due to wavelengths parallel to the equilibrium magnetic \feld,\nand we \frst ask whether dissipation due to parallel wavelengths is dominated by the linear \u00112contribution that has\nbeen widely recognised since Braginskii (1965), or the nonlinear \u00110contribution. The ratio of the two terms inside the\ncurly brackets in Eq. (22) is\n3h2\nx\u00110\n\u00112\u00141\u0000h2\nx\n(1\u00002h2x)2+1\n4h2x(1\u0000h2x)\u0015\n\u00193h2\nx\u00110\n\u00112\u00192:4h2\nx(\ni\u001ci)2; (23)\nwhere the \frst step simpli\fes using h2\nx\u001c1 (for the observed value of h2\nx\u001910\u00002, retaining the terms in the square\nbracket increases the ratio by 2.8%, so this approximation is both accurate and conservative) and the substitution for\n\u00110=\u00112is by Eq. (21). For the coronal wave amplitudes and \n i\u001civalues noted in §3.2, this ratio exceeds 109, with\nthe nonlinear damping via \u00110dominating the heating rate by that factor. In other words, the viscous dissipation of\nAlfv\u0013 en waves via derivatives aligned with the equilibrium magnetic \feld is a factor 109stronger than predicted by\nlinear theory.\nWe now evaluate the role of derivatives perpendicular to the equilibrium magnetic \feld by comparing the nonlinear\n\u00110term in Eq. (22) to the term involving @yVx. The ratio of these heating rate terms is\n12h2\nx\u00110\n\u00112\u0012\u0015?\n\u0015jj\u00132\u00141\u0000h2\nx\n1 + 3h2x\u0015\n\u001912h2\nx\u00110\n\u00112\u0012\u0015?\n\u0015jj\u00132\n\u00199:6h2\nx(\ni\u001ci)2\u0012\u0015?\n\u0015jj\u00132\n; (24)\nwhere\u0015?and\u0015jjare the wavelengths perpendicular and parallel to the equilibrium magnetic \feld (for the observed\nvalue ofh2\nx\u001910\u00002, the approximation of the terms in h2\nxis accurate to 3.9%). For the coronal parameters noted\nabove, if\u0015?\u0019\u0015jjthen the nonlinear \u00110term again dominates by a factor that exceeds 109. For smaller transverse\nwavelengths, the nonlinear \u00110term dominates whenever \u0015?&10\u00005\u0015jj. If one considers a wave speed of 400 km s\u00001\nand a frequency of 3 mHz, consistent with the observations by Morton et al. (2015), the condition that the nonlinear \u00110\nterm dominates becomes \u0015?&800 m. Given that CoMP has imaged Alfv\u0013 enic waves using 3 Mm pixels, this condition\nappears to be met by a very large margin, making the nonlinear \u00110dissipation dominant over the \u00112linear dissipation.\nThe purpose of deriving Eq. (22) and the ratios on the left hand sides of Eq. (23) and (24) such that they include\nall appearances of hxandhzis that they can be evaluated exactly for a given value of hx. This makes it explicit that\nour conclusions are insensitive to the precise value of hx, only that the value of hxis broadly consistent with coronal\nobservations. While that approach is most comprehensive, the same conclusions can also be reached by separately\nsimplifying each term in Eq. (22) using h2\nx\u001c1 andh2\nz= 1\u0000h2\nx\u00191 to obtain the less cumbersome formula\nQvisc=\b\n3\u00110h2\nx+\u00112\t\n(@zVx)2+\u00111(@yVx)2; (25)\nand comparing terms to reach the same conclusions.\n3.4. Compressible plasma with large Re\nUnder typical coronal conditions, the thermal pressure is too small to prevent compression of the plasma by nonlinear\nmagnetic pressure forces, thus a nonlinear Vzdevelops that is known as the ponderomotive \row (Hollweg 1971). This\n\row component a\u000bects the viscous heating rate via the parallel viscosity coe\u000ecient \u00110, hence compressible theory is\nrequired for nonlinear viscous damping of Alfv\u0013 en waves in plasma.\nWe de\fne the Alfv\u0013 enic Reynolds number as\nRe =\u001avA\nkjj\u00110: (26)\nThis dimensionless parameter di\u000bers from the traditional Reynolds number since it refers to the Alfv\u0013 en speed vA=\nB=p\u00160\u001ainstead of a typical \ruid velocity. This distinction mirrors that between the Lundquist number and magnetic\nReynolds number in resistive MHD. Justi\fcation for de\fning Re according to Eq. (26) will be found in the detailedNonlinear Alfv \u0013en Waves with Viscosity Tensor 9\nmathematical solutions in §5, in which it is found to be a natural parameter of the system (also see Nocera et al.\n1986).\nIn this section, the ponderomotive Vzwill be related to Vxusing expansions in the amplitude of the primary wave\n\felds. Several assumptions are used to accomplish this. First, we make use of the result that a travelling wave solution\npropagating in the positive z-direction has\n@\n@t\u0011\u0000vA@\n@z; (27)\nwherevAis the wave speed. For simplicity it is assumed that derivatives of background quantities are su\u000eciently weak\nto play a higher-order role on the dynamics. We also simplify here by replacing full treatment of thermal conduction\nwith two thermodynamic cases: adiabatic and isothermal. Finally, it is assumed that Re is large enough that viscous\nforces can be neglected at leading order when evaluating Vz, which makes it possible to obtain an algebraic relationship\nbetweenVzandb2. This assumption will be removed for §5, in which the e\u000bect of viscous forces on VxandVzis\nincluded.\nThex-components of the momentum and induction equations are una\u000bected by the ponderomotive \row at leading\norder in the wave amplitude. From them one recovers the Wal\u0013 en relation for propagating Alfv\u0013 en waves, b=B 0=\u0000Vx=vA.\nAt leading order, the z-component of the momentum equation is\n\u001a0@Vz\n@t+@\n@z\u0012\n\u000ep+b2\n2\u00160\u0013\n= 0; (28)\nwhere the viscous force has been neglected since we currently consider the limit of large Re. Using Eq (27) and\nintegrating yields an algebraic relationship between Vz,\u000epandb2. In an adiabatic treatment, the energy equation\nyields\u000ep=\rp0Vz=vA, hence we obtain\nVz\nvA=1\n2(1\u0000\f)\u0012b\nB0\u00132\n; (29)\nwhere\n\f=\u0012cs\nvA\u00132\n=\r\n2\u0012p\nB2\n0=2\u00160\u0013\n: (30)\nIn an isothermal treatment, the ideal gas law p=\u001aRT yields\u000ep=p 0=\u000e\u001a=\u001a 0=Vz=vA. This does not change the\nform of Eq. (29); instead, the isothermal case is recovered simply by setting \r= 1 in the de\fnition of \f.\nDe\fning\fas the square of the ratio of the sound speed ( cs=p\n\rp0=\u001a0) to the Alfv\u0013 en speed di\u000bers slightly from the\nconvention of de\fning \fas the ratio of thermal pressure to magnetic pressure, due to the factor \r=2, which is 5/6 for\nan adiabatic monoatomic gas, and 1 =2 for an isothermal model. De\fning \fas the speed ratio squared leads to cleaner\nmathematics for many MHD wave problems, including this one, and it has therefore become established practice in\nMHD wave theory.\nThe\f= 1 singularity in Eq. (29) arises because cs=vAimplies resonance between the Alfv\u0013 en wave and an acoustic\nwave, which resonantly transfers energy between the waves. In this speci\fc case, Eq. (27) does not apply because it\ndoes not account for evolution due to resonance. Similarly, Eq. (28) assumes that Vz\u001cvAto simplify the convective\nderivative, and the solution in Eq. (29) does not satisfy this condition in the immediate vicinity of \f= 1. The\f= 1\nresonance and the singularity in Eq. (29) are not of concern for most coronal applications, which typically have \f <0:2,\nbut there are special cases in which it is of interest, such as waves propagating towards coronal magnetic nulls or across\nthe\f= 1 layer in the lower solar atmosphere. Russell et al. (2016) have previously applied such nonlinear resonant\ncoupling to the problem of sunquake generation by magnetic \feld changes during solar \rares.\nDi\u000berentiating Eq. (29) and employing the relation b=B 0=\u0000Vx=vAyields\n@\u000bVz=\u00001\n(1\u0000\f)hx\nhz@\u000bVx; (31)\nwhich can be used to eliminate Vzfrom Eq. (20). To leading order in h2\nxin each viscosity coe\u000ecient, we \fnd\nQvisc=\b\nC\u00110h2\nx+\u00112\t\n(@zVx)2+\u00111(@yVx)2; (32)\nwhere\nC=1\n3\u00121\u00003\f\n1\u0000\f\u00132\n: (33)10 Russell\nSome special cases are noteworthy. The incompressible results of §3.3 are recovered for \f!1 , which gives Vz!0\nandC!3. Similarly, the cold plasma solution is recovered by setting \f= 0, which gives Vz= (b=B 0)2=2 and\nC= 1=3.\nThe nonlinear heating rate for cold plasma ( \f= 0) is a factor nine smaller than for incompressible plasma ( \f!1 ),\nwhich demonstrates the importance of compressibility for this problem. Furthermore, C(\f) is monotonically decreasing\nbetween\f= 0 and\f= 1=3. Since 0< \f < 1=3 for most coronal applications, the dissipation rate due to nonlinear\nBraginskii viscosity in these environments is reduced compared to the cold plasma solution. For example, given \f= 0:1,\nthe heating rate is approximately 60% of the value for cold plasma. It is therefore evident that compressibility and\n\fnite-beta e\u000bects must be treated when assessing viscous dissipation of Alfv\u0013 en waves.\nAnother important feature is that Chas a zero for \f= 1=3. This is one circumstance in which\nVz\nvA=3\n4\u0012Vx\nvA\u00132\n=3\n4\u0012b\nB0\u00132\n; (34)\nwhich causes cancellation within the \u00110contribution to Qvisc. That a particular organisation of Vz=vAcan suppress\nnonlinear viscous dissipation is an important novel \fnding that §4 explores further in the context of low Re.\nThe \fnal feature of Cis the singularity at \f= 1. As noted earlier in this section, cs=vAimplies that the Alfv\u0013 en\nwave is in resonance with a sound wave, which transfers energy between the Alfv\u0013 en wave and the sound wave. Caution\nis needed around the resonance, since resonant energy transfer cannot be described using Eq. (27), which was used to\nderive Eq. (33).\nEvaluation of ratios of heating terms from Eq. (32) proceeds as for the comparison in Sec. 3.3, but with \u00110multiplied\nbyC=3. The top-level conclusions remain intact: heating by the Braginskii viscous stress tensor is dominated by an\n\u00110term that is nonlinear in the wave amplitude, and for coronal values, heating due to the nonlinear \u00110term is many\norders of magnitude larger than the heating due to the linear \u00111and\u00112terms.\n3.5. What wave amplitude is linear?\nAn important implication of the preceding analysis is that nonlinear e\u000bects become signi\fcant for anisotropic viscosity\nat far lower wave amplitudes than they do for other terms in the MHD equations. Linearizing the Braginskii viscous\nstress tensor is only appropriate when h2\nx\u0018(b=B 0)2\u001c(\ni\u001ci)\u00002, which in the corona corresponds to a requirement\nthat the wave energy density is less than 10\u000011times the energy density of the background magnetic \feld, far too\nsmall to be relevant to coronal energetics. Waves that have small enough amplitudes to be governed by linear viscous\ndamping theory would be unobservable and have no e\u000bect on the coronal energy balance. Thus, for coronal Alfv\u0013 en\nwaves, viscosity must be treated nonlinearly in the wave amplitude, as well as anisotropically due to the magnetic \feld.\nInterestingly, this linearization condition is far more stringent than the linearization condition for other terms in the\nMHD equations, whereby h2\nxis normally compared to unity. The extreme di\u000berence in these linearization conditions\nis due to the large \u00110=\u00112ratio produced by the strong magnetization.\n3.6. Damping scales for large Re (energy derivation)\nIt is of major interest to know the time and length scales over which waves damp. This section provides a relatively\nsimple derivation of the decay scales for nonlinear viscous damping of propagating shear Alfv\u0013 en waves for large Re,\nusing energy principles.\nDropping the \u00111and\u00112terms from Eq. (32), the heating rate due to the \u00110parallel viscosity coe\u000ecient for large Re\nis\nQvisc=\u00110\n3\u00121\u00003\f\n1\u0000\f\u00132\u0012b\nB0\u00132\n(@zVx)2: (35)\nA wave energy decay time can be de\fned according to \u001cd=hEwi=hQvisci, whereh:idenotes the time average over\na wave period, and Ewis the wave energy density. The corresponding decay length is Ld=vA\u001cd.\nFor forward propagating Alfv\u0013 en waves, Ew\u0019\u001a0V2\nx, andVx=acos(\u001e) where\u001e=kjj(z\u0000vAt). Hence,\nEw=\u001a0a2(1 + cos(2\u001e))\n2: (36)\nSimilarly, using Eq. (35) with ( b=B 0)2\u0019(Vx=VA)2,\nQvisc=\u00110k2\njj\n3v2\nA\u00121\u00003\f\n1\u0000\f\u00132\na4(1\u0000cos(4\u001e))\n8; (37)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 11\nwhich give the fast time averages\nhEwi=\u001a0a2\n2; (38)\nhQi=\u00110k2\njja4\n24v2\nA\u00121\u00003\f\n1\u0000\f\u00132\n: (39)\nThe wave energy decay scales are therefore\n\u001cd=12\u001a0\n\u00110k2\njj(a=vA)2\u00121\u0000\f\n1\u00003\f\u00132\n; (40)\nLd=12\u001a0vA\n\u00110k2\njj(a=vA)2\u00121\u0000\f\n1\u00003\f\u00132\n: (41)\nEquations (40) and (41) show that waves with larger kjj(equivalently, higher frequencies) are damped on shorter\nscales. We also remark that since Lddepends on the amplitude of Vx(the constant a), the decay envelope is non-\nexponential. The damping properties are elaborated on more fully in §5, in which the assumption of large Re is\nremoved and the functional form of the wave envelope is determined.\n4.SELF-ORGANISED VISCOUS FLOW\nIn the limit Re!0, the viscous force in the z-component of the momentum equation risk becoming extremely large,\nunless the \row self-organises to prevent this. Correspondingly, in the limit Re !0, strong dissipation will prevent\nwaves from propagating, unless Vzis determined by viscosity. One can therefore expect self-organisation of the \row\npattern for Alfv\u0013 en waves in highly viscous plasma (small values of Re), which is a concept previously advanced by\nMontgomery (1992).\nTo investigate quantitatively, we analyze the highly magnetised regime \n i\u001ci\u001d1, simplifying the stress tensor\nand heating rate by retaining only the \u00110parallel viscosity coe\u000ecient. Inspecting Eq. (5), components of \u0019\u000b\fare\nproportional to ( h\u0016h\u0017\u0000\u000e\u0016\u0017=3)@\u0016V\u0017, andQviscis proportional to the square of this expression. Applying the shear\nAlfv\u0013 en wave geometry of Eqs. (8) and (9) and simplifying by h2\nx\u001c1,\n\u0012\nh\u0016h\u0017\u0000\u000e\u0016\u0017\n3\u0013@V\u0017\n@x\u0016\u0019hx@Vx\n@z+2\n3@Vz\n@z: (42)\nViscous forces and heating can be suppressed, allowing Alfv\u0013 en wave propagation, if the \row self-organises to keep this\nexpression close to zero. Using the Alfv\u0013 en wave relation to substitute hx\u0019b=B 0\u0019\u0000Vx=vAand integrating, we \fnd\nthat for small Re\nVz\nvA=3\n4\u0012Vx\nvA\u00132\n=3\n4\u0012b\nB0\u00132\n: (43)\nThe relation speci\fed by Eq. (43) appeared previously in the di\u000berent context of §3.4, where it was seen that viscous\ndissipation of Alfv\u0013 en waves in high Re plasma is suppressed for the special case of \f= 1=3. The \row pattern required\nto produce cancellation within the \u00110part ofQviscis independent of Re and \f, but it occurs for di\u000berent reasons\nin the two cases: in §3.4 it arose as a special case of ponderomotive \row with \fnite \f; when Re is small, it occurs\nbecause of self-organisation through viscous forces.\nThis novel result demonstrates that decay scales and other properties derived in §3 should not be extrapolated to\nsmall Re. Instead, we expect that as Re !0, the viscous force organises the \row such that Vzobeys Eq. (43), for\nwhich dissipation is suppressed by cancellation within the \u00110part ofQvisc.\n5.MULTIPLE SCALE ANALYSIS\nSection 3 used methods of analysis based on heating rates and energy principles. Section 5 now takes a complementary\napproach of solving the full set of governing equations using multiple scale analysis, to reinforce the results of §3,\nextend to general Re by including the e\u000bect of the viscous force on V, and obtain additional results including the\nfunctional form of the nonlinear decay.12 Russell\n5.1. Comparison to Nocera et al. (1986)\nWe preface the multiple scale analysis part of this paper with some remarks about related calculations by Nocera\net al. (1986). Their work and ours both concentrate on \u00110viscosity as the main source of wave damping, treating this\nnonlinearly in the wave amplitude (the horizontal branch of Figure 1). Also in common, both treat ponderomotive\nand \fnite\fe\u000bects.\nThe previous work of Nocera et al. (1986) derived a version of the viscous stress tensor that includes the leading\norder e\u000bect of hx6= 0 in the \u00110term. Terms in the viscosity tensor were then compared, concluding like our §3\n(but by di\u000berent arguments) that the nonlinear \u00110term exceeds contributions from other viscosity coe\u000ecients when\n(b=B 0)2\u001d(\ni\u001ci)\u00002(their Eq. (3.13)). The two studies thus agree on the dominance of nonlinear \u00110viscosity.\nNocera et al. (1986) then found a decay length using the following strategy. A self-consistent perturbation ordering\nwas introduced, then the x-components of the momentum and induction equations were combined to obtain a single\nequation for Vx, which at linear order is a wave equation. Next, all variables apart from Vxwere eliminated from the\nleading-order nonlinear term. Finally, they concluded from a stability analysis that waves with k?= 0 are damped\nnonlinearly, with a decay time that has the same form as our Eq. (40) (their Eq. (5.7), given in terms of normalised\nvariables).\nThe detailed derivation that follows in §5.2 draws inspiration from the framework developed by Nocera et al. (1986).\nWe have also taken the opportunity to make several changes that we regard as improvements, most importantly:\n1. Nocera et al. (1986) assumed that the fast time average of Vzis zero, which necessitated adding a non-zero\nconstant of integration to Vz. By contrast, we will set the constant of integration to zero, which is the only choice\nfor which an Alfv\u0013 en wave driver switching on at one boundary does not unphysically send an instantaneous signal\nto in\fnity. Additional support for our choice comes from simulations of nonlinear longitudinal \rows produced\nby Alfv\u0013 en waves (e.g. McLaughlin et al. 2011), which are consistent with the constraint used in our work.\n2. The stability analysis in §5 Nocera et al. (1986) is replaced with a multiple scale analysis of the type covered in\nChapter 11 of Bender & Orszag (1978).\n3. Nocera et al. (1986) made their wave envelope a function of z+vAt. We treat the envelope as time-independent\nand thus explicitly investigate damping of a propagating wave with respect to distance.\n4. Our derivation provides the envelope of Vxas well as the decay length.\n5. Our solution is valid for general Re, whereas Nocera et al. (1986) solved for the decay scales in the low-viscosity\nlimit of high Re only.\nEqually, Nocera et al. (1986) treated cases that we do not, including the possibility of k?large enough for coupling\nbetween the Alfv\u0013 en and fast modes to alter the wave properties (referred to in their paper as the case of phase mixed\nwaves).\n5.2. Detailed solution\n5.2.1. Geometry and perturbations\nWe assume the Alfv\u0013 en wave geometry of Eqs. (8) and (9), set @=@y\u00110 to concentrate on waves without short\nperpendicular scales, and introduce density and pressure perturbations \u000e\u001aand\u000eptogether with a self-consistent\nperturbation ordering that has Vx=vA\u0018b=B 0\u0018\u000f1=2andVz=vA\u0018\u000e\u001a=\u001a 0\u0018\u000ep=p 0\u0018\u000f. The viscosity \u00110and\nbackground quantities B0,\u001a0andp0are treated as locally homogeneous for simplicity.\n5.2.2. Nonlinear wave equation\nStarting from the ideal induction equation,\n@B\n@t=r\u0002(V\u0002B); (44)\nwe have@b\n@t\u0000B0@Vx\n@z=\u0000@\n@z(bVz) (exact); (45)\nwhere the linear terms have been grouped on the left hand side and the nonlinear term on the right hand side.Nonlinear Alfv \u0013en Waves with Viscosity Tensor 13\nThe momentum equation is\n\u001a\u0012@V\u000b\n@t+ (V\u0001r)V\u000b\u0013\n=\u0000@\n@x\u000b\u0012\np+B2\n2\u00160\u0013\n+1\n\u00160(B\u0001r)B\u000b\u0000@\u0019\u000b\f\n@x\f: (46)\nWhen\u00110contributions dominate the viscous force, Eqs. (13) and (17) give\n\u0000\u0019xz= 3\u00110b\nB0\u0012b\nB0@Vx\n@z+2\n3@Vz\n@z\u0013\n+O(\u000f5=2) (47)\nso thex-component of Eq. (46) becomes\n@Vx\n@t\u0000B0\n\u00160\u001a0@b\n@z=\u0000\u000e\u001a\n\u001a0@Vx\n@t\u0000Vz@Vx\n@z+3\u00110\n\u001a0@\n@z\u0012b\nB0\u0014b\nB0@Vx\n@z+2\n3@Vz\n@z\u0015\u0013\n+O(\u000f5=2); (48)\nwhere linear terms and nonlinear terms have again been placed on opposite sides of the equation.\nTaking the time derivative of Eq. (48) and using Eq. (45) to eliminate bfrom the linear terms,\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nVx=\u0000v2\nA@2\n@z2\u0012b\nB0Vz\u0013\n\u0000@\n@t\u0012\u000e\u001a\n\u001a0@Vx\n@t+Vz@Vx\n@z\u0013\n+3\u00110\n\u001a0@2\n@t@z\u0012b\nB0\u0014b\nB0@Vx\n@z+2\n3@Vz\n@z\u0015\u0013\n+O(\u000f5=2):\n(49)\nInterpreting Eq. (49), the linear terms (on the left hand side) correspond to a wave equation with wave speed vA. The\nleading nonlinear terms (those shown explicitly on the right hand side) include the leading-order e\u000bect of the anisotropic\nviscosity, which enters at the same order as the leading nonlinear terms that appear in perturbative nonlinear theory\nof ideal Alfv\u0013 en waves.\nNext, we eliminate band\u000e\u001afrom theO(\u000f3=2) nonlinear terms in Eq. (49). Equations (45) and (48) are solved at\nlinear order by the Alfv\u0013 en wave relation\nb\nB0=\u0006Vx\nvA+O(\u000f3=2): (50)\nWe choose the negative sign so waves travel in the positive zdirection, giving\nb\nB0=\u0000Vx\nvA+O(\u000f3=2): (51)\nThe travelling wave behaviour of the linear solution together with assumption that the wave envelope changes over a\ndistance controlled by the leading order nonlinear terms in Eq. (49) allows replacement\n@\n@t=\u0000vA@\n@z+O(\u000f): (52)\nThe density perturbation is governed by the mass continuity equation\n@\u001a\n@t+r\u0001(\u001aV) = 0; (53)\nwhich gives for our shear Alfv\u0013 en wave\n@\u000e\u001a\n@t=\u0000\u001a0@Vz\n@z+O(\u000f2): (54)\nThen, using Eq. (52) and integrating,\n\u000e\u001a\n\u001a0=Vz\nvA+O(\u000f2): (55)\nThe constant of integration has been set to zero, for reasons discussed in §5.1.\nUsing these results, Eq. (49) becomes\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nVx=vA@2\n@z2(VxVz) +3\u00110\n\u001a0@2\n@t@z\u0012Vx\nvA\u0014Vx\nvA@Vx\n@z\u00002\n3@Vz\n@z\u0015\u0013\n+O(\u000f5=2): (56)14 Russell\nNow that the problem has been reduced to the two variables VxandVz, it is convenient to make the \u000fdependence\nexplicit by introducing dimensionless variables vandwde\fned by\nVx(z;t) =\u000f1=2vAv(z;t); Vz(z;t) =\u000fvAw(z;t): (57)\nExpressing Eq. (56) in the dimensionless variables vandw, and dropping the non-explicit higher order terms from the\nright hand side, we seek solutions to\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nv=\u000f\u0012\nv2\nA@2\n@z2(vw) +\u00110\n\u001a0@2\n@t@z\u0012@v3\n@z\u00002v@w\n@z\u0013\u0013\n: (58)\n5.2.3. Multiple scale analysis\nEquation (58) is now solved using multiple scale analysis (e.g. Bender & Orszag 1978). Applying this technique, one\nintroduces a new variable Z=\u000fzthat de\fnes a long length scale, and the perturbation expansions\nv(z;t) =v0(z;Z;t ) +\u000fv1(z;Z;t ) +::: (59)\nw(z;t) =w0(z;Z;t ) +\u000fw1(z;Z;t ) +:::: (60)\nDerivatives are treated using the chain rule as though zandZare independent variables and setting d Z=dz=\u000f.\nThus,\n@v\n@z=@v0\n@z+\u000f\u0012@v0\n@Z+@v1\n@z\u0013\n+O(\u000f2); (61)\n@2v\n@z2=@2v0\n@z2+\u000f\u0012\n2@2v0\n@Z@z+@2v1\n@z2\u0013\n+O(\u000f2); (62)\nwith equivalent expressions for derivatives of w.\nSubstituting into Eq. (58), collecting \u000f0terms, and thus solving the homogeneous wave equation\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nv0= 0; (63)\nobtains d'Alembert's solution\nv0(z;Z;t ) =f(z\u0000vAt;Z) +g(z+vAt;Z): (64)\nFor forward propagating waves, the function gis zero.\nThe corresponding w0is obtained by integrating the z-component of the momentum equation, Eq. (46), which gives\nVz\nvA=1\n\u001a0v2\nA\u0012\n\u000ep+b2\n2\u00160+\u0019zz\u0013\n+O(\u000f2): (65)\nFrom Eqs. (13) and (17), we have\n\u0000\u0019zz= 2\u00110\u0012b\nB0@Vx\n@z+2\n3@Vz\n@z\u0013\n+O(\u000f2): (66)\nA substitution for the pressure perturbation \u000epis obtained by integrating the energy equation,\n@p\n@t+V\u0001rp+\rpr\u0001V= (\r\u00001)Qvisc: (67)\nThe viscous heating Qviscis of orderO(\u000f2), so integration gives the adiabatic relation\n\u000ep\np0=\rVz\nvA+O(\u000f2): (68)\nAlternatively, one can consider isothermal conditions using \u000ep=p 0=Vz=vAfrom the ideal gas law, which is recovered\nfrom Eq. (68) by setting \r= 1.Nonlinear Alfv \u0013en Waves with Viscosity Tensor 15\nUsing Eqs. (66) and (68), and eliminating bterms using Eq. (51), Eq. (65) can be expressed as\n\u0012\n1\u0000\f+4\n3\u00110\n\u001a0vA@\n@z\u0013\nw=\u00121\n2+\u00110\n\u001a0vA@\n@z\u0013\nv2; (69)\nwhere\f= (cs=vA)2and dropped terms are O(\u000f). Whenvandware expanded according to Eqs. (59) and (60), an\nequation identical to (69) connects w0andv0.\nInspecting Eq. (69), it is evident that obtaining wfor a known vin general requires solving a \frst order linear partial\ndi\u000berential equation. In the limit where the viscous terms can be neglected, the problem simpli\fes to the algebraic\nw=v2=2(1\u0000\f) relation used in Sec. 3.4. Similarly, when the viscous terms dominate, one obtains the w= (3=4)v2\nrelation for viscously self-organised parallel \row discussed in Sec. 3.4. For the detailed solution in this section, we\nretain the complete set of forces that determine Vz, solving the full Eq. (69).\nSolution is facilitated by considering the special case where v0oscillates sinusoidally in time. For the rest of this\nderivation we therefore set\nv0=A(Z)ei\u001e+A\u0003(Z)e\u0000i\u001e; \u001e =kjj(z\u0000vAt); (70)\nwhereA(Z)2Cand\u0003denotes the complex conjugate. Representing Ain polar form,\nA(Z) =R(Z)ei\u0012(Z); (71)\nEq. (70) is equivalent to\nv0(z;Z;t ) = 2R(Z) cos(kjj(z\u0000vAt) +\u0012(Z)): (72)\nFrom inspection, 2 R(Z) is the local amplitude and \u0012(Z) is a phase shift. We have found the complex form in Eq. (70)\nmore convenient to work with in the following.\nWe now solve for the corresponding w0. Noting that\nv2\n0=A2e2i\u001e+ 2jAj2+A\u00032e\u00002i\u001e; (73)\nwherejAj2=AA\u0003, we seek a solution of the form\nw0=De2i\u001e+D\u0003e\u00002i\u001e+ ^w0: (74)\nSubstituting into Eq. (69), terms in e0give\n^w0=jAj2\n1\u0000\f; (75)\nwhile terms in e2i\u001eand e\u00002i\u001eindependently give\nD=\u000bA2; \u000b =1\n2(1 + 4ikjj\u00110=\u001a0vA)\n(1\u0000\f+ (8=3)ikjj\u00110=\u001a0vA): (76)\nThe solution for w0can also be expressed without complex numbers. Making explicit the real and imaginary parts\nof\u000b=\u000br+i\u000bi, we have the real constants\n\u000br=1\n2(1\u0000\f+ (32=3)(Re)\u00002)\n((1\u0000\f)2+ (64=9)(Re)\u00002); (77)\n\u000bi=2\n3(1\u00003\f)(Re)\u00001\n((1\u0000\f)2+ (64=9)(Re)\u00002); (78)\nwhere Re = \u001a0vA=kjj\u00110consistent with Eq. (26). It is then easily shown that\nw0\n2R(Z)2=\u000brcos(2(kjj(z\u0000vAt) +\u0012(Z)))\u0000\u000bisin(2(kjj(z\u0000vAt) +\u0012(Z))) +1\n2(1\u0000\f): (79)\nTo deduceR(Z) and\u0012(Z), we return to analysing Eq. (58). The \u000f1terms in Eq. (58) give the inhomogeneous partial\ndi\u000berential equation\n\u0012@2\n@t2\u0000v2\nA@2\n@z2\u0013\nv1= 2v2\nA@2v0\n@Z@z+v2\nA@2\n@z2(v0w0) +\u00110\n\u001a0@2\n@t@z\u0012@v3\n0\n@z\u00002v0@w0\n@z\u0013\n: (80)16 Russell\nThev0andw0terms drive v1, and the solution for v1will have a secular contribution (i.e. one or more terms that grow\nrelative to corresponding solutions of the homogeneous equation) if terms on the right hand side resonate with the\nsolution to the undriven wave equation. In the speci\fc case where v0is given by Eq. (70), secular terms in the solution\nforv1will restrict the domain for which v0is a valid approximation if the right hand side of Eq. (80) contains ei\u001eor\ne\u0000i\u001eterms. The central idea in multiple scale analysis is to solve for the A(Z) that makes the resonance disappear,\nmakingv0a durable approximation for v.\nUsing Eqs. (70) and (74){(76), ei\u001eterms vanish from the right hand side of Eq. (80) if and only if\n1\nAdA\ndZ=\u0000kjjjAj2\n2\u0014\ni\u0012\n\u000b+1\n1\u0000\f\u0013\n+kjj\u00110\n\u001a0vA(3\u00004\u000b)\u0015\n: (81)\nThe same condition also removes the e\u0000i\u001eterms.\nChanging to polar form, Eq. (71) implies\n1\nAdA\ndZ=1\nRdR\ndZ+id\u0012\ndZ: (82)\nHence, the real and imaginary parts of Eq. (81) yield the real ordinary di\u000berential equations\ndR\ndZ=\u0000\u00141\n2R3; (83)\nd\u0012\ndZ=\u00142R2; (84)\nwhere\n\u00141=kjj\u0012kjj\u00110\n\u001a0vA(3\u00004\u000br)\u0000\u000bi\u0013\n; (85)\n\u00142=\u0000kjj\n2\u0012\n\u000br+1\n1\u0000\f\u0000kjj\u00110\n\u001a0vA4\u000bi\u0013\n: (86)\nEqs. (83) and (84) govern the local amplitude and phase drift of the Alfv\u0013 en wave respectively ( c.f.Eq. (72)).\nOur main interest is in R(Z), which determines how the waves decay. Equation (83) is a separable \frst order\ndi\u000berential equation. The solution is\nR(Z) =R(0)p\n1 +\u00141R(0)2Z: (87)\nFor\u00141>0 the wave envelope decays non-exponentially, over a damping length that is inversely proportional to the\nsquare of the initial wave amplitude. Having obtained R(Z), the solution for \u0012(Z) is obtained by directly integrating\nEq. (84). Using Eq. (87),\n\u0012(Z) =\u0012(0) +\u00142\n\u00141ln\f\f1 +\u00141R(0)2Z\f\f: (88)\n5.2.4. Solution in original variables\nHaving ensured corrections to v\u0019v0remain of order \u000f\u0018(b=B 0)2\u001c1, the multiple scale analysis is concluded by\nusingv0as the approximation for v. Returning to the original variables,\nVx(z;t) =a0p\n1 +z=Ldcos\u0000\nkjj(z\u0000vAt) + (\u00142=\u00141) lnj1 +z=Ldj+\u00120\u0001\n; (89)\nwherea0is the amplitude of Vx(0;t),\u00120sets the initial phase of the wave (at z= 0,t= 0) and\nLd=4\n\u00141(a0=vA)2(90)\nis the decay length. Using Eqs. (77), (78) and (85),\n\u00141=kjj\n3Re(1\u00003\f)2\n(1\u0000\f)2+ (64=9)(Re)\u00002; (91)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 17\nwhere Re is the Reynolds number for the wave, as de\fned by Eq. (26). Thus,\nLd=12Re\nkjj(a0=vA)2(1\u0000\f)2+ (64=9)(Re)\u00002\n(1\u00003\f)2: (92)\nIf one neglects the Re\u00002term in the numerator of Eq. (92), then Ldagrees exactly with the formula in Eq. (41) that\nwe derived from energy principles. The formula for Ldin Eq. (92) is more general since it was derived without direct\nassumptions about the value of Re = kjj\u00110=\u001a0vA, although the multiple scale analysis requires that the combination\nof parameters kjj,a2\n0and Re are such that waves damp over a signi\fcantly longer scale than the wavelength.\n5.3. Non-exponential decay and interpretation of damping length\nAs a general principle, the Alfv\u0013 en wave energy density Ew=\u001aV2\nxdecays more rapidly than the perturbation Vx, due\nto the quadratic power. For exponential decay this is re\rected in a factor two di\u000berence in the respective e\u00001decay\nlengths. For the non-exponential decay produced by nonlinear viscous damping, the situation is handled di\u000berently.\nThe sameLddescribesVxandEw, however they have di\u000berent functional forms. The velocity amplitude decays as\n(1 +z=Ld)\u00001=2(see Eq. (89)), while the wave energy density decays as (1 + z=Ld)\u00001. Therefore, over a distance Ld,\nthe velocity amplitude reduces by a factorp\n2 and the energy density halves.\n5.4. Inclusion of thermal conduction\nThe multiple scale analysis can also be modi\fed to include explicit thermal conduction. Since thermal conduction\nis highly anisotropic, we include the parallel thermal conduction, setting the heat \row vector to\nq=\u0000Kjj(h\u0001rT)h; (93)\nwhereKjjis the coe\u000ecient of parallel thermal conduction, and temperature T=p=\u001aRwhereRis the gas constant.\nThe energy equation with heat \row is\n@p\n@t+V\u0001rp+\rpr\u0001V= (\r\u00001)(Qvisc\u0000r\u0001 q); (94)\nwhich replaces Eq. (67).\nIt follows that \u000ep=p 0is related to vz=vAby the partial di\u000berential equation\n\u0012\n1 + \u0003@\n@z\u0013\u000ep\np0=\u0012\n\r+ \u0003@\n@z\u0013Vz\nvA+O(\u000f2): (95)\nwhere\n\u0003 =(\r\u00001)Kjj\n\u001a0RvA(96)\nis a conductive length scale. In the limit of weak thermal conduction, \u0003 !0 gives\u000ep=p 0=\rVz=vA, recovering the\nadiabatic case treated above. Similarly, for strong thermal conduction, \u0003 !1 gives\u000ep=p 0=Vz=vA, recovering the\nisothermal case.\nIntroducing a dimensionless pressure variable cde\fned by\u000ep=\u000fp0c(z;t), expanding c(z;t) =c0(z;Z;t )+\u000fc1(z;Z;t )+\n:::and setting c0=Ce2i\u001e+C\u0003e2i\u001e+ ^c0, terms in e0in Eq. (95) yield ^ c0=\r^w0, and terms in e2i\u001eyieldC= \u0000D,\nwhere\n\u0000 =\r+ 2ikjj\u0003\n1 + 2ikjj\u0003: (97)\nSolving further, an equation D=\u000bA2analogous to Eq. (76) is obtained but with \freplaced by the complex-valued\n\u0000p0=\u001a0v2\nAin the formula for \u000b. Meanwhile, (75) and (81) are unchanged, retaining the real-valued \f=\rp0=\u001a0v2\nA. The\nwave amplitude is therefore governed by results identical to Eqs. (83) and (85), with the aforementioned change in the\nde\fnition of \u000b.18 Russell\n6.DISCUSSION\n6.1. Optimum damping\nInspecting Eq. (92), the formula for Ldkjjhas a minimum with respect to the Alfv\u0013 enic Reynolds number at Re =\n8=(3j1\u0000\fj). Thus, shear Alfv\u0013 en waves with kjj= 3\u001a0vAj1\u0000\fj=3\u00110are damped in the fewest number of wavelengths,\nwhich we refer to as optimum damping. The optimally damped waves have\nLd\n\u0015jj=32\n\u0019j1\u0000\fj\n(a0=vA)2(1\u00003\f)2: (98)\nWhen\f\u001c1, the right hand side of Eq. (98) is approximately ten divided by the square of the normalised wave\namplitude. Hence, while nonlinear viscous damping can in principle damp Alfv\u0013 en waves in a small number of wave-\nlengths, this requires large amplitudes a=vA\u00181, or\f\u00181. For a more typically encountered amplitudes a=vA\u001810\u00001\nand low\f, one \fndsLd=\u0015jj&1000, making nonlinear viscous damping negligible for many coronal situations.\n6.2. Viscous self-organisation\nThe suppression of nonlinear viscous damping for small Re (highly viscous plasma) does not mean that viscous\ne\u000bects are unimportant in this regime. On the contrary, nonlinear damping is suppressed for small R ebecause viscous\nforces organise the parallel \row associated with the Alfv\u0013 en wave to approach the relationship Vz=vA= (3=4)(Vx=vA)2.\nThis modi\fcation of the parallel \row plays a crucial role in avoiding signi\fcant nonlinear damping in highly viscous\nplasma, when modelled using Braginskii MHD.\n6.3. Validity constraints\nThroughout this paper, we have assumed that \f6= 1 to avoid resonant wave coupling. This condition holds\nthroughout most of the corona, so it is appropriate for our primary applications. Additionally, the multiple scale\nanalysis in §5 uses\u000f\u0018(Vx=vA)2\u0018(b=B 0)2as a small parameter, one consequence of which is that the nonlinearly\ndamping occurs over a distance considerably greater than the parallel wavelength. As noted in §3.2, transverse coronal\nwaves are observed in open-\feld regions with \u000f\u001810\u00002, making weakly nonlinear theory appropriate for such situations.\nObtaining nonlinear viscous solutions in the resonant and strongly nonlinear regimes nonetheless remain interesting\nfuture challenges for plasma theory.\nApplicability of this paper's results to physical problems is also constrained to conditions under which Braginskii\nMHD can be rigorously applied. As discussed in §2, the traditional derivation of Braginskii MHD assumes that the\ncollisional mean free path is less than the macroscopic scales. Comparing the mean free path parallel to the magnetic\n\feld to the parallel wavelength, this condition can be given as kjj\u0015mfp<1, where\u0015mfp=vTi\u001ci,vTi=p\nkBT=miand\n\u001ciis the ion collision time. Using the formula (Braginskii 1958, 1965; Hollweg 1985)\n\u00110= 0:96nkBT\u001ci; (99)\nand the de\fnition of Re in Eq. (26), one can show that\nkjj\u0015mfp<1,\f1=2Re =\u001acs\nkjj\u00110>1: (100)\nIn other words, Braginskii MHD requires that the Reynolds number based on the sound speed is greater than unity. One\nshould therefore be cautious about applying small Alfv\u0013 enic Reynolds number results such as viscous self-organisation\nto real low- \fplasmas.\n6.4. Formulas for applications\nFor applications to real plasmas, the following formulas are convenient. In cases where the parallel viscosity coe\u000ecient\nis determined by Coulomb collisions,\n\u00110= 0:96nkBT\u001ci=22\n\u0015C\u000210\u000017T5=2; (101)\nwhere this formula is stated in S.I. units with Tin kelvin, and \u0015Cis the Coulomb logarithm (e.g. Hollweg 1985). The\nReynolds number de\fned in Eq. (26) can then be expressed as\nRe = 5:8\u00021020\u0015CB2f\u00001T\u00005=2; (102)Nonlinear Alfv \u0013en Waves with Viscosity Tensor 19\nalso in S.I. units, where f=vAkjj=2\u0019is the wave frequency. This formula makes explicit the dependences on\nfrequency, magnetic \feld strength and temperature. The Alfv\u0013 enic Reynolds number is smallest when the plasma has\nhigh temperature and low magnetic \feld strength, and for higher frequency waves. Finally, we express the damping\nlength in Eq. (92) as a function of frequency and the mean square velocity\nV2\nx\u000b\n=a2\n0=2, which gives\nLd=3\n\u0019v3\nARe\nfhV2xi(1\u0000\f)2+ (64=9)(Re)\u00002\n(1\u00003\f)2: (103)\n6.5. Waves in a coronal open-\feld region\nOutgoing transverse waves in the magnetically open solar corona contain su\u000ecient energy to heat the open corona and\naccelerate the fast solar wind (McIntosh et al. 2011; Morton et al. 2015), and they are observed to damp signi\fcantly\nwithin a solar radius above the Sun's surface (Bemporad & Abbo 2012; Hahn et al. 2012; Hahn & Savin 2013; Hahn\net al. 2022). Heating at these altitudes is also thought to be important for producing the observed rapid acceleration of\nthe fast solar wind (Habbal et al. 1995; McKenzie et al. 1995). The problem of how the outgoing waves damp has not\nbeen conclusively solved, although one leading hypothesis is turbulent cascade driven by interactions with downgoing\nAlfv\u0013 en waves (Hollweg 1986; Heyvaerts & Priest 1992; Matthaeus et al. 1999; Cranmer et al. 2007; Verdini et al. 2010;\nMiki\u0013 c et al. 2018) produced either by re\rection from density inhomogeneities (van Ballegooijen & Asgari-Targhi 2016;\nPascoe et al. 2022) or by parametric decay instability (Galeev & Oraevskii 1963; Derby 1978; Goldstein 1978; Shoda\net al. 2019; Hahn et al. 2022).\nHere, we demonstrate that Braginskii viscosity does not cause signi\fcant damping of Alfv\u0013 en waves at the altitudes\nat which the traditional derivation of Braginskii MHD holds. For concreteness, we consider the Sun's northern polar\nopen-\feld region on 27 March 2012, using observational values reported by Morton et al. (2015). Enhanced wave\npower was present around f= 5 mHz, which suggests Alfv\u0013 enic waves produced by p-modes (Morton et al. 2019). We\nwill calculate damping lengths for this frequency, noting that Re and Lddepend on f, withLd\u0018f\u00002in the limit\nof high Re. Morton et al. (2015) inferred that the Alfv\u0013 en speed was nearly constant with vA= 400 km s\u00001on their\ndomain ofr= 1:05 to 1:20R\f. For temperature, we set T= 1:6\u0002106K, the formation temperature of the Fe XIII\nlines used by the CoMP instrument, which implies the proton thermal speed VTi=p\nkBT=miis 115 km s\u00001. Hence,\nin for an isothermal equation of state \f= 0:083 and\f1=2= 0:29. For the wave velocity amplitude, Morton et al.\n(2015) recommended that the reported non-thermal line width should be used, which varies with altitude.\nStarting with lowest altitude observed by Morton et al. (2015), r= 1:05R\f, we setn= 1014m\u00003,B= 2\u000210\u00004T\nand take the rms value of Vxas 35 km s\u00001. We therefore \fnd \u0015C= 19 and Re = 28. Since \f1=2Re = 8>1, Braginksii\nMHD applies and we evaluate Ld= 4:2\u0002108km\u0011600R\f.\nAtr= 1:20, we set n= 1013m\u00003,B= 6\u000210\u00005T and take the rms value of Vxas 50 km s\u00001. The observed\nparameters therefore give \u0015C= 21 and Re = 2 :7. Since\f1=2Re = 0:8\u00191, this altitude is close to the maximum at\nwhich the assumptions by which Braginskii MHD is traditionally derived remains valid (for this particular open \feld\nregion, and assuming Eq. (101)). Evaluating the damping length here returns Ld= 4:2\u0002107km\u001161R\f.\nWe conclude that Braginskii viscosity does not cause signi\fcant wave damping below r= 1:2R\f, which is consistent\nwith observational results that Alfv\u0013 enic wave amplitudes in coronal holes follow ideal WKB scaling out to around this\naltitude (Cranmer & van Ballegooijen 2005; Hahn & Savin 2013).\nBetween the altitudes we have examined, Ldreduces by an order of magnitude. If one were to extrapolate using high\nRe or incompressible results, it would appear that viscous damping becomes important near the altitudes at which the\nwaves are observed to damp. We are cautious about making such an assertion for two reasons. First, as discussed in\n§6.1, our results show that for Re <8=(3j1\u0000\f)) the damping length in a Braginskii MHD model increases again as the\n\feld-aligned \row self organises to supress viscous damping. Secondly, as the plasma becomes increasingly collisionless\n(\f1=2Re<1) the traditional derivation of Bragniskii MHD falters.\nIntriguingly, it may be signi\fcant that the onset of wave damping broadly coincides with the altitude at which\nBraginskii MHD can no longer be con\fdently applied if one invokes the \u00110expression for Coulomb collisions given in\nEq. (101). This correspondence is suggestive that the wave damping observed in coronal holes may involve collisionless\nand heat \row e\u000bects not found in the most common \ruid models.\n6.6. Future work\nThe present types of analyses should be extended in future to other types of propagating transverse MHD waves.\nThe nonlinear longitudinal \row that accompanies propagating torsional Alfv\u0013 en waves di\u000bers from its counterpart for20 Russell\npropagating shear Alfv\u0013 en waves (Vasheghani Farahani et al. 2011) and it will be of interest to investigate how this\ndi\u000berence a\u000bects the nonlinear viscous damping. It is similarly desirable to determine how nonlinear viscosity a\u000bects\npropagating kink waves (Edwin & Roberts 1983).\nFor propagating shear Alfv\u0013 en waves, viscous damping appears most promising near the cs=vAsingularity, which\nmust be treated using di\u000berent methods to those used in this paper. The solar wind frequently has \f\u00181, while\f= 1\noccurs in the lower solar atmosphere and in the vicinity of coronal nulls points. Hence, this case is of considerable\nphysical interest. One challenge for application to magnetic nulls is that the magnetic \feld unit vector h=B=jBj\nis not de\fned at the null itself, so one must be careful to evaluate the Braginskii stress tensor using appropriate\ncalculations, e.g. see recent discussion by MacTaggart et al. (2017).\nA further challenge is to develop a theory of nonlinear viscous damping applicable to strongly nonlinear waves with\namplitudes b\u0018B0and greater. The results of the multiple-scale analysis in §5 are rigorous only for the weakly\nnonlinear case, in which \u000f\u0018(b=B 0)2can be treated as a small parameter and it is assumed that the damping length is\nsigni\fcantly longer than the wavelength. Strongly nonlinear Alfv\u0013 en waves with b\u0018B0are a feature of the solar wind,\nand while the low collisionality of the solar wind means that Braginskii MHD may not be an appropriate framework\nfor that application, extending the current work to strongly nonlinear waves remains an interesting problem.\nThere is a diverse collection of MHD wave problems beyond wave damping for which viscous e\u000bects are likely\nto be signi\fcant. Prime among these are nonlinear phenomena involving Alfv\u0013 en waves, for which the nonlinear\nviscosity tensor enters the equations at the same order as the e\u000bect of interest. For example, standing Alfv\u0013 en waves\ndrive signi\fcantly stronger \feld-aligned \rows than occur for propagating waves because standing Alfv\u0013 en waves create\ninhomogeneous time-averaged magnetic pressure. There could also be signi\fcant value in investigating how viscosity\nmodi\fes wave interactions, including Alfv\u0013 en wave collisions and parametric decay instability (Galeev & Oraevskii 1963;\nDerby 1978; Goldstein 1978), which are central to leading hypotheses of wave heating in the magnetically open solar\ncorona.\nFinally, we point to the continuing need for basic plasma physics research to provide increasingly rigorous derivation\nand validation of the appropriate \ruid equations for weakly collisional and collisionless plasma, in the face of the\nclosure problem summarised in §2. As discussed in §2 and 6.3, Braginskii MHD breaks down at higher altitudes\nin the corona as the plasma becomes increasingly collisionless (see Eqs. (100) and (102)). The CGL double-adiabatic\nequations and other models that evolve the stress tensor may provide a more suitable framework in these conditions.\nHunana et al. (2019a,b, 2022) provide recent discussions of such models and their limitations. Alternatively, it may\nbe necessary for the solar waves community to more widely adopt non-\ruid plasma models. However, tractability of\nkinetic models remains a limiting factor, especially in light of the large separations between kinetic and macroscopic\nscales that are characteristic of the Sun's corona. Eloquent comments on these matters can be found in Montgomery\n(1996).\n7.CONCLUSIONS\nThis paper has investigated the properties of propagating shear Alfv\u0013 en waves subject to the nonlinear e\u000bects of the\nBraginskii viscous stress tensor. The main points are as follows:\n1. For many plasma environments, including the low-altitude solar corona, Braginskii MHD provides a more accurate\ndescription of plasma than classical MHD does, by rigorously treating the stress tensor and thermal conduction.\nStress tensor e\u000bects nonetheless remain relatively unexplored for many solar MHD phenomena.\n2. The dominant viscous e\u000bects for propagating shear Alfv\u0013 en waves are nonlinear in the wave amplitude and occur\nthrough the \\parallel\" viscosity coe\u000ecient, \u00110. Theoretical results based on linearizing the stress tensor with\nrespect to the wave amplitude are only valid for amplitudes satisfying ( b=B 0)2\u001c(\ni\u001ci)\u00002. Such waves would\nbe energetically insigni\fcant under normal coronal conditions, hence nonlinear treatment is required.\n3. Compressibility and pressure a\u000bect the nonlinear \feld-aligned \row associated with shear Alfv\u0013 en waves, hence\nthey impact nonlinear wave damping. Both must be included to produce accurate coronal results.\n4. Braginskii viscosity damps propagating shear Alfv\u0013 en waves nonlinearly, such that the primary wave \felds band\nVxdecay as (1 + z=Ld)\u00001=2, where the decay length\nLd=12Re\nkjj(a0=vA)2(1\u0000\f)2+ (64=9)(Re)\u00002\n(1\u00003\f)2:Nonlinear Alfv \u0013en Waves with Viscosity Tensor 21\nHere,a0is the initial velocity amplitude of the wave, \f= (cs=vA)2and Re =\u001avA=kjj\u00110is the Alfv\u0013 enic Reynolds\nnumber of the wave. The energy density decays as (1 + z=Ld)\u00001.\n5. Optimal damping (the minimum normalised damping length kjjLd) is obtained when Re = 8 =(3j1\u0000\fj). For low\n\fplasma and ( a0=vA).10\u00001, one \fnds Ld=\u0015jj&1000, indicating that nonlinear viscous damping is negligible\nfor many coronal situations.\n6. The asymptotic behaviour that Ld!1 in the highly viscous regime Re !0 is attributed to self-organisation\nof the parallel \row by viscous forces such that Vz=vA\u0019(3=4)(Vx=vA)2, which suppresses dissipation.\n7. Applicability of the Braginskii MHD solutions to real plasmas is constrained by the traditional derivation of\nBraginskii MHD assuming that kjj\u0015mfp<1 which is equivalent to \f1=2Re =\u001acs=kjj\u00110>1. In other words,\nBraginskii MHD requires that the Reynolds number based on the sound speed is greater than unity. We therefore\nrecommend that only the damping results for large Alfv\u0013 enic Reynolds number should be applied to real coronal\nplasma, using the simpli\fed formula Ld= 12Re(1\u0000\f)2=(kjj(a0=vA)2(1\u00003\f)2)) that has been derived in this\npaper by two di\u000berent techniques: energy principles and multiple scale analysis.\n8. Application to transverse waves observed in a polar open-\feld region concludes that nonlinear Braginskii viscosity\ndoes not cause signi\fcant damping of the waves at the altitudes at which the assumptions by which Braginskii\nMHD is traditionally derived remain valid ( r.1:2R\ffor the considered region and wave properties). Intrigu-\ningly, the observed onset of wave damping broadly coincides with the altitude at which Braginskii MHD can no\nlonger be con\fdently applied if one invokes the \u00110expression for Coulomb collisions given in Eq. (101).\nThis work was prompted by and bene\fted from conversations with Paola Testa, Bart De Pontieu, Vanessa Polito,\nGraham Kerr, Mark Cheung, Wei Liu, David Graham, Joel Allred, Mats Carlsson, Iain Hannah and Fabio Reale\nduring a research visit to LMSAL funded by ESA's support for the IRIS mission (August 2018) and meetings of\nInternational Space Science Institute (Bern) International Team 355 (November 2018). I am grateful to Peter Cargill,\nAndrew Wright, Bart De Pontieu and Paola Testa for comments on an early draft (June 2019), and Declan Diver for\nencouragment to explore connections with pressure anisotropy. I thank the reviewer for considered and constructive\nsuggestions, and several unnamed individuals for comments that also improved the manuscript. 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P. 2014, Transport Processes in Space Physics and\nAstrophysics, Vol. 877 (Springer),\ndoi: 10.1007/978-1-4614-8480-6"    },    {        "title": "1701.02475v1.Magnetic_properties_in_ultra_thin_3d_transition_metal_alloys_II__Experimental_verification_of_quantitative_theories_of_damping_and_spin_pumping.pdf",        "content": "1  Magnetic properties in ultra -thin 3d transition metal alloys \nII: Experimental verification of quantitative theories of \ndamping and spin -pumping    \n \nMartin A. W. Schoen,1,2* Juriaan Lucassen,3 Hans T. Nembach,1 Bert Koopmans,3 T. J. Silva,1 Christian \nH. Bac k,2 and Justin M. Shaw1 \n \n1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO \n80305 , USA  \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, \nGermany  \n3Department of Applied P hysics, Eindhoven University of Technology, 5600 MB Eindhoven, The \nNetherlands  \n \n \nDated: 01/05/2017  \n \n *Corresponding author: martin1.schoen@physik.uni -regensburg.de  \n \n \n \n \nAbstract  \nA systematic experimental study of Gilbert damping is performed via ferromagnet ic \nresonance for the disordered  crystalline  binary 3 d transition metal alloys Ni -Co, Ni -Fe and \nCo-Fe over the full range of alloy compositions. After accounting for inhomogeneous \nlinewidth  broadening , the damping shows clear evidence of both  interfacial da mping \nenhancement (by spin pumping) and radiative damping. We quantify these two extrinsic  \ncontributions  and thereby determine the intrinsic damping. The comparison of the intrinsic \ndamping to multiple theoretical calculation s yields good qualitative  and q uantitative  \nagreement  in most cases . Furthermore, the values of the damping obtained in this study are \nin good agreement with  a wide range of published experimental and theoretical values . \nAdditionally, we find a compositional dependence of the spin mixing  conductance.  \n \n \n \n \n 2  \n \n \n \n1  Introduction  \nThe magnetization dynamics in f erromagnetic films are phen omenologically well described \nby the Landau -Lifshitz -Gilbert formalism (LLG)  where  the damping  is described by a \nphenomenological damping parameter α.4,5 Over the past four decades, there have been \nconsiderable efforts to derive the phenomenological d amping parameter from first principles \ncalculations and to do so in a quantitative manner. One of the early promising  theories was that of \nKamberský, who introduced the so -called breathing Fermi surface model6–8. The name “breathing \nFermi  surface ” stems from the picture that the precessing magnetization, due to spin -orbit coupling, \ndistorts the Fermi surface. Re -populating the Fermi surface is delayed by the scattering time, \nresulting in a phase lag between the precession  and the Fermi sur face distortion. This  lag leads to  a \ndamping that is proportional to the scattering time. Although this approach describes the so-called \nconductivity -like behavior of the damping at low temperatures, it fails to describe  the high \ntemperature behavior of so me materials . The high temperature or resistivity -like behavior is \ndescribed by the so-called  “bubbling Fermi surface ” model. In the case of energetically shifted  \nbands , thermal  broaden ing can lead to a significant  overlap  of the spin-split bands  in 3d \nferromagnets.  A precessing magnetization can induce elec tronic transitions between such \noverlapping bands , leading to spin-flip process es. This process scales with the amount of band \noverlap. Since s uch overlap is further increased with the band broadening th at results from the finite \ntemperature of the sample, this contribution  is expected to increase as the temperature is increased. \nThis model for interband transition mediated damping describes the resistivity -like behavior of the \ndamping at higher temperatu res (shorter scattering times).  These two damping processes are  \ncombined in a torque correlation model by Gilmore , et al.9, as well as Thonig , et al.10, that  describes \nboth the low -temperature  (intra band transitio ns) and high -temperature  (inter band transitions) \nbehavior  of the damping . Another app roach via  scattering theory was successfully implemented by \nBrataas , et al.11 to describe damping in transition metals. Starikov , et al. ,2 applied the scattering \nmatrix approach to calculate the damping  of NixFe1-x alloys and  Liu, et al. ,12 expanded the formalism \nto include the influence of electron -phonon interaction s.  \nA numerical realization of the torque correlation model was performed  by Mankovsky , et \nal., for NixCo1-x, Ni xFe1-x, CoxFe1-x, and FexV1-x1. More recently , Turek , et al. ,3 calculated the \ndamping for NixFe1-x and CoxFe1-x alloys with the torque -correlation model, u tilizing non -local \ntorque correlators. It is important to stress that a ll of these  approaches  consider  only the intrinsic \ndamping. This complicates the quantitative comparison of calculated values for t he damping to \nexperimental data  since there are many extrinsic contributions to the damping that result from \nsample  structure , measurement geometr y, and/or sample properties . While some extrinsic \ncontributions to the damping  and linewidth  were discovered in the 1960 ’s and 1970 ’s, and are well \ndescribed  by the ory, e.g. eddy -current damping13,14, two -magnon scattering15–17, the slow rel axer \nmechanism18,19, or radiative damping20,21, interest in these mechanisms has been re -ignited \nrecently22,23. Further contributions, such as  spin-pumping, both extrinsic24,25 and intrinsic24,26, have 3 been discovered more recently and a re subject to extensive research27–31 for spintronics application.  \nTherefore , in order to allow a quantitative comparison to theoretical calculations  for intrins ic \ndamping, both the measurement and sample geometry must  be designed  to allow both the \ndetermination and possibl e minimization  of all additional contributions to the measured damping.  \nIn this study , we demonstrate methods to determine significant extrins ic contributions to the \ndamping , which  includ es a measurement of the effective spin mixing conductance for both the pure \nelements and select alloys. By precisely accounting for all of these extrinsic contributions, we \ndetermine the intrinsic damping  parame ters of the binary alloys Ni xCo1-x, Ni xFe1-x and Co xFe1-x and \ncompare them  to the calculation s by Mankovsky , et al. ,1, Turek , et al. , and Starikov , et al.2. \nFurthermore, we present the concentration -dependence of the  inhomogeneous linewidth \nbroadening, which for most alloys shows exceptionally small values, indicative  of the high \nhomogeneity of our samples.  \n2 Samples and method  \nWe deposited  NixCo1-x, Ni xFe1-x and Co xFe1-x alloys of varying composition (all compositio ns \ngiven in atomic percent)  with a thickness of 10 nm  on an oxidized (001)  Si substrate with  a Ta(3 \nnm)/Cu(3 nm) seed layer and  a Cu(3 nm) /Ta(3 nm) cap layer. In order to investigate interface \neffects , we also deposited multiple thickness series at 10 nm, 7 nm, 4 nm, 3 nm , and 2 nm of  both \nthe pure elements and select alloy s.  Structural characterization  was performed using X-ray \ndiffraction (XRD).  Field swept vector -network -analyzer  ferromagnetic resonance spectroscopy \n(VNA -FMR) was used in the out -of-plane geometry  to determine the total damping parameter αtot. \nFurther d etails of the deposition conditions, XRD, FMR measurement and fitting of the complex \nsusceptibility to the measured S21 parameter are reported in  Ref [66]. \nAn example of susceptibility fits t o the complex S21 data is shown in Fig.  1 (a) and (b). All \nfits were constrained  to a 3×  linewidth ΔH field window around the resonance field in order to \nminimize the influence of measurement drifts on the error in the susceptibility fits. The total \ndampin g parameter αtot and the inhomogeneous linewidth broadening Δ H0 are then determined  from \na fit to  the linewidth Δ H vs. frequency f plot22, as shown in Fig.  1 (c). \n ∆𝐻= 4𝜋𝛼𝑡𝑜𝑡𝑓\n𝛾𝜇0+ ∆𝐻0, (1) \nwhere γ=gμB/ħ is the gyro -magnetic ratio, μ0 is the vacuum permeability, μB is the Bohr -magneton , \nħ is the reduced Planck constant,  and g is the spectroscopic g-factor reported in Ref [ 66].  \n  4  \nFigure 1: (a) and (b) show respectively the real and imaginary part of the S21 transmission \nparameter (black squares)  measured at 20 GHz  with the complex susceptibility fit (red lines)  for \nthe Ni 90Fe10 sample . (c) The linewidths  from the suscept ibility fits  (symbols) and linear fits (solid \nlines) are plotted against frequency for different Ni -Fe compositions. Concentrations are denoted \non the right -hand  axis. The damping α and the inhomogeneous linewidth broadening Δ H0 for each \nalloy can be extracted from the fits via Eq.  (1). \n3 Results  \nThe first contribution t o the linewidth we discuss  is the  inhomogeneous linewidth broadening \nΔH0, which  is presumably  indicative of  sample inhomogeneity32,33. We plot Δ H0 for all the alloy \nsystem s against the respective concentrations in Fig.  2. For all alloys , ΔH0 is in the range of a few \nmT to 10  mT . There are only a limited  number of reports  for ΔH0 in the literature with which to \ncompare . For Permalloy (Ni 80Fe20) we measure Δ H0 = 0.35 mT, which is close to other reported \nvalues .34 For the  other  NixFe1-x alloys , ΔH0 exhibits a significant peak  near the fcc-to-bcc (face -\ncentered -cubic to body -centered -cubic) phase transition  at 30  % Ni , (see Fig.  2 (b)) which is easily \nseen in the raw data  in Fig. 1 (c). We speculate  that this  increase of  inhomogeneous broadening  in \nthe NixFe1-x is caused by the coexistence of the bcc and fcc  phases at the phase transition.  However,  \nthe CoxFe1-x alloys do not exhibit an increase in ΔH0 at the equivalent phase transition  at 70  % Co . \nThis suggests that the bcc and fcc phases of NixFe1-x tend to segregate near the phase transition, \nwhereas the same phases for CoxFe1-x remain intermixed throughout the transition.  \n5 One possible explanation for inhomogeneous broadening is magnetic anisotropy , as originally \nproposed  in Ref. [35]. However, this explanation does not account for our measured dependence of \nΔH0 on alloy concentration, since the perpendicular magnetic anisotropy , described in Ref [ 66] \neffectively  exhibits opposite behavior with alloy concentration.  For our alloys Δ H0 seems to roughly \ncorrelate to the inverse exchange constant36,37, which co uld be a starting point for future \ninvestigation of a quantitative theory of inhomogeneous broadening.  \n \n \n \n  \nFigure 2: The inhomogeneous linewidth -broadening Δ H0 is plotted vs. alloy composition for (a) \nNi-Co, (b) Ni -Fe and (c) Co -Fe. The alloy phases are denoted by color code described in Ref [ 66] \n \nWe plot the total measured damping αtot vs. composition for NixCo1-x, NixFe1-x and CoxFe1-x in \nFig. 3 (red crosses). The total damping of the NixCo1-x system increases monotonically  with \nincreased Ni content . Such smooth  behavior in the damping is  not surprising  owing to the absence \nof a phase transition for this alloy . In the NixFe1-x system , αtot changes very little from pure Fe to \napproximately 25 % Ni  where the bcc to fcc phase transition occurs . At the phase  transition,  αtot \nexhibits a step, increasing  sharply by approximately 30  %. For higher Ni concentrations , αtot \nincreases monotonically  with increasing Ni concentration . On the other hand, t he CoxFe1-x system \nshows a different behavior in the damping and d isplays a sharp minimum of (2.3 ± 0.1)×10-3 at 25 \n6 % Co as previously reported38. As the system changes to an fcc phase ( ≈ 70 % Co), αtot become \nalmost  constant.  \nWe compare our data to previously published values in Table I. However, direct c omparison \nof our data to previous report s is not trivial , owing to the variation in measurement conditions  and \nsample characteristic s for all the reported measurements . For example , the damping  can depend  on \nthe temperature .9,39 In addition, multiple intrinsic and intrinsic contributions to the total damping  \nare not always accounted for in the literature . This can be seen in the fact that the reported  damping \nin Ni80Fe20 (Permalloy) varies  from α=0.0 055 to α=0.04  at room tem perature  among studies . The \nlarge variation  for these reported data  is possibly the result of  different uncontrolled contributions  \nto the extrinsic damping  that add  to the total damping in the different experiments , e.g. spin-\npumping40–42, or roughness41. Therefore , the value for the intrinsic damping of Ni 20Fe80 is expected \nto be at the low end of this scatter . Our measured value of α=0.007 2 lies within the range of reported \nvalues. Similarly , many of our measured damping values for different alloy compositions lie within \nthe range  of reported values22,43 –48. Our measured  damping of the pure elements and the Ni 80Fe20 \nand Co 90Fe10 alloys is compared to room temperature values found in literature in Table 1, Col umns  \n2 and 3 . Column 5 contains theoretically calculated values .   \n \nTable 1: The total measured damping α tot (Col. 2)  and the intrinsic damping (C ol. 4) f or Ni 80Fe20, \nCo90Fe10, and the pure elements are compared to both experimental (Col. 3) and theoretical (Col. \n5) values from the literature . All values of the damping  are at room temperature if not noted \notherwise . \nMaterial  αtot (this study)  \n Liter ature values  αint (this study)  Calculated literature \nvalues  \nNi 0.029 (fcc)  0.06444 \n0.04549 0.024 (fcc)  0.0179 (fcc) at 0K  \n0.02212 (fcc) at 0K  \n0.0131 (fcc)  \nFe 0.0036 (bcc)  0.001944 \n0.002746 0.0025 (bcc)  0.00139 (bcc) at 0K  \n0.001012 (bcc)  at 0K  \n0.00121 (bcc) at 0K  \nCo 0.0047 (fcc)  0.01144  \n 0.0029 (fcc)  0.00119 (hcp)  at 0K  \n0.0007312 (hcp)  at 0K  \n0.0011 (hcp)  \nNi80Fe20 0.0073 (fcc)  0.00844 \n0.008 -0.0450 \n0.007848 \n0.00751 \n0.00652 \n0.00647 \n0.005553 0.0050 (fcc)  0.00462,54 (fcc) at 0K  \n0.0039 -0.00493 (fcc)  at 0K  \nCo90Fe10 0.0048 (fcc)  0.004344 \n0.004855 0.0030 (fcc)   7  \n \n \n \n  \nFigure 3: (color online) The measured damping αtot of all the alloys is plotted against the alloy \ncompositi on (red crosses) for (a) Ni -Co, (b) Ni -Fe and (c) Co -Fe (the data in (c) are taken from \nRef.[38]). The black squares  are the intrinsic damping  αint after correction for spin pumping and \nradiative contributions to the measured damping. The blue line is the intr insic damping calculated \nfrom the Ebert -Mankovsky theory ,1 where the blue circles are the values for the pure elements  at \n300K . The green  line is the calculated damping for the Ni -Fe alloys  by Starikov , et al.2 The inset \nin (b) depicts the damping in a smaller concentration window in order to better depict the small \nfeatures in the damping around the ph ase transition. The damping  for the Co -Fe alloys, calculated \nby Turek et al.3 is plotted as the orange  line. For the Ni -Co alloys the damp ing calculated by th e \nspin density of the respective alloy weighted bulk damping55 (purple dashed line).  \n8  \n \nThis scatter in the experimental data reported in the literature and its divergence from calculated \nvalues of the damping shows th e necessity to determine  the intrinsic damping  αint by quantification \nof all extrinsic contributions to the measured total damping α tot.  \nThe first extrinsic  contribution to the damping  that we consider  is the radiative damping α rad, \nwhich is caused by ind uctive coupling between sample and  waveguide , which results in energy \nflow from the sample back into the microwave circuit.23 αrad depends directly  on the measurement \nmethod and geometry. The effect is easily understood , since the strength of the inductive coupling \ndepends on the inductance of the FMR mode itself , which is in turn determined by the saturation \nmagnetization, sampl e thickness, sample  length, and waveguide width.  Assuming a homogeneous \nexcitation field , a uniform magnetization profile throughout the sample , and negligible spacing \nbetween the waveguide and sample , αrad is well approximated by23  \n 𝛼rad=𝛾𝑀𝑠𝜇02𝛿𝑙\n16 𝑍0𝑤𝑐𝑐, (2) \nwhere l (= 10 mm  in our case) is the sample length on the waveguide, wcc (= 100µm) is the width \nof the co -planar wave guide center conductor and Z0 (= 50 Ω), the impedance of the waveguide. \nThough inh erently small for most thin films , αrad can become  significant  for alloys with \nexceptionally small intrinsic damping and /or high saturation magnetization.  For example, it plays \na significant role (values of αrad ≈ 5x10-4) for the whole composition range of  the Co -Fe alloy system \nand the Fe -rich side of the Ni -Fe system. On the other hand, for pure Ni and Permalloy (Ni 80Fe20) \nαrad comprises only 3 % and 5  % of αtot, respectively.  \nThe second non -negligible contribution to the damping  that we consider  is the interfacial \ncontribution to the measured  damping , such as  spin-pumping into the adjacent Ta/Cu bilayers . Spin \npumping  is proportional to the reciprocal sample thickness as described in24 \n 𝛼sp= 2𝑔eff↑↓𝜇𝐵𝑔\n4𝜋𝑀s𝑡. (3) \nThe spectroscopic  g-factor and the saturation magnetization Ms of the alloys were reported  in \nRef [66] and the factor of 2 accounts for the presence of two nominally identical interfaces of the \nalloys in the cap and seed layers.  In Fig.  4 (a)-(c) we plot the damping dependence on reciprocal \nthickness  1/t for select alloy concentrations, which allows us to determine  the effective spin mixing \nconductance 𝑔eff↑↓  through fits to Eq. (3) . The effective spin m ixing conductance contains details of \nthe spin transport in the adjacent non -magnetic layers, such as the interfacial spin mixing \nconductance, both the conductivity and spin diffusion for all the non -magnetic layers with a non -\nnegligible spin accumulation,  as well as the details of the spatial profile for the net spin \naccumulation .56,57 The values of 𝑔eff↑↓, are plotted versus the alloy concentration in Fig. 4 (d), and are \nin the range of previously reported values  for samples prepare d under similar growth conditions55–\n59. Intermediate values of 𝑔eff↑↓ are determined by a guide to the eye interpolation [ grey lines, Fig.  4 \n(d)] and αsp is calculated for all alloy concentrations utilizing  those interpolated values.  \nThe data for 𝑔eff↑↓  in the NixFe1-x alloys shows approximately a factor two  increase of 𝑔eff↑↓ between \nNi concentrations of 30  % Ni and 50  % Ni, which we speculate to occur at the fcc to bcc phase \ntransition around 30  % Ni. According to this line of speculation , the previously mentioned step  \nincrease in the measured total damping at the NixFe1-x phase transition can be fully attributed to the \nincrease in spin pumping at the phase transition.  In CoxFe1-x, the presence of a step in  𝑔eff↑↓   at the \nphase transition is not confirmed, given the measurement precision, although  we do observe an \nincrease in the effective spin mixing conductance when transitioning from the bcc to fcc phase.  The 9 concentration dependence of 𝑔eff↑↓  requires further thorough investigation and we therefore restrict \nourselves to reporting the expe rimental findings.  \n \n \n           \n \nFigure 4: The damping for the thickness series at select alloy compositions vs. 1/ t for (a) Ni -Co, \n(b) Ni -Fe and (c) Co -Fe (data points, concentrations denoted in the plots), with linear fits to Eq. \n(3) (solid lines). (d) The extracted effective spin mixing conductance 𝑔eff↑↓  for the measured alloy \nsystems, where the gray lines show the  linear  interpolations for intermediate alloy concentrations. \nThe data for the Co -Fe system are taken from Ref.[38].  \n \n \n Eddy -current damping13,14 is estimated  by use of  the equations given in Ref.  [23]  for films  \n10 nm thick or less . Eddy currents are neglected because they are found to be less than 5  % of the \ntotal damping. Two -magnon scattering  is disregarded  because the mechanism is largely e xcluded \nin the out -of-plane measurement geometry15–17. The total measured damping  is therefore well \napproximated as the sum   \n 𝛼tot≅𝛼int+𝛼rad+𝛼sp, (4) \nWe determine  the intrinsic damping of the material by subtracting α sp and α rad from the measured \ntotal damping , as shown in Fig. 3 .  \n10 The intrinsic damping increases monotonically with Ni concentration  for the NixCo1-x alloys . \nIndicative of the importance of extrinsic sources of damping, αint is approximately 40 % smaller \nthan αtot for the Fe -rich alloy, though the difference decreases to only 15 % for pure Ni. This \nbehavior is expected, given that both αrad and αsp are proportional to  Ms. A comparison of αint to the \ncalculations by Mankovsky , et al. ,1 shows excellent quantitative agreement  to within 30 %.  \nFurthermore, w e compare αint of the NixCo1-x alloys to the spin density weighted average of the \nintrinsic damping of Ni and C o [purple dashed line in Fig. 3 (a)] , which gives good agreement with \nour data, as previously reported .55  \nαint for NixFe1-x (Fig. 3 (b)) also increases with Ni concentration after a small initial decrease \nfrom pure Fe to the first NixFe1-x alloys. The step increase found in αtot at the bcc to fcc phase \ntransition is fully attributed to αsp, as detailed in the previous section, and therefore does not occur \nin αint. Similar to the NixCo1-x system αint is significantly lower than αtot for Fe -rich alloys. With in \nerror bars, a  comparison to the calculations by Mankovsky , et al.1 (blue line) and Starikov , et al.2 \n(green line) exhibit excellent agreement in the fcc phase, with marginally larger deviations in the \nNi rich regime. Starikov , et al.2 calculated the damping over the ful l range of compositions, under \nthe assumption of continuous  fcc phase. This calcu lation deviates further from  our measured αint in \nthe bcc phase exhibiting qualitatively different behavior.  \nAs previously reported, t he dependence of αint on alloy compositio n in the CoxFe1-x alloys \nexhibits strongly non -monotonic behavior, differing from the two previously discussed alloys.38  \nαint displays a minimum at 25 % Co concentration with a, for conducting ferromagnets \nunprecedented, low value of int (5±1.8)  × 10-4. With increasing Co concentration , αint grows up \nto the phase transition, at which point  it increases by 10  % to 20 % unt il it reaches  the value for \npure Co.  It was shown  that αint scales with the density of states  (DOS)  at the Fermi energy n( EF) in \nthe bcc phase38, and the DOS also exhibits a sharp  minimum for Co 25Fe75. This scaling is \nexpected60,61 if the damping is dominated by the breathing Fermi surface process. With the \nbreathing surface model,  the intraband scattering  that leads to damping  directly scales with n( EF). \nThis scaling is particularly pronounc ed in the Co -Fe alloy system due to the small concentration  \ndependence of the spin -orbit coupling on alloy composition. The special properties of the CoxFe1-x \nalloy system are discussed  in greater detail in Ref.[38]. \nComparing αint to the calculations by Mankovsky  et al.1, we find good quantitative \nagreement with the value of the minimum. However, t he concentration of the minimum is \ncalculated to occur  at approximately 10 % to  20% Co, a slightly lower value than 25 % Co t hat we \nfind in this study. Furthermore , the strong concentration dependence around the minimum is not \nreflected in the calculations. More recent calculations by Turek et al.3, for the bcc CoxFe1-x alloys \n[orange line in Fig. 3 (c)] find the a minimum of the damping of 4x10-4 at 25 % Co concentration \nin good agreement with our experiment, but there is some deviation in concentration dependenc e \nof the damping around the minimum. Turek et al.3 also reported on the damping in the NixFe1-x \nalloy system, with similar qualitative and  quantitative results as the other two presented quantitative \ntheories1,2 and the results are therefore not plotted  in Fig.  3 (b) for the sake of comprehensibility of \nthe figure.  For both NixFe1-x and the CoxFe1-x alloys , the calculated spin density weighte d intrinsic \ndamping of the pure elements (not plotted) deviates significantly from the determined intrinsic \ndamping of the alloys, in contrary to the  good agreement  archived  for the  CoxNi1-x alloys. We \nspeculate that this difference between the alloy syste ms is caused by the non -monotonous \ndependence of the density of states at the Fermi Energy in the CoxFe1-x and NixFe1-x systems.  \nOther  calculated damping values for the pure elements and the Ni80Fe20 and Co 90Fe10 alloys \nare compared to the determined intr insic damping in Table 1. Generally , the calculations \nunderestimate the damping significantly, but our data are in good agreement with more recent \ncalculations for Permalloy ( Ni80Fe20).   11 It is important to point out that n one of the theories  considered he re include thermal \nfluctuations . Regardless, we find exceptional agreement with the calculations to αint at intermediate \nalloy concentrations . We speculate that the modeling of atomic disorder in the alloys in the \ncalculations, by the coherent potential approximation (CPA)  could be responsible for this \nexceptional agreement. The effect of disorder on the  electronic band structure possibly dominates \nany effect s due to nonzero temperature. Indeed, both effects cause a broadening of the bands due \nto enhanced  momentum scattering rates. This directly correlates to a change of the damping \nparameter  according to  the theory of Gilmore and Stiles9. Therefore , the inclusion of the inherent  \ndisorder  of solid -solution alloys  in the calculations  by Mankovsky et al1 mimic s the effects of \ntemperature on damping  to some extent . This argument is corroborated by the fact that the \ncalculations  by Mankovsky et al1 diverge for diluted alloys and pure elements  (as shown in Fig. 2 \n(c) for pure Fe) , where no  or to little disorder is introduced to account for temperature effects. \nMankovsky et al.1 performed temperature dependent calculations of the damping for pure bcc Fe, \nfcc Ni and hcp Co  and the values for 300 K are shown in Table 1 and Fig. 3. These calculations  for \nαint at a temperature of 300 K  are approximately a factor of two  less than our measured values , but \nthe agreement is  significantly improved relative to those  obtained by calculations that neglect \nthermal fluctuations . \n \n \n Figure 5: The intrinsic damping α int is plotted against ( g-2)2 for \nall alloys. We do not observe a proportionality between α int and \n(g-2)2. \n12 Finally, i t has been  reported45,64 that there is a  general proportionality between αint and (g-\n2)2 , as contained in the original microscopic BFS model proposed by  Kambersky .62 To examine \nthis relationship, w e plot αint versus (g-2)2 (determined in Ref [66]) for all samples measured here  \nin Figure 5 . While some samples with large values for ( g-2)2 also exhib it large αint, this is not a \ngeneral trend for all the measured samples . Given  that the damping is not purely a function of the \nspin-orbit strength, but also depends on the details of the band structure , the result  in Fig.  5 is \nexpected .  For example , the amount of  band overlap  will determine the amount of interband \ntransition leading to that damping channel.  Furthermore, the density of states at the Fermi energy \nwill affect the intraband contribution to the damping9,10.   Finally , the ratio of inter - to intra -band \nscattering that mediate s damping contributions at a fixed temperature (RT for our measurements) \nchanges  for different elements9,10 and therefore with alloy concentration. None of these f actors are \nnecessarily proportional to the spin -orbit coupling .  Therefore , we conclude that this simple \nrelation, which originally traces to an order of magnitude estimate  for the case of spin relaxation \nin semiconductors65, does not hold for  all magnetic systems  in general.   \n \n4  Summary  \nWe determined  the damping for the full compositi on range of the binary  3d transition  metal all oys \nNi-Co, Ni -Fe, and Co -Fe and showed that the measured damping can be explained by three \ncontributions to the damping: Intrinsic damping, radiative damping and damping due to spin \npumping. By quantifying all  extrinsic contributions to the measured damping, we determine  the \nintrinsic damping over the whole range of alloy compositions .  These values are compared  to \nmultiple theoretical calculations and yield excellent qualitative and good quantitative agreement for \nintermediate alloy concentrations. 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Schoen, Martin A. W. , Lucassen, Juriaan , Nembach, Hans T. , Silva, T. J. , Koopmans, Bert , Back, Christian H. , \nShaw, Justin M.  Magnetic properties of ultra -thin 3d transition -metal binary alloys I: spin and orbital mo ments, \nanisotropy, and confirmation of Slater -Pauling behavior . arXiv:1701.02177  (2017 ).  \n \n \n "    },    {        "title": "1007.0856v2.Magneto_elastic_oscillations_and_the_damping_of_crustal_shear_modes_in_magnetars.pdf",        "content": "Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 26 October 2018 (MN L ATEX style file v2.2)\nMagneto-elastic oscillations and the damping of crustal shear modes\nin magnetars\nMichael Gabler1;2;3, Pablo Cerd ´a-Dur ´an1, Jos´e A. Font2, Ewald M ¨uller1\nand Nikolaos Stergioulas3\n1Max-Planck-Institut f ¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany\n2Departamento de Astronom ´ıa y Astrof ´ısica, Universidad de Valencia, 46100 Burjassot (Valencia), Spain\n3Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece\n26 October 2018\nABSTRACT\nIn a realistic model of magneto-elastic oscillations in magnetars, we find that crustal shear\noscillations, often invoked as an explanation of quasi-periodic oscillations (QPOs) seen af-\nter giant flares in soft gamma-ray repeaters (SGRs), are damped by resonant absorption on\ntimescales of at most 0.2s, for a lower limit on the dipole magnetic field strength of 5\u00021013G.\nAt higher magnetic field strengths (typical in magnetars) the damping timescale is even\nshorter, as anticipated by earlier toy-models. We have investigated a range of equations of state\nand masses and if magnetars are dominated by a dipole magnetic field, our findings exclude\ntorsional shear oscillations of the crust from explaining the observed low-frequency QPOs. In\ncontrast, we find that the Alfv ´en QPO model is a viable explanation of observed QPOs, if the\ndipole magnetic field strength exceeds a minimum strength of about several times 1014G to\n1015G. Then, Alfv ´en QPOs are no longer confined to the fluid core, but completely dominate\nin the crust region and have a maximum amplitude at the surface of the star.\nKey words: MHD - stars: magnetic field - stars: neutron - stars: oscillations - stars: flare -\nstars: magnetars\n1 INTRODUCTION\nThe observation of giant flares in soft gamma-ray repeaters (SGRs;\ncompact objects with very strong magnetic fields or magnetars\n(Duncan & Thompson 1992)) may open a gateway towards the ex-\nciting field of neutron star seismology. In the decaying X-ray tail\nof two such events, SGR 1900+14 and SGR 1806-20, a number\nof long-lasting, quasi-periodic oscillations (QPOs) have been ob-\nserved (see Israel et al. (2005) and Watts & Strohmayer (2007) for\nrecent reviews). Early models interpreted the observed QPO fre-\nquencies as directly related to torsional shear oscillations of the\nsolid crust of a neutron star excited during a giant flare event (see\nDuncan (1998); Strohmayer & Watts (2005); Piro (2005); Sotani\net al. (2007); Samuelsson & Andersson (2007), and references\ntherein) raising hopes that through their identification, the physi-\ncal properties of the crust could be probed (Steiner & Watts 2009).\nDue to the extremely strong magnetic fields present in magnetars,\nhowever, a self-consistent model that includes global magnetohy-\ndrodynamic (MHD) oscillations interacting with the shear oscilla-\ntions of the crust is required (Levin (2006), Glampedakis & An-\ndersson (2006), Levin (2007), Lee (2007, 2008)). In a highly sim-\nplified model, Levin (2007) showed that shear oscillations will be\nabsorbed by an MHD continuum of Alfv ´en oscillations, while long-\nlived QPOs may still appear at the turning points or edges of the\ncontinuum.Sotani et al. (2008b) and subsequently Cerd ´a-Dur ´an et al.\n(2009) (see also Colaiuda et al. (2009)), using a more realis-\ntic, general-relativistic MHD model but still ignoring an extended\ncrust, found two families of Alfv ´en QPOs related to turning points\nof the frequency of torsional Alfv ´en waves near the magnetic pole\nand inside a region of closed magnetic field lines near the equa-\ntor. Each QPO family consists of two sub-families differing by\ntheir symmetry behaviour with respect to the equatorial plane. The\nresults of the numerical simulations were explained by a semi-\nanalytic model based on standing waves in the short-wavelength\nlimit (Cerd ´a-Dur ´an et al. 2009). The Alfv ´en QPO model is very\nattractive, because it reproduces the near-integer-ratios of the ob-\nserved 30, 92 and 150 Hz frequencies in SGR 1806-20, at magnetic\nfield strengths expected for magnetars. In this model, the observed\nSGR QPOs, rather than probing the crust, yield information on the\nmagnetic field and the compactness of the star.\nThe omission of an extended crust in the previous studies of\nSotani et al. (2008b), Cerd ´a-Dur ´an et al. (2009), and Colaiuda et al.\n(2009) can be considered as a limiting case of a very strong mag-\nnetic field. For intermediate magnetic field strengths, however, an\nunderstanding of magnetar oscillations requires the inclusion of\ncrust-core coupling. In this Letter, we present the first such real-\nistic simulations of coupled, magneto-elastic oscillations. We use a\ngeneral-relativistic framework, a dipolar magnetic field, and a tabu-\nc\r0000 RASarXiv:1007.0856v2  [astro-ph.HE]  24 Nov 20102 Michael Gabler, Pablo Cerd ´a-Dur ´an, Jos ´e A. Font, Ewald M ¨uller and Nikolaos Stergioulas\nlated equation of state (EOS) for dense matter. The numerical sim-\nulations are based on state-of-the-art Riemann solver methods for\nboth the interior MHD fluid and the crust. A recent study by van\nHoven & Levin (2010) also takes entanglement of magnetic field\nlines into account, thereby generalising the toy model of Levin\n(2007). Some first results on coupled crust-core oscillations also\nappeared in Kokkotas et al. (2010).\nWe use units where c=G= 1 withcandGbeing the speed\nof light and the gravitational constant, respectively. Latin (Greek)\nindices run from 1 to 3 (0 to 3).\n2 THEORETICAL FRAMEWORK\nThe present study of torsional oscillations of magnetars is based on\na numerical integration of the general relativistic MHD equations.\nAs in Cerd ´a-Dur ´an et al. (2009), who considered purely Alfv ´en\noscillations of the fluid core, assume (i) a zero temperature EOS,\n(ii) axisymmetry, (iii) a purely poloidal magnetic field configura-\ntion, (iv) the Cowling approximation, (v) a spherically symmetric\nbackground, and (vi) small amplitude oscillations. Because of as-\nsumptions (ii) and (iii) polar oscillations decouple from axial ones\nin the linear regime. Therefore, we concentrate on purely axial os-\ncillations and evolve the '-component of the evolution variables\nonly. We assume a conformally flat metric1\nds2=\u0000\u000b2dt2+\u001e4\u0000\ndr2+r2d\u00122+r2sin\u00122d'2\u0001\n; (1)\nwhere\u000bis the lapse function and \u001ethe conformal factor, and con-\nsider a stress-energy tensor T\u0016\u0017of the form\nT\u0016\u0017=T\u0016\u0017\n\ruid+T\u0016\u0017\nmag+T\u0016\u0017\nelas\n=\u001ahu\u0016u\u0017+Pg\u0016\u0017+b2u\u0016u\u0017+1\n2b2g\u0016\u0017\u0000b\u0016b\u0017\n\u00002\u0016S\u0006\u0016\u0017; (2)\nwhere\u001ais the rest-mass density, hthe specific enthalpy, Pthe\nisotropic fluid pressure, u\u0016the 4-velocity of the fluid, b\u0016the mag-\nnetic field measured by a co-moving observer (with b2:=b\u0016b\u0016),\n\u0006\u0016\u0017the shear tensor, and \u0016Sthe shear modulus. The latter is ob-\ntained according to Sotani et al. (2007).\nThe conservation of energy and momentum r\u0017T\u0016\u0017= 0, and\nthe induction equation lead to the following system of evolution\nequations\n1p\u0000g\u0012\n@p\rU\n@t+@p\u0000gFi\n@xi\u0013\n= 0; (3)\nwheregand\rare the determinants of the 4-metric and 3-metric,\nrespectively. The two-component state and flux vectors are given\nby\nU= [S'; B']; (4)\nFr=h\n\u0000b'Br\nW\u00002\u0016S\u0006r\n';\u0000v'Bri\n; (5)\nF\u0012=\u0014\n\u0000b'B\u0012\nW\u00002\u0016S\u0006\u0012\n';\u0000v'B\u0012\u0015\n; (6)\nwhereBiare the magnetic field components as measured by an Eu-\nlerian observer (Ant ´on et al. 2006), and W=\u000butis the Lorentz\nfactor. The shear tensor \u0006i'= 1=2gii\u0018'\n;icontains the spatial\nderivatives (denoted by a comma) of the fluid displacement \u0018'due\n1This provides a very good approximation as our neutron star models are\nalmost perfectly spherically symmetric except for very small deviations due\nto the presence of an axisymmetric magnetic field.to the oscillations, which are related to the fluid 4-velocity accord-\ning to\u0018'\n;t=\u000bv'=u'=ut, wherev'is the'-component of the\nfluid 3-velocity. Hence, the evolution of the spatial derivatives \u0018'\n;r\nand\u0018'\n;\u0012is given by\n(\u0018'\n;k);t\u0000(\u000bv');k= 0 with k2fr;\u0012g: (7)\nWe also need to provide boundary conditions. At the surface\nthe radial derivative of the displacement has to vanish ( \u0018'\n;r= 0),\nas we assume a continuous traction and vanishing surface currents.\nAt the crust-core interface we demand the continuity of the par-\nallel electric field, which implies a continuous displacement \u0018'.\nTogether with the continuity of the traction the latter leads to a re-\nlation between the radial derivatives of the displacement in the core\nand in the crust: \u0018'\ncore;r= (1 +\u000e)\u0018'\ncrust;rwith\u000e=\u0016S=(brbr).\nTo construct equilibrium models we choose between differ-\nent barotropic EOSs for the core that are matched to an EOS for\nthe crust. The available models for the core are the soft EOS A\n(Pandharipande 1971), the intermediate EOS WFF3 (Wiringa et al.\n1988), EOS APR (Akmal et al. 1998) and the stiff EOS L (Pand-\nharipande & Smith 1975). For the low density region of the crust\nwe choose between EOS NV (Negele & Vautherin 1973) and EOS\nDH (Douchin & Haensel 2001). Details about the different combi-\nnations of EOSs can be found in Sotani et al. (2007). We use a ref-\nerence model with a mass of 1:4M\fand a circumferential radius\nRstar= 12:26km, described by the APR +DH EOS. In contrast\nto Sotani et al. (2007) we compute magnetised equilibrium models\nusing the LORENE library ( www.lorene.obspm.fr ).\nOur simulation code is an extended version of the GRMHD\ncode presented in Cerd ´a-Dur ´an et al. (2009). It includes the shear\nterms as they appear in (2), and the evolution of the displacements\n(7). The proper working of the MHD part of the code was demon-\nstrated in Cerd ´a-Dur ´an et al. (2008). To test the extended code we\ncompared its results obtained for two limiting cases, zero magnetic\nfield and zero shear modulus, with those of previous studies. The\npurely crustal shear oscillations presented in Sotani et al. (2007) are\nrecovered with an agreement of 1%, and the Alfv ´en continuum is\nobtained naturally as in Cerd ´a-Dur ´an et al. (2009). Further details\non the derivation of the model equations, the numerical methods,\nand the code tests will be discussed in Gabler, Cerd ´a-Dur ´an, Font\n& Stergioulas (in preparation).\n3 DAMPING OF CRUSTAL SHEAR MODES\nTo study the behaviour of coupled crust-core oscillations we per-\nturb the equilibrium stellar model by imposing a velocity pertur-\nbation and then follow the time evolution of the system (3) - (7).\nUnless stated otherwise, we use a grid of 150 \u0002100 zones in our\nsimulations covering a domain [0;Rstar]\u0002[0;\u0019]. The angular grid\nis equidistant, while the radial grid is equidistant only in the crust,\nwhere 40 per cent of the zones are located, and coarsens towards\nthe centre of the star. Symmetries are exploited whenever a pertur-\nbation is of purely odd or even parity with respect to the equatorial\nplane. We use the term damping in the following to refer to resonant\nabsorption of crustal shear oscillations by the Alfv´en continuum of\nthe core (unless we explicitly refer to numerical damping caused\nby finite-differencing).\nWe investigated perturbations of different radial extent encom-\npassing only the crust, only the core, or the whole star. Since all\nthree types of perturbations give qualitatively similar results, we\nfocus on whole star perturbations in the following, as this is the\nmost generic case. We first consider initial perturbations consist-\ning of a single torsional, spherical vector harmonics l-mode. To\nc\r0000 RAS, MNRAS 000, 000–000Magneto-elastic oscillations and the damping of crustal shear modes in magnetars 3\n0 250 500 750 1000\ntime  [ms]10-410-310-210-1100normalised overlap integralsl=2\nl=3\nl=9\n0 100 200 300 400 500\ntime [ms]-5e-1505e-15overlap integral l=20G\n5×1013G\n5×1014G\n050100150200250\ntime [ms]-4e-15-2e-1502e-154e-15overlap integral l=90G\n5×1013G\n5×1014G\nFigure 1. Time evolution of overlap integrals with the eigenmodes of the crust. Left panel : Damping of l= 2;3, and 9initial perturbations due to resonant\nabsorption of the fundamental ( n= 0) crustal shear mode for a magnetised model with 5\u00021013G (dots). In the corresponding unmagnetised models (solid\nlines) only numerical damping occurs which increases with the angular order lof the mode. Middle panel : Overlap integrals for the l= 2 mode, where\nresonant absorption of the crustal modes becomes stronger with increasing magnetic field strength. Right panel : same as middle panel, but for l= 9.\nl 2 3 9 10\n\u001c[s] forB= 0 5.500 4.620 0.260 0.170\n\u001c[s] forB= 5\u00021013G 0.072 0.080 0.087 0.081\n\u001c[s] forB= 1014G 0.040 0.040 0.022 0.021\nTable 1. Damping timescales \u001cdue to resonant absorption of crustal shear\nmodes by the Alfv ´en continuum for various initial perturbation modes l.\ninvestigate the damping of a single crustal mode we compute over-\nlap integrals of the evolved variables with mode eigenfunctions\n(the latter are found by solving the linear eigenvalue problem for\nthe crustal modes, see Schumaker & Thorne (1983); Messios et al.\n(2001); Sotani et al. (2007)). Because the eigenmodes of the crust\nform a complete orthonormal set we can expand any perturbation\nin terms of the corresponding eigenfunctions. The expansion fac-\ntors, which provide a measure of how strong each crustal mode\ncontributes to the perturbation, are obtained via the overlap inte-\ngrals with the eigenfunctions. For more details on this method see\nGabler et al. (2009). In the left panel of Fig. 1 we show the max-\nimum (absolute) amplitudes of the overlap integrals for different\ninitial perturbations and for simulations both without magnetic field\n(solid lines) and with a polar magnetic field of 5\u00021013G (dots) .\nIn the field-free case the lines represent the numerical damping of\ncrustal modes due to finite-differencing. When a magnetic field is\npresent, the damping (now due to resonant absorption) increases\nwith the magnetic field strength.\nFor all modes, the timescale of resonant absorption is much\nshorter than that of numerical damping (see Table 1). After about\n500 ms, the overlap integrals no longer sample the crust oscilla-\ntions, but instead the magneto-elastic oscillations which then dom-\ninate the evolution (see below).\nThe middle and right panels of Fig. 1 show the overlap inte-\ngrals forl= 2 andl= 9 modes of the crust as a function of time\nfor different magnetic field strengths, respectively. For l= 2 and\nB= 5\u00021013G we find almost complete damping of the crustal\nmode after\u00180:5s. For a stronger magnetic field ( B= 5\u00021014G)\nthe crustal mode becomes already damped after less than one oscil-\nlation, and only the dominant magneto-elastic oscillations remain.\nSimilar statements hold for the evolution of the l= 9perturbation.\nHowever, in that case it takes a few oscillations before the crustal\nmode is damped, even for B= 5\u00021014G.\nWe also analysed a more general initial perturbation consist-\ning of a mixture of l= 2 up tol= 10 modes, which excites a\nlarge number of crustal modes of different angular order land ra-EOS \u001c[ms] atB= 1014G\nn= 0,l= 2n= 0,l= 3n= 0,l= 9\nA+DH 1.6 23 23 15\nA+NV 1.6 39 43 75\nAPR+DH 2.0 32 31 17\nAPR+NV 2.0 50 54 80\nL+DH 1.6 50 52 24\nL+NV 1.6 79 86 53\nL+DH 2.0 44 46 21\nL+NV 2.0 73 80 87\nW+DH 1.6 29 30 14\nW+NV 1.6 53 58 69\nTable 2. Damping timescales \u001cdue to resonant absorption of crustal shear\nmodes by the Alfv ´en continuum for initial perturbation modes l= 2,l= 3\nandl= 9 and for different combinations of equations of state at B=\n1014G. The number in the labelling of the EOS represents the mass of the\nneutron star model in M \f\n0 200 400 600800 1000\ntime [ms]-4e-17-3e-17-2e-17-1e-1701e-172e-17velocity ( αvϕ)\n0G\n5×1013G\n5×1014G\nFigure 2. Evolution of \u000bv'in the crust at \u0012=\u0019=4for initial data contain-\ning a large number of different perturbation modes. For vanishing magnetic\nfield, only crustal modes are excited. For B= 5\u00021013G the crustal os-\ncillations are strongly damped, and for a ten times stronger magnetic field\n(B= 5\u00021014G) magneto-elastic oscillations dominate the evolution from\nthe start.\ndial ordern. Fig. 2 shows the resulting evolution of the velocity in\nthe crust at\u0012=\u0019=4. For the unmagnetised model, low frequency,\nfundamental ( n= 0) oscillations can easily be distinguished from\nhigher frequency overtones with n>1. When increasing the mag-\nnetic field strength to 5\u00021013G the lower frequency modes are\ncompletely damped after \u0018250ms, whereas the higher frequency\nc\r0000 RAS, MNRAS 000, 000–0004 Michael Gabler, Pablo Cerd ´a-Dur ´an, Jos ´e A. Font, Ewald M ¨uller and Nikolaos Stergioulas\n0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 2 4 6 8 100 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10\nc d\ni j k l1.5Hz 1.7Hz\n3.4Hz 3.8Hz 5.1Hz 5.1Hz\n6.8Hz 7.8Hz 10Hzf g h2.7Hzb\neay [km] y [km] y [km]\nx [km] x [km] x [km] x [km]2.9Hz\n7.7HzU0(−)\n0(±)L E1(+)\nU1(−)U0(+)\n1(±)L U1(+)E2(+)\nU2(+)E3(+)U2(−)E4(+)\nFigure 3. Spatial distribution of Fourier amplitudes at several QPO frequencies. Three different types of QPOs are present: (i) upper QPOs in panels a, d, f, g,\ni, and k; (ii) lower QPOs in panels bande; (iii) edge QPOs in panels c, h, j , and l. The two black dashed lines mark the location of the crust, while the blue\nlines represent magnetic field lines. The colour scale ranges from zero amplitude (white) to maximum amplitude (black).\novertones survive for a longer time. In the long run, a low frequency\n(\u00180:3Hz) magneto-elastic oscillation dominates. At the largest\nmagnetic field strength shown here, 5\u00021014G, there is no sign\nof either low or high frequency crustal modes, the evolution being\ncompletely dominated by magneto-elastic oscillations.\nDamping timescales of n= 0,l=f2;3;9gmodes for dif-\nferent EOSs are listed in Table 2. For all EOSs and masses stud-\nied here, we find significant damping of the crustal shear modes at\nB= 1014G. Up to the highest-order l= 9 mode, the damping\ntimescales are shorter than 0.1s. From the results in Tables 1 and\n2, we deduce that even at B= 5\u00021013G, which is up to two or-\nders of magnitude weaker than assumed magnetar field strengths,\nthe damping timescale is shorter than 0.2s , which is more than two\norders of magnitude shorter than the duration of observed QPOs.\n4 LONG-TERM QPOS\nBesides the damping of crustal modes, we observe long-lasting os-\ncillations in the fluid core of the magnetar. These long-term QPOs\nare identified by local maxima in Fourier space. Let us consider\nan intermediate magnetic field strength of 4\u00021014G, where both\nthe magnetic field and the crust influence the dynamics. As in the\ncase without crust (Sotani et al. 2008b; Cerd ´a-Dur ´an et al. 2009)\nwe find two different families of long-term QPOs, as demonstrated\nin Fig. 3, which shows the spatial distribution of Fourier amplitudes\nat several QPO frequencies. The lower QPOs (L(\u0006)\nn) are located in-\nside the region of closed field lines, while the upper QPOs (U(\u0006)\nn)\nconcentrate along open magnetic field lines closer to the poles. We\ncomputed QPOs of either odd (\u0000)or even (+) parity w.r.t. the\nequatorial plane, which allows for a better identification of QPOs\nof similar frequency but opposite parity.\nThe lower QPOs (Fig. 3, panels bande) appear to be similar to\nthose found for models without crust, except that they are limited tothe region of closed magnetic field lines inside the core. The upper\nQPOs are influenced by the presence of the crust in several ways.\nFirst, they are limited to the fluid region, and become vanishingly\nsmall at the base of the crust (Fig. 3, panels a, d, f, g, i , and k), where\nthey are reflected. This behaviour is similar to that caused by the\nboundary conditions in Sotani et al. (2008b), and differs from that\nof the pure fluid case considered in Cerd ´a-Dur ´an et al. (2009). In\nthe latter work the continuous traction boundary condition imposed\nat the surface of the star resulted in a strong displacement there.\nTheir different behaviour at the base of the crust (w.r.t. the pure\nfluid case in Cerd ´a-Dur ´an et al. (2009)) causes a rearrangement of\nthe QPOs. Here, the lowest frequency QPO (panel a) is symmetric\nw.r.t. the equatorial plane (even parity) - while it did not exist at all\nin Cerd ´a-Dur ´an et al. (2009) - and the lowest-frequency QPO has\nodd parity.\nWhile QPOs are located close to the symmetry axis of the field\n(polar axis) in models without crust (Sotani et al. 2008b; Cerd ´a-\nDur´an et al. 2009), they are attached to field lines crossing the\nequator at around 4km in our models including the crust. Appar-\nently the strong coupling introduced by the crust complicates the\noscillatory behaviour of the field lines, such that the interaction be-\ntween neighbouring polar field lines prevents the QPOs from being\nestablished.\nFurthermore, we find a new family of QPOs (Fig. 3, panels c,\nh, j, and l) connected to the last open field line of the fluid core,\neach member representing the lower-frequency edge of an Alfv ´en\ncontinuum along the open field lines. QPOs at similar locations\nwere also identified in simulations by Colaiuda et al. (2009) without\na crust.\nc\r0000 RAS, MNRAS 000, 000–000Magneto-elastic oscillations and the damping of crustal shear modes in magnetars 5\n5 DISCUSSION\nIn this Letter we have presented the first numerical simulations of\naxisymmetric, torsional, magneto-elastic oscillations in a realistic\nmagnetar model, including an extended crust.\nWe focus on the timescales for resonant absorption of low\nfrequency,n= 0 crustal shear modes by the Alfv ´en continuum\nof the core and find that even at B= 5\u00021013G, which is up\nto two orders of magnitude weaker than assumed magnetar field\nstrengths, the damping timescale is shorter than 0.2s, which is more\nthan two orders of magnitude shorter than the duration of observed\nQPOs. Furthermore, the crust EOSs NV and DH which we have\nused have a very high shear modulus, compared to a range of other\nproposed EOSs (Steiner & Watts 2009). Comparing the damping\ntimescales obtained here for the NV and DH EOSs, we find that,\nat a given magnetic field strength, a lower shear modulus results\nin a shorter damping timescale due to resonant absorption. Thus,\nfor lower values of the shear modulus than considered here, the de-\nduced damping timescales would be even shorter than our current\nfindings. In addition, no significant excitation of crustal modes by\nthe Alfv ´en continuum was observed at the end of our simulations,\nwhich reached up to several seconds. The above results do not leave\nmuch room for shear modes of the crust to be able to sustain oscil-\nlations that lead to significant modulations in the X-ray tail of giant\nSGR bursts lasting for several tens of seconds, when the magnetic\nfield is global. The case of magnetic field configurations confined\nto the crust, which may be realised if the core is a type I supercon-\nductor, requires further investigation.\nIn contrast to the shear modes, the Alfv ´en QPO model (see\nLevin (2007); Sotani et al. (2008b); Cerd ´a-Dur ´an et al. (2009); Co-\nlaiuda et al. (2009)) has several attractive features, matching to ob-\nserved frequencies for a dipole field strength up to several times\n1015G and explaining the observed integer ratios for the frequen-\ncies of QPOs. Here we find that for magnetic field strengths less\nthan about 1015G, the Alfv ´en QPOs are mostly confined to the\nfluid core. For EOSs with a smaller shear modulus, this could re-\nduce to several times 1014G. Above this minimum field strength ,\nAlfv ´en oscillations completely dominate in the crust and the result-\ning QPOs have a maximum amplitude at the surface of the star.\nIt is thus likely, that the Alfv ´en QPO model can operate efficiently\nonly above such a minimum field strength. In that case, the possible\nstrength of magnetar magnetic fields is limited to a narrow range,\nbetween the minimum strength discussed here and the upper bound\ndetermined in Cerd ´a-Dur ´an et al. (2009). Furthermore, the empir-\nical formulas for the QPO frequencies presented in Cerd ´a-Dur ´an\net al. (2009) could be used to directly constrain the strength of the\nmagnetic field and the compactness of the star.\nA more extended investigation of the minimum magnetic field\nstrength for the Alfv ´en QPO model will appear in Gabler et al.\n(2010). We are also planning to extend our model by taking into\naccount additional effects, such as different magnetic field topolo-\ngies (see also Sotani et al. (2008a)), the coupling of the interior\ndynamics to an external magnetosphere, and the effect of field-line\nentanglement in the core.\nACKNOWLEDGEMENTS\nWe are grateful to the anonymous referee for useful comments\nwhich helped to improve the final version of this letter. This work\nwas supported by the Collaborative Research Center on Gravita-\ntional Wave Astronomy of the Deutsche Forschungsgemeinschaft(DFG SFB/Transregio 7), the Spanish Ministerio de Educaci ´on y\nCiencia (AYA 2007-67626-C03-01), the ESF grant COMPSTAR\nand a DAAD exchange grant. Computing time was provided by the\nServicio de Inform ´atica de la Universidad de Valencia .\nREFERENCES\nAkmal A., Pandharipande V . R., Ravenhall D. G., 1998,\nPhys. Rev. C, 58, 1804\nAnt´on L., Zanotti O., Miralles J. A., Mart ´ı J. M., Ib ´a˜nez J. M.,\nFont J. A., Pons J. A., 2006, ApJ, 637, 296\nCerd ´a-Dur ´an P., Font J. A., Ant ´on L., M ¨uller E., 2008, A&A, 492,\n937\nCerd ´a-Dur ´an P., Stergioulas N., Font J. 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C, 38, 1010\nc\r0000 RAS, MNRAS 000, 000–000"    },    {        "title": "1710.03585v1.A_four_field_gyrofluid_model_with_neoclassical_effects_for_the_study_of_the_rotation_velocity_of_magnetic_islands_in_tokamaks.pdf",        "content": "A four-\feld gyro\ruid model with neoclassical e\u000bects for the study of\nthe rotation velocity of magnetic islands in tokamaks\nA. Casolari\nJune 11, 2022\n1 Introduction\nAt equilibrium, in a tokamak, magnetic \feld lines lie on surfaces forming a family of nested tori, named\nmagnetic surfaces. This structure of nested magnetic surfaces can be a\u000bected by instabilities. One of the most\nimportant ones is the so-called tearing mode, which is an instability \"tearing\" and reconnecting magnetic \feld\nlines. Magnetic reconnection locally breaks the topology of magnetic surfaces leading to a more energetically-\nfavorable con\fguration. Magnetic islands result from the nonlinear evolution of tearing modes and represent a\nserious obstacle for obtaining nuclear fusion in magnetic con\fnement devices. In fact, the breaking of magnetic\nsurfaces causes an increase in the heat and particle \ruxes. The uncontrolled growth of magnetic islands can\nalso lead to major disruptions, causing serious damage to the device.\nMany e\u000borts have been made in the past decades to develop a theory of magnetic island dynamics in tokamaks.\nThe interest in this kind of studies is to understand the conditions for the onset of the islands in the tokamak\nexperiments and to control them to prevent their growth to large amplitudes and the consequent negative e\u000bects\non con\fnement. Magnetic islands arise from the nonlinear evolution of tearing modes [1]. In the presence\nof an equilibrium density and temperature gradient, the tearing mode acquires a propagation frequency and\nthe instability is said a drift-tearing mode [2]. According to the linear drift-tearing dispersion relation, the\npropagation frequency of the instability should be close to the electron diamagnetic frequency, !\u0000!E\u0019!\u0003e\n[3, 2], where the frequency is related to the velocity through the wave vector k,!=k\u0001v. The tearing\nmode is an instability characterized by a long wavelength, which corresponds to a small wavevector. !Eis\ntheE^B-drift frequency, due to the equilibrium electric \feld. In fact, the plasma as a whole rotates with\ntheE^Bvelocity, so that this contribution must be subtracted from the island rotation velocity (Doppler\nshift). Experimental observations of magnetic islands in tokamaks, under speci\fc conditions, show a rotation\nfrequency closer to the ion diamagnetic frequency, !\u0000!E\u0019!\u0003i[4, 5, 6]. This disagreement between the\npredictions of the linear theory and the experimental observations raises doubts on the validity of the most\ncredited theoretical models describing magnetic island dynamics. According to recently developed models, in\nthe presence of signi\fcant electron temperature gradients, the introduction of the so-called \"mode inductivity\"\n[7] in the Ohm's law permits the existence of modes propagating with the ion diamagnetic frequency. This\ne\u000bect arises naturally in the linear regime, but the experimental observations of the island rotation concern\nnonlinear islands, thus a direct check of the validity of this model is not currently possible. Another widely\naccepted interpretation of the observed rotation velocity is that, when the island width becomes larger than\nthe ion-acoustic radius, the ion \ruid cannot cross the island separatrix, thus the island is forced to propagate\nwith the velocity of the ion \row [8]. This explanation works for islands which are large enough, but it cannot\naccount for the transition from one diamagnetic velocity to the other. Nonlinear island dynamics is still not\nfully understood, and the processes that determine the island rotation velocity are under investigation.\nAttempts to study the stationary rotation of magnetic islands have been made by Fitzpatrick & Waelbroeck\nin a series of papers on the subject [8, 9, 10, 11] by solving an improved version of the four-\feld model,\npreviously deduced by Hazeltine, Kotschenreuther and Morrison [12], which is a reduction of the two-\ruid\nplasma description. The result of their studies is that both the island width and the neoclassical e\u000bects in\ruence\nthe island rotation. In particular, the critical parameters which determine the island dynamics are the ratio\nbetween the island width wand the ion-acoustic radius \u001ason one side, and the ratio between the collision\nfrequency\u0017iand the bounce frequency !bon the other side. The \frst parameter determines if the island is\nin the sonic or hypersonic regime, which is related to the relative role of ion-acoustic waves on the \rattening\nof the density pro\fle inside the separatrix. The second parameter determines if the plasma is in the weak or\nin the intermediate damping regime, which is related to the relative strength of the neoclassical e\u000bects. The\n1arXiv:1710.03585v1  [physics.plasm-ph]  10 Oct 2017simultaneous presence of both the e\u000bects in a tokamak plasma makes it particularly di\u000ecult to determine the\nislands rotation velocity.\nThe purpose of this work is to investigate the magnetic island dynamics in tokamaks, in particular as regards\nisland rotation. The attempts to study the island rotation by Fitzpatrick & Waelbroeck rested on the inclusion\nof the neoclassical e\u000bects in their equations by using simpli\fed expressions for the neoclassical terms, together\nwith the possibility to keep the island-size e\u000bects by using an appropriate normalization for the \felds. Although\ntheir work shows results consistent with the experimental observations, their results come from a system of\n\ruid equations which did not include the Finite Larmor Radius (FLR) corrections from the start. In this work\nwe attempt to improve their results by starting from a set of gyro\ruid equations, which result from taking\nthe moments of the gyrokinetic equation [13, 14], and then reducing them to a four-\feld model analogous\nto that used by Fitzpatrick & Waelbroeck. The neoclassical e\u000bects are included in the model by using the\nsame simpli\fed expressions, with an important di\u000berence. To be consistent with the gyro\ruid equations, we\ncompute the lowest order FLR corrections to the poloidal \row damping by solving the gyrokinetic equation in an\nappropriate limit and then computing the poloidal \row damping by following the same approach adopted in the\nbook \"Collisional transport of impurities in plasmas\" by Helander & Sigmar [15]. The equations thus obtained\nhave been solved by adopting a series of perturbative expansions introduced by Fitzpatrick & Waelbroeck in\ntheir works and based on the multiple-scale approach [16]. The \fnal equations have been solved, in two di\u000berent\nregimes of collisionality, together with the torque balance condition, imposing that the total electromagnetic\nforce acting on the freely-rotating islands is zero. The solution of this system of equations provides the \feld\npro\fles and the self consistent phase velocity of the islands. Attempts to study both analitically and numerically\nthe FLR e\u000bects on magnetic island evolution have been done [17, 18]. In these works the focus of the authors\nwas mainly on the analysis of the island dynamics and related phenomena, such as the emission of drift waves\nand the \rattening of the density pro\fle, given the island phase velocity, which was just a parameter of their\nmodels. The approach we choose, which is the same used by Fitzpatrick & Waelbroeck, is to deduce the island\nrotation frequency consistently with the \feld pro\fles in a stationary regime.\nThe paper is organized as follows. In Section 2 an analytical solution of the gyrokinetic equation is deduced and\nthe lowest-order FLR corrections to the poloidal \row damping are calculated. In Section 3 a four-\feld gyro\ruid\nmodel with neoclassical e\u000bects is deduced and a series of simpli\fcations is performed on these equations to apply\nthem to the study of the stationary rotation of a chain of magnetic islands. In Section 4 the torque balance\ncondition is introduced and its explicit form is deduced for the case being considered. In Section 5 the model\nis applied to the study of the weak-damping regime. In Section 6 the model is applied to the study of the\nintermediate-damping regime, where a new term appears which contains the lowest-order FLR corrections to\nthe poloidal \row damping. In Section 7 the results of the numerical integration of the system of equations in\nthe weak and the intermediate regime are displayed. Conclusions are drawn in Section 8.\n2 A particular solution of the gyrokinetic equation\nThe \frst step in our calculation is to deduce a particular solution for the gyrokinetic equation\n@f\n@t+ (b0vk+vd)\u0001rf\u0000\u001a1\nm\u0012\n1 +\u0016B1?\nB0\u0013\u0000\nZerk\u0016\u001e1+\u0016rk(B0+\u0016B1k)\u0001\n+\n+1\nmvkvd\u0001\u0000\nZer\u0016\u001e1+\u0016r(B0+\u0016B1k)\u0001\u001b@f\n@vk=C(f)(1)\nunder speci\fc simplifying hypotheses. This solution will be then used to \fnd the FLR corrections to the\nneoclassical e\u000bects which occur in a tokamak. The resolution will follow the method outlined in [15, 19].\n2.1 FLR expansion of the gyrokinetic equation\nBy starting from Eq.1, the distribution function fis expanded in an equilibrum Maxwellian part plus a small\nperturbation, ordered with \u000e=\u001ai=L\u001c1:f=FM+f1.\u001aiis the ion Larmor radius, while Lis a macroscopic\nlength scale. The equilibrium solution is assumed a stationary \rux function, that is: @FM=@t= 0,rkFM= 0.\nThe following orderings are used\n@\n@t=O(\u000e2vth=L);Ze\u0016\u001e1\nT=O(\u000e);vd\nvth=O(\u000e) (2)\n2With these orderings, the gyrokinetic equation Eq.1 to order \u000ebecomes\nvkrkf1+vd\u0001r(f1+FM)\u0000\u0016\nmrkB0@f1\n@vk\u00001\nm\u0014\nZerk\u0016\u001e1+\u0016\u0012\nrk\u0016B1k+\u0016B1?\nBrkB0\u0013\u0015@FM\n@vk\u0000\n\u00001\nmvkvd\u0001[Zer\u0016\u001e1+\u0016r(B0+\u0016B1k)]@FM\n@vk+Ze\nmE(A)\nk@FM\n@vk= 0(3)\nThe term proportional to the parallel induced electric \feld E(A)\nkhas been introduced to include the e\u000bect of the\nmagnetic \rux variation in a tokamak. We can meake the further assumption that rB0\u001cr\u001e1;rB1k, meaning\nthat the equilibrium magnetic \feld is almost uniform. Using the following identity to express vdin terms of vk:\nvd\u0001rf=Ivkrk\u0010vk\n\n\u0011@f\n@ (4)\nwhereI=RB'is a \rux function, the gyrokinetic equation becomes:\nvkrk\u0014\nf1+Ivk\n\n@\n@ (f1+FM)\u0015\n=\u0000FM\nTvkn\n[Zerk\u0016\u001e1\u0000ZeE(A)\nk+\u0016rk\u0016B1k]+\n+rk\u0014Ivk\n\n\u0012\nZe@\u0016\u001e1\n@ +\u0016@\u0016Bk1\n@ \u0013\u0015\u001b (5)\nGyrokinetic theory is usually used to study turbulent transport, which is typically much larger than the col-\nlisional one. For this reason the gyrokinetic equation we started from didn't have the collisional term on the\nright-hand side. To deal with neoclassical e\u000bects, we need to include the e\u000bect of collisions by using an appro-\npriate collision operator. The contribution from E(A)\nkcan be absorbed in a Spitzer function fs, as customary\nin the drift-kinetic case. This one is neglected in respect to f1because, for the ions, fs\u001cf1. For the \felds\nperturbations caused by the onset of a magnetic island, the leading term is \u0016B?1, so that we can neglect \u0016Bk1.\nWith these simpli\fcations:\nvkrk\u0014\nf1+Ivk\n\n@\n@ (f1+FM)\u0015\n=\u0000FMvk\nT\u001a\nZerk\u0016\u001e1+rk\u0014Ivk\n\n\u0012\nZe@\u0016\u001e1\n@ \u0013\u0015\u001b\n+C(f1) (6)\nIn the low-collisional regime, we can expand f1in a power series of the collisionality \u0017\u0003, so that we can write\nf1=f(0)\n1+f(1)\n1+\u0001\u0001\u0001[15, 19]. To the two lowest orders\nrk\u0014\nf(0)\n1+Ivk\n\n@\n@ (f(0)\n1+FM)\u0015\n+FMZe\nTrk\u0014\n\u0016\u001e1+Ivk\n\n@\u0016\u001e1\n@ \u0015\n= 0\nvkrk\"\nf(1)\n1+Ivk\n\n@f(1)\n1\n@ #\n=C(f(0)\n1)(7)\nUsing the fact that rkFM=rkT= 0 (neglecting signi\fcant perturbations to the temperature), the lowest\norder equation becomes\nrk\u0014\nf(0)\n1+Ivk\n\n@\n@ (f(0)\n1+FM) +FMZe\nT\u0012\n\u0016\u001e1+Ivk\n\n@\u0016\u001e1\n@ \u0013\u0015\n= 0 (8)\nBy integrating once, we \fnd the following equation for f(0)\n1:\n\u0012\n1 +Ivk\n\n@\n@ \u0013\nf(0)\n1=g\u0000Ivk\n\n@FM\n@ \u0000FMZe\nT\u0012\n1 +Ivk\n\n@\n@ \u0013\n\u0016\u001e1 (9)\nwithgan unknown function such that rkg= 0. Eq.9 can be solved formally, by writing the solution f(0)\n1in an\nintegral form. Every time we deal with an equation of this form\n\u0012\n1 +ad\ndx\u0013\nf(x) =K(x) (10)\nthe particular solution takes the form [20]:\nf(x) =e\u0000x=a\naZx\nx0ey=aK(y)dy (11)\n3Eq.9 is in the form Eq.10, so the solution for f(0)\n1becomes\nf(0)\n1=c1(v)e\u0000 = s+e\u0000 = s\n sZ \n 0e\u001f= s\u0014\ng(\u001f)\u0000 s@FM\n@\u001f\u0000ZeFM\nT\u0012\n1 + s@\n@\u001f\u0013\n\u0016\u001e1\u0015\nd\u001f (12)\nwhere s=Ivk=\n has the dimensions of a magnetic \rux. c1(v)e\u0000 = sis the solution of the homogeneus\nequation. Once Eq.12 has been solved, we can multiply both members of the second equation of Eq.7 by B=vk\nand take the \rux surface average, so that we are left with the equation\n\u001cB\nvkC(f(0)\n1)\u001d\n= 0 (13)\n2.2 Analytical solution\nTthe collision operator Ccan be chosen in the following form [15, 19]:\nCii(fi) =\u0017ii\nD(v)\u0012\nL(fi1) +mivkui\nTifMi\u0013\n(14)\nBy introducing the following de\fnition of the Lorentz operator\nL=2hvk\nv2@\n@\u0015\u0015vk@\n@\u0015(15)\nwhereh\u0011B0=Bis the toroidal metric coe\u000ecient and \u0015is related to the particles pitch angle by \u0015\u0011hsin2\u000b,\nEq.13 becomes:\n*\nB(\n2h\nv2@\n@\u0015\u0015vk@\n@\u0015 \ng\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s@\u001fg+e\u0000 = sc1!\n+I\n\n[@ FM+\n+ZeFM\nT@ \u0016\u001e1\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+miui\nTFM)+\n= 0(16)\nWe de\fne the auxiliary function J:\nJ=g\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s@\u001fg+e\u0000 = sc1 (17)\nThe equation for Jis\n@\n@\u0015\u0015\nvk\u000b@\n@\u0015J=\u0000v2\n2\u001aI\nh\n\u0014\n@ FM+ZeFM\nT@ \u0016\u001e1\u0000\n\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+Dui\nhEmi\nTFM) (18)\nFrom the form of Eq.18, we can deduce that Jplays the role of the function which, in the drift-kinetic equation,\nvanishes in the trapped particle space, so that the solution of Eq.18 is\nJ=H(\u0015c\u0000\u0015)v2\n2Z\u0015c\n\u0015d\u00150\n\nvk(\u00150)\u000b\u001aI\nh\n\u0014\n@ FM+ZeFM\nT@ \u0016\u001e1\u0000\n\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+Dui\nhEmi\nTFM) (19)\n4whereHis the Heaviside function. By using a few results from drift-kinetik theory, we \fnd the following\nsolution:\nf(0)\n1=\u0000Ivk\n\n\"\n@ FM+ZeFM\nT@ \u0016\u001e1\u0000e\u0000 = sZ \n 0d\u001fe\u001f= s\u0012\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0013#\n+\n+IHVk\nh\n\u0012mv2\n2T\u00001:33\u0013dlogT\nd FM\u0000Ze\u0016\u001e1\nTFM+e\u0000 = sZ \n 0d\u001fe\u001f= sZe@\u001f\u0012FM\nT\u0016\u001e1\u0013\n\u0000\n\u0000IHVk\nh\ne\u0000 = sZ \n 0d\u001fe\u001f= s\u0014\n@2\n\u001fFM+Ze@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013\u0015\n\u0000\n\u0000IHVk\nh\nFM\b\n\u0017ii\nD\t(\n\u0017ii\nDe\u0000 = sZ \n 0d\u001fe\u001f= s\"\n@2\n\u001fFM\nFM+Ze\nFM@\u001f\u0012FM\nT@\u001f\u0016\u001e1\u0013#)(20)\nThe curly braces in the last line of Eq.20 represent the velocity-space average, which is de\fned as:\nfFg=Z\nd3vFmv2\nnTFM (21)\nThe solution Eq.20 still contains terms in an integral form. However, it contains the FLR corrections which\nprovide, after velocity-space integration, the modi\fed transport coe\u000ecients in the di\u000berent collisionality regimes.\n2.3 Poloidal \row damping\nEq.20 can be used to compute the neoclassical e\u000bects, in particular the poloidal \row damping, which comes\nfrom the toroidal geometry, through the equation [15, 19]\nhB\u0001r\u0001\u0019i=D\nB(Fk+nZeE(A)\nk)E\n(22)\nwhere\u0019is the stress tensor and Fkis the parallel component of the friction force, which is de\fned in terms of\nthe distribution function\nFk\u0011Z\nmvkC(f1)d3v (23)\nWe can now use the Spitzer function to eliminate the term proportional to the inductive electric \feld and remind\nthat, for the ions, the Spitzer function is negligible in respect to the function f1. Using the particular form for\nthe collision operator Eq.14, together with Eq.23, Eq.22 becomes\nhB\u0001r\u0001\u0019i=\u001c\nBZ\nmvk\u0017ii\nD\u0012\nL(f(0)\n1) +mvkUki\nTFM\u0013\nd3v\u001d\n(24)\nUkiis the parallel \row velocity of the ions, which is de\fned as\nUki=Z\nvkf(0)\n1d3v (25)\nEq.20 contains the FLR e\u000bects in terms of integral expressions. Such quantities can be expanded in a power\nseries in respect to  sperforming an integration by parts\ne\u0000 = sZ\nd\u001fe\u001f= sF= sF\u0000 2\ns@ F+O( 3\ns@2\n F) (26)\nThis expansion is made possible by the smallness of the ion Larmor radius: in fact, after velocity integration,\n s= =\u001ai=L\u001c1. From this result we notice that, when taking the velocity moments of the distribution\nfunction, only the terms which have the correct parity will remain and the others will be zero. This is particularly\nimportant because, from the lowest order expansion Eq.26, only terms proportional to \u001a2\niwill remain. When\napplying this expansion to Eq.20 and applying it to Eq.25, we \fnd:\nUk=\u0000I\n\nZ\nd3vv2\nk\u0012\n@ FM+ZeFM\nT@ \u0016\u001e1\u0013\n+I\n\nZ\nd3vv2\nk\u0012mv2\n2T\u00001:33\u0013dlogT\nd FM+\n+I\n\nZ\nd3vv2\nkZe@ \u0012FM\nT\u0016\u001e1\u0013\n+I3\n\n3Z\nd3vv2\nkFM\nf\u0017ii\nDg(\n\u0017ii\nDv2\nk\"\n@3\n FM\nFM+Ze\nFM@2\n \u0012FM\nT@ \u0016\u001e1\u0013#) (27)\n5Since the radial derivatives are steep, we only keep the highest order derivatives in Eq.27;\n@3\n FM+Ze@2\n \u0012FM\nT@ \u0016\u001e1\u0013\n=\u00141\nPd3P\nd 3+ 2Ze\nTd3\u0016\u001e1\nd 3+\u0012mv2\n2T\u00005\n2\u00131\nTd3T\nd 3\u0015\nFM (28)\nInserting the solution Eq.20 expanded according to Eq.26 into Eq.24, and by using Eq.28:\nhB\u0001r\u0001\u0019i\u0019B\u001601minft\u0017iiITi\nmi\ni\u001a\n1:17dlogTi\nd +\n+I2Ti\nmi\n2\ni\u0014\n2:70\u00121\nPd3P\nd 3+ 2Ze\nTd3\u0016\u001e1\nd 3\u0013\n\u00000:701\nTd3T\nd 3\u0015\u001b (29)\nwhere\u00160i=f\u0017ii\nDg,ftis the fraction of trapped particles and the following properties have been used [15]:\nf\u0017ii\nDg\u00190:53f\u0017ii\nDx2g\u00190:71f\u0017ii\nDx4g\u00191:59 (30)\nwherex2=v2=v2\nth. The \frst term in Eq.29 is the result from drift-kinetic theory. The additional terms are the\n\frst order FLR corrections, which are proportional to \u001a2\ni.\n3 Four-\feld gyro\ruid model\nThe system of gyro\ruid equations originally developed by P. B. Snider [21] consists of six equations, evolving\nthe density, the parallel velocity, the parallel and perpendicular pressure and the parallel and perpendicular\nheat \rux for each particle species. Here we just need the \frst two of them, together with the vorticity equation,\nwhich can be deduced from the quasi-neutrality condition\nne= \u00001=2\n0ni+n0Ze\nTi(\u00000\u00001)\u001e (31)\nwhere \u0000 0=\nJ2\n0\u000b\nis the velocity average of the zero-order Bessel function, with argument k2\n?\u001a2\ni, which comes\nfrom the gyroaverage involved in the gyrocenter transformation. The second term on the right-hand side of\nEq.31 is the so-called polarization density. The momentum equation for the two species electrons and ions can\nbe written as:\n@\u0016ni\n@t+v\b\u0001r\u0016ni+n0\u0016rk\u0016uki= 0\n@ne\n@t+v\u001e\u0001rne+n0rkuke= 0\nmin0\u0012@\u0016uki\n@t+v\b\u0001r\u0016uki\u0013\n=\u0000T0i\u0016rk\u0016ni+en0\u0012@\t\n@t\u0000rk\b\u0013\n+Fie\nmen0\u0012@uke\n@t+v\u001e\u0001ruke\u0013\n=\u0000T0erkne\u0000en0\u0012@ \n@t\u0000rk\u001e\u0013\n+Fei(32)\nwhere \b = \u00001=2\n0\u001e, \t = \u00001=2\n0 are the gyroaveraged \felds, FieandFeiare the collisional friction forces, whose\nparallel component is de\fned in Eq.23. \u0016rkis the parallel gradient performed along the gyroaveraged magnetic\n\feld. \u0016niand \u0016ukiare the ion density and parallel velocity expressed in the gyrocenter coordinates. Owing to\nmomentum conservation in Coulomb collisions, the property Fie=\u0000Feiholds. We used P=nT0, withT0\nuniform and constant. The common equilibrium density n0multiplies the electric force term in the momentum\nequations because the electric \feld is perturbative. By proceeding similarly to the quasi-neutrality calculation,\nwe can \fnd the gyrokinetic de\fnition of the current:\nJk=\u0000en0(uke\u0000\u00001=2\n0\u0016uki) (33)\nIntroducing the Debye length \u0015Di=p\nTi=(nie2), Eq.31 becomes:\n1\n\u00152\nDi(\u00000\u00001)\u001e=e(ne\u0000\u00001=2\n0\u0016ni) (34)\n6Taking the time derivative of Eq.34 and using the equations above, we obtain the following vorticity equation:\n\u0012@\n@t+v\u001e\u0001r\u0013\u00121\n\u00152\nDi(\u00000\u00001)\u001e+e(\u00001=2\n0\u00001)\u0016ni\u0013\n=rkJk+e(\u00001=2\n0\u00001)v\u001e\u0001r\u0016ni\u0000erk(\u00001=2\n0\u00001)\u0016uki\u0000e(\u0016rk\u0000rk)\u0016uki\n(35)\nWe can further simplify this system of equations by neglecting the electron inertia in the electron momentum\nequation, which becomes the generalized Ohm law. Then we use the quasi-neutrality condition Eq.34 to express\nnein terms of \u0016 niand we sum the momentum equations of the ions and the electrons.\nThe system of equations we get is:\n@\u0016ni\n@t+v\b\u0001r\u0016ni+n0\u0016rk\u0016uki= 0\n@ \n@t\u0000rk\u001e+T0e\nen0rk\u0014\n\u00001=2\n0\u0016ni+n0(\u00000\u00001)e\u001e\nT0i\u0015\n=\u0011Jk\nmi\u0012@\n@t+v\b\u0001r\u0013\n\u0016uki=\u0000T0e\nn0\u0016rk\u0014\n(\u001c+ \u00001=2\n0)\u0016ni+ (\u0000 0\u00001)e\u001e\nT0i\u0015\n\u0012@\n@t+v\u001e\u0001r\u0013\u00121\n\u00152\nDi(\u00000\u00001)\u001e+e(\u00001=2\n0\u00001)\u0016ni\u0013\n=rkJk+e(\u00001=2\n0\u00001)v\u001e\u0001r\u0016ni(36)\nwhere\u001c=T0i=T0e. We neglected in the parallel momentum equation the electric force coming from the di\u000berence\nbetween the \felds  ,\u001eand their gyroaverage. v\band \u0016rkare theE^Bvelocity and the parallel gradient on\nthe \felds calculated with the gyroaveraged \felds: B0v\b\u0001r= (^ez^r\u00001=2\n0\u001e)\u0001r,B0\u0016rk= (^ez^r\u00001=2\n0 )\u0001r.\nThe quantity 1 =\u00152\nDi(\u00000\u00001)\u001e+e(\u00001=2\n0\u00001)\u0016niis the gyrokinetic vorticity. In the limit of large wavelengths, this\nquantity reduces to \u001a2\nie(noe=Tir2\n?\u001e+ 1=2r2\n?ni). The \frst term corresponds to the E^Bdrift velocity, while\nthe second one represents the contribution from the diamagnetic velocity. The factor 1 =2 appearing in front of\nthis term comes from the expansion of the gyroverage operator [22]. Attempts to study an Hamiltonian version\nof these equations, both analytically and numerically, has been done by di\u000berent authors [23, 24].\nGyro\ruid equations surpass the \ruid equations because they include the FLR e\u000bects which come from the\ngyrokinetic theory. However, these FLR e\u000bects are present in the form of nonlinear di\u000berential operators, quite\ndi\u000ecult to deal with both analytically and numerically. Several attempts have been made by di\u000berent authors to\ndeal with these operators by approximating them with power expansions and elementary functions. An overview\nof these attempts is provided in [14]. The operators \u0000 0and \u00001=2\n0involve all the even powers of b=k2\n?\u001a2\ni:\n\u00000=\nJ2\n0\u000b\n=I0(b)e\u0000b= 1\u0000b+O(b2)\n\u00001=2\n0\u0019\nJ2\n0\u000b1=2=I1=2\n0(b)e\u0000b=2= 1\u0000b=2 +O(b2)(37)\nwhereI0is the modi\fed Bessel function. The Taylor expansion of these operators provides the FLR corrections\nto all orders in b. In the limit of large wavelengths k2\n?\u001a2\ni\u001c1 the power expansion can be truncated to a low\norder (usually the second order is already a good approximation). However, in the limit of small wavelengths\nk2\n?\u001a2\ni\u001d1 (or more realistically, k2\n?\u001a2\ni=O(1)), the power expansion isn't a good approximation any longer.\nIf we introduce the following normalization for the \felds [11]:\n^ =Ls\nB0w2 ; ^n=\u0000Ln\nw\u0016ni\nn0;^\u001e=\u0000\u001e\nwV\u0003eB0+ ^xVp\n^V=\u000f\nqVki\nV\u0003e; ^J=LS\u00160\nB0\u000eeJk;^\u0011=\u0011\n\u00160k\u0012V\u0003ew2(38)\nwherewis the island width, \u000ee=\f=\u000b2,\u000b2=w2=\u001a2\nsL2\nn=L2\nsand\u001as=cs=\niis the ion-acoustic radius. The\nx-derivatives are normalized to w, they-derivatives are normalized to Ln= 1=k\u0012and the time derivatives are\nnormalized to k\u0012V\u0003e.Vpis the unknown island rotation velocity. The additional term ^ xVpin the normalized\nelectrostatic potential represents the contribution from the island-induced electric \feld. With this choice for\nthe normalization, the gradients length-scale of the \felds in the radial direction is comparable with the island\nwidthw.\n3.1 Neoclassical e\u000bects\nThe neoclassical e\u000bects come from the inhomogeneity of the magnetic \feld and the low collisionality of the\nplasma. Neoclassical theory and the poloidal \row damping have been thoroughly described in [19, 15]. In\n7addition to the poloidal damping caused by the toroidal shape of the tokamak, there is a similar phenomenon\ncaused by non-axisymmetric e\u000bects, such as the magnetic islands. The broken poloidal symmetry of the torus\ncauses the travelling particles to experience a magnetic-mirror e\u000bect, which leads to the phenomenon of banana\norbits and the consequent poloidal \row damping. Analogously, the broken axisymmetry caused by the presence\nof magnetic islands leads to a situation of \"helically-trapped particles\", causing a braking e\u000bect on the plasma\nrotation called \"island-induce \row damping\" [25, 26]. The procedure to obtain the non-axisymmetric e\u000bects\non the plasma rotation is analogous to that used in the axisymmetric case but the calculations are much more\ninvolved because of the complex shape of the \rux surfaces. The island-induced \row damping is proportional\nto the island width wsquared [25, 27], so that its e\u000bect becomes signi\fcative only when the FLR e\u000bects are\nnegligible. For this reason we chose not to compute the FLR corrections to this term by solving the gyrokinetic\nequation. The only correction we are going to keep is the usual lowest-order expansion of the gyroaveraged\nelectrostatic potential. That said, let us consider the divergence of the ion stress tensor we deduced in the\nsection above: according to [27], the \row damping can be included in the system of equations we are using by\nimposing that, under the e\u000bect of this damping, the poloidal rotation velocity tends to relax to its neoclassical\nvalue. This amounts to introducing the following damping term:\nmini\u0017\u0012(Vi\u0001e\u0012\u0000Vnc\n\u0012) (39)\nAfter switching from  tor, simplifying a few terms and introducing the notation T0=dT=dr , Eq.39 becomes:\nmini\u0017\u0012\u001a\nVki^b\u0001e\u0012+1\nB0\u0016\u001e0\n1+Ti\nen0B0\u0016n0\ni(1\u0000c\u0012)\u0000Ti\nen0B0\u0014\nc1n000\ni\nn0+c2Ze\nTi\u0016\u001e000\n1\u0015\u001b\n(40)\nwherec\u0012= 1:17\u0011i,c1= 2:70 + 2\u0011i,c2= 5:40 and\u0011i=Ln=LTis the ratio between the length scales of the\ndensity and temperature gradients. If we use the normalization for the \felds introduced in Eq.38, we \fnd the\nfollowing adimensional expression:\n\u0000^\u0017\u0012n\n^V+Vp\u0000@^x[\u00001=2\n0^\u001e+\u001c^n(1\u0000c\u0012)] +\u001c\u001a2@^x[c1@2\n^x^n+c2@2\n^x^\u001e]o\n(41)\nwhere ^\u0017\u0012=\u0017\u0012=(k\u0012V\u0003e) is the poloidal damping coe\u000ecient, which is determined by the kinetic theory. An analo-\ngous expression exists for the island-induced perpendicular \row damping [25, 27]. When using the normalization\nEq.38, it takes the following form:\n^\u0017?@^x[\u00001=2\n0^\u001e+\u001c^n(1\u0000c?)] (42)\n^\u0017?=\u0017?=(k\u0012V\u0003e) is the perpendicular damping coe\u000ecient, which is proportional to w2, andc?= 2:37. As\nemphasized in [27], the island-induced \row damping acts in the perpendicular direction, so that it doesn't\ncontribute to the parallel momentum equation. When including these e\u000bects in Eq.36, the normalized system\nof equations becomes:\n@^n\n@^t= [^\u001e;^n] + [^V;^ ] +\u001a2\n2f[@2\n^x^\u001e;^n] + [^V;@2\n^x^ ]g\n@^ \n@^t= [^\u001e\u0000^n;^ ] + ^\u0011\u000ee^J\u0000\u001a2[@2\n^x^\u001e=\u001c+@2\n^x^n=2;^ ]\n@^V\n@^t= [^\u001e;^V] +\u000b2(1 +\u001c)[^n;^ ] +\u001a2\u001a1\n2[@2\n^x^\u001e;^V] +\u000b2\n2\u0010\n(1 +\u001c)[^n;@2\n^x^ ] + [@2\n^x^n;^ ]\u0011\n+1\n\u001c[^ ;@2\n^x^\u001e]\u001b\n\u0000\n\u0000^\u0017\u0012\u0012\u000f\nq\u00132\u001a\n^V+Vp\u0000@^x\u0014\u0012\n1 +\u001a2\n2@2\n^x\u0013\n^\u001e+\u001c^n(1\u0000c\u0012)\u0000\u001cc1\u001a2@2\n^x^n\u0000\u001cc2\u001a2@2\n^x^\u001e\u0015\u001b\n@\n@^t@2\n^x\u0010\n^\u001e+\u001c\n2^n\u0011\n=h\n^\u001e;@2\n^x\u0010\n^\u001e+\u001c\n2^n\u0011i\n+\u001c\n2[^n;@2\n^x^\u001e] + [^J;^ ] +\u001a2nh\n^\u001e;@4\n^x\u0010\n^\u001e+\u001c\n4^n\u0011i\n+\u001c\n4[^n;@4\n^x^\u001e]o\n\u0000\n\u0000\u0017?@2\n^x\u0014\u0012\n1 +\u001a2\n2@2\n^x\u0013\n^\u001e+\u001c^n(1\u0000c?)\u0015\n+\u0017\u0012@^x\u001a\n^V\u0000@^x\u0014\u0012\n1 +\u001a2\n2@2\n^x\u0013\n^\u001e+\u001c^n(1\u0000c\u0012)\u0000\u001cc1\u001a2@2\n^x^n\u0000\u001cc2\u001a2@2\n^x^\u001e\u0015\u001b\n(43)\nNote that the FLR corrections coming from the analytical resolution of the gyrokinetic equation are consistent\nwith those coming from the the small-Larmor-radius expansion of the gyro\ruid equations. In fact, although the\nperturbation to the distribution function f1was assumed ordered with \u000e=\u001ai=L\u001c1, the distinction between\nthe parallel and the perpendicular length scales was appropriately addressed, so that the FLR corrections have\nnaturally emerged from our calculations.\n83.2 Simpli\fcation of the system of equations\nEq.43 provides a system of equations describing a plasma in the presence of an island whose width wis larger\nthan the ion Larmor radius \u001ai, so that the FLR corrections enter only to order \u001a2. In this approximation, we can\nreasonably assume \u000b2=O(1) [11, 27]. If the ordering \u001a2\u001c1 holds, we can expand the \felds in the following\nway:\n\u001e=\u001e0+\u001a2\u001e1+O(\u001a4) (44)\nIn the following calculations we are going to use the constant-  approximation, which holds as long as j\u00010wj;\u000ee\u001c\n1, where \u00010is the tearing mode stability parameter and \u000eewas de\fned previously. If this condition holds, the\nmagnetic \rux function takes the form:\n (x;y) =x2\n2+ cosy (45)\nEq.45 describes a magnetic island centered in x= 0, with the O-point in y= 0. The region inside the separatrix\ncorresponds to\u00001<  < 1 and the region outside the separatrix corresponds to  \u00151. From now on, the\nmagnetic \rux function  will no longer be an unknown and, wherever possible, we will express the \felds as\nfunctions of  or its derivatives. For consistency with the results of Fitzpatrick [11, 27], we assume the zero-order\n\felds to be \rux functions, so that the \frst-order \felds are going to be the lowest order FLR corrections. We\nde\fne the following functions:\nM\u0011d\u001e0\nd ; L\u0011dn0\nd ; V0\n0\u0011dV0\nd (46)\nThe dissipative terms represented by the resistivity and the neoclassical viscosity are generally small, so that\nwe can neglect them in \frst instance. We introduce the \rux-surface average operation, which is de\fned as [11]:\nhf(\u001b; ;y )i\u00118\n<\n:1\n2\u0019H\ndyf(\u001b; ;y )p\n2( \u0000cosy)( \u00151)\n1\n2\u0019P\n\u001bRy0\n\u0000y0dyf(\u001b; ;y )p\n2( \u0000cosy)(\u00001< < 1)(47)\nwhere\u001b= sign(x) andy0= cos\u00001 . The \rux surface average is the annihilator of the parallel gradient, so that\nevery term in the form [ A; ] in the equations is deleted by this operator. By using the small-Larmor-radius\nexpansion, we can \fnd explicit expressions of the \frst-order \felds in terms of the zero-order quantities M,Land\nV0\n0. To \fnd the zero-order \felds we need to introduce a second ordering which involves the transport coe\u000ecients.\nThis new ordering assumes that the \frst order \felds are as small as the transport coe\u000ecients, which are in turn\nmuch smaller than the FLR parameter \u001a2.\nTo recover the correct form of the equations [11], we introduce a phenomenological perpendicular viscosity \u0016\nand a di\u000busion coe\u000ecient D(which is related to resistivity through the parallel compressibility [8]). The \felds\nappearing in the \fnal equations obey the following boundary conditions for x! 1 [11]:\nn!x\n\u001e!xVp\nV!V1(48)\nThe \frst condition means that the density gradient becomes constant far from the island. The gradient of the\nelectrostatic potential tends to a constant value which is the electric \feld induced by the island rotation. The\nasymptotic velocity V1is determined by the neoclassical theory.\n4 Torque balance\nThe linear stability index \u00010comes from the equilibrium current which causes the mode to be unstable, but\nevery other contributions to the current a\u000bect the mode growth. It can be easily shown [28] that, with the\nchoice Eq.45 for the magnetic \rux function, the contributions to the mode growth can be parametrized by this\nquantity :\nJc= 4Z+1\n\u00001hJcosyid (49)\n9where the angular brackets represent the \rux-surface average operation. Eq.49 means that the only currents\nthat contribute to the mode growth are those which have the cos y-symmetry. Analogously, there is a similar\nexpression parametrizing the contributions to the torque which is exerted on the island by external currents:\nJs= 4Z+1\n\u00001hJsinyid (50)\nEq.50 means that the only currents that contribute to the torque on the magnetic island are those which have\nthe siny-symmetry. By solving the lowest-order vorticity equation:\n[J(0)\n0; ]\u0000[\u001e(0)\n0;@2\nx(\u001e(0)\n0+\u001c=2n(0)\n0)]\u0000\u001c=2[n(0)\n0;@2\nx\u001e(0)\n0] = 0 (51)\nwe \fnd out that the solution for the current is:\nJ(0)\n0=\u0010\nM0\u0010\nM+\u001c\n2L\u0011\n+\u001c\n2ML0\u0011\nfx2=1\n2[M(M+\u001cL)]0fx2 (52)\nSincefx2=x2\u0000\nx2\u000b\n, this term doesn't contribute to the torque. To \fnd the lowest-order contribution to the\ntorque, we have to consider the following \frst-order vorticity equation:\n[J(1)\n0; ] + [\u001e(1)\n0;@2\nx(\u001e(0)\n0+\u001c=2n(0)\n0)] +\u001c=2[n(1)\n0;@2\nx\u001e(0)\n0] + [\u001e(0)\n0;@2\nx(\u001e(1)\n0+\u001c=2n(1)\n0)] +\u001c=2[n(0)\n0;@2\nx\u001e(1)\n0]+\n+\u0016@4\nx(\u001e(0)\n0+\u001c=2n(0)\n0)\u0000\u0017?@2\nx[\u001e(0)\n0+\u001cn(0)\n0(1\u0000c?)] +\u0017\u0012@xfV(0)\n0\u0000@x[\u001e(0)\n0+\u001cn(0)\n0(1\u0000c\u0012)]g= 0(53)\nIt follows by just performing the calculations and applying the boundary conditions, that the following identity\nholds: Z+1\n\u00001h[J; ]xid =\u0000Z+1\n\u00001hJsinyid (54)\nEq.54 enables us to compute the lowest order contribution to the torque by just multiplying Eq.53 by x, solving\nit for [J(1)\n0; ]xand operating on it with the \rux-surface average and the  -integration. For an isolated island,\nwhich is not interacting with an external electromagnetic \feld, the total torque is zero. By doing this and\napplying again the boundary conditions, the torque-balance condition becomes:\nZ+1\n\u00001d f\u0017\u0012[(V0\u0000V1+Vp+\u001c(1\u0000c\u0012))h1i+ (M+\u001cL(1\u0000c\u0012))] +\u0017?[(Vp+\u001c(1\u0000c?))h1i+ (M+\u001cL(1\u0000c?))]g= 0\n(55)\nBy using the system of equations for the \felds we have deduced above, with their boundary conditions, together\nwith the torque balance condition, we can \fnd the phase velocity of the island in the following way: we \frst\nchoose a value for the phase velocity Vpand we solve the di\u000berential equations for the \felds by the shooting\nmethod, we substitute these solutions in the torque balance condition and we \fnd a new value for Vp, we use\nthis new value in the equations again and we iterate until convergence is reached.\nV1represents the velocity of the plasma far from the island, which depends on the damping e\u000bects. We will\nsee in the following sections that a solution can be found in two di\u000berent collisionality regime, namely the weak\ndamping and the intermediate damping regimes.\n5 Weak damping regime\nIn the weak-damping regime, the following ordering holds:\n1\u001dD;\u0016;\u0011;\u0017\u0012\u001d\u0017? (56)\nIn this case, we can disregard the terms where the products between the FLR parameter \u001a2and the transport\ncoe\u000ecients appear, as well as the perpendicular damping coe\u000ecient \u0017?. Also the product ( \u000f=q)2\u0017\u0012is small and\n10can be neglected. In the weak-damping regime, the equations become:\nDD\n@2\nxn(0)\n0E\n+\u001a2=2D\n[x2M0;n(1)\n0]E\n= 0\n\u0016D\n@2\nxV(0)\n0E\n+\u001a2nD\n[\u001e(0)\n1;V(1)\n0]E\n+D\n[\u001e(1)\n0;V(0)\n1]E\n+ 1=2D\n[x2M0;V(1)\n0]Eo\n= 0\n\u0016D\n@4\nx(\u001e(0)\n0+\u001cn(0)\n0)E\n+D\n[\u001e(1)\n0;x2(M0+\u001c=2L0)]E\n\u0000\u0017\u0012D\n@xfV(0)\n0\u0000@x[\u001e(0)\n0+\u001cn(0)\n0(1\u0000c\u0012)]gE\n+\n+\u001c=2D\n[n(1)\n0;x2M0]E\n+\u001a2nD\n[\u001e(1)\n0;@2\nx(\u001e(0)\n1+\u001c=2n(0)\n1)]E\n+D\n[\u001e(1)\n0;x4(M000+\u001c=2L000) + 6x2(M00+\u001c=2L00)]E\n+\n+\u001c=2D\n[n(1)\n0;@2\nx\u001e(0)\n1]E\n+\u001c=4D\n[n(1)\n0;x4L000+ 6x2L00]Eo\n= 0\n(57)\nEqs.57 neglect the FLR corrections to the neoclassical \row damping. After many mathematical steps, the\nsystem of equations reduces to\nd\nd \u0000\nL\nx2\u000b\u0001\n= 0;d\nd \u0012\nx2\u000bdV0\nd \u0013\n= 0\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\u0012D\n\u0016ML0+V00\n0\u0013D\nfx2fx2E\n\u0000\n\u0000\u0017\u0012\n\u0016\b\n(V0\u0000V1+Vp+\u001c(1\u0000c\u0012)) + [M+\u001cL(1\u0000c\u0012)]\nx2\u000b\t\n= 0(58)\nwhere we used V0=V(0)\n0. Note that the solution of the \frst equation, compatible with the boundary condition\nV!V1, is [11]:\nL=\u00001=\nx2\u000b\n(59)\nThe solution for V0, compatibly with the boundary condition V0!V1, isV0=V1.\nWe can neglect the island-induced \row damping in the torque-balance condition Eq.55, which becomes:\nZ+1\n\u00001d f(V0\u0000V1+Vp+\u001c(1\u0000c\u0012))h1i+ (M+\u001cL(1\u0000c\u0012))g= 0 (60)\nNow we consider the fact that \u001e0andn0are \rux functions, so that they must have the same x-symmetry of\nthe \rux function  , which is an even function. However, for the tearing symmetry, both \u001e0andn0are even in\nrespect tox. The only way to solve this contradiction is by imposing that they must be zero inside the separatrix\n[11], that is for\u00001< < 1. By using this result in the parallel momentum equation, we \fnd out that V0=\u0000Vp\ninside the separatrix. Furthermore, the quantity c\u0012which drives the intrinsic poloidal rotation, depends on the\ntemperature gradient and it is thus zero inside the separatrix. Outside the separatrix, instead, the system of\nequations Eq.58 hold. The \frst two equations bring to the solutions we have already seen L=\u00001=\nx2\u000b\nand\nV0=V1. SinceV0is a constant, the relation V0=V1holds in all the region 1 <  < +1. By imposing\nthatV0is continuous across the separatrix, we also \fnd that Vp=\u0000V1. By putting these results in the torque\nbalance condition Eq.60, we \fnd an equation for Vp, whose solution is\nVp=\u0000\u001c\u0014\n1\u0000c\u0012\nI1(I2\u0000I3)\u0015\n\u00001\nI1Z+1\n1d (M+\u001cL) (61)\nwhere we introduced the quantities I1=R+1\n\u00001d h1i,I2=R+1\n1d h1iandI3=\u0000R+1\n1d L. The remaining\nunknown function Mmust be determined by solving the following equation:\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000D\n\u0016\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\nML0D\nfx2fx2E\n\u0000\u0017\u0012\n\u0016(Vp+M\nx2\u000b\n) = 0 (62)\nwhereH=M(L\u0000M)+\u000b2(1+\u001c). By solving Eq.62 with the boundary condition M!Vp=p2 and computing\nVpby using Eq.61 iteratively, we can obtain the radial pro\fle of Mand the phase velocity Vp.\n6 Intermediate damping regime\nIn the intermediate-damping regime, the following ordering holds:\n1\u001d\u0017\u0012\u001dD;\u0016;\u0011;\u0017? (63)\n11In this case, we have to keep the terms where the product between \u001a2and the poloidal damping coe\u000ecient \u0017\u0012\nappear, together with the non-axisymmetric contributions \u0017?. By neglecting again the products \u001a2D,\u001a2\u0016and\n\u001a2\u0017\u0012(\u000f=q)2, the set of surface-averaged equations becomes:\nd\nd \u0012\nx2\u000bdV0\nd \u0013\n\u0000\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\nf(V0+Vp)h1i+ [M+\u001cL(1\u0000c\u0012)]g= 0\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000\u0017\u0012\n\u0016(V0\u0000V1+Vp+M\nx2\u000b\n)\u0000\u0017?\n\u0016(Vp+M\nx2\u000b\n)\u0000\n\u0000\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\"\u0012D\n\u0016ML0+V00\n0\u0013D\nfx2fx2E\n+\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\n[M+\u001cL(1\u0000c\u0012)]D\nfx2~xE#\n+\n+\u001a2\u0017\u0012\n\u0016\u001c\nxd\nd \u0014A\n\u001cHfx2\u0000\u00121\n2\u0000\u001cc2\u0013\nM0x2\u0000B\nHfx2(1\u0000c\u0012) +\u001cc1L0x2\u0015\u001d\n= 0(64)\nwhereH\u0011V0\n0+M(L\u0000M) +\u000b2(1 +\u001c),A\u0011M2(M0+\u001c=2L0) +\u001c=2MLM0\u0000\u000b2(1 +\u001c) (M0+\u001c=2L0) +\n\u001c=2(V0\n0M0+\u000b2L0)\u0000M0andB\u0011A+H(M0+\u001c=2L0). In the torque balance, we have to include the island-\ninduced damping terms proportional to \u0017?. In the internal region, the same considerations hold as before. In\nthe external region, however, we can impose the further condition that the plasma velocity tends to the intrinsic\npoloidal velocity far from the island. This condition is equivalent to imposing that the poloidal \row damping\ntends to zero far from the island, that is:\n\u0017\u0012fV+Vp\u0000@x[\u001e+\u001cn(1\u0000c\u0012)]g ! 0 (65)\nBy imposing the boundary conditions, Eq.65 becomes V1=\u001c(1\u0000c\u0012). By using this result, the torque-balance\ncondition Eq.55 becomes an equation for Vp, whose solution is\nVp=\u0000\u001c\u0014\n1\u0000\u0017\u0012c\u0012+\u0017?c?\n(\u0017\u0012+\u0017?)I1(I2\u0000I3)\u0015\n\u0000\u0017\u0012\n(\u0017\u0012+\u0017?)I1Z+1\n1d h1i(\u001c(1\u0000c\u0012)\u0000V0)\u00001\nI1Z+1\n1d (M+\u001cL) (66)\nIn the limit \u0017?!0 andV0=\u001c(1\u0000c\u0012), which is the case of the weak-damping regime, Eq.66 reduces to Eq.61.\nThe remaining unknown functions V0andMmust be determined by solving the system Eqs.64. By taking a\nfew more steps, the term containing the FLR corrections to the poloidal \row damping can be written more\nexplicitly, so that Eqs.64 become:\nd\nd \u0012\nx2\u000bdV0\nd \u0013\n\u0000\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\nf(V0+Vp)h1i+ [M+\u001cL(1\u0000c\u0012)]g= 0\nd\nd \u0014\nx4\u000bd\nd (M+\u001cL)\u0015\n\u0000\u0017\u0012\n\u0016(V0+Vp\u0000\u001c(1\u0000c\u0012) +M\nx2\u000b\n)\u0000\u0017?\n\u0016(Vp+M\nx2\u000b\n)\u0000\n\u0000\u0014\u0010\n1 +\u001c\n2\u0011M0\nH+\u001c\n2L0\nH\u0015\"\u0012D\n\u0016ML0+V00\n0\u0013D\nfx2fx2E\n+\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\n[M+\u001cL(1\u0000c\u0012)]D\nfx2~xE#\n+\n+\u001a2\u0017\u0012\n\u0016d\nd \u0014\u00121\n\u001cH[A+\u001cB(1\u0000c\u0012)] (1\u0000h1i)\u0000\u00121\n2\u0000\u001cc2\u0013\nM0+\u001cc1L0\u0013\nx2\u000b\u0015\n= 0(67)\nwhereH=V0\n0+M(L\u0000M) +\u000b2(1 +\u001c),AandBhave been de\fned above. Just as in the weak-damping case,\nby solving Eq.67 with the boundary conditions M!Vp=p2 ,V0!\u001c(1\u0000c\u0012) and computing Vpby using\nEq.66 iteratively, we can obtain the radial pro\fles of MandV0and the phase velocity Vp.\n6.1 Further simpli\fcation\nEqs.67 can be further simpli\fed by considering the limit of small-Larmor-radius, \u001a2!0, and the ordering\nEq.63. By using these simpli\fcations, V0outside the separatrix can be deduced by solving the second equation\nof Eqs.67:\nV0=\u001c(1\u0000c\u0012)\u0000(Vp+M\nx2\u000b\n)\u0017\u0012+\u0017?\n\u0017\u0012(68)\nFor the reasons explained above, V0=\u0000Vpinside the separatrix. Substituing Eq.68 in Eq.55\nVp=\u0000\u001c\u0014\n1 +(\u0017\u0012+\u0017?)I2\n\u0017?(I1\u0000I2)\u0000(\u0017\u0012c\u0012+\u0017?c?)\n\u0017?(I1\u0000I2)(I2\u0000I3)\u0015\n\u0000(\u0017\u0012+\u0017?)\n\u0017?(I1\u0000I2)Z+1\n1d [M(1\u0000\nx2\u000b\nh1i) +\u001cL] (69)\n12By substituing Eq.68 in Eqs.67, we get:\n\u0017\u0012+\u0017?\n\u0017\u0012d\nd \u0014\nx2\u000bd\nd \u0000\nM\nx2\u000b\u0001\u0015\n\u0000\u0017?\n\u0016\u0012\u000f\nq\u00132\n[Vp+M\nx2\u000b\n]h1i+\n+\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\b\nM(1\u0000\nx2\u000b\nh1i) +\u001c(1\u0000c\u0012)(h1i+L)\t\n= 0(70)\nBy solving simultaneously Eq.69 and Eq.70, with the appropriate boundary conditions, we can obtain the radial\npro\fle ofMand the phase velocity Vp. The systems of equations Eq.61+Eq.62 and Eq.69+Eq.70 represent\nlimit cases which can be easily solved numerically, and they can be both deduced from Eq.66+Eq.67 under\nappropriate limits. To further simplify the calculations, we introduce the variable k=p\n(1\u0000 )=2. Then we\nde\fne the following quantities, E=\nx2\u000b\n=(2k),F= 2kh1iand we de\fne the variable Qso as to absorb the\nfactorVp+\u001c(1\u0000c\u0012):\nQ=\u001c(1\u0000c\u0012)\u00002kEM\n\u001c(1\u0000c\u0012) +Vp(71)\nBy using these quantities, Eq.69 and Eq.70 become:\nVp=\u0000\u001c\u0014\n1 +I6\nI4+I5(c\u0012\u0000c?)\u0000c\u0012I5\nI4+I5\u0015\n(72)\n\u0017\u0012+\u0017?\n4\u0017\u0012d\ndk\u0014\nEdQ\ndk\u0015\n\u0000\u0017?\n\u0016\u0012\u000f\nq\u00132\nF(Q\u00001)\u0000\u0017\u0012\n\u0016\u0012\u000f\nq\u00132\nQ\u0012\nF\u00001\nE\u0013\n= 0 (73)\nwhereI4=R1\n02kh1idk,I5= (\u0017\u0012+\u0017?)=\u0017?R+1\n1Q(F\u00001=E)dkandI6=R+1\n1(F\u00001=E)dk.\n7 Numerical results\nIt is possible to determine the magnetic islands rotation velocity by numerically integrating the system of\nequations consisting in Eqs.61,62 for the weak damping regime and Eqs.72,73 for the intermediate damping\nregime, in the limit of small Larmor radius. Unfortunately, the numerical integration of Eqs.66,67 still represents\na challenge too di\u000ecult to solve. We leave the analysis of the more general case to a future work.\nThe weak damping regime is characterized by the parameters \u000b, which enters the denominator H, and\u0017\u0012=\u0016.\n\u000bis proportional to the ratio between the island width wand the ion-acoustic radius \u001as, and it is a measure\nof the importance of the ion-acoustic waves on the \rattening of the density pro\fle inside the separatrix. \u0017\u0012=\u0016\ndepends on the plasma collisionality and measures the importance of the poloidal \row damping. The result of\nthe numerical integration of Eqs.61,62 for di\u000berent values of \u000band\u0017\u0012=\u0016is displayed in Fig.1. The presence of\nFigure 1: Island phase velocity Vpversus the ion-acoustic parameter \u000bfor di\u000berent choices of the poloidal \row\ndamping parameter \u0017\u0012=\u0016\nthe resonant denominator Hin Eq.62 prevents the solution from converging for the smaller values of \u000b, which\n13correspond to the small island width limit. The interesting feature of Fig.1 is the transition of the value of Vp\nfrom positive values to negative values as the parameter \u0017\u0012=\u0016is increased from values much smaller than one to\nvalues close to one. A positive phase velocity corresponds to an island rotating in the direction of the electron\n\ruid, while a negative value corresponds to the direction of the ion \ruid. The hypotheses of zero equilibrium\nelectric \feld means that the E^Bdrift has been subtracted from the plasma velocity.\nThe intermediate damping regime, in the limit of small Larmor radius, is characterized by the parameters\nw=\u001as, which enters the perpendicular damping coe\u000ecient, and \u0017i, which enters both the poloidal and the\nperpendicular damping coe\u000ecients. From Eq.72, it is evident that the island phase velocity Vpis determined\nby the neoclassical velocities c\u0012andc?, which are proportional to the radial temperature gradient through the\nparameter\u0011i=Ln=LT. The result of the numerical integration of Eqs.61,62 for di\u000berent values of w=\u001asand\u0017i\nis displayed in Figs.2,3,4 for the choices of \u0011i= 0:5,\u0011i= 1 and\u0011i= 2.\nFigure 2: Island phase velocity Vpversus the normalized island width w=\u001asfor di\u000berent choices of the collision\nfrequency\u0017iwith\u0011i= 0:5\nFigure 3: Island phase velocity Vpversus the normalized island width w=\u001asfor di\u000berent choices of the collision\nfrequency\u0017iwith\u0011i= 1\nBecause of the absence of a resonant denominator, the solution converges even in the limit of small island\nwidth. However, the equations we integrate are valid only in the limit of small Larmor radius, so that the results\nlose validity for w=\u001as<1. The interesting feature of these pictures is the transition of the value of Vpfrom\nnegative values to positive values as the parameter \u0011iis increased from values less than one to values larger than\none. Note that LnandLTare of the same order in realistic tokamak plasmas, so that very large values and\nvery small values of \u0011are unrealistic. The di\u000berent slopes of the curves corresponding to the di\u000berent values of\n14Figure 4: Island phase velocity Vpversus the normalized island width w=\u001asfor di\u000berent choices of the collision\nfrequency\u0017iwith\u0011i= 2\n\u0017ishow that, the smaller the collisionality, the more e\u000bective the neoclassical \row damping is in relaxing the\nisland velocity towards the neoclassical value, which is determined by the parameters c\u0012andc?. Note that the\nnumerical integration of Eqs.61,62 for the weak damping regime was performed with the choice \u0011i= 1, but the\nresults are not signi\fcantly a\u000bected by the choice of \u0011i. In all our integrations we chose \u001c= 1.\n8 Conclusions\nIn this paper we addressed the issue of determining the phase velocity of a chain of freely rotating magnetic\nislands by using a four \feld gyro\ruid system of equations which includes the neoclassical \row damping e\u000bects\nand the lowest order FLR corrections. To do that, we \frst solved the gyrokinetic equation under some simplifying\nhypotheses and we computed the FLR corrections to the poloidal \row damping. Then we deduced a four \feld\ngyro\ruid model by starting from a set of gyro\ruid equations and we closed it by using a simpli\fed form for the\ndivergence of the stress tensor, which provides the neoclassical \row damping e\u000bects. By following the method\ndescribed by Fitzpatrick & Waelbroek [8, 9, 10, 11], we managed to obtain a system of equations whose solution\nprovides the islands rotation velocity consistently with the \felds radial pro\fles close to the resonant surface.\nWe applied this system of equations to the investigation of two collisionality regimes, namely the weak damping\nregime and the intermediate damping regime. In the second case, which corresponds to the low collisionality\nregime, an additional term, containing the lowest order FLR corrections to the poloidal \row damping, appeared\nin the equations. The numerical integration of Eqs.61,62 in the weak damping regime shows that the island\nphase velocity moves from positive values to negative values as the poloidal damping parameter \u0017\u0012=\u0016is increased\nfrom values much smaller than one to values close to one. A positive phase velocity is associated with a magnetic\nisland rotating in the direction of the electron \ruid, while negative values means that the island rotates in the\ndirection of the ions. The numerical integration of Eqs.72,73 in the intermediate damping regime shows that\nthe phase velocity Vpmoves from negative values to positive values as the parameter \u0011iis increased from values\nless than one to values larger than one. These results are in agreement with what was already known about\nthe subject, but they are valid only within the limitations of their hypotheses. Unfortunately, the numerical\nintegration of Eqs.66,67 still represents a challenge too di\u000ecult to solve. We leave the analysis of the more\ngeneral case to a future work. In the case of large, saturated islands, we expect \u001a2to be small, so that the\nFLR corrections we found should be small as well. However, if the islands are not much smaller than the ion\nacoustic radius, or if we are in the case of a high temperature plasma, the product \u001a2\u0017\u0012=\u0016multiplying the\nadditional term in Eq.67 might be comparable with the other terms entering the equations. The procedure we\nused to deduce the \fnal equations and to numerically solve them is thorughly described in [10, 11]. Although\nthis procedure is based on the assumption that the island width is much larger than the ion-acoustic radius, the\nso called sonic regime, the extension of this approach to the hypersonic regime would require a few changes in\nthe initial hypotheses. In particular, when we deal with hypersonic islands, the hypothesis that the lowest order\n\felds are \rux functions is no longer valid, and the e\u000bect of the drift-acoustic waves must be kept into account\nto determine the radial pro\fles of density and electrostatic potential inside the separatrix.\n15References\n[1] H. P. Furth, J. 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Sabbagh, \\Theory for island induced neoclassical toroidal plasma\nviscosity in tokamaks,\" Nuclear Fusion , vol. 51, no. 4, p. 043013, 2011.\n[27] R. Fitzpatrick and F. Waelbroeck, \\Spontaneous healing and growth of locked magnetic island chains in\ntoroidal plasmas,\" Physics of Plasmas (1994-present) , vol. 19, no. 11, p. 112501, 2012.\n[28] R. Fitzpatrick and F. Waelbroeck, \\E\u000bect of drift-acoustic waves on magnetic island stability in slab\ngeometry,\" Physics of Plasmas (1994-present) , vol. 12, no. 12, p. 122511, 2005.\n17"    },    {        "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": "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": "2404.00845v1.Harnessing_Interlayer_Magnetic_Coupling_for_Efficient__Field_Free_Current_Induced_Magnetization_Switching_in_a_Magnetic_Insulator.pdf",        "content": " \n 1 Harnessing Interlayer Magnetic Coupling for Efficient, Field -Free  \nCurrent -Induced Magnetization Switching in a Magnetic Insulator  \n \nLeran Wang1, Alejandro O. Leon2*, Wenqing He 3, Zhongyu Liang1,  \nXiaohan L i3, Xiaoxiao Fang1, Wenyun Yang1, Licong Peng4,  \nJinbo Yang1,5*, Caihua Wan3, Gerrit E. W. Bauer6,7, Zhaochu Luo1,5* \n1State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, Institute of Condensed \nMatter Physics and Materials, School of Physics, Peking University, 100871 Beijing, China.  \n2Departamento de Física, Facultad de Ciencias Naturales, Matemática y del Medio Ambiente, \nUniversidad Tecnológica Metropolitana, Las Palmeras 3360, Ñuñoa 780- 0003, Santiago, Chile.  \n3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese \nAcademy of Sciences, Chinese Academy of Sciences, 100190 Beijing, China.  \n4School of Materials Science and Engineering, Peking University, 100871 Beijing, China.  \n5Beijing Key Laboratory for Magnetoelectric Materials and Devices, 100871 Beijing, China  \n6Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, 100864 \nBeijing, China.  \n7Advanced Institute for Materials Research (AIMR), Tohoku University, 980- 8576 Sendai, Japan.  \n \n*Correspondence to: zhaochu.luo@pku.edu.cn (Z.Luo); aleonv@utem.cl  (A.L.);  \njbyang@pku.edu.cn  (J.Y .)  \n \nKeywords: Magnetic coupling; magnetic insulator; spin- orbit torques; spin -mixing \nconductance; spin -Hall magnetoresistance  \nAbstract: Owing to the unique features of low Gilbert damping, long spin -diffusion lengths \nand zero Ohmic losses, magnetic insulators are pro mising candidate materials for next -\ngeneration spintronic applications. However, due to the localized magnetic moments and the \ncomplex metal -oxide interface between magnetic insulators and heavy metals, spin -functional \nDzyaloshinskii –Moriya interactions or  spin Hall and Edelstein effects are weak, which \ndiminishes the performance of these typical building blocks for spintronic devices. Here, we \nexploit the exchange coupling between metallic and insulating magnets for efficient electrical \nmanipulation of heavy metal/magnetic insulator heterostructures. By inserting a thin Co layer, \nwe enhance the spin -orbit torque efficiency by more than 20 times, which significantly reduces \nthe switching current density. Moreover, we demonstrate field -free current -induced \nmagnetization switching caused by a symmetry- breaking non -collinear magnetic texture. Our \nwork launches magnetic insulators as an alternative platform of low -power spintronic devices.    \n 2 1. Introduction  \nEngineering the magnetic coupling plays a crucial role in determining the functionalities \nof spintronic devices. The interlayer Ruderman –Kittel–Kasuya –Yosida (RKKY) interaction[1] \nand exchange -bias effect[2] stabilize the magnetic reference layer[3,4] in the magnetic tunnel \njunctions of non- volatile magnetoresistive random -access memories (MRAM), while the \nintralayer dipolar and chiral couplings are essential for scalable spin logic[5-7] and neuromorphic \ncomputing devices[8-10]. Research on the physics and applications of these couplings focusses \non electric conductors such as magnetic/ non-magnetic and ferro magnetic/antiferromagnetic \nmetal bilayers, but  much of the cor responding phenomenology in magnetic insulators remains \nto be explored. Taking advantages of low Gilbert damping[11,12], long spin- diffusion lengths[13-\n15] and zero Ohmic losses, magnetic insulators, particularly the iron garnets, are promising \ncandidate materials for low -power and high -speed spintronic applications. Pioneering studies \non magnetic metal/magnetic insulator bilayers demonstrated injection and modulation of spin \nwaves[16-20], but due to the complexity of polycrystalline metal- oxide interfaces[21-23] a full \nunderstanding of the underlying mechanisms remains elusive.  \nHere, we report experiments on a high- quality magnetic heterostructure comprising of an \nultrathin Co film on an epitaxial terbium iron garnet (Tb 3Fe5O12, TbIG ) layer, which shows a \nstrong ferromagnetic exchange coupling of ~69.0 μ J/m2. By growing a heavy metal layer on \ntop of the Co/TbIG bilayer, we electrically detect and manipulate the TbIG magnetization via \nthe direct and inverse spin Hall and/or Edelstein effects. The interlayer exchange coupling \nsignificantly enhances the interfac ial spin -mixing conductance, leading to a large spin -Hall \nmagnetoresistance (SMR) and a high spin -orbit torque (SOT) efficiency. Moreover, the non-\ncollinear magnetization of the Co  layer breaks the mirror symmetry of the device, thereby \nfacilitating current -induced switching of the TbIG magnetization without external magnetic \nfields. Our work demonstrates an interplay between magnetic coupling and SOTs that can help to design reliable and efficient magnetic memory devices and pave a new pathway for magnetic \ninsulator spintronics.  \n2. Structural and magnetic properties of Pt/TbIG   \n 3 The insulating rare -earth iron garnet TbIG was epitaxially grown on (111) -oriented single -\ncrystalline gadolinium gallium garnet (Gd 3Ga5O12, GGG) substrates by pulsed laser deposition \n(PLD) (see Method s). The structure of TbIG is constructed with three magnetic sublattices: an \noctahedral Fe, a tetrahedral F e, and a dodecahedral Tb sublattice[24] (Figure 1a ). TbIG films \nwith various thicknesses ranging from 5 to 53 nm were deposited for different experiments. The \n5 nm -thick TbIG was used for the electrical tr ansport measurement, whereas the thick TbIG \nyielding a stronger structural signal was used for X -ray diffraction (XRD) analysis that gives \ninsight into the microstructure and strain state. As shown in Figure 1b, we present the symmetric \nXRD scans of a 53 n m-thick TbIG around the (444) peak of GGG substrate. The rocking curve \ngives a full -width at half -maximum (FWHM) of ~0.02 ° (see Supplementary Information S1), \nindicating a high- quality single crystallinity. We can further extract the lattice cons tant of d 444 \n= 0.1815 nm for the epitaxial TbIG  film which is slightly larger than that (~0.1795 nm) in a \nTbIG bulk[25], implying a strained state originating from the TbIG/GGG interface[26,27]. We then \nconducted the magnetization measurements using vibrating- sample magnetometer (VSM) and \nmagneto -optical Kerr microscopy (MOKE) to study the magnetic properties of TbIG films \n(Figure 1c). The magnetization hysteresis loops with out -of-plane magnetic fields reveal a \nstrong perpendicular magnetic anisotropy and a small saturation magnetization ( MS ~23 emu/cc ) \nas a result of the partial compensation of sublattice magnetizations.  \nWe sputtered a 5 nm -thick Pt layer on TbIG (5 nm)/GGG with a low deposition power (20 \nW) to mitigate the bombardment damage of Pt atoms on the TbIG surface. The cross- sectional \nhigh- resolution transmission electron microscopy (HR -TEM) image confirms the cl ean and \nsharp interface in the heterostructure that is p rerequisite to efficiently manipulate the \nmagnetization by spin currents (Figure 1d). The Pt/TbIG heterostructure was then patterned \ninto Hall bar structures with the channel width of 10 µ m using UV p hotolithography combined \nwith the ion milling process. We measured the electrical transport with out -of-plane magnetic \nfields and observed an anomalous Hall resistance of - 1.3 mΩ at room temperature (Figure 1e).  \nBy varying the measurement temperature, the coercivity exhibits a peak at a critical temperature \nat which the polarity of the anomalous Hall resistance is reversed, indicating magnetic moment compensation in TbIG around 270 K  (T\ncomp) (Figure 1f). Since the sign of the anomalous Hall \nresistance depends on the direction of the magnetic moment of  the tetrahedral Fe  sublattice[28],  \n 4 the sign is reversed  when the temperature varies across the compensation state due to the fact \nthat the alignment relation ship between the magnetic field and the magnetic moment of the \ntetrahedral Fe sublattice reverses at T comp (Figure 1e).  \n3. Spin Hall magnetoresistance and magnetic coupling in Pt/Co/TbIG  \nTo exploit the rich interfacial magnetic effect of heavy metal/metallic magnet such as high \nspin-mixing conductance and large Dzyaloshinskii –Moriya interaction (DMI), we inserted a \nthin Co layer ( tCo = 0.3, 0.6 and 0.9 nm as calculated by multiplying the deposition rate and \ntime) between the Pt and TbIG layers by DC magnetron sputtering ( Figure 2 a). With the \ninsertion of Co layers, a linear Hall resistance with positive slope emerges on top of the square -\nlike anomalous Hall loop, indicating an in- plane magnetic anisotropy of the Co layer (Figure \n2b). The Co magnetization gets fully perpendicular at  large out -of-plane magnetic fields (Figure \n2c). Interestingly, the magnitude of the square -like anomalous Hall loop is significantly \nenhanced for thicker Co films, accompanying a change of sign of the Hall resistance from \nnegative to positive (Figure 2d). Moreover, the anomalous Hall loop in Pt/Co ( 0.9 nm)/TbIG of \n52.0 mΩ is more than an order of magnitude larger than that in Pt/TbIG ( -1.3 mΩ) with the \nsame TbIG thickness. An ultrathin Co insertion therefore offers a sensitive method to \nelectrically detect the magnetization direction of insulators.  \nTwo mechanisms may cause the large anomalous Hall loop. On one hand, the exchange \ncoupling between Co and TbIG may force a non- collinear alignment of the Co and TbIG \nmagnetizations such that the out -of-plane component of the Co magnetization contributes to  \nthe anomalous Hall effect. In th is scenario, the longitudinal ( Rxx) and transverse Hall ( Rxy) \nresistance can be written as:  \n𝑅𝑅𝑥𝑥𝑥𝑥=𝑅𝑅0+∆𝑅𝑅AMR sin2𝜃𝜃Cosin2𝜑𝜑Co,                      (1)  \n𝑅𝑅𝑥𝑥𝑥𝑥=−∆𝑅𝑅AMR sin2𝜃𝜃Cosin2𝜑𝜑Co+𝑅𝑅AHECo𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃Co+𝑅𝑅OHE𝐻𝐻𝑧𝑧,             (2)  \nwhere R0 and ΔRAMR represent the magnetization -independent longitudinal resistance and the \nanisotropic magnetoresistance (AMR), respectively. 𝑅𝑅AHECo (ROHE) is the anomalous (ordinary) \nHall resistance. θ Co (φCo) is the angle between Co magnetization and z-axis ( y-axis) as defined \nin Figure 2a. We disregard the very small magnetic- proximity effect in Pt (see Supplementary \nInformation S2). The second mechanism would follow the spin Hall magnetoresistance (SMR)  \n 5 scenario as widely used to extract spin transport parameters in heavy metal/magnetic insulator \nbilayers[29,30] such as the interfacial spin -mixing conductance G. In such measurements, an \nelectric current is applied to the heavy metal layer and due to the spin Hall effect, it can generate a transverse spin current that will either be transmitted or reflected at the heavy metal/magnetic \ninsulator interface depending on the relative orientations between the spin current polarization \nσ and the magnetization m . The ratio of spin transmission/reflection at the interface will further \nmodulate the electric current in the heavy metal layer due to the inverse spin Hall effe ct. This \nleads to the presence of a resistance change that has a distinct angle dependence. In the high \nfield limit, the magnetizations of Co and TbIG are aligned along the field direction with angle θ (φ) with the z -axis ( y-axis). The longitudinal ( R\nxx) and transverse Hall ( Rxy) resistance then \nread[29,30]:  \n𝑅𝑅𝑥𝑥𝑥𝑥=𝑅𝑅0−∆𝑅𝑅SMR sin2𝜃𝜃cos2𝜑𝜑,                     (3) \n𝑅𝑅𝑥𝑥𝑥𝑥=−∆𝑅𝑅SMR sin2𝜃𝜃sin2𝜑𝜑+𝑅𝑅AHESMRcos𝜃𝜃+𝑅𝑅OHE𝐻𝐻𝑧𝑧,            (4)  \nwhere ΔRSMR and 𝑅𝑅AHESMR are the SMR and the SMR -induced anomalous Hall resistances of the \nTbIG/Co interfaces adjacent to Pt, respectively, which are a function of the Co thickness.  \nNote that the transverse SMR and the SMR -induced anomalous Hall effect (AHE) have \nthe same symmetry as the AMR -induced transverse (planar Hall) resistance and anomalous Hall \nresistance in Co, respectively. Hence, we cannot quantitively distinguish the cont ribution of \nanomalous Hall resistance from the SMR -induced AHE and conventional AHE by the \ntransverse resistance only (see Supplementary Information  S3). By contrast, the contributions \nof the AMR and SMR to the longitudinal resistance changes distinctly de pend on the \nmagnetization angles and can be resolved by angular -dependent measurements. As shown in \nFigures 2e -2g, we measured Rxx with a rotating magnetic field of 60 kOe in zx, zy and xy planes. \nAccording to the symmetry of SMR and AMR, a fit to the angular -dependent  data in zy plane \nyields ΔRSMR = 61.1, 87.7, 225.9 and 555.7 m Ω for different Co thickness of tCo=0, 0.3, 0.6 and \n0.9 nm, respectively, while a fit to the angular -dependent  data in zx  plane yields Δ RAMR = -0.9, \n1.4, 13.2 and 72.6 m Ω. The sizeable ΔRSMR implies a substantial absorption of the spin currents \nby the heavy metal/magnets interface.  \nInterfacial spin transmission/reflection can be quantified by the complex spin -mixing \nconductance G with a real ( Gr) and an imaginary ( Gi) part[29,30]. Gr is related to the damping - \n 6 like torques acting upon m, which is proportional to m ×(m×σ) and manifests itself as a \nlongitudinal SMR resistance Δ RSMR. On the other hand, the imaginary part G i is associated with \na field -like torque, which is proportional to m×σ and contributes a Hall resistance 𝑅𝑅AHESMR in Pt. \nThe experimental values of ΔRSMR and 𝑅𝑅AHESMR, and assuming spin Hall angle θ Pt = 0.08 and spin \ndiffusion length λPt =1.4 nm for Pt[30,32], we can estimate Gr = 4.3×1014 Ω-1m-2 and Gi = -1.2×1013 \nΩ-1m-2, which is consistent with reported values for Pt/magnetic garnet bilayers[30-32]. The \ninsertion of the Co layer greatly enhances the spin -mixing conductance up to one order of \nmagnitude to a G r for tCo of 0.6 and 0.9 nm of 3.9 ×1015 Ω-1m-2 and -5.8×1015 Ω-1m-2, respectively, \nmuch larger than that in simple Pt/TbIG or Pt/Co heterostructures[33], implying that the \nexchange coupling plays a critical role in the spin -dependent transport.  \nTo unravel the effect of the Co spacer, we recorded the hysteresis loop by sweeping the in -\nplane magnetic field. In Pt/TbIG, the Hall resistance exhibits the usual shape for a perpendicular \nmagnetization under an in -plane magnetic field ( Figure 3a ). By fitting the in -plane hysteresis \nloop, we can obtain the effective magnetic anisotropy field HK = 8 kOe and magnetic anisotropy \nenergy Ean = HKMS/2 = 9.2× 103 J m-3 (see Supplementary Information  S6). In contrast, Pt/Co \n(0.9 nm)/TbIG  shows a distinctly different shape with a sharp peak at small magnetic fields and \nsaturation at a higher magnetic field  (Figure 3b). This curve  contains information about the \nexchange coupling parameter J  between Co and TbIG. The magnetic energy per unit area of \nPt/Co/TbIG can be written as:  \n𝐸𝐸=−𝐽𝐽𝒎𝒎�Co∙𝒎𝒎�TbIG−𝐾𝐾TbIG𝑡𝑡TbIG𝒎𝒎�TbIG ,𝑧𝑧2−𝐾𝐾Co𝑡𝑡Co𝒎𝒎�Co,𝑧𝑧2−𝑯𝑯∙(𝒎𝒎Co+𝒎𝒎TbIG ),  (5)  \nwhere KCo(TbIG)  is the magnetic anisotropy coefficient and 𝒎𝒎�Co(TbIG )  is the unit vector of the \nmagnetization 𝒎𝒎Co(TbIG )  of Co (TbIG) . At zero magnetic field, the Co and TbIG \nmagnetization unit vectors 𝒎𝒎�Co and 𝒎𝒎�TbIG  are in -plane and out-of-plane , respectively . \nMinimizing Eq. (5) for H =0 and leading order in the interface exchange J  leads to:  \n𝑚𝑚�Co,𝑧𝑧=±𝐽𝐽(2|𝐾𝐾Co|𝑡𝑡Co) ⁄ ,                               \n𝑚𝑚�TbIG ,𝑥𝑥=𝐽𝐽(2|𝐾𝐾TbIG |𝑡𝑡TbIG ) ⁄ ,                         (6)   \nwhere the sign of ± holds for up - and downward TbIG  magnetization (see Supplement S4). A \nrelatively weak in -plane magnetic field overcomes  the exchange coupling and pulls the Co \nmagnetization into the plane, lead ing to a sharp decrease of the Hall resistance around zero \nmagnetic field (Figure 3c). Due to its large perpendicular magnetic anisotropy,  the TbIG  \n 7 magnetization persists to be perpendicular up to higher in- plane magnetic fields. The gradual \npull into the plane is accompanied by a reduced and ultimately vanishing Hall resistance. The \neffective magnetic anisotropy field H K = 53.0 kOe of TbIG in Co (0.9 nm)/TbIG is even larger \nthan that in pure TbIG (see Figure 3b). It has been reported that the interfaces of Pt/Co and \nCo/oxide can give large interfacial perpendicular magnetic anisotropy due to spin- orbit \ncoupling[34]. Owing to the interlayer exchange co upling and proximity effect, the perpendicular \nmagnetic anisotropy of TbIG may get enhancement from Pt/Co interfaces.  We can then deduce \na Hall resistance caused by the out -of-plane tilt of the Co magnetization ( 𝑅𝑅AHECo = 71.0 m Ω), \nwhich is larger than the high -field TbIG/Co SMR ( 𝑅𝑅AHESMR = -19.0 mΩ). We hence estimate G i \nin Pt/Co (0.9 nm)/TbIG  to be -7.2×1014 Ω-1 m-2, which is 60 times larger than that in Pt/TbIG. \nFrom the ratio of 𝑅𝑅AHECo at zero and that at the saturation out -of-plane magnetic fields (Figure \n2c) the competition between Co/TbIG exchange coupling and magnetic anisotropies leads to a \ntilt angle of the Co magnetization of 83.2 ° at zero magnetic fields, i.e. pulled out of the plane \nby 6.8° . Substituting the observed magnetic anisotropies into the macrospin model, we arrive \nat an  exchange coupling strength between Co and TbIG of J = 69.0 μ J m-2 (see Supplementary \nInformation S4 ).  \n4. Efficient current -induced magnetization switching  \nThe enhanced spin -mixing conductance implies efficient SOTs in current -biased \nPt/Co/TbIG structures. We measured the SOT efficiency by recording magnetic hysteresis loops as a function of a charge current bias and in -plane and out -of-plane magnetic fields,  a common \ntechnique for both metallic and insulating magnet/heavy metal bilayers\n[35-37]. As shown in \nFigure 4a , we can measure the damping -like SOTs by the current -induced shifts of the \nhysteresis loops since they give rise to out -of-plane effective fields H eff that act on the DMI -\nstabilized Néel -type domain walls[36]. We may define an SOT efficiency χ as: \n𝜒𝜒=𝐻𝐻eff/𝑗𝑗.                                 (7)  \nRepresentative hysteresis loops of Pt/Co (0.6 nm)/TbIG with H x = 300 Oe and j = ±4.3×1010 \nAm-2 are shown in Figure 4b. Changing the direction of the applied currents cause opposite \nshifts of the hysteresis loops as expected for a current -induced Heff caused by damping -like \ntorques. The hysteresis loops shifts increase with an in -plane magnetic field and saturate at a  \n 8 critical value that is governed by the DMI[32,33,35] (Figure 4c). In Pt/TbIG, the SOT efficiency \nsaturates to 0.9 ×10-14 TA-1m2 at Hx = 340 Oe. We can estimate the modulus of the effective DMI \nconstant |𝐷𝐷|=𝜇𝜇0𝑀𝑀sΔ|𝐻𝐻𝑥𝑥| from the saturation field H x, where  μ0 is the vacuum permeability \nand Δ = (A/Ku)1/2 the width of the domain wall. Here, A = 2.3 pJ m-1 is the exchange stiffness \nand Ku is the anisotropy energy obtained from the experiment , leading to |DPt/TbIG | =8.6 μJ m-2 \nconsistent with other  Pt/magnetic garnet bilayers[35,36,38 -41],. Interestingly, the insertion of a thin \nCo layer between TbIG and Pt significantly enhances the SOT efficiency as well as the effective \nDMI. For Pt/Co (0.6 nm)/TbIG 𝜒𝜒=24.0× 10-14 TA-1m2 at Hx = 500 Oe, more than an order of \nmagnitude larger than that of other heavy metal/magnetic insulator systems[35,42].  \nThe model of magnetization reversal by the motion and annihilation of Neel domain walls \nimplies that the spin -orbit torque is most efficient at the domain wall center (see Figure 4a) \nwhere the magnetizations of Co and TbIG are strictly parallel. We may the refore carry over the \nanalysis of the SMR in the collinear limit of high magnetic fields to understand the observed \nenhancement of the damping- like torque. Here the Co capping layer enhances the effective spin \nmixing conductance G r from a small value for p ure TbIG to the large one of Co. Since Gr \nmeasures the absorption of the transverse spin current by the ferromagnet, this result directly \nexplains the enhanced spin transfer torque. The interface exchange coupling subsequently \ncommunicates the torque to the TbIG which leads to the motion of the e ntire domain wall. The \nCo overlayer ensures non -chiral domain walls required for the magnetization reversal. An \ninteresting subject for future research is an analysis of the Co -thickness dependence of the spin \ntransfer, since partial trapping of the spins into quantum wells[43,44] and the associated multiple \nscattering at the TbIG/Co interface could additionally increase 𝐻𝐻eff.  \nSince large damping -like torques in Pt/Co/TbIG should reduce the critical currents that \nswitch the magnetization, we measure the changes in the transverse resistance under current \npulses with a duration of 0.2 ms and a DC current of up to 1 mA corresponding to a current \ndensity of 2.5 ×1010 Am-2 in the same direction (see Supplementary Information S5). In the \npresence of an in -plane magnetic field of 400 Oe  along the current direction, pulses with an \namplitude up to j  = 6.1×1010 Am-2 change the sign of the t ransverse resistance which indicates \na complete TbIG magnetization switching (Figure 4d). Reversing the direction of the in- plane \nmagnetic field leads to an anti -clockwise hysteresis loop (Figure 4e), which agrees with the  \n 9 sign of the spin Hall angle of Pt[45]. Increasing the in- plane magnetic fields reduces the energy \nbarrier of magnetization reversal and the critical switching current density of T bIG/Co (0.6 \nnm)/P t from 6.9×1010 to 5.0×1010 Am-2 (Figure 4f) half of that in Pt/TbIG (1.2 ×1011 Am-2 at H x \n= 204 Oe). The suppression of the critical switching current density is less than expected from \nthe enhanced SOT efficiency between Pt/Co/TbIG and Pt/TbIG deduced above, which suggests that Joule heating plays a role in the magnetization switching at high  current densities\n[46,47].  \n5. Field-free magnetization switching  \nCurrent -induced switching of perpendicular magnetization without the need to apply \nmagnetic fields is highly desirable in high -density magnetic memories. To this end, various \nschemes have been proposed in metallic systems, e.g., by breaking the mirror symm etry in \nasymmetric lateral designs[48,49] and non- collinear magnetic alignment[50,51]. These invoke \ncomplex  devices incorporating multiple exotic materials that might be difficult to realize with \nmagnetic insulators. Here we may take advantage of the exchange coupling between Co and TbIG that leads to a non- collinear magnetic texture that allows switching of a perpendicular \nmagnetization by electric currents without the assistance of magnetic fields ( Figure 5a ).  \nAs in Pt/Co (0.6 nm)/TbIG, we observed a full current -induced switching of magnetization \nin Pt/Co (0.9 nm)/TbIG  with the same switching polarity cycle as function of the in- plane \nmagnetic field sweeps (Figures 5b -5c). Taking the device out of the ele ctromagnet set -up, where \nthe residual magnetic field is less than 1.0 Oe, does not deteriorate the switching performance \n(Figures 5d -5e). The switching polarity is determined by the history of the applied in -plane \nmagnetic fields. When initialized with a p ositiv e (negative) in -plane magnetic field of 1 kOe, \nthe electric current can switch the magnetization with anti -clockwise (clockwise) polarity. To \nverify the reliability of current -induced field- free switching, we repeated measurements by \napplying alternating c urrent pulses of ± 1.43×10\n11 Am-2 in the absence of magnetic fields (Figure \n5f). The Hall resistance jumps between the resistance levels corresponding to up and down \nmagnetizations after every pulse. The switching polarity reverses when the direction of the  pre-\nset in -plane magnetic field changes from positive to negative, which is consistent with the \ncurrent -driven hysteresis loops.  \nThe SOT efficiency measurement in Pt/Co (0.9 nm)/TbIG  led to several interesting  \n 10 observations (Figure 5g). First, in contrast to Pt/Co (0.6 nm)/TbIG, the magnitude of SOT \nefficiency in Pt/Co (0.9 nm)/TbIG saturates at a low magnetic field, implying the existence of \nan effective in -plane magnetic field due to the coupling with the in- plane magnetized Co layer. \nSecond, the saturation SOT efficiency ( 2.6×10-14 TA-1m2) in Pt/Co (0.9 nm)/TbIG is lower than \nthat ( 24.0× 10-14 TA-1m2) in Pt/Co (0.6 nm)/TbIG, though the spin-mixing conductance in Pt/Co \n(0.9 nm)/TbIG is higher than that in Pt/Co (0.6 nm)/TbIG. However, a thicker Co layer also \ndissipates more spin currents, leading to a reduced SOT efficiency. Therefore, there is a trade -\noff of Co thickness to obtain the  optimum SOT efficiency. The SOT efficiency can be further \nenhanced by optimizing the thickness of TbIG[52] and tuning its magnetization compensation \nstate[53]. Moreover, there is a substantial  non-zero SOT efficiency at zero magnetic fields and \nits sign depe nds on the history of sweeping in- plane magnetic fields, supp orting the \nperformance of field -free magnetization switching.  \n6. Conclusion  \nIn summary, we  exploit the magnetic coupling between metallic and insulating magnets, \nto efficiently manipulate the magnetism in an insulator with perpendicular magnetization. The itinerant conduction electrons in metallic magnets provide rich and strong spin -related \ninterfacial effects in magnetic trilayers. By harnessing the magnetic coupling between metallic \nand insulating magnets, these interfacial effects can be imprinted into magnetic insulators, \nallowing for efficient electrical de tection and manipulation of magne tism. In addition, coating  \nthe interface with  a few metallic magnetic atoms significantly enhances the SOT efficiency as \nwell as the DMI. Furthermore, we demonstrate the performance of field -free current -induced \nmagnetization switching that results from symmetry -breaking non -collinear magnetic textures, \npaving the way for scalable magnetic memory devices. Therefore, our work offers a new avenue to engineer  efficient  spin memory and logic devices based on magnetic insulators.  \n \n  \n 11 Methods  \nGrowth of TbIG films : TbIG thin films were deposited on 5 mm × 5 mm Ga 3Gd5O12 (111) single -\nside-polished substrates via pulsed laser deposition (PLD) at a laser fluence of ~1.4 Jcm-2, and a \ntarget -to-substrate distance of ~6 cm. During deposition, the substrate temperature was heated to \n800°C and the oxygen pressure was 30 mtorr. After deposition, annealing process was performed to \npromote the epitaxial growth of TbIG films with high quality and the cooling rate of the chamber \nwas 20 °C min-1. Epitaxial growth of the films was confirmed via a high- resolution X -ray diffraction \n2θ scan of the (444) reflection. The thickness of thick TbIG films was determined by X -ray \nreflectometry, whereas the thickness of thin TbIG films (~5 nm) was calculated by the number of \nlaser pulses.  \nDevice fabrication and measurement : After the deposition of the TbIG thin film via PLD, metallic \nlayers (such as Pt and Co/P t) were deposited by DC magnetron sputtering at room temperature with \na base pressure <5 ×  10-8 torr. The deposition rate was 0.026 nms-1 for P t and 0.011 nms-1 for C o. \nThe thickness of Co and Pt layers was calculated according to the deposition rate. These multilayers \nwere patterned into Hall bars using a combination of UV photolithography and Ar ion milling \ntechnique with lateral dimensions of 10 µ m × 35 µm (width ×  length). The magnetic properties of \nTbIG films  were  measured by superconducting quantum interference device (SQUID) \nmagnetometer. 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( c) Magnetization as a function of out -of-plane magnetic field Hz \nobtained by SQUID VSM (red curves) and MOKE (black curves) mea surement. ( d) Cross -\nsectional HR -TEM image of the T bIG/P t heterostructure on a GGG substrate. Scale bar: 10 nm. \n(e) AHE hysteresis loops measured at different temperatures in Pt/T bIG. (f) Measured \ncoercivity as a function of temperature measured in Pt/T bIG.  \n  \n \n 17  \nFigure 2. Transport properties of Pt/Co/TbIG heterostructures.  (a) Schematics of \nPt/Co/TbIG/GGG multilayer (not on scale) and optical image of a TbIG Hall bar device in a \nthree -dimensional rendering of the measurement setup with white s cale bar of 20 µm.  The \ncoordinate system and magnetic field angles are indicated. ( b) AHE hysteresis loops measured \nin Pt/Co/T bIG for different thicknesses of the Co layer. ( c) AHE hysteresis loops measured in \nPt/Co (0.9 nm)/T bIG up to large out -of-plane magnetic fields. The AHE hysteresis loops at \nsmall magnetic fields are shown in the inset. ( d) Magnitude of the anomalous Hall loops as a \nfunction of thickness of the Co layer. ( e-g) Angular -dependent longitudinal resistance in \nPt/Co/T bIG with an applied ma gnetic field of 60 kOe rotating in the xy, xz  and yz planes.  \n  \n \n 18  \nFigure 3. AHE hysteresis loops with in -plane magnetic fields. AHE hysteresis loops as a \nfunction of in- plane magnetic fields in ( a) Pt/TbIG and ( b) Pt/Co (0.9 nm)/TbIG. A fit to the \ndata according to the experimental parameters and the model of magnetic coupling is indicated. \n(c) Schematics illustrating the magnetization when the in -plane magnetic field is swept from \nnegative to positive in P t/Co (0.9 nm)/TbIG.  \n  \n \n 19  \nFigure 4. Efficient current -induced SOTs in Pt/Co (0.6 nm)/TbIG.  (a) Current -induced \ndomain- wall motion (domain expansion) with an in- plane magnetic field Hx used to realign \ndomain- wall moments. ( b) AHE measurement with currents of ±4.3×1010 Am-2 under an in-\nplane magnetic field Hx of 300 Oe. The horizontal shift of the hysteresis loops corresponds to \nthe current -induced effective field H eff. (c) SOT efficiency calculated from horizontal shifts for \ndifferent current densities at different in -plane magnetic fields in Pt/ TbIG and Pt/Co (0.6 \nnm)/TbIG. ( d) and ( e) Current -induced hysteresis loops with in- plane magnetic fields of H x = \n±400 Oe in Pt/Co ( 0.6 nm)/TbIG. The switching polarity is indicated by the arrows. ( f) Critical \nswitching current densit ies as a function of in -plane magnetic fields in Pt/ TbIG and Pt/Co ( 0.6 \nnm)/TbIG.  \n  \n \n 20  \nFigure 5. Field -free magnetization switching in Pt/Co (0.9 nm)/TbIG. (a) Mirror symmetry \nof the degenerate P t/Co/TbIG  magnetization configurations underlying field- free \nmagnetization switching. ( b) and ( c) dc current -induced hysteresis loops with in- plane magnetic \nfields of H x = ±140 Oe in Pt/Co ( 0.9 nm)/TbIG. ( d) and ( e): Current -induced hysteresis loops \nin the absence of magnetic fields in Pt/Co ( 0.9 nm)/TbIG after the pre -set with positive/negative \nin-plane magnetic fields of 1 kOe. ( f) Current -induced magnetization switching with 0.2 ms -\nlong current pulses of ±1.4×1011 Am-2 in the absence of an external magnetic field after the pre-\nset with positive/negative in -plane magnetic fields of 1 kOe. The magnetic field -induced \nhysteresis loops are shown in left, giving the resistance reference for magnetizations pointing \nup and down. ( g) SOT efficiency as a function of in -plane magnetic fields in Pt/Co (0.9 \nnm)/TbIG.   \n \n"    },    {        "title": "1302.6884v1.Resonantly_damped_oscillations_of_elliptically_shaped_stratified_emerging_coronal_loops.pdf",        "content": "arXiv:1302.6884v1  [astro-ph.SR]  27 Feb 2013Resonantly damped oscillations of elliptically shaped\nstratified emerging coronal loops\nK. Karami1∗, S. Amiri1†, K. Bahari2‡, Z. Ebrahimi1§\n1Department of Physics, University of Kurdistan, Pasdaran Stree t, Sanandaj, Iran\n2Physics Department, Faculty of Science, Razi University, Kerman shah, Iran\nJune 26, 2018\nAbstract\nThe effects of both elliptical shape and stage of emergence of the c oronal loop on the\nresonant absorption of standing kink oscillations are studied. To do so, a typical coronal\nloop is modeled as a zero-beta longitudinally stratified cylindrical magn etic flux tube. We\ndeveloped the connection formulae for the resonant absorption o f standing transversal oscil-\nlations of a coronal loop with an elliptical shape, at various stages of its emergence. Using\nthe connection formulae, the dispersion relation is derived and solve d numerically to obtain\nthe frequencies and damping rates of the fundamental and first- overtone kink modes. Our\nnumerical results show that both the elliptical shape and stage of e mergence of the loop\nalter the frequencies and damping rates of the tube as well as the r atio of frequencies of the\nfundamental and its first-overtone modes. However, the ratio o f the oscillation frequency to\nthe damping rate is not affected by the tube shape and stage of its e mergence and also is\nindependent of the density stratification parameter.\nKey words: Sun: corona — Sun: magnetic fields — Sun: oscillations\n∗KKarami@uok.ac.ir\n†Sirwanamiri@yahoo.co.uk\n‡K.Bahari@razi.ac.ir\n§zanyar.ebrahimi@gmail.com\n11 Introduction\nSolar corona and its extraordinary high temperature has bee n the topic of various debates and\nstudies from several decades ago. The origin and the source o f coronal continual heating and\nhigh temperature have been related to coronal loops. The cla im that the coronal loops and their\nbehaviors such as their damping oscillations may be one of th e main reasons of coronal heating,\nhas been investigated through several studies so far.\nTransverse oscillations of coronal loops have been observe d by the Transition Region and\nCoronal Explorer (TRACE) for several years (see e.g. Aschwa nden et al. 1999; Schrijver &\nBrown 2000). Nakariakov et al. (1999) interpreted these osc illations as fast kink modes with the\nperiod ranging from 2 .3 to 10.8 min and decay time from 3 .2 to 20.8 min. The observed values\nof the periods and decay times make it possible to obtain indi rect information on the conditions\nof the plasma and magnetic field in coronal loops.\nOfman & Aschwanden (2002) used the data deduced by Aschwande n et al. (2002) to inves-\ntigate the oscillations of 11 coronal loops. They argued tha t the observed TRACE loops consist\nof multiple unresolved thin loop threads which produce inho mogeneous internal structure of the\nobserved loop. They adopted 1-dimensional cartesian slabs of plasma with the magnetic field\nlines in the z-direction and the direction of the inhomogeneity along the x-axis normal to the\nmagnetic surfaces, as a simple model for the oscillating loo ps. They found that the dependence\nof the decay time on both the length Land the width wof the loop is in excellent agreement\nwith the power law damping predicted by phase mixing.\nThe property of resonant absorption as a non-thermal mechan ism makes it possible to de-\nscribe the heating of magnetic loops in solar corona as well a s rapid decaying of magnetohydro-\ndynamics (MHD) waves even in weakly dissipative plasmas (se e e.g. Ionson 1978; Poedts et al.\n1989; Ofman et al. 1994; Erd´ elyi & Goossens 1994, 1995; Tirr y & Goossens 1996; Andries et al.\n2005b; Safari et al. 2006; Dymova & Ruderman 2006; Goossens e t al. 2009).\nVerwichte et al. (2004), using the observations of TRACE, ha ve identified the fundamental\nand its first harmonic of the transverse kink mode in two coron al loops. The period ratios\nobserved by Verwichte et al. (2004) are 1 .81±0.25 and 1.64±0.23. However, these values were\ncorrected with the improvement of the observational error b ars to 1.82±0.08 and 1.58±0.06,\nrespectively, by Van Doorsselaere et al. (2007). Also Verth et al. (2008) added some further\ncorrections by considering the effects of loop expansion and e stimated a period ratio of 1.54. All\nthese values clearly are lower than 2. This may be caused by di fferent factors such as the effects\nof density stratification (see e.g. Andries et al. 2005a; Erd ´ elyi & Verth 2007; Karami & Asvar\n2007; Safari et al. 2007; Karami et al. 2009) and magnetic twi st (see Erd´ elyi & Carter 2006;\nErd´ elyi & Fedun 2006; Karami & Barin 2009; Karami & Bahari 20 10, 2012) in the loops.\nKarami et al. (2009, hereafter Paper I) investigated the effec t of longitudinally stratification\non resonant absorption of MHD waves for both kink ( m= 1) and fluting ( m= 2) modes. They\nfound that the frequencies and damping rates of both the fund amental and first-overtone modes\nincrease when the stratification parameter increases. Also for stratified loops they obtained the\nratio of the frequencies ω2/ω1of the first overtone and its fundamental mode less than 2.\nMorton & Erd´ elyi (2009) studied the effects of both the ellipt ical shape and stage of emer-\ngence of the loops on the period ratio P1/P2for the minor and major elliptical cases. Their\nresults showed that the parameter characterising the stage of emergence does affect the value of\nperiod ratio P1/P2. Particularly, the greatest contribution from emergence t o the period ratio\noccurs when the loop is fully emerged. Also they showed that t he ellipticity of the loop has an\nimportant role in the value of P1/P2for minor elliptical case but the major ellipse was found to\nhave a less effect on the period ratio of standing oscillations .\n2Here we combine the two models considered in Paper I and Morto n & Erd´ elyi (2009) to\ninvestigate of the effects of both elliptical shape and stage o f emergence of the coronal loop on\nthe resonant absorption of standing transversal kink oscil lations observed by the TRACE. This\npaper is organized as follows. In Sections 2 and 3, we combine the two techniques of Paper I and\nMorton & Erd´ elyi (2009) to derive the equations of motion, i ntroduce the relevant connection\nformulae and obtain the dispersion relation. In Section 4, w e give numerical results. Section 5\nis devoted to conclusions.\n2 Equations of motion and modeling of the flux tube\nThe linearized MHD equations for a zero-beta plasma are give n by\n∂δv\n∂t=1\n4πρ{(∇×δB)×B+(∇×B)×δB}+η\nρ∇2δv, (1)\n∂δB\n∂t=∇×(δv×B)+c2\n4πσ∇2δB, (2)\nwhereδvandδBare the Eulerian perturbations of velocity and magnetic fiel ds;B,ρ,σ,ηand\ncare the background magnetic filed, the mass density, the elec trical conductivity, the viscosity\nand the speed of light, respectively.\nThe simplifying assumptions are the same as in Karami & Asvar (2007). According to\nAndries et al. (2005b) and Paper I, one can expand the perturb ed quantities δvandδBas\nfollows\nδB(r,z) =∞/summationdisplay\nk=1δB(k)(r)ψ(k)(z),\nδv(r,z) =∞/summationdisplay\nk=1δv(k)(r)ψ(k)(z), (3)\nwhereψ(k)(z)s form a complete set of orthonormal eigenfunctions and sat isfy the eigenvalue\nrelation\nLAψ(k)=ζkψ(k), (4)\nwhereLAis the Alfv´ en operator,\nLA=ρω2+B2\n4π∂2\n∂z2=ρ/parenleftigg\nω2+v2\nA∂2\n∂z2/parenrightigg\n, (5)\nwith Alfv´ en velocity vA=B√4πρand straight constant background magnetic filed B=Bˆz.\nWe further assume there is a density stratification along the tube axis in z-direction.Since\nwe are interested in resonantly damped oscillations, it imp lies that the density profile must be\nradially structured too. Following Paper I, we consider the density profile given by\nρ(r,z) =ρ0(r)ρ(z), (6)\nwhere\nρ0(r) =\n\nρin, (r<R1),/bracketleftig\nρin−ρex\nR−R1/bracketrightig\n(R−r)+ρex,(R1<r<R),\nρex, (r>R).(7)\n3Here,Ris the loop radius and R1< Ris the radius of the homogeneous part of the tube.\nThe radius at which resonant absorption occurs is between R1andR. The thickness of the\ninhomogeneous layer, l=R−R1, will be assumed to be small. Here, ρinandρexare the\nfootpoint densities of the interior and exterior regions of the tube, respectively.\nAccording to Morton & Erd´ elyi (2009) we consider two types o f elliptical loop that can occur,\nthe minor elliptical loop where minor axis of the ellipse is t he vertical axis of the loop, and the\nmajor elliptical loop where major axis of the ellipse is the v ertical axis of the loop. Note that\nthe minor ellipse is a situation that occurs most plausibly u nder coronal conditions.\nFor the minor elliptical case, the longitudinally stratifie d density profile takes the form\nρ(z) = exp\n−µcos/parenleftig\nα(z)/parenrightig/bracketleftig\n1−ǫ2sin2/parenleftig\nα(z)/parenrightig/bracketrightig−1\n2−λ\n1−λ\n, (8)\nand for the major one, it is given by\nρ(z) = exp\n−µcos/parenleftig\nα(z)/parenrightig/parenleftig\n1−ǫ2/parenrightig1/2/bracketleftig\n1−ǫ2cos2/parenleftig\nα(z)/parenrightig/bracketrightig−1\n2−λ\n1−λ\n, (9)\nwhere\nǫ=/parenleftigg\n1−b2\na2/parenrightigg1/2\n, (10)\nis the ellipticity of the loop with minor half-axis of length b, and major half-axis of length a.\nAlsoµ:=L\nπHis defined as stratification parameter, where HandLare the density scale height\nand length of the loop, respectively. The parameter λdescribes the stage of emergence of the\nloop from the photosphere. It is defined as the ratio of the dis tance of the photosphere from\ncenter of the ellipse to the vertical half-axis. A positive v alue ofλrefers to the situation in\nwhich the center of ellipse is sitting below the photosphere (early stage emergence), and thus,\nthe negative λfor the center above the photosphere (late stage emergence) . A zero value of λ\ncorresponds to a loop having a semi-elliptical shape. For th e minor elliptical case, λis given by\nλ= 1−µH\nb, (11)\nwhereµHis the distance of the loop apex from the photosphere. For the major one, λis defined\nas\nλ= 1−µH\na. (12)\nNote we have considered a tube of length Lwhich its footpoints are in the two points z= 0 and\nz=L, and also note that in Eqs. (8) and (9), α(z) is the angle between the vertical axis of the\nloop and the line joining the center of the ellipse to the plas ma element located at distance z\nalong the tube. Following Morton & Erd´ elyi (2009) for the mi nor elliptical case, one can obtain\nthe value of α(z) by calculating the ellipse arc length defined as\n/integraldisplayt1\n0/parenleftig\n1−ǫ2sin2(t)/parenrightig1\n2dt=/parenleftbigg2z\nL−1/parenrightbigg/integraldisplayt2\n0/parenleftig\n1−ǫ2sin2(t)/parenrightig1\n2dt, (13)\n4wheret1andt2are parametric angles given by\nt1= arctan/parenleftbiggb\natan(α)/parenrightbigg\n, t2= arctan/parenleftbiggb\natan(θ)/parenrightbigg\n, (14)\nand\nθ= arctan\n1\nλ/parenleftigg\n1−λ2\n1−ǫ2/parenrightigg1\n2\n, (15)\nis the angle between the vertical axis of the loop and a line th at joins the ellipse center to the\nloop foot-point (see Fig. 2 in Morton & Erd´ elyi 2009).\nFor the major elliptical case, we have\n/integraldisplayπ\n2\nt1/parenleftig\n1−ǫ2sin2(t)/parenrightig1\n2dt=/parenleftbigg2z\nL−1/parenrightbigg/integraldisplayπ\n2\nt2/parenleftig\n1−ǫ2sin2(t)/parenrightig1\n2dt, (16)\nwhere\nt1= arctan/parenleftbiggb\nacot(α)/parenrightbigg\n, t2= arctan/parenleftbiggb\nacot(θ)/parenrightbigg\n, (17)\nand\nθ= arctan\n/parenleftig\n(1−ǫ2)(1−λ2)/parenrightig1\n2\nλ\n. (18)\nAccording to Paper I, in the absence of dissipation, in the in terior region ( r<R1), solutions of\nEqs. (1) and (2) are\nδB(in)\nz(r,z) =+∞/summationdisplay\nk=1A(in,k)Jm(|kin,k|r)ψ(in,k)(z), (19)\nδv(in)\nr(r,z) =−iωB\n4π+∞/summationdisplay\nk=1kin,k\nζin,kA(in,k)J′\nm(|kin,k|r)ψ(in,k)(z), (20)\nwhere\nk2\nin,k=ζin,k\nB2/4π. (21)\nHereJmis the Bessel function of the first kind and a prime on Jmand hereafter on each function\nindicates a derivative with respect to their appropriate ar guments. Thesolutions for the exterior\nregionr>R, are the same as equation (20) except that Jm, index ”in”, and |kin,k|are replaced\nbyKm, ”ex” and kex,k=−ζex,k\nB2/4π, respectively, everywhere. Where Kmis the modified Bessel\nfunction of the second kind and shows that the wave amplitude vanishes in large distance away\nfrom the tube boundary.\n53 Boundary conditions, Connection formulae and dispersion re-\nlation\nIn the absence of dissipation effects, an appropriate dispers ion relation is obtained by requiring\nthat the solutions for perturbed quantities are continues a t the tube surface. When a dissipative\nlayer is considered, the solutions may experience jumps acr oss the layer. An appropriate relation\nconnecting the solutions of outside and inside the tube, is c alled the “connection formulae”.\nFollowing Paper I, the jump across the boundary (resonance l ayer) forδBzandδvris\n[δBz] = 0, (22)\n[δvr] =−+∞/summationdisplay\nk=1B˜ωm2/angbracketleftig\nφ(in,k)/vextendsingle/vextendsingle/vextendsingleδB(in,k)\nz/angbracketrightig\n4r2\nA/angbracketleftig\nφ(in,k)/vextendsingle/vextendsingle/vextendsingleLA1/vextendsingle/vextendsingle/vextendsingleφ(in,k)/angbracketrightigφ(in,k), (23)\nwhere\nLA1=∂LA\n∂r/vextendsingle/vextendsingle/vextendsingle\nr=rA= ˜ω2[1+Skk]∂ρ0(r)\n∂r/vextendsingle/vextendsingle/vextendsingle\nr=rA, (24)\nand\nφ(in,k)=/radicalbigg2\nL+∞/summationdisplay\nj=1φ(in,k)\njsin/parenleftbiggjπ\nLz/parenrightbigg\n, (25)\nwith\nφ(in,k)\nj=/braceleftiggk2Skj\nρin(1+Skk)(j2−k2)j∝ne}ationslash=k\n1j=k. (26)\nHereφ(in,k)satisfiesLAφ(in,k)= 0 and ˜ω=ω+iγwhereγis damping rate. Also R1<rA<R\nR1<rA<Ris the radius at which the singularity occurs. Note that l=R−R1is the thickness\nof the inhomogeneous layer and Davila (1987) showed that in t heresonance absorption, however,\nthe damping rate is independent of the dissipation coefficien t values. But the resonance layer\nwidth scales as δA∝(ν\nρ+c2\n4πσ)1/3. Karami & Bahari (2010) showed that for the Reynolds\nR=/parenleftig\nR2ρi\nν/parenrightig\n//parenleftig\n2πR\nvAi/parenrightig\n= 560 and Lundquist S=/parenleftig\n4πσR2\nc2/parenrightig\n//parenleftig\n2πR\nvAi/parenrightig\n= 104numbers given by Ofman\net al. (1994), and taking L= 105km,R/L= 0.01,a/R= 0.08,ρex/ρin= 0.1, and interior\nAlfv´ en velocity vAin= 2000 km s−1for a typical coronal loop, then one can get δA≃85 km\nwhich is very close to the thickness of the inhomogeneous lay erl≃87 km. This suggests that\none can use the thin boundary approximation which assumes th at the thickness of the resonance\nlayer is the same as the inhomogeneous layer width (see also G oossens et al. 2009).\nNote that the effects of elliptical shape and stage of emergenc e of the loop on the resonant\nabsorption appear in jump conditions via the function Skjwhich is obtained as\nSkj=/radicalbigg2\nL/integraldisplayL\n0sin/parenleftbiggkπ\nLz/parenrightbigg\nln/parenleftig\nρ(z)/parenrightig\nsin/parenleftbiggjπ\nLz/parenrightbigg\ndz. (27)\nSubstituting the fields of equation (19) and (20) in jump cond itions (22) and (23) gives\n\nΠ(ex,1)\n1 −Π(in,1)\n1 Π(ex,2)\n1 −Π(in,2)\n1...\nΞ(ex,1)\n1Ξ(in,1)\n1+D(in,1)\n1Ξ(ex,2)\n1Ξ(in,2)\n1+D(in,2)\n1...\nΠ(ex,1)\n2 −Π(in,1)\n2 Π(ex,2)\n2 −Π(in,2)\n2...\nΞ(ex,1)\n2Ξ(in,1)\n2+D(in,1)\n2Ξ(ex,2)\n2Ξ(in,2)\n2+D(in,2)\n2...\n...............\n\nA(ex,1)\nA(in,1)\nA(ex,2)\nA(in,2)\n...\n= 0,(28)\n6where all the definitions in Eq. (28) are as those of Paper I. No te that the dispersion relation is\nobtained by requiring that the system (28) has non-trivial s olutions, i.e. its determinant is zero.\nIn the next section, we solve the dispersion relation numeri cally to obtain the frequencies ωand\ndamping rates γof the resonantly damped MHD kink oscillations of longitudi nally stratified\nelliptical emerging coronal loops.\n4 Numerical Results\nTo solve the dispersion relation (28) numerically, we chose the physical parameters L= 105km,\nR/L= 0.01,l/R= 0.02,B= 100 G,ρin= 2×10−14g cm−3andρex/ρin= 0.1. For such a loop\none findsvAin=B√4πρin= 2×103km s−1andωAin:=vAin\nL= 0.02 rad s−1. In what follows, we\nillustrate our numerical studies in the three separate equi librium cases containing (i) circle-arc\nemerged loop (ii) minor elliptical semi-emerged loop and (i ii) minor and major elliptical loops.\n4.1 Circle-arc emerged loop ( ǫ= 0,λ∝ne}ationslash= 0)\nThe effect of stage of emergence of the tube on both the frequenc iesωand damping rates γare\ncalculated by numerical solution of the dispersion relatio n, i.e. Eq. (28). In Figs. 1 and 2, the\nfrequencies, damping rates and their ratio for the fundamen tal and first-overtone kink ( m= 1)\nmodesareplottedversusthestratificationparameter µforacircle-arcfluxtube( ǫ= 0)atvarious\nstages of emergence containing early stage emergence ( λ= 0.75), semi-emerged ellipse ( λ= 0)\nand late stage emergence ( λ=−0.75). Figures 1 and 2 show that (i) for a given loop shape\nparameterλ, both frequencies ω1,ω2and their corresponding damping rates |γ1|,|γ2|increase\nwhen the stratification parameter µincreases. (ii) For a given µ, both frequencies and damping\nrates increase when λincreases. For instance, for µ= 0.9, the early stage emergence λ= 0.75 in\ncomparison with late stage emergence λ=−0.75 would cause ω1,ω2,|γ1|and|γ2|to increase by\nabout 10.2%,11.7%,10% and 14 .5%, respectively. (iii) The ratio of the oscillation freque ncy to\nthe damping rate, ω/|γ|, is independent of stratification. Also the stage of emergen ce of the loop\ndoes not affect ω/|γ|. This means that the ratio ω/|γ|is independent of how the loop center is\nsituated with respect to the photosphere.\nIn Fig. 3, the ratio of the frequencies ω2/ω1of the first overtone and its fundamental mode is\nplotted versus the stratification parameter. Figure 3 shows that (i) for a given λ, the frequency\nratio decreases from 2 (for an unstratified loop) with increa sing density stratification. (ii) For a\ngivenµ, the frequency ratio decreases when λdecreases. The results of ω2/ω1are in agreement\nwith those obtained by Morton & Erd´ elyi (2009).\n4.2 Minor elliptical semi-emerged loop ( ǫ∝ne}ationslash= 0,λ= 0)\nFigures 4 and 5 as Figs. 1 and 2 display the result of frequency , damping rate and ratio ω/|γ|\nbut for a minor semi-emerged loop ( λ= 0) with different ellipticity parameters ǫ= 0.0, 0.4 and\n0.6. Figures 4 and 5 show that (i) for a given ellipticity parame terǫ, both frequencies ω1,ω2\nand their corresponding damping rates |γ1|,|γ2|increase when the stratification parameter µ\nincreases. (ii) For a given µ, both frequencies and damping rates increase with increasi ngǫ. For\ninstance, for µ= 0.9 considering a loop with ǫ= 0.6 would cause to increase ω1,ω2,|γ1|and|γ2|\nup to 3.5%, 4%, 3.5% and 5.3%, respectively, in comparison with a semi-circular loop ( ǫ= 0).\n(iii) The ratio ω/|γ|remains unchanged by increasing the stratification paramet er. Also this\nratio is independent of the elliptical shape of the loop.\n7Figure 6 illustrates the frequency ratio ω2/ω1as a function of stratification parameter for a\nsemi-emerged loop ( λ= 0) with different ellipticity parameters. Figure 6 presents that (i) for\na given ellipticity parameter, the ratio ω2/ω1decreases from 2 by increasing the stratification\nparameter and approaches below 1.5. (ii) For a given µ, the frequency ratio ω2/ω1decreases\nwhenǫdecreases. This is in good concord with the result of Morton & Erd´ elyi (2009).\n4.3 Minor and major elliptical emerged loops ( ǫ∝ne}ationslash= 0,λ∝ne}ationslash= 0)\nFigures 7, 8 and 9 make it possible to do some comparisons betw een major and minor elliptical\ncases. Figures 7 and 8 show that: (i) the frequencies and damp ing rates of both fundamental\nand first-overtone kink ( m= 1) modes of a minor elliptical case are greater than those of a\nmajor elliptical case and also of a circular-arc semi-emerg ed loop (ǫ=λ= 0) studied in Paper\nI. For instance, for µ= 0.9 for a minor elliptical loop with ǫ= 0.6 andλ= 0.75, the values of\nω1,ω2,|γ1|and|γ2|are 19.1%, 26.2%, 18.4% and 22.4% greater than those of a major elliptical\ncase, respectively. (ii) The frequencies and damping rates of a major elliptical loop are slightly\ngreater than the circular-arc semi-emerged loop. (iii) The ratioω/|γ|of both fundamental and\nfirst-overtone modes do not affected by stratification for mino r/major elliptical and circular-arc\nsemi-emerged loops.\nIn Fig. 9, the frequency ratio ω2/ω1versus the stratification parameter is plotted for major\nand minor cases as well as a circular-arc semi-emerged loop. Figure 9 shows that: (i) for a given\nµ, the frequency ratio ω2/ω1of the minor elliptical loop is greater than the major one. Fo r\ninstance, for µ= 0.9, the ratio ω2/ω1of a minor elliptical case is 5 .9% greater than that of a\nmajor elliptical one. (ii) The ratio ω2/ω1in a major elliptical loop is slightly greater than the\ncircular-arc semi-emerged loop. Therefore, the major elli pse has a lesser effect on the frequency\nratio of standing kink oscillations. This is in good accorda nce with that obtained by Morton &\nErd´ elyi (2009).\n5 Conclusions\nHere we investigated the effects of both ellipticity and stage of emergence on the resonant\nabsorption of standing kink waves in longitudinally strati fied coronal loops. We considered a\ncoronal loop as a pressureless cylindrical flux tube embedde d in a straight magnetic field that\nundergoes a longitudinal density stratification and radial density structuring. We extended\nthe relevant connection formulae for the resonant absorpti on of transverse kink oscillations of a\ncoronal loop with an elliptical shape, at various stages of i ts emergence fromthe sub-photosphere\ninto the solar corona. We studied three stages of emergence o f the loop at which the center of\neclipse is sitting below ( λ>0), on (λ= 0), and above ( λ<0) the photosphere. We considered\ntwo types of elliptical loop that can occur, the minor ellips e and the major ellipse. Note that the\nminor ellipse is a situation that occurs most plausibly unde r coronal conditions. By numerically\nsolvingthedispersionrelation, weobtainedthefrequenci esanddampingratesofthefundamental\nand first-overtone kink ( m= 1) modes. Our numerical results show the following.\ni) By increasing the density stratification parameter µin the loop, both frequencies and\ndamping rates increase while the frequency ratio ω2/ω1decreases.\nii) In a circle-arc emerged loop ( ǫ= 0,λ∝ne}ationslash= 0) for a given µ, the frequencies, damping rates\nand the frequency ratio increase when the stage of emergence parameterλincreases.\niii) In a minor elliptical semi-emerged loop ( ǫ∝ne}ationslash= 0,λ= 0) for a given µ, the frequencies,\ndamping rates and the frequency ratio increase when the elli pticity parameter ǫincreases.\n8iv) In a minor elliptical emerged loop ( ǫ∝ne}ationslash= 0,λ∝ne}ationslash= 0) for a given µ, the frequencies, damping\nrates and frequency ratio are greater than those of a major on e. However, the results obtained\nfor the aforementioned quantities in the major elliptical e merged loop are slightly greater than\nthose of a circular-arc semi-emerged loop ( ǫ=λ= 0).\nv) The ratio of the oscillation frequency to the damping rate ,ω/|γ|, in minor/major elliptical\nand circular-arc semi-emerged loops, is not affected by makin g changes in density stratification\nparameter, ellipticity and stage of emergence of the loop.\nAcknowledgements\nThe work of K. Karami has been supported financially by Depart ment of Physics, University of\nKurdistan, Sanandaj, Iran under research project No. 1/139 0.\nReferences\n[1] Andries J., Arregui I., Goossens M., 2005a, ApJ, 624, L57\n[2] Andries J., Goossens M., Hollweg J.V., Arregui I., Van Do orsselaere T., 2005b, A&A, 430,\n1109\n[3] Aschwanden M.J., De Pontieu B., Schrijver C.J., Title A. M., 2002, Sol. Phys., 206, 99\n[4] Aschwanden M.J., Fletcher L., Schrijver C.J., Alexande r D., 1999, ApJ, 520, 880\n[5] Davila J.M., 1987, ApJ, 317, 514\n[6] Dymova M.V., Ruderman M.S., 2006, A&A, 457, 1059\n[7] Erd´ elyi R., Goossens M., 1994, Ap&SS, 213, 273\n[8] Erd´ elyi R., Goossens M., 1995, A&A, 294, 575\n[9] Erd´ elyi R., Verth G., 2007, A&A, 462, 743\n[10] Goossens M., Terradas J., Andries J., Arregui I., Balle ster J.L., 2009, A&A, 503, 213\n[11] Ionson J.A., 1978, ApJ, 226, 650\n[12] Karami K., Asvar A., 2007, MNRAS, 381, 97\n[13] Karami K., Barin M., 2009, MNRAS, 394, 521\n[14] Karami K., Nasiri S., Amiri S., 2009, MNRAS, 394, 1973 (P aper I)\n[15] Karami K., Bahari K., 2010, Sol. Phys., 263, 87\n[16] Karami K., Bahari K., 2012, ApJ, 757, 186\n[17] Morton R.J., Erd´ elyi R., 2009, A&A, 502, 315\n[18] Nakariakov V.M., Ofman L., DeLuca E.E., Roberts B., Dav ila J.M., 1999, Science, 285, 862\n[19] Ofman L., Aschwanden M.J., 2002, ApJ, 576, L153\n9[20] Ofman L., Davila J.M., Steinolfson R.S., 1994, ApJ, 421 , 360\n[21] Poedts S., Goossens M., Kerner W., 1989, Sol. Phys., 123 , 83\n[22] Safari H., Nasiri S., Karami K., Sobouti Y., 2006, A&A, 4 48, 375\n[23] Safari H., Nasiri S., Sobouti Y., 2007, A&A, 470, 1111\n[24] Schrijver C.J., Brown D.S., 2000, ApJ, 537, L69\n[25] Tirry W.J., Goossens M., 1996, ApJ, 471, 501\n[26] Van Doorsselaere T., Nakariakov V.M., Verwichte E., 20 07, A&A, 473, 959\n[27] Verth G., Erd´ elyi R., Jess D.B., 2008, ApJ, 687, L45\n[28] Verwichte E., Nakariakov V.M., Ofman L., Deluca E.E., 2 004, Sol. Phys., 223, 77\n1000.10.20.30.40.50.60.70.80.90.040.050.060.070.080.090.1ω1\n  \nλ=0\nλ=0.75\nλ=−0.75\n00.10.20.30.40.50.60.70.80.90.020.030.040.050.06−γ1(×10−2)\n00.10.20.30.40.50.60.70.80.911.21.41.61.82\nµ (ε=0)−ω1/γ1(×102)\nFigure 1: Frequency of the fundamental kink ( m= 1) mode and its damping rate as well as the\nratio of theoscillation frequency tothedampingrateas a fu nctionof thestratification parameter\nµfor a circle-arc flux tube ( ǫ= 0) at various stages of emergence λ= 0 (solid), 0 .75 (dashed)\nand−0.75 (dash-dotted). The loop parameters are L= 105km,R/L= 0.01,l/R= 0.02,\nρex/ρin= 0.1,ρin= 2×10−14gr cm−3andB= 100 G. Both frequencies and damping rates are\nin units of the interior Alfv´ en frequency, ωAin= 0.02 rad s−1.\n1100.10.20.30.40.50.60.70.80.90.080.10.120.140.16ω2\n  \nλ=0\nλ=0.75\nλ=−0.75\n00.10.20.30.40.50.60.70.80.90.40.50.60.70.8−γ2(×10−2)\n00.10.20.30.40.50.60.70.80.900.10.20.30.4\nµ (ε=0)−ω2/γ2(×102)\nFigure 2: Same as Fig. 1, for the first-overtone kink modes.\n1200.10.20.30.40.50.60.70.80.91.551.61.651.71.751.81.851.91.952\nµ (ε=0)ω2/ω1\n  \nλ=0\nλ=0.75\nλ=−0.75\nFigure 3: Ratio of the frequencies ω2/ω1of the first-overtone and its fundamental kink ( m= 1)\nmode versus µfor a circle-arc flux tube ( ǫ= 0) at various stages of emergence λ= 0 (solid),\n0.75 (dashed) and −0.75 (dash-dotted). Auxiliary parameters as in Fig. 1.\n1300.10.20.30.40.50.60.70.80.90.040.050.060.070.080.090.1ω1\n  \nε=0\nε=0.4\nε=0.6\n00.10.20.30.40.50.60.70.80.90.020.030.040.050.06−γ1(×10−2)\n00.10.20.30.40.50.60.70.80.911.21.41.61.82\nµ (λ=0)−ω1/γ1(×102)\nFigure 4: Frequency of the fundamental kink ( m= 1) mode and its damping rate as well as\nthe ratio of the oscillation frequency to the damping rate as a function of the stratification\nparameterµfor a minor semi-emerged loop ( λ= 0) with different ellipticity parameters ǫ= 0.0\n(solid), 0.4 (dashed) and 0 .6 (dash-dotted). Auxiliary parameters as in Fig. 1.\n1400.10.20.30.40.50.60.70.80.90.080.10.120.140.16ω2\n  \nε=0\nε=0.4\nε=0.6\n00.10.20.30.40.50.60.70.80.90.40.50.60.70.8−γ2(×10−2)\n00.10.20.30.40.50.60.70.80.900.10.20.30.4\nµ (λ=0)−ω2/γ2(×102)\nFigure 5: Same as Fig. 4, for the first-overtone kink modes.\n1500.10.20.30.40.50.60.70.80.91.551.61.651.71.751.81.851.91.952\nµ (λ=0)ω2/ω1\n  \nε=0\nε=0.4\nε=0.6\nFigure 6: Ratio of the frequencies ω2/ω1of the first-overtone and its fundamental kink ( m= 1)\nmodeversus µforaminorsemi-emerged loop ( λ= 0) withdifferent ellipticity parameters ǫ= 0.0\n(solid), 0.4 (dashed) and 0 .6 (dash-dotted). Auxiliary parameters as in Fig. 1.\n1600.10.20.30.40.50.60.70.80.90.040.060.080.10.12ω1\n  \nminor case, ε =0.6, λ=0.75\nε =0, λ=0\nmajor case, ε =0.6, λ=0.75\n00.10.20.30.40.50.60.70.80.90.020.030.040.050.060.07−γ1(×10−2)\n00.10.20.30.40.50.60.70.80.911.21.41.61.82\nµ−ω1/γ1(×102)\nFigure 7: Frequency of the fundamental kink ( m= 1) mode and its damping rate as well as the\nratio of theoscillation frequency tothedampingrateas a fu nctionof thestratification parameter\nµfor circular-arc semi-emerged (solid), minor (dash-dotte d) and major (dashed) elliptical loops.\nAuxiliary parameters as in Fig. 1.\n1700.10.20.30.40.50.60.70.80.90.080.10.120.140.160.18ω2\n  \nminor case, ε =0.6, λ=0.75\nε =0, λ=0\nmajor case, ε =0.6, λ=0.75\n00.10.20.30.40.50.60.70.80.90.40.50.60.70.80.9−γ2(×10−2)\n00.10.20.30.40.50.60.70.80.900.10.20.30.4\nµ−ω2/γ2(×102)\nFigure 8: Same as Fig. 7, for the first-overtone kink modes.\n1800.10.20.30.40.50.60.70.80.91.551.61.651.71.751.81.851.91.952\nµω2/ω1\n  \nminor case, ε =0.6, λ=0.75\nε =0, λ=0\nmajor case, ε =0.6, λ=0.75\nFigure 9: Ratio of the frequencies ω2/ω1of the first-overtone and its fundamental kink ( m= 1)\nmode versus µfor circular-arc semi-emerged (solid), minor (dash-dotte d) and major (dashed)\nelliptical loops. Auxiliary parameters as in Fig. 1.\n19"    },    {        "title": "1309.2248v3.Characterization_of_the_International_Linear_Collider_damping_ring_optics.pdf",        "content": "arXiv:1309.2248v3  [physics.acc-ph]  4 Sep 2014Preprinttypeset inJINST style-HYPER VERSION\nCharacterizationof theInternationalLinearCollider\ndampingringoptics\nJ. Shanks,D.L.Rubin,andD.Sagan\nCLASSE,CornellUniversity, Ithaca,NewYork 14853,USA\njs583@cornell.edu\nABSTRACT : Amethodispresented for characterizing theemittance dil ution anddynamic aperture\nforanarbitraryclosedlatticethatincludesguidefieldmag neterrors,multipoleerrorsandmisalign-\nments. This method, developed and tested at the Cornell Elec tron Storage Ring Test Accelerator\n(CesrTA), has been applied to the damping ring lattice for th e International Linear Collider (ILC).\nTheeffectiveness ofbeambasedemittancetuningislimited bybeampositionmonitor(BPM)mea-\nsurement errors, number of corrector magnets and their plac ement, and correction algorithm. The\nspecifications for damping ring magnet alignment, multipol e errors, number of BPMs, and preci-\nsion in BPM measurements are shown to be consistent with the r equired emittances and dynamic\naperture. The methodology is then used to determine the mini mum number of position monitors\nthat isrequired toachieve theemittance targets, andhow th at minimum depends onthelocation of\ntheBPMs. Similarly,themaximumtolerablemultipoleerror sareevaluated. Finally,therobustness\nof each BPMconfiguration with respect torandom failures is e xplored.\nKEYWORDS : Accelerator modelling and simulations (single-particle dynamics); Beam Optics.Contents\n1. Introduction 1\n2. DTC04Lattice 3\n3. Misalignment andCorrection Procedure 4\n4. Error Tolerance of DTC04 7\n4.1 Nominal Lattice and Errors 7\n4.2 Reduced BPMSchemes 8\n4.3 Effect of Random BPMFailures 11\n5. DynamicAperture 11\n6. SummaryandFutureWork 13\nAppendix: DTC04Magnet Misalignments andMultipoles 14\n1. Introduction\nThe International Linear Collider (ILC) design utilizes da mping rings to cool beams delivered by\nthe electron and positron sources before transferring to th e main linacs [1]. Three of the primary\nrequirements of the baseline damping rings are: 1) they must accept an injected bunch from the\npositron source with normalized phase-space amplitude 0 .07 m·rad andδE/E=0.75%; 2) the\nbeams must be cooled to an equilibrium zero-current geometr ic vertical emittance ≤2 pm; and 3)\nthe damping timemust beshort enough toprovide fully damped bunch trains atarepetition rateof\n5Hz.\nThe above requirements must be met in a real machine that incl udes magnet misalignments,\nguidefieldmultipoleerrors(bothsystematicandrandom), a ndbeampositionmonitor(BPM)mea-\nsurement errors. Emittance tuning will be essential to achi eve the target zero-current emittance.\nQuadrupoles andcorrector magnetswillnecessarily beinde pendently powered, allowing forlocal-\nized corrections through the use of beam-based measurement s.\nThe damping rings cannot be characterized with respect to th e exact set of errors they will\nhave, as the rings are not yet built. For the purposes of this s tudy, alignment errors in the damping\nrings are assumed to be randomly distributed with amplitude s dictated by survey tolerances. By\nsimulating a large number of lattices with random distribut ions of errors at the appropriate levels,\nthedamping ringscanbecharacterized withastatistical an alysis ofthelikelihood that therequired\nemittances anddynamicaperturewillbeachieved. Aconfigur ation isdeemedacceptable if95%of\n– 1–the randomly-misaligned and corrected lattices meet the re quired vertical emittance and dynamic\naperture, asper the Technical Design Report specifications .\nSeveral methods for optics correction have been developed a t storage ring light sources. By\nfar the most widespread method is response matrix analysis ( RMA), and specifically, orbit re-\nsponse matrix (ORM)analysis [17]. The method involves taki ng difference orbits where corrector\nstrengths are varied. Both the Swiss Light Source and Austra lian Synchrotron have demonstrated\nthe capability of correcting vertical emittance below 1 pm u sing ORM [3, 9], below the required\nvertical emittance for the ILC damping rings. However, the t ime required for ORM data acquisi-\ntion scales linearly with the number of correctors in the rin g. For large rings such as the proposed\nILC damping rings with over 800 steering correctors, acquir ing difference orbits becomes a time-\nconsuming process.\nAnother increasingly-common class of optics correction al gorithms is based on turn-by-turn\nBPM measurements, where the beam is pinged with a short-dura tion kick and allowed to oscillate\nfreely [6, 2]. Typically this data is then processed as a reso nance-driving term (RDT) correction\n[13]. Thiscorrection technique hasthebenefit that correct ion timesareroughly independent ofthe\nsize of the ring. For the ILC this represents a considerable s avings in time. While this method has\nshown promise, there are limitations. The beam decoheres du e to amplitude dependent tune shift,\nlimitingthenumberofusefulturnsofdata. Thedecoherence maybepartiallymitigatedbyreducing\nthe chromaticity to near zero, however measurements are sti ll limited to a few thousand turns. As\nmentioned in [2], the resolution of the betatron phase measu rement scales roughly inverse with\nthe number of turns (1 /N). The limitations on the number of turns in a data set therefo re directly\ncorrespond to alimitation on accuracy of betatron phase mea surements.\nA turn-by-turn-based alternative to RDT has been developed for use at the Cornell Electron\nStorage Ring (CESR) for CESR Test Accelerator program (Cesr TA) [10, 22]. The CESR beta-\ntron phase and coupling measurement relies on phase-lockin g turn-by-turn kickers to resonantly\nexciteasinglebunch toanamplitude ofseveral millimeters , thusavoiding limitations duetobunch\ndecoherence and allowing for data sets of over 40,000 turns. The effectiveness of the betatron\nphase/coupling correction procedure has been evaluated at CesrTA through measurements and\nsimulations, demonstrating good agreement and validating our understanding of the limitations\nof optics correction.\nInthispaper,themethodformodelingopticscorrectiondev elopedatCesrTAisusedtocharac-\nterizetheILCdampingrings. Severalscenarios areevaluat ed, including thepossibility ofreducing\nthetotal BPMcount andrelaxing theconstraints onguidefiel dmultipoleerrors, withrespect tothe\nbaseline design. The effect of random BPM failures is analyz ed to benchmark the robustness of\neach of the proposed BPMdistributions.\nThe characterization is comprised of two parts: generating a randomly misaligned lattice and\nsimulating the optics correction procedure, and analyzing the dynamic aperture of the corrected\nlattice as a function of guide field multipole errors. The sof tware used in all analyses is based on\ntheBmadaccelerator code library [18].\nOnlysingle-particle dynamicsfromthebeamopticsarecons ideredintheworkpresentedhere.\nThereportedemittancesthereforerepresentthezero-curr entlimit. Othercurrent-dependent sources\nof emittance dilution, such as intra-beam scattering (IBS) , electron cloud, and fast-ion instability\nwill increase the vertical emittance beyond this lower limi t.\n– 2–2. DTC04 Lattice\nThe baseline ILC damping ring design is the DTC04 lattice dev eloped by Rubin et al.[21]. The\nlattice is a 3.2 km racetrack with a modified TME-type arc cell . The zero-dispersion straights are\nbased on the work of Korostelev and Wolski [14]. A schematic o f the ring is shown in figure 1.\nOpticsfunctionsareshowninfigure2. Tables1and2summariz ethelatticeparametersandnumber\nofmagnetsofeachclass. Thequotedemittances εa,barethehorizontal-likeandvertical-likenormal\nmodes, respectively. In the absence of coupling, these will correspond to εx,y. Decomposition into\nnormal modes isdiscussed elsewhere [25, 16,4],and as such, aderivation isomitted here.\nInjection Extraction Chicane \nDamping Wigglers RF Phase Trombone \nx (m) y (m) \nFigure 1: Layout of DTC04lattice.\nTable 1: Summaryof theDTC04lattice parameters.\nParameter Value Units\nCircumference 3239 m\nEnergy 5.0 GeV\nBetatron Tunes ( Qx,Qy) (48.850, 26.787)\nTune Advance per Arc (16.418, 6.074)\nChromaticity ( ξx,ξy) (1.000, 0.302)\nChromaticity per Arc (9.074, 10.896)\nTrainRepetition Rate 5 Hz\nBunch Population 2 ×1010\nExtracted εgeometric\na 0.6 nm\nExtracted εgeometric\nb<2 pm\nExtracted Bunch Length 6 mm\nExtracted σE/E 0.11 %\nDamping Time 24 ms\nWigglerBmax1.5 T\n– 3–βa,b  (m) ηx (cm) \ns (m) βa\nβb\nηxInjection Arc Phase \nTrombone RF Damping \nWigglers Arc Extraction Chicane \nFigure 2: Horizontal andvertical βfunctions andhorizontal dispersion for theDTC04lattice. The\nripple inβbinthe arcs isdue to aliasing, and is not physical.\nTable2: Summaryof elements in the DTC04lattice.\nClass Count\nBeam Position Monitor 511\nDipole 164\nHorizontal Steering 150\nVertical Steering 150\nCombined H+VSteering 263\nQuadrupole 813\nSkewQuadrupole 160\nSextupole 600\nDamping Wigglers 54\nThearccelllayoutisshowninfigure3. Arccellsarecomprise dofonedipole,threequadrupoles,\nfour sextupoles, one horizontal and one vertical steering c orrector, one skew quadrupole, and two\nbeam position monitors.\n3. MisalignmentandCorrection Procedure\nThe effects of misalignments, BPM errors, multipoles, and c hoice of correction procedure are\nevaluated by theprogram ring_ma2 [22]. Thering_ma2 workflow isstructured as follows:\n1. Assign random misalignments and BPM errors with user-defi ned amplitudes to the ideal\nlattice to create arealistic machine model.\n2. Simulate beam based measurements of optics functions inc luding the effects of BPM mea-\nsurement errors.\n3. Compute and apply corrections for each iteration based on the simulated measurements.\n4. After each correction iteration, record the effectivene ss of the correction in terms of emit-\ntances and optics functions.\n– 4–Figure 3: DTC04 arc cell. Two BPMsare available in each arc ce ll, and are either odd-indexed or\neven-indexed.\nTheentireprocedure isrepeatedtypically100times,eachw itharandomlychosendistribution\nof errors in order to generate statistics sufficient for anal ysis. Misalignments are allowed on any\nelement parameter described by Bmad, including but not limited to: position offsets and angles,\nrandom field strength errors, and systematic and random mult ipole errors. BPM measurement\nerrorsallowedinthesimulationincludeBPM-to-quadrupol etransverseoffset,tilt,button-by-button\ngain error, and button-by-button timing errors.\nTheparticular configuration of the guide fieldmagnets, corr ectors, and BPMsisreferred toas\na scenario. For each scenario, 100 lattices are generated wi th random errors and misalignments.\nEach misaligned lattice is referred to as a seed. As noted abo ve, typically 100 seeds are used\nfor the evaluation of each scenario, and the same 100 seeds ar e used when evaluating multiple\nscenarios. That is, for agiven seed, each element in alattic e for each scenario starts withthe same\nmisalignments and field errors, regardless of whether a subs et of elements or detectors are vetoed\nor disabled during the correction.\nThe method used for simulating measurements with BPM errors is based on the method de-\nveloped for diagnosing emittance dilution at CesrTA [22]. I n the original analysis, modeling the\nfull correction procedure includes turn-by-turn tracking for several damping times in order to ac-\ncurately simulate measurements of betatron phase and coupl ing. The turn-by-turn measurements\nare then post-processed identically to actual machine data into orbit, dispersion, or phase and cou-\npling data. Though this method for simulation is as close to m odeling the actual measurement\nprocess, it is very time-consuming. In the case of the ILC dam ping ring lattice, this method is\nprohibitively slow due to the large number of elements. Twom odifications have been made to the\nmeasurement simulation procedure in order to improve the ef ficiency of the calculations such that\nthe characterization method developed in[22] may beapplie d.\nFirst, there is a modification to tracking in the damping wigg lers. When accounting for wig-\ngler nonlinearities isrequired, afully-symplectic Liema pisusedfortrackingthrough thedamping\nwigglers[7,8]. However,asitispresentlyimplementedin Bmad,thistrackingmethodiscomputa-\ntionally intensive. Whenthefullnonlinearities ofthedam pingwigglersareknowntonot affectthe\n– 5–resultingsimulation, asimplifiedMAD-style“bend-drift- bend” wigglermodelwhichpreservesthe\nradiation integrals can be used to reduce the timerequired f or simulations. Radiation integrals and\nhorizontal emittancesthetwowigglermodelsareshowninta ble3. Thehorizontal emittancescom-\nputed with the two models differ by less than 3%. The level of a greement is more than sufficient\nfor the studies presented here.\nTable 3: Comparison of emittance and radiation integrals fo r DTC04 lattice with two wiggler\nmodels: afully-symplectic Liemap, and asimplified “bend-d rift-bend” model.\nParameter LieMap Bend-Drift-Bend Δ\nεa 0.553 nm 0.567 nm -2.51%\nI1 1.08 1.08 0.00%\nI2 0.512 m−20.515 m−2-0.59%\nI33.39×10−2m−33.40×10−2m−3-0.44%\nI41.98×10−4m−22.01×10−4m−2-1.82%\nI57.71×10−6m−27.95×10−6m−2-3.13%\nComparisons of simulations using the full wiggler model and the reduced MAD-style “bend-\ndrift-bend” wigglerhaveshownthatwigglernonlinearitie s donotsignificantlyaffectthecorrection\nof optics functions or the final emittance. Therefore, the si mplified “bend-drift-bend” wiggler\nmodel is used for all ring_ma2 studies on the ILC damping ring lattice. This reduces the tim e\nrequiredtosimulatethecorrectionprocedureforonerando mly-misaligned latticebyafactorof15.\nThesecondmodificationaffectstheprocessthroughwhichop ticsmeasurementsaresimulated.\nIn[22],measurementsaresimulatedonaturn-by-turn basis ,applyingBPMmeasurementerrorson\nevery turn. Simulating one betatron phase measurement requ ires particle tracking for over 40,000\nturns,applyingsimulatedBPMmeasurementerrorsoneveryt urn. Evenwiththesimplifiedwiggler\nmodel, this is prohibitively slow on a lattice as large as DTC 04. An alternative is used in these\nstudies where BPM errors are applied directly to the Bmad-computed optics functions. Although\nnot as rigorous as the full measurement simulation method, s ide-by-side comparisons of the two\nmethods have shown minimal difference in the resulting opti cs functions. As a result, the amount\nof time to simulate the correction procedure for one randoml y-misaligned and corrected lattice is\nfurther reduced by another factor of 25, allowing the charac terization of each configuration to be\ncompleted inareasonable amount of time.\nThe optics correction procedure used in this characterizat ion was developed at CesrTA [22],\nand relies on the measurement and correction of the betatron phase and coupling [19] using beam\nposition monitors (BPMs) with turn-by-turn capabilities [ 15]. The BPM modules are capable of\npre-processing phase data in parallel, therefore this meth od has the benefit that data acquisition\ntime is roughly independent of the number of BPMs, and is enti rely independent of the number of\ncorrectors.\nThecorrection procedure used here isas follows:\n1. Measuretheclosedorbit. Fitthedatausingall826steeri ngcorrectors(150eachofdedicated\nhorizontal andvertical steerings, and263combined horizo ntal/vertical correctors) andapply\n– 6–corrections to thelattice model.\n2. Measure the betatron phase/coupling and dispersion. Cor rect betatron phase and horizontal\ndispersion to the design values using all 813 normal quadrup oles; simultaneously correct\nbetatron coupling using all 160 skew quadrupole correctors , and apply all corrections.\n3. Remeasure the closed orbit, coupling, and vertical dispe rsion; simultaneously correct all\nmachinedatausingall513steeringcorrectorsand160skewq uadrupolecorrectorsandapply\ncorrections.\nThecorrection procedure described aboveisroutinely used atCesrTA,therefore adirect com-\nparison of simulation with measurement is available to vali date the emittance tuning method. De-\ntails of the CesrTA characterization are available in [24, 2 2]. Although CesrTA has not achieved\na vertical emittance below 2 pm as required for the ILC dampin g rings, simulated optics functions\nafter correction are in agreement with measurements of the o ptics at CesrTA, and demonstrate ac-\ncuracyinthemodelingofthecorrectionprocedure. Inpract ice,correctionsbasedonbetatronphase\nand coupling have yielded coupling and dispersion on par wit h that achieved using orbit response\nmatrix analysis at CesrTA[23].\n4. Error Tolerance ofDTC04\nThe magnet misalignments, guide field errors, multipole err ors, and BPM measurement error tol-\nerances specified in the ILC Technical Design Report are appl ied in order to demonstrate that the\nbaseline design satisfiestherequirements for achieving ve rtical emittance. Aswillbeseen, thefull\ncomplement of BPMsis morethan sufficient to achieve the requ isite vertical emittance.\n4.1 NominalLattice and Errors\nMisalignments andBPMmeasurement errortolerances aresum marizedintable7intheAppendix,\nand are based on errors used in previous studies [21]. In addi tion, quadrupole k1and sextupole\nk2errors are now included. Multipole error coefficients are ta ken from measurements of PEP-II\nguide fieldmultipole errorsbyCai[5],andaresummarizedin table8intheAppendix. Theseerror\namplitudes and multipole coefficients, along with the full c omplement of 511 BPMs, define the\nnominal ILC-DRscenario.\nResults of ring_ma2 studies for this scenario are shown in figure 4 and are summari zed\nin table 4. The coupling in model lattices is characterized u sing the ¯Ccoupling matrix [20], an\nextension oftheEdwardsand Tengformalism [11]. Inparticu lar, theout-of-phase coupling matrix\nelement¯C12is used, asthe ¯C12measurement is insensitive toBPMrotation.\nIncluding misalignments, guide field errors, multipole err ors, and BPM measurement error\ntolerances, 95% of the resulting lattices have a vertical em ittance below εb=0.224 pm after cor-\nrections. This is well below the emittance budget of εb=2 pm, though it is important to note that\ncollective effects and other non-static sources of emittan ce dilution have not been accounted for in\nthe calculation of the simulated emittance.\nThe fundamental lower limit for the vertical emittance is de termined by the finite opening\nangle of synchrotron radiation, and for the ILC damping ring s is around 0.1 pm. The zero-current\n– 7–10-1100101102103\u0000b (pm)0204060No. Seeds\n10-310-210-1\nRMS \u0001y (m)0153045No. Seeds\n10-310-210-1\nRMS Cbar12015304560No. Seedsiter 0\niter 1\niter 2\niter 3\nFigure 4: Distributions for emittance, dispersion, and cou pling with misalignments, guide field\nmultipoleerrors, andBPMerrorsasspecifiedintheILCTDR,u singthefullcompliment ofBPMs.\nBefore correction (red), and after the first, second, and thi rd corrections (blue, green, and black,\nrespectively) are shown. Notethat the horizontal axis ison alog-scale.\nemittances achieved after correction for the nominal scena rio is close to this limit, with 95% of\nseeds well within a factor of three of the lower bound.\n4.2 ReducedBPMSchemes\nItisclearthatthebaselinespecifications formisalignmen ts, multipoleerrors, andBPMandcorrec-\ntordistributions, asspecifiedintheTechnicalDesignRepo rt,aremorethansufficienttocontainthe\nstaticopticscontribution totheverticalemittancewithi nthe2pmemittancebudget. Itisofinterest\nto consider whether the number of BPMs can be reduced without compromising the effectiveness\nof the emittance correction procedure.\nThe ILC damping ring lattice is wiggler-dominated, with >80% of synchrotron radiation\nbeing generated by the damping wigglers. As such, residual v ertical dispersion in the wiggler\nstraight will contribute significantly more to the vertical emittance than a comparable residual in\nthe arcs. Preserving the 52 BPMs in the damping wiggler strai ght is therefore essential. The\nremaining 459 BPMsare either in other straight sections (15 9) or inthe arcs (300).\nThe baseline arc cell design has two BPMs, indicated in figure 3. The optics functions at\nthe two BPMs are summarized in table 5. The betatron phase adv ance per arc is roughly 15 ( ≈\n94.25rad),corresponding toabout10BPMsperbetatronwavelen gthinthearcs. Thelargenumber\n– 8–Table 4: 95th-percentile vertical emittance, RMSvertical dispersion, and RMScoupling after each\nround of corrections, for the nominal lattice defined inSec.\nIteration Parameter 999555thPercentile\nInitial εb 1010 pm\nRMSηy64.5 mm\nRMS¯C12121×10−3\nx+y εb 12.3 pm\nRMSηy6.37 mm\nRMS¯C1223.0×10−3\nφa,b+¯C12+ηxεb 10.9 pm\nRMSηy6.92 mm\nRMS¯C121.11×10−3\ny+¯C12+ηy εb 0.224 pm\nRMSηy0.502 mm\nRMS¯C120.704×10−3\nofBPMsandlowphaseadvanceperarccellimplythatnotevery BPMinthearcsmaybenecessary\nin order to maintain the ability tocorrect the lattice tobel ow 2pm vertical emittance.\nTable 5: Optics functions evaluated at each of the two period ic BPM locations in the DTC04 arc\ncell, denoted by“odd-indexed” and “even-indexed”.\nParameter Odd-Indexed Even-Indexed\nβa 15.1 m 6.5m\nβb 14.9 m 23.5 m\nηa 0.312 m 0.185 m\nFivescenarios for BPMdistributions areexamined. Theyare :\n1. Full complement of BPMs, for atotal of 511; 10 BPMsper beta tron wavelength. Thisis the\nnominal scenario from Sec. 4.1.\n2. Remove odd-indexed arc BPMs, for a total of 361; 5 BPMs per b etatron wavelength, at\nlocations with βb>βa\n3. Remove even-indexed arc BPMs, for a total of 361; 5 BPMs per betatron wavelength, at\nlocations with βb≈βa\n4. Removeallodd-indexedandeveryothereven-indexedarcB PM,foratotalof287;2.5BPMs\nper betatron wavelength, at locations with βb>βa\n5. Removealleven-indexedandeveryotherodd-indexedarcB PM,foratotalof287;2.5BPMs\nper betatron wavelength, at locations with βb≈βa\n– 9–For each BPM distribution, the misalignment and correction procedure described in Sec. 4.1\nis repeated. Magnet and BPM errors are identical for the five s cenarios, therefore a direct com-\nparison of resulting corrections ispossible. Correction l evels achieved for lattices withthese BPM\ndistributions are shown infigure 5, and aresummarized intab le 6.\n0.10 0.15 0.20 0.25 0.30 0.35 0.40\u0002b (pm)0612182430No. Seeds\n0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014\nRMS \u0003y (m)010203040No. Seeds\n0.0005 0.0010 0.0015 0.0020 0.0025 0.0030\nRMS Cbar1201530456075No. SeedsScenario 1Scenario 1\nScenario 2Scenario 1\nScenario 2\nScenario 3Scenario 1\nScenario 2\nScenario 3\nScenario 4Scenario 1\nScenario 2\nScenario 3\nScenario 4\nScenario 5\nFigure5: εb,ηy,and¯C12after thefull correction procedure forvarious BPMdistrib utions. Seetext\nfor definitions of each scenario. Dashed lines indicate a sce nario with 50% of arc BPMsremoved;\ndash-dotted linesindicate ascenario with75%ofarcBPMsre moved. Notethat thehorizontal axis\nis now linear.\nTable 6: 95thpercentile of εb, RMSηy, and RMS ¯C12after final correction with reduced BPM\ncounts. Seetext for definitions of each BPMcount scenario.\nBPMScheme εεεbbb(pm) RMS ηηηyyy(mm) RMS ¯CCC12(%)\nScenario 1 0.224 0.502 0 .0704\nScenario 2 0.224 0.547 0 .0790\nScenario 3 0.225 0.670 0 .0863\nScenario 4 0.263 0.892 0 .216\nScenario 5 0.269 0.838 0 .239\nWithhalf ofthearcBPMsremoved(scenarios 2-3), thecorrec tion procedure achieves vertical\nemittance below 0.224-0.225pm for 95% of the lattices. With the removal of 3/4 of BPMs in the\narcs (scenarios 4-5), there is aslight increase in the 95thpercentile εbto 0.263-0.269 pm, still well\n–10–below the 2 pm requirement. There is a weak preference for ret aining the “even-indexed” BPMs\n(scenarios 2 and 4) as compared to the “odd-indexed” BPMs (sc enarios 3 and 5). Scenarios 3 and\n5 will therefore beomitted from further discussion.\n4.3 Effect of RandomBPMFailures\nFrom the proceeding discussion, it is clear that the BPM coun t can be reduced substantially with\nrespecttothebaselinedesignspecificationwithoutcompro misingtheeffectivenessoftheemittance\ntuning procedure. However, this assumes all remaining BPMs are fully functional. This is rarely\nthecaseinarealmachine,andassuch,theeffectsofrandomB PMfailuresmustbeexploredbefore\ndeeming aBPMdistribution acceptable.\nAgainusingthesamerandom seedsasinprevious simulations for generating latticeandBPM\nerrors, a random subset of BPMs is flagged as not functional, a nd therefore not used in correc-\ntions. Asthere are multiple BPMdistributions under evalua tion, only the fraction of BPMstagged\nas “failed” is constant between BPM distribution scenarios , rather than failing a fixed number of\nBPMs. The exact subset of BPMs for each test will therefore no t remain the same between each\nBPM distribution scenario. It is not possible to have a truly random distribution of BPM failures\nwhile requiring that the same BPMs are tagged as “failed” bet ween the three BPM distribution\nscenarios evaluated here, since the total number and distri bution of BPMs varies from one sce-\nnario to the next. As such, BPM failure tests on each individu al BPM distribution scenario are\ndirectly comparable seed-by-seed, but there is a small stat istical variation between the different\nBPMdistribution scenarios for afixedpercentage of BPMfail ures.\nIt is assumed that in practice, the damping rings would not be operated if more than 10% of\nBPMs have failed. In this study, the fraction of BPMs tagged a s “failed” is increased from 0% to\n10%. Figure 6 demonstrates the evolution of the 95th-percentile εb, RMSηy, and RMS ¯C12with\nrespect tothefractionoffailedBPMsforthenominalcase(s cenario1)andBPMdistributions with\n1/2 and 3/4 of arc BPMsremoved (scenarios 2and 4, respectively).\nThe resulting emittance, dispersion, and coupling of scena rios 1 and 2 are nearly identical,\nwithin theexpected statistical variation. Scenario 4,wit honly 1/4of arcBPMswithrespect tothe\nbaseline design, shows more rapid growth in emittance, disp ersion, and coupling as the fraction\nof failed BPMsincreases. Vertical emittance growth from in tra-beam scattering will increase with\nvertical dispersion [12], thus making scenario 4 unattract ive. It is therefore concluded that one of\nthe two BPMs in every arc cell may be removed without affectin g the robustness of the emittance\ntuning procedure.\n5. DynamicAperture\nAfter a lattice model has been misaligned and corrected usin g the simulated emittance tuning pro-\ncedure, thedynamic aperture isevaluated through tracking . Thedynamic aperture isdefined asthe\nmaximum stable amplitude in the transverse plane. Trajecto ries with initial coordinates inside the\ndynamic aperture boundary will remain within that boundary for at least 1000 turns. The ampli-\ntude for trajectories with initial coordinates outside the boundary will be lost within 1000 turns.\nThetracking isrepeated for off-energy particles tocomput e theenergy dependence of thedynamic\naperture.\n–11–% BPMs Failed0.00.20.40.60.81.0\u0004b (pm)\n0.00000.00030.00060.00090.0012RMS \n\u0005y (m)\n0 2 4 6 8 10\n% BPMs Failed0.0000.0020.0040.0060.008RMS Cbar12Scenario 1\nScenario 2\nScenario 4\nFigure6: 95th-percentile εb,RMSηy,andRMS ¯C12asafunctionofthefractionofBPMsrandomly\ndisabled, for three potential BPM distributions. See text f or definitions of each BPM distribution.\nThe dashed black line indicates εb=1pm.\nThe full wiggler map is required for evaluating the dynamic a perture in order to account for\nwiggler nonlinearities. This greatly increases the requir ed computation time for a dynamic aper-\nture study, however individual jobs are easily parallelize d. Analysis has shown minimal variation\nof dynamic aperture between different sets of misalignment s and corrections, given the same mis-\nalignment amplitudes. Thereforeonlyoneseedforeachtest configuration isevaluatedfordynamic\naperture. Here, the random seed corresponding to the 95th-percentile lattice after misalignments\nand correction isused.\nFigure 7a shows the dynamic aperture for the nominal scenari o, as defined in Sec. 4.1. It is\nevident that the systematic and random multipole errors as s pecified in the ILC Technical Design\nReportaremorethansufficient toachievetherequireddynam icapertureforaccepting anincoming\nbunch from the positron source.\nThepossibility that magnet manufacturing requirements ma yberelaxed byincreasing thetol-\nerance on magnet multipole errors isexplored. Starting wit hthe nominal DTC04lattice character-\nized in Section 4.1, both systematic and random multipole co efficients are increased by a constant\nmultiplier from 5x-20x with respect to the values in table 8. All other misalignments and BPM\nmeasurement errors areidentical to thenominal scenario. R esults are shown infigures 7b–7d.\nThe multipole error constraints as specified in the ILC TDR ar e evidently over-specified by a\nfactor of 20.\n–12– 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\n-6 -4 -2  0  2  4  6Vertical amplitude (mm1/2)\nHorizontal amplitude (mm1/2)ΔE/E=0   \nΔE/E=0.75%   \nΔE/E=-0.75%         Ax+Ay=0.01 m-rad\n   Ax+Ay=0.04 m-rad\n   Ax+Ay=0.07 m-rad\n(a)NominalscenarioasdefinedinSec. 4.1. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\n-6 -4 -2  0  2  4  6Vertical amplitude (mm1/2)\nHorizontal amplitude (mm1/2)ΔE/E=0   \nΔE/E=0.75%   \nΔE/E=-0.75%         Ax+Ay=0.01 m-rad\n   Ax+Ay=0.04 m-rad\n   Ax+Ay=0.07 m-rad\n(b)Multipoleerrorsincreasedbya factorof5x.\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\n-6 -4 -2  0  2  4  6Vertical amplitude (mm1/2)\nHorizontal amplitude (mm1/2)ΔE/E=0   \nΔE/E=0.75%   \nΔE/E=-0.75%         Ax+Ay=0.01 m-rad\n   Ax+Ay=0.04 m-rad\n   Ax+Ay=0.07 m-rad\n(c)Multipoleerrorsincreasedbyafactorof10x. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\n-6 -4 -2  0  2  4  6Vertical amplitude (mm1/2)\nHorizontal amplitude (mm1/2)ΔE/E=0   \nΔE/E=0.75%   \nΔE/E=-0.75%         Ax+Ay=0.01 m-rad\n   Ax+Ay=0.04 m-rad\n   Ax+Ay=0.07 m-rad\n(d)Multipoleerrorsincreasedbyafactorof20x.\nFigure 7: Dynamic aperture for the DTC04 lattice, while vary ing the amplitude of systematic\nand random multipole errors. All scenarios shown include th e full complement of BPMs. The\noverlayed black ellipse represents the maximum expected am plitude for bunches transferred from\nthe positron source [1].\n6. Summary andFuture Work\nAmethod hasbeenpresented for determining theeffectivene ss ofcorrections for theILCdamping\nrings and the resulting dynamic aperture after correction, using the emittance tuning algorithm\ndeveloped and used at CesrTA. Lattice models include misali gned magnets, magnet strength and\nmultipole errors, andBPMmeasurement errors. Themethodis versatile andmaybeappliedtoany\nclosed lattice to evaluate the sensitivity of the effective ness of emittance corrections to changes in\nBPMand corrector distributions, multipole tolerances, an d more.\nBasedonthis method, theILCdamping ring design, misalignm ent tolerances, andBPMmea-\nsurement errors as presented in the ILC Technical Design Rep ort (TDR) have been shown to\nachieve the required zero-current vertical emittance and d ynamic aperture. Note that collective\neffects such as intra-beam scattering, electron cloud, and fast-ion instability can only increase the\nvertical emittance above this minimum.\nOne of the two BPMs in every arc cell may be removed without imp act on the correction\ncapabilities, even when accounting for up to a10% BPMfailur e rate. Thisleaves approximately 5\nBPMs per betatron wavelength in the arcs. A further reductio n of BPMs in the arcs to 1/4 of the\n–13–originalcount( ≈2.5BPMsperbetatronwavelength)causesasubstantial incr easeindispersionand\ncoupling, subsequently fueling emittancegrowthfromcoll ective effects. Specifications formagnet\nmultipole errors can be relaxed bya factor of twenty without compromising thedynamic aperture.\nAn infrastructure exists for evaluating further lattice mo difications, such as sextupole opti-\nmizations, increased magnet or BPM errors, changes in corre ctor distributions, or a change in\nworking point. Further lattice developments inthese areas cantherefore beeasily evaluated for the\nILC damping rings or anyarbitrary closed lattice.\nAcknowledgments\nThisworkwassupported bytheNationalScienceFoundationg rantPHY-1002467andDepartment\nof Energy grant DE-SC0006505.\nAppendix: DTC04 Magnet MisalignmentsandMultipoles\nMultipole coefficients are defined in the following way:\n(By+ıBx)=B(r)∑\nn=1(bn+ıan)/parenleftBigx\nr+ıy\nr/parenrightBign−1\n(1)\nwhere nisthe multipole order, and bn,anare the normal and skew components, respectively. Mul-\ntipolesareevaluatedatareferenceradiusof r=3cmfordipoles, 5cmforquadrupoles, and3.2cm\nfor sextupoles.\nTable7: Misalignments and errors introduced into themodel ILC-DRlattice.\nElement Error Amplitude Units\nDipole Roll 50 µrad\nQuadrupole x,yOffset 25 µm\nTilt 50 µrad\nk1 0.1% %\nSextupole x,yOffset 25 µm\nTilt 25 µrad\nk2 1% %\nWiggler Tilt 100 µrad\nyOffset 100 µm\nBPM Diff. Resolution 1 µm\nAbs. Resolution 50 µm\nTilt 10 mrad\nButton Gains 0.5% %\nButton Timing 10 ps\n–14–Table 8: Nominal multipoles (systematic and random) introd uced into the model lattice. Coeffi-\ncients are taken from multipole measurements by Y.Cai at PEP -II[5].\nElement Multipole Systematic Random\nDipole b3 1 .6×10−48×10−5\nb4 −1.6×10−58×10−6\nb5 7 .6×10−53.8×10−5\nQuadrupole a3 −1.15×10−57.25×10−5\na4 1 .41×10−51.27×10−4\na5 6 .2×10−71.62×10−5\na6−4.93×10−53.63×10−4\na7−1.02×10−66.6×10−6\na8 3 .8×10−76.6×10−6\na9 −2.8×10−74.9×10−6\na10 −5.77×10−52.33×10−4\na11 −3.8×10−73.5×10−6\na12 −6.53×10−63.66×10−5\na13 1 .2×10−68.6×10−6\na14 −7.4×10−74.46×10−5\nb3−1.24×10−57.61×10−5\nb4 2 .3×10−61.32×10−4\nb5 −4.3×10−61.5×10−5\nb6 3 .4×10−41.65×10−4\nb7 3 ×10−76.7×10−6\nb8 6 ×10−78.9×10−6\nb9 6 ×10−74.6×10−6\nb10 −6.17×10−52.46×10−4\nb11 −2×10−74.2×10−6\nb12 3 .6×10−63.48×10−5\nb13 6 ×10−79.2×10−6\nb14 1 ×10−64.76×10−5\nSextupole b4 1 ×10−41×10−4\nb5 5 ×10−53×10−5\nb6 3 .5×10−41×10−4\nb7 5 ×10−53×10−5\nb8 5 ×10−53×10−5\nb9 5 ×10−53×10−5\nb10 5 ×10−53×10−5\nb11 5 ×10−53×10−5\nb12 1 .6×10−31×10−4\nb13 5 ×10−53×10−5\nb14 5 ×10−53×10−5\n–15–References\n[1]TheInternationalLinearColliderTechnicalDesignReport ,Tech.report,ILCGlobalDesignEffort,\n2013.\n[2] M.Aiba,M. 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It is shown that the chemical potential oscillations,\ninvolved in the frequency combinations observed in the case of uncompensated orbits, are strongly\ndamped and can even be suppressed when the effective masses of the electron- and hole-type or-\nbits are the same. When magnetic breakdown between bands occ urs, this damping is even more\npronounced and the Lifshits-Kosevich formalism accounts f or the data in a wide field range.\nPACS numbers: 71.18.+y,71.20.Rv,74.70.Kn\nI. INTRODUCTION\nIn large enough magnetic field, the Fermi surface (FS)\nof multiband quasi-two-dimensional metals, is liable\nto give rise to networks of orbits coupled by magnetic\nbreakdown (MB). The most studied type of network is\nthe linear chain of coupled orbits introduced by Pippard1\nand illustrated by several quasi-two-dimensional (q-2D)\norganic conductors such as κ-(BEDT-TTF) 2Cu(NCS) 2.\nAs discussed, in Ref.2, magnetic oscillations spectra\nof such networks contain many frequencies that are\nlinear combinations of two basic frequencies. In ad-\ndition to those linked to MB-induced orbits, other\nfrequencies are observed that are not accounted for by\nthe semi-classical theory of Falicov and Stachowiak3.\nThey can be attributed to quantum interference (as\nfar as magnetoresistance oscillations are concerned),\nMB-induced modulation of the density of states4,5,6and\noscillation of the chemical potential7,8,9,10, even though\nthe actual respective contribution of these three phenom-\nena to the oscillatory behavior remains to be established.\nAnother type of network is provided by q-2D metals of\nwhich the FS is composed of compensated electron- and\nhole-types tubes. This is the case of the family of or-\nganic metals (BEDT-TTF) 8Hg4Cl12(C6H5X)2(X = Cl,\nBr) whose FS, which originates from two pairs of cross-\ning q-1D sheets, is composed of one electron and one\nhole tube with the same area11. As it is the case of the\nabove-mentioned linear chains of coupled orbits, magne-\ntoresistance oscillations spectra in this type of network\nreveal frequencies that are linear combinations of three\nbasic frequencies, linked to the compensated orbits and\nto the two FS pieces located in-between12,13,14. How-\never, in striking contrast to the data relevant to linear\nchains of orbits, de Haas-van Alphen (dHvA) oscillations\nspectra recorded in the case of the compound with X\n= Br only exhibit oscillations, the field and tempera-\nture dependence of which can be consistently interpreted\non the basis of the semiclassical model of Falicov andStachowiak3,14. Analogous networks are observed in or-\nganic metals, with two carriers per unit cell. In this case,\nthe FS originates from the overlapping in two directions\nof hole tubes with an area equal to that of the First Bril-\nlouin zone (FBZ) and from the resulting gap openings15.\nAs reported in the case of (BEDO) 4Ni(CN) 4·4CH3CN16,\nsuch a FS yields a network consisting in two hole- and\none electron-type tubes (see Fig. 1(a)). Closely related\nnetwork is obtained in the case where the large hole or-\nbit come close to the FBZ boundary, as it is the case of\n(BEDT-TTF) 4NH4[Fe(C2O4)3]17. In this latter case, a\nlarge MB gap is observed at this point and the resulting\nnetwork only consists in one electron- and one hole-type\norbit,asdisplayedinFig. 1(b). Linearchainofsuccessive\nelectron-hole tubes might also be observed in Bechgaard\nsalt (TMTSF) 2NO318,19.\nThe aim of this paper is to explore the field and tem-\nperature dependence of the dHvA oscillations spectra of\nan ideal 2D metal whose FS is composed of one electron-\nand one hole-type compensated orbit. It is demonstrated\nthat the field-induced oscillations of the chemical poten-\ntial are strongly damped for such a FS and can even\nbe suppressed in the case where electron- and hole-type\norbits have the same effective masses. The chemical po-\ntential oscillations can be even more damped when the\ntwo orbits are coupled by MB. In this case, the Lifshits-\nKosevich (LK) formalism accounts for the data up to\nhigh magnetic field and low temperature.\nII. MODEL\nWe firstconsidera2Dmetal whoseelectronicstructure\nconsistsoftwoparabolicbandsofhole-andelectron-type,\nrespectively. The bottom of the electron band is set at\nzero energy while the top of the hole band is at ∆ >0.\nThe total number of electrons in the system is such that\nthe hole band is completely filled. Since the lower part of\nthe electron band is lower in energy than the top of the\nhole band, some quasiparticlesmove to the electron band2\nFIG. 1: (color online) Calculated Fermi surface (FS) of q-\n2D organic metals (a) (BEDO) 4Ni(CN) 4·4CH3CN16and (b)\n(BEDT-TTF) 4NH4[Fe(C2O4)3]17leading to networks of com-\npensated electron- and hole-orbits (solid red and blue line s,\nrespectively). Ellipses in dotted blue lines correspond to the\nhole orbits, with an area equal to that of the FBZ, from which\nthe FS is built (see text).\nin order to lower the total energy. The effective masses\nlinked to the two bands, m∗\ne≡1/Ceandm∗\nh≡1/Ch>0,\ncan be different. It is useful to define the physical units\nof the problem. The reduced field b=eBA0/his in\nunits of the characteristic field ˜B=h/eA0, the effective\nmasses are in units of the electron mass me, the energies\nare in units of ˜E= 2π/planckover2pi12/meA0and the temperature tin\nunits of˜T=˜E/kB. Given a unit cell area A0=197.6˚A2,\nwhich stands for (BEDT-TTF) 8Hg4Cl12(C6H5Cl)220, we\nobtain˜B= 2093 T and ˜T= 2812 K. Therefore, real-\nistic experimental conditions yield small values of band\ntcompared to ˜Band˜T, respectively. The semi-classical\nquantizationoftheenergylevelsinthepresenceofamag-\nnetic field leads to the Landau equations:\nEe(n) =Ceb(n+1\n2), Eh(n) = ∆−Chb(n+1\n2) (1)\neachLandau level (LL) havinga degeneracy bper sam-\nple area. The zero field Fermi energy is simply given\nbyEF=m∗\nh/(m∗\ne+m∗\nh)∆. At finite temperature, the\ntotal free energy is given by the difference between the\ncontribution of the electron and of the hole band, with\nthe condition that the total number of quasiparticles\nNehis fixed. In addition, the compensation condition\nimposes that the number of quasiparticles in the elec-\ntron (Ne) and in the hole ( Nh) band are the same.\nFrom the thermodynamical relations, we can define a\nfree energy for the system based on the difference be-\ntween the free energy for the electrons in the electron\nband and the free energy for the holes in the hole band\n∆F= Ωe−Ωh+(Ne−Nh)µ, where Ω e(h)is the Grand\nPotential for the electrons (holes) and µis the chemical\npotential [ µ(t= 0,b) =EF(b)]:Ωe(t,b) =−tb/summationdisplay\nn≥0log(1+exp[ β(µ−Ee(n))])\nΩh(t,b) =tb/summationdisplay\nn≥0log(1+exp[ β(Eh(n)−µ)]) (2)\nSinceNe=Nh, we conclude that F= Ωe−Ωhin com-\npensated metals. µis evaluated from the self-consistent\nequation ∂∆F/∂µ= 0. At zero temperature, the above\nexpressions reduce simply to the ground state (GS) en-\nergy ∆E0. The Fermi energy EF(b) is given by the con-\nditionneb=nhb, whereneandnhare the (integer) num-\nbers of LL filled, and\n∆E0=−bne−1/summationdisplay\nn=0(EF(b)−Ee(n))−bnh−1/summationdisplay\nn=0(Eh(n)−EF(b))\n=Ee−Eh+b(nh−ne)EF(b) =Ee−Eh(3)\nThe special cases where the Fermi energy goes trough\none Landau level, or where this Landau level is partially\nfilled, correspond to singular points in the energy spec-\ntrum as a function of the inverse field that do not modify\nthe thermodynamical quantities. The exact expression\nfor ∆E0is simply\n∆E0=1\n2(Ce+Ch)b2n2\ne−b∆ne (4)\nwithne= [EF/Ceb+1/2]i, the notation [ .]istanding\nfor the integer part of the argument. The GS energy\noscillates around the limit of zero field, where ∆ E0(q=\n1) =−∆2/2(Ce+Ch). For example, taking ∆ = 1,\nCe= 1 and Ch= 2/5, we obtain ∆ E0≃ −0.357. We\ndeduce the oscillating part of the magnetization Mosc=\n−∂∆E0/∂bfrom the latter expression, using succesively\nthe Fourier transforms of the periodic functions [ x]i−x\nand ([x]i−x)2:\nMosc=−(Ce+Ch)F0/summationdisplay\nk≥1(−1)k\nπksin(2πkF0/b) (5)\nwhereF0= ∆/(Ce+Ch) =m∗\nem∗\nh∆/(m∗\ne+m∗\nh) is the\nfundamental frequency corresponding to the FS area of\nthe electron and hole band. At zero temperature and for\nfixed number of electrons, the magnetization oscillates\nwith characteristic frequency F0, and the amplitudes Ak\nof thekthharmonics are given by the LK formula with\n1/kdependence21:\nAk= (−1)k+1F0\nkπ(Ce+Ch) (6)\nThe sum Ce+Chmeans that the 2 orbits circulating\naround the hole and electron bands contribute individu-\nally to the magnetization.3\nIII. SELF-CONSISTENT EQUATION FOR THE\nCHEMICAL POTENTIAL\nThe oscillatory parts of the Grand Potentials Eqs. (2)\ncan be extracted using Poisson’sformula for any function\nF(x) (see Ref.21,22for details):\n1\n2F(0)+/summationdisplay\nn≥1F(n) =/integraldisplay∞\n0F(x)dx\n+ 2ℜ/summationdisplay\nn≥1/integraldisplay∞\n0F(x)exp(2iπnx)dx (7)\nwhereF(n) is equal to log(1 + exp[ β(µ−Ee(n))]) or\nlog(1 + exp[ β(Eh(n)−µ)]) for electrons and holes, re-\nspectively. The last series in Eq. (7) gives the oscil-\nlatory part of the Grand Potential in terms of Fourier\ncomponents. The index nis expressed as a function of\nthe energy n=n(E) by relations (1), then expanded\naround the chemical potential µat low temperature\nwhere the energyderivatives of the distribution functions\n1/(1+exp[ β(Ee(n)−µ)]) or 1/(1+exp[ β(µ−Eh(n))])\nare strongly peaked. After some algebra, we obtain for\neach band\nΩe≃ −1\n2Ceµ2(8)\n+b2Ce\n2/bracketleftBigg\n1\n12+∞/summationdisplay\nn=1(−1)n\nπ2n2R(nm∗\ne)cos(2πnµ\nCeb)/bracketrightBigg\nΩh≃1\n2Ch(∆−µ)2(9)\n−b2Ch\n2/bracketleftBigg\n1\n12+∞/summationdisplay\nn=1(−1)n\nπ2n2R(nm∗\nh)cos(2πn∆−µ\nChb)/bracketrightBigg\nwhere\nR(nm∗\ne(h)) =2π2nm∗\ne(h)t/b\nsinh(2π2nm∗\ne(h)t/b)(10)\nis the temperature reduction factor for effective\nmassesnm∗\ne(h). A Dingle term RD(nm∗\ne(h),t∗\ne(h)) =\nexp(−2π2nm∗\ne(h)t∗\ne(h)/b) can also be added in the case\nwhere the relaxation times t∗\neandt∗\nhfor electron- and\nhole-band have to be taken into account. In the follow-\ning we will assume that these two relaxation times are\nnegligible for convenient purpose. It is always possible\nto add those terms in final expressions. The chemical\npotential satisfies therefore the self-consistent relation:\nµ=EF+b\nm∗e+m∗\nh∞/summationdisplay\nn=1(−1)n\nπn[R(nm∗\nh)sin(2πn∆−µ\nChb)\n−R(nm∗\ne)sin(2πnµ\nCeb)] (11)\nmh= 5/2 mh= 1.1m1= 5/2 m1= 1.1\nFIG. 2: (color online) Field dependence of the chemical po-\ntential normalized to its value in zero-field at t = 0.001. Sol id\nanddashed lines correspond toone electron andone hole com-\npensated orbits (with m∗\ne= 1 and m∗\nh= 1.1 or 5/2) and two\nelectron orbits (with m∗\n0= 1 and m∗\n1= 1.1 or 5/2), respec-\ntively (see text).\nAt a first order approximation, which will be discussed\nin the following sections, we can replace µin the sine\nfunctions of the previous expression by EFwhen the os-\ncillations of the chemical potential are small compared to\nEF, in the small field and high temperature regime ( t/b\nlarge). This gives:\nµ≃EF+b\nm∗e+m∗\nh× (12)\n∞/summationdisplay\nn=1(−1)n\nπn[R(nm∗\nh)−R(nm∗\ne)]sin(2πnF0\nb)\nIt can be remarked first that, in the case where the ef-\nfectivemassesandrelaxationtimeslinkedtotheelectron-\nand hole-type orbits are the same, there is an exact solu-\ntionforEq. (11)with µ=EF. Inthiscasetheoscillatory\npart of Eq. (11) or Eq. (12) vanishes and the chemical\npotential remains constant in magnetic field and temper-\nature. This is due to the fact that the energy levels are\nsymmetric around EF. More generally, the amplitude of\nthe chemical potential oscillations can be compared to\nthe case of two electron bands10. In Fig. 2, the field-\ndependent chemical potential Eq. (11) is calculated for\ncompensated orbits with m∗\ne= 1 and m∗\nh= 1.1 or 5/2\nand compared to the case of two electronic orbits with\neffective masses m∗\n0= 1 and m∗\n1= 1.1 or 5/2. It can\nbe observed that the chemical potential oscillations are\nstronglydamped forcompensated orbits, evenin the case\nwhere m∗\nhand m∗\nehave strongly different values23.\nIV. DE HAAS-VAN ALPHEN OSCILLATIONS\nThe oscillatory magnetization can be obtained putting\nthe solution of the chemical potential given by Eq. (11)4\nFIG. 3: (color online) Fourier spectrum of the magnetizatio n\nat t = 0.001 for m∗\ne= 1 and m∗\nh= 1.1 or 5/2. The field\nrange is from b = 0.015 to b = 0.033. F 0is the fundamental\nfrequency (see text).\nback into the expression for the free energy Fusing Eqs.\n(1) and (2). As discussed above, the chemical potential\nis field-independent for m e= mhand the LK formula21\nholds in this case. However, for m∗\ne/negationslash= m∗\nh, the oscillatory\nmagnetization differs from the LK theory. Examples\nof Fourier analysis of the magnetization are given in\nFig. 3. It should be noted that, contrary to the case of\nelectron-hole systems away from exact compensation24,\nfrequency combinations are not observed directly in\nthe oscillatory spectra. However, as discussed later on,\ndeviations from the LK behavior are observed for the\nsecond harmonics.\nItisusefulforexperimentalists,tochecktowhatextent\nit is possible to determine a temperature and field range\nin which the LK formalism provides a satisfactory ap-\nproximation of the oscillatory behavior in the case where\nm∗\ne/negationslash= m∗\nh. The b/t dependence of the Fourier compo-\nnents of the magnetization with frequencies F 0and 2F 0\naregiveninFig. 4. The LKformulaaccountsforthefield\nand temperature dependence of the first harmonics. Fur-\nthermore, it can be noticed that in the case where m∗\nhis\nstrongly different from m∗\ne, the contribution of the orbit\nwith the smallest effective mass dominates in a largefield\nrange (see the dashed line in Fig. 4a). It has also been\nchecked that, in the opposite case where m∗\neis close to\nm∗\nh, the data can be accounted for by the contribution of\nonlyoneorbitwithameaneffectivemass,namely, m∗\nmean\n≃(m∗\ne+ m∗\nh)/2. In contrast, the LK formula cannot ac-\ncount for the second harmonics in the case where m∗\neand\nm∗\nhsignificantly differ. The observed behavior of the sec-\nond harmonics in this latter case is reminiscent to that\nobserved for two electronic orbits10.\nV. MAGNETIC BREAKDOWN\nIn this section, we consider the case where the mag-\nnetic field is large enough for the quasiparticles can tun-FIG. 4: t/b dependence of the Fourier components of the\nmagnetization with frequencies F 0and 2F 0. The effective\nmass of the electronic-type orbit is m∗\ne= 1. The effective\nmass of the hole-type orbit is m∗\nh= 5/2 and 1.1 in (a) and (b),\nrespectively. Solid lines are obtained from the LK formula.\nThe dashed line in (a) corresponds to the contribution of the\nelectronic orbit, only.\nnel through MB between the two bands.\nThe topology of the new FS is depicted in Fig. 5:\nthe quasiparticles can tunnel between the junctions α′\nandβ(β′andα) with probability amplitude ipor be\nreflected between the junctions α′andα(β′andβre-\nspectively) with probability amplitude q. This surface is\na simplification of Fig. 1b in the sense that it does not\ninclude the 1D network feature. pandqare related by\np2+q2= 13,21, and the field dependence of pis given by\nthe Chambers formula p= exp(−b0/2b)25,26, where b0\nis a characteristic MB field. b0comes in the field range\ncovered by experiments in the case where the two bands\nare located closely enough in the FBZ. Depending on\nthe band type (electron or hole), the quasiparticle wave-\nfunction acquires a phase either σe=Se/borσh=Sh/b5\nσhσeσhσe−ie\nαα′ β′\nβe/2 /2\n−iβ iαi\nFIG. 5: Fermi surface of a two-band model with magnetic\nbreakdown in the FBZ. The quasiparticles orbit clockwise fo r\nthe Hole surface and counterclockwise for the Electron band .\nThey can tunnel between the points (which represent also the\namplitudes of the wave-function) α′andβ(orβ′andα) with\na probability amplitude ip, or be relected between α′andα\n(orβ′andβ) with a probability amplitude q. At the turning\npoints (cross symbols), the wavefunction acquires a factor i\n(or−i) depending wether the quasiparticle orbits an electron\nor hole surface.\nwhen orbiting around each part of the Fermi surface (be-\ntweenαandα′counterclockwise for the electron band or\nβandβ′clockwise for the hole band). The semiclassical\nactionsSe= 2πm∗\neEandSh= 2πm∗\nh(∆−E) represent\nthe areas delimited by each Fermi surface in the FBZ.\nAssuming that the wavefunction of the quasiparticle is α\nat the point α, we obtain that after one orbit around the\nelectron band, α′=−αexp(iσe) (the quasiparticle goes\nthrough 2 turning points where it acquires a factor ior\n−ieach time, depending on the electron or hole type26).\nSimilarly, itfollows β′=−βexp(−iσh)forthe holeband.\nThe four amplitudes α,α′,βandβ′are related by the\ntransfer matrix5,26\n/parenleftbigg\nα\nβ/parenrightbigg\n=/parenleftbigg\nq ip\nip q/parenrightbigg/parenleftbigg\nα′\nβ′/parenrightbigg\n(13)\nThis set of equations have non-zero solution when the\nenergyEsatisfies the implicit equation:\n(1+qexpiσe)(1+qexp(−iσh))\n+p2expi(σe−σh) = 0 (14)\nIn the case where the bands are disconnected ( p=\n0), Eq. (14) can be factorized and reduces simply to\n1 + exp( iσe) = 0 or 1 + exp( −iσh) = 0, which corre-\nsponds to the two independent sets of discrete energy\nlevels given by Eq. (1). In general Eq. (14) can be\nsolved numerically for p/negationslash= 0: the field dependence of the\nLLs energy is plotted in Figs. 6 and 7 for q=0.6 and\nfor 2 different sets of effective masses. In the case where\np/negationslash= 0, gaps open at the former LL intersections, and the\nstructure of the two bands is modified. Alike in the case\nwherep= 0, a quasi-hole and a quasi-electron band can\nbe defined. Indeed, increasing the magnetic field, levels\nthat go upwards above the Fermi energy EF(b) (see dia-\nmond symbol lines) and downwards below EF(b) repre-\nsentquasi-electronandquasi-holebands,respectively. At\nzero temperature, the compensation condition Ne=Nh\nis replaced by the condition that the lower quasi-hole5 10 15\n1/b0.60.8Energy Levels\nFIG. 6: (color online) Energy levels for a reflection amplitu de\nq= 0.6 (or, equivalently, tunnelling amplitude p=p\n1−q2),\n∆ = 1, m∗\ne= 1 and m∗\nh= 5/2. Levels that belong to a quasi-\nelectron band (going upwards in the limit of large field, blac k\nlines) above the Fermi energy EF(b) (diamond symbols, green\nline) can be distinguished from levels that belong to a quasi -\nhole band (respectively going downwards, red lines). The\nFermi energy at zero field is shown as a dashed line.\nband is filled completely, the upper electron-band being\nempty. The two bands are therefore always separated by\nthe chemical potential energy, as shown in Figs. 6 and\n7. As the magnetic field decreases, the field-dependent\nLandau levels near the Fermi energy become flatter since\nthe levels of each band do not intersect the Fermi level.\nIn the case where q= 0 (p= 1), the quasiparticles tunnel\ndirectly through the 2 bands at each revolution, and the\nexact solution of Eq. (14) is given by the set of LLs:\nEeh(q= 0,n) =EF−bCeCh\nCe+Ch(n+1\n2)\n=Ce\nCe+ChEh(n) (15)\nwithn≥0. Within the semi-classical approach, the\ncross-section area of the FS corresponding to this in-\nverted parabolic quasi-hole band, which is completely\nfilled, is zero. Since it is proportional to the frequency of\nthe oscillations, there are no oscillations at q= 0. This\nsimple feature is in line with the prediction of the semi-\nclassical model for two compensated orbits, keeping in\nmind that the area of these orbits has opposite signs3.\nWe assume that Ch=Cep0/q0wherep0andq0are co-\nprimeintegers5,10. Indeed, it isuseful, alsoforcomputing\ntime reasons, to approximate any real value of Ch/Ceby\na close rational ratio since in this case the spectrum (14)\nis periodic in energy, with periodicity TE=Cebp0\nIn order to estimate the total free energy in the gen-\neral case, it is necessary to solve numerically Eq. 14. We\nassume that Ch=Cep0/q0wherep0andq0are coprime\nintegers5,10. Indeed, it is useful, also for computing time\nreasons, to approximate any real value of Ch/Ceby a6\n5 10 15\n1/b0.30.40.50.60.7Energy Levels\nFIG. 7: (color online) Energy levels for a reflection amplitu de\nq= 0.6, ∆ = 1, m∗\ne= m∗\nh= 1. Levels that belong to a quasi-\nelectron band above the Fermi energy located at EF(b)=0.5\n(diamond symbols, green line) can be distinguished from lev -\nels that belong to a quasi-hole band (red lines). The 2 bands\nare fully symmetric with respect with the chemical potentia l\nenergy.\nclose rational ratio since in this case the spectrum (14) is\nperiodic in energy, with periodicity TE=Cebp0Within\neach interval TE, there are exactly p0+q0discrete so-\nlutions (there is indeed conservation of the number of\nLLs when qdecreases from unity). By successive energy\ntranslations, it is then easy to reproduce all the spec-\ntrum. The GS energy ∆ E0given by Eq. (3) is however\nno more well defined, due to the mixing between electron\nand hole levels. Indeed, the GS energy should corre-\nspond to the sum of all the quasi-hole energies below the\nFermi energy EF(b). These energies being not bounded\nby below, the GS energy is formally infinite. We propose\nto account numerically for this problem by introducing a\ncutoff function in the density of states with the necessary\ncondition that the magnetization should not depend on\nvariations of the cutoff parameters.\nGiven a set of LL energies Eeh(q,n),n= 0,···,p0+\nq0−1, solutions of (14) in any energy interval of width\nTE, we can introduce a cutoff function ϕc(E) such as\nϕc(E) = 1 for Elarger than a characteristic energy Ec\nand equal to exp[ −c(E−Ec)2δ] forE≤Ec, whereδis\nany positive integer greater than 1 (we take δ= 4 in the\nsimulations which gives a very smooth cutoff function).\nThis function has the required property of conserving\nthe important physical properties near the Fermi surface\nand making in particular the GS energy finite. The LL\ndensity of states ρc(E) takes the following form\nρc(E) =bp0+q0−1/summationdisplay\nn=0∞/summationdisplay\nk=−∞ϕc(E)δ(E−Eeh(q,n)−kTE)\nGivenEc, the parameter cis found to be solution of\nthe equation of conservation at zero temperature:Neh=/integraldisplayEF(b)\n−∞dEρc(E)\n=b/integraldisplayEF(b)\n−∞dEp0+q0−1/summationdisplay\nn=0∞/summationdisplay\nk=−∞ϕc(E)\nδ(E−Eeh(q,n)−kTE) (16)\nwhereNehis, as mentioned before, the total number\nof quasiparticles in the Canonical Ensemble. Once the\nparameter cis obtained, we can define for example the\nGS energy ∆ E0\n∆E0=b/integraldisplayEF(b)\n−∞dEp0+q0−1/summationdisplay\nn=0∞/summationdisplay\nk=−∞ϕc(E)\nEδ(E−Eeh(q,n)−kTE) (17)\nor the free energy ∆ F\n∆F=−tb/integraldisplay∞\n−∞dEp0+q0−1/summationdisplay\nn=0∞/summationdisplay\nk=−∞ϕc(E)\nlog[1+exp β(µ−E)]δ(E−Eeh(q,n)−kTE)\n+Nehµ (18)\nThe potential µ(t,b) is calculated from Eq. (18) by ex-\ntremizing the free energy ∂∆F/∂µ= 0, and compelling\nthe magnetization Mosc=−∂∆F/∂bto be independent\nof the parameter EcforEcfar away from the chemi-\ncal potential or at energies large compared to the Lan-\ndau gap. We have checked, for different values of Ec\nin units of ∆, for example, Ec=−1,−2,−4, and for\na large range of fields, that the resulting magnetization\ndoes not change. We choose Neh, which is arbitrary, as\na multiple of the characteristic zero field energy density\nm∗\ne+m∗\nh= (Ce+Ch)/CeChtimes the zero field Fermi\nenergyEF. ForEc=−2, we take in the following nu-\nmerical simulations Neh= 8EF(Ce+Ch)/CeCh. Then,\nfor each value of the field b, the parameter cdefined from\nEq. (16) is unique. Also, in the case q= 1, the numerical\nsolution for the magnetization is fully consistent with the\nresults of the first section in absence of magnetic break-\ndown. In Fig. 8, the oscillationsofthe chemicalpotential\nµ(t,b) obviously decrease with quntil the total tunneling\noccurs where only orbits with frequency zero are allowed.\nA. Analytical amplitudes for small field\nIn this section, we extract the analytical expression\nfor the first harmonics amplitude (corresponding to the\nfrequency F0) in the small b/tregime, and compare it\nto the numerical results of the previous section. We first\nassume that the oscillations can be described by an effec-\ntive free energy which is constructed by adding reflection7\n30 32 34 36 38 40\n1/b0.7120.7130.7140.7150.716Chemical potentialq=1\nq=0.8\nq=0.6\nq=0.4\nq=0.2\nq=0.01\nFIG. 8: (color online) Oscillations of the chemical potenti al\natt= 10−3for different values of q; ∆ = 1, m∗\ne= 1 and m∗\nh\n= 5/2.\namplitudes qnto the temperature reduction factors in\nthe expression of the Grand Potentials Eqs. 8 and 9, for\norbits that circulate ntimes around Fermi surfaces Se\norSh. In this approximation, it is indeed clear that the\nquasi-particles orbit the same surface if the field is small\nenough, and do not tunnel through the junction. The ef-\nfective free energy can then be derived directly from Eqs.\n(8) and (9) as\n∆Feff= Ωe−Ωh≃ −µ2\n2Ce−(∆−µ)2\n2Ch(19)\n+b2Ce\n2∞/summationdisplay\nn=1(−q)n\nπ2n2R(nm∗\ne)cos(2πnµ\nCeb)\n+b2Ch\n2∞/summationdisplay\nn=1(−q)n\nπ2n2R(nm∗\nh)cos(2πn∆−µ\nChb)\nand the chemical potential is derived as in Eq. 11\nµ=EF+b\nm∗e+m∗\nh∞/summationdisplay\nn=1(−q)n\nπn[R(nm∗\nh)sin(2πn∆−µ\nChb)\n−R(nm∗\ne)sin(2πnµ\nCeb)] (20)\nNumerically, we can measure the deviations between\nthis approximation and the numerical result µ=\n∂∆F/∂b, where ∆ Fis defined by Eq. (18), for m∗\ne= 1\nandm∗\nh= 5/2 (see Fig. 9), and find that Eq. (20) is\nvalid for b/tsmaller than approximatively 12 (approxi-\nmation (a)), and 8 in the case where µis replaced by EF\nin the right hand side of Eq. (20) (approximation (b)),\nas in Eq. (12) for q= 1.\nThe complete derivation of the first harmonic ampli-\ntude is done in Appendix A. If we keep the dominant\nterms in Eq. (A7), which are those with the dominant15 20 25\n1/b0.710.715 Chemical Potentialt=0.004\nt=0.004, approximation (a)\nt=0.004, approximation (b)\nt=0.008                          \nt=0.008, approximation (a)\nt=0.008, approximation (b)\nFIG. 9: (color online) Comparison between the numerical so-\nlution for µ(Eq. 18) with approximation (a) from Eq. (20),\nand (b) from Eq. (20) with µin the sine functions replaced\nbyEF.m∗\ne= 1 and m∗\nh= 5/2, andq= 0.6. The 2 sets\nt= 0.004 and t= 0.008 are shifted along the vertical axis for\nclarity.\nreduction factor R(m∗\ne(h)), and discard the other expo-\nnentially small ones R(nm∗\ne(h)) forn≥2 or products\nR(m∗\ne(h))2, we obtain\nA1≃2F0\nπq/braceleftbiggR(m∗\ne)\nm∗eJ1(αe)\nαe+R(m∗\nh)\nm∗\nhJ1(αh)\nαh/bracerightbigg\n(21)\nwhereαe(h)= 2m∗\ne(h)q(R(m∗\ne)−R(m∗\nh))/(m∗\ne+m∗\nh) as\ndefined in Appendix A, and J1is the Bessel function of\norder 1. In the limit where αe(h)are small, the functions\n2J1(αe(h))/αe(h)are very close to unity and we recover\nthe LK formula:\nALK\n1≃F0\nπq/braceleftbiggR(m∗\ne)\nm∗e+R(m∗\nh)\nm∗\nh/bracerightbigg\n(22)\nFor example, for q= 0.6,m∗\ne= 1 and m∗\nh= 5/2,αe\nvanishes when b/tis either large or small, αebeing al-\nways less than 0.2 elsewhere. At this extremum value,\nthe function 2 J(x)/xis approximatively equal to 0.995.\nTherefore we do not expect, as in the case q= 1, a sig-\nnificant deviation from the LK formula at small b/t, as it\ncan be seen in Fig. 10 in this range of fields for different\nvalues of qstrongly different from unity.\nB. LK formula for large field\nIn the large b/tlimit, the previous approximationis no\nmore valid (the reduction factors are close to unity). We\nshow hereafter that there are an infinite number of orbits\ncontributingtothe frequency F0. Theseorbits, whichare\ndescribed in details in Appendix B, have large effective\nmassesme(n) = (n+1)m∗\ne+nm∗\nhormh(n) =nm∗\ne+(n+\n1)m∗\nh, wherenisa positiveinteger. Theircontributionto8\nthe total amplitude are therefore negligible at low field\n(or high temperature) since their reduction factors are\nexponentially small. An example of orbit of mass me(1)\ncontributing to frequency F0is shown in figure 11.\nWe can however compute the exact amplitude A1\nwithin the LK theory, and show that there is no dis-\ncernible difference with the numerical solution of Eq.\n(18) in the Canonical Ensemble.\nAt low fields, the first harmonics amplitude A1con-\ntains the contribution of the 2 single orbits of mass m∗\ne\nandm∗\nh, as in Eq. (22). At higher fields, we can replace\nthe previous quantity by an exact expression\nALK\n1=F0\nπ(Te+Th) (23)\nwhere the amplitudes Te(h)includes all the orbits that\ncontribute to the first harmonics\nTe(h)=q\nm∗\ne(h)R(m∗\ne(h))\n+∞/summationdisplay\nn=1te(h)(n)\nme(h)(n)R(me(h)(n)) (24)\nThe combinatorial coefficients te(h)(n) defining the to-\ntal amplitude for nonequivalent orbits of mass me(h)(n)\nare computed in Appendix B. By definition, we set\nte(h)(0) =q. For comparison, we have solved numeri-\ncally the magnetization from Eq. (18) for different values\nofqand extracted the first harmonics A1from Fourier\nanalysis. In Fig. 10 the amplitude of the first harmon-\nicsA1is plotted versus b/tforq= 0.6,0.8,0.98 and 1\n(symbols). Since qdepends on bthrough the Chambers\nformula, these plots should merely be regarded as the\ntemperature dependence of A1at a given magnetic field\nvalue. The data are compared to the predictions of the\nLK theory(solid lines) given by Eqs. (23), (24) and (B5),\nform∗\ne= 1 and m∗\nh= 5/2. The deviations with the LK\nformula are negligible in all the b/trange explored, what-\never the qvalue. In particular, the contributions of the\nhigher mass orbits which become important in the large\nfield range are still well described by the LK formula.\nVI. SUMMARY AND CONCLUSION\nThe field-dependent chemical potential oscillations in\nFScomposedoftwocompensatedelectronandholeorbits\nis strongly damped when compared to the case of a FS\nwith only electronic orbits7,8,9,10. It is even suppressed\nin the case where the effective mass of electron (m∗\ne) and\nhole (m∗\nh) band are the same (assuming the relaxation\ntimest∗\ne(h)are either identical or negligibly small). In\naddition, the LK formula accounts for the field and tem-\nperature dependence of the first harmonic’s dHvA oscil-\nlationsamplitude in allcases. Asforthe amplitude ofthe\nsecond harmonics, it is accounted for by the LK formula,\nFIG. 10: (color online) First harmonics amplitude A1for dif-\nferent values of q; ∆ = 1, m∗\ne= 1 and m∗\nh= 5/2. The filled\ncircle symbols represent the numerical analysis from the fr ee\nenergy expression (18), and the lines represent the LK for-\nmula (23, 24), with elements te(h)(n) of Eq.(B10) computed\nup ton= 10 from Eq. (B5).\nOe\nSh Se\nFIG. 11: Example of orbit contributing to first harmonics F0.\nThe effective mass here is me(1) = 2m∗\ne+m∗\nh. This orbit can\nberepresentedbytheoperator ˆQeˆPˆPorˆPˆPˆQe(seeAppendix\nB for details). The point Oeon surface Seis crossed twice by\nthe trajectory.\nprovided m∗\neand m∗\nhare not strongly different. Besides,\nas MB develops, the chemical potential oscillations are\nfurther damped and the previous conclusion on the LK\nvaliditystill holdsforthe amplitude ofthe firstharmonic.\nIn this case, as well as in the general case of compensated\nelectron-hole orbits, the contributions of an infinite num-\nber of orbits to the main frequency become relevant at\nhigh fields, due to the existence of closed trajectories in\nthe FBZ which have zero frequency and make the ex-\npression of the LK amplitude rather complex. Within\nthe FS topology that we have considered here, we have\ndemonstrated that it can nevertheless be computed ex-\nactly. We expect these additional”zerofrequency”orbits\nto be reminiscent in other systems as well.9\nFinally, it should be noticed that the Landau energy\nspectrumof1Dand2Dperiodicnetworksofelectron-hole\ncompensatedorbits(seeFigs. 1(b)and(a), respectively),\nis likely a set of Landau bands rather than discrete Lan-\ndau levels as observed in Figs. 6 and 7. This feature\ncould induce a MB-induced modulation of the density\nof states, as observed in non-compensated networks,4,5,6\nand hence frequency combinations in dHvA oscillatory\nspectra. In that respect, the reported method for solving\nEq. 14 can be used for the study of such orbits networks.\nAPPENDIX A\nFrom Eq. (20), we can define the periodic function\nG(x) =∞/summationdisplay\nn=1(−q)n\nπn[R(nm∗\nh)−R(nm∗\ne)]sin(2πnF0x) (A1)\nwherex= 1/b. The chemical potential Eq. (20) be-\ncomesµ=EF+bG(x)/(m∗\ne+m∗\nh). Replacing µin\n(19) by the previous expression, we compute the oscil-\nlating part of the magnetization Mosc=−∂∆Feff/∂b=\nx2∂∆Feff/∂x:\nMosc≈ −CeCh\n(Ce+Ch)G(x)G′(x) (A2)\n+1\n2∞/summationdisplay\nn=1(−q)n\nπ2n2/bracketleftbigg\nCeR(nm∗\ne)∂\n∂xℜexp(2iπnµ\nCex)\n+ChR(nm∗\nh)∂\n∂xℜexp(2iπnµ−∆\nChx)/bracketrightbigg\nWe also define the Fourier coefficients Be(h)(n,m) as\ne2iπnwe(h)G(x)=+∞/summationdisplay\nm=−∞Be(h)(n,m)e2iπmF 0x(A3)\nwherewe(h)=Ch(e)/(Ce+Ch). Then, we perform the\nderivatives in expression (A2) by noticing for example\nthat\n∂\n∂xℜexp(2iπnµ\nCex) =ℜ/bracketleftBigg+∞/summationdisplay\nm=−∞Be(n,m)\n×2iπ(m+n)F0exp(2iπ(m+n)F0x)] (A4)\nand\n∂\n∂xℜexp(2iπnµ−∆\nChx) =ℜ/bracketleftBigg+∞/summationdisplay\nm=−∞Bh(n,m)\n×2iπ(m−n)F0exp(2iπ(m−n)F0x)] (A5)For extracting the first amplitude, we select the\nintegers such as m+n=±1 in (A4) and m−\nn=±1 in (A5). In the product G(x)G′(x) of ex-\npression (A2), the terms can be rearranged noticing\nthat cos(2 πmF0x)sin(2πnF0x) = [sin(2 π(n+m)F0x) +\nsin(2π(n−m)F0x)]/2 withm,n≥1. For the first har-\nmonic, only terms n−m=±1 contribute. A further\napproximation is to keep the first term of the series (A1),\ni.e.G(x)≈ −q(R(m∗\nh)−R(m∗\ne))sin(2πF0x)/π, in (A3),\nso that the coefficients Be(h)(n,m) can be computed ex-\nactly, from the explicit relation\nexp/bracketleftbig\n2inwe(h)q(R(m∗\ne)−R(m∗\nh))sin(2πF0x)/bracketrightbig\n=\n∞/summationdisplay\nm=−∞Jm(nαe(h))exp(2iπmF0x) (A6)\nwhereJmis the Besselfunction oforder mandαe(h)=\n2we(h)q(R(m∗\ne)−R(m∗\nh)). Inthiscaseitiseasytoidentify\nBe(h)(n,m) =Jm(nαe(h)). After some algebra, we find\nthat the amplitude A1for the first harmonic F0is given\nby the expression, in the large t/blimit\nπA1\nF0≈∞/summationdisplay\nn=12(−q)n\nn/braceleftbigg\n(−1)nCeR(nm∗\ne)Jn(nαe)\nnαe\n−ChR(nm∗\nh)Jn(nαh)\nnαh/bracerightbigg\n+CeCh\nCe+Ch×\n∞/summationdisplay\nn=11\nn[R(nm∗\nh)−R(nm∗\ne)]× (A7)\n/braceleftbig\n[R((n−1)m∗\nh)−R((n−1)m∗\ne)]q2n−1\n−[R((n+1)m∗\nh)−R((n+1)m∗\ne)]q2n+1/bracerightbig\nAPPENDIX B\nInthis appendix, wecomputethe amplitude ofthefirst\nhamonics in the LK theory for any given reflection am-\nplitudeq. In the lowest approximation, the contribution\nto the amplitude is given by one single orbit around the\nelectron or hole band (see Fig. 5). Each contributes with\na mass equal to m∗\ne= 1/Ceandm∗\nh= 1/Chrespectively\nand a reflection amplitude qat the junction point. At\nfinite temperature, we add a reduction factor R(m∗\ne) or\nR(m∗\nh) so that we obtain (with a F0factor overall) Eq.\n(22).\nHowever, there are other contributions to the F0har-\nmonics, coming from more complex orbits. For example\nin Fig. 11, from a starting point Oenear the junction on\nthe Fermi surface Se, we can reach the junction point\nand go through to the other band Sh, then complete\nan entire orbit and come back through the same junc-\ntion to the band Se, and finally perform 2 orbits around\nSeuntil we reach Oeagain. The frequency will be pro-\nportional to 2 Se−Sh. For a compensated system, this10\nfrequency is just F0sinceSe=Sh. The total mass is\nhowever proportional to the derivative of 2 Se−Shwith\nrespect to the energy, which is 2 m∗\ne+m∗\nh(the derivative\nofSh= 2πm∗\nh(∆−E) is indeed negative). Therefore we\nshould take into account all possible trajectories. This\ncan be done exactly by summing up all the contributions\nfor a given mass me(n) = (n+1)m∗\ne+nm∗\nh(n≥0). As\nbefore we start from a point Oe(Oh) on the surface Se\n(Sh) and turn until we reach the junction. Here we note\nˆQe(ˆQh) the reflection operator which keep the quasipar-\nticleonthesurface Sewithamplitude q(Shrespectively),\nandˆPthe operator which take the quasiparticle through\nthe junction (with amplitude ip). Their electron-hole\nrepresentations are given by the matrices:\nˆQe=ˆQh=/parenleftbigg\nq0\n0q/parenrightbigg\n,ˆP=/parenleftbigg\n0ip\nip0/parenrightbigg\n(B1)\nIn the simplest case, the orbit is described only by the\noperator or graph ˆQe, whereas in the second example,\nthe orbit is described by the graph ˆPˆPˆQe. The orbit\nˆQeˆPˆPis equivalent and has the same mass 2 m∗\ne+m∗\nh.\nIndeed, the orbit goes trough the same point Oetwice,\nand therefore there are 2 different ways of writing the\ngraph above, depending on which branch of the orbit we\nplace the point Oe. Equivalent graphs are described by\na cyclic permutation of their operator elements. We also\nobserve that multiple orbits ( ˆPˆP)nhave zero frequency\nand mass n(m∗\ne+m∗\nh). We will note in the following te(n)\n(th(n)) the sums of the amplitudes of all the trajectories\nwith the same mass me(n) (mh(n) =nm∗\ne+(n+ 1)m∗\nh\nrespectively), which are constructed starting from an ar-\nbitrary point Oe(orOhrespectively). Te(Th) will be the\ntotal amplitude for the frequency F0(−F0) starting from\npointOe(Oh) on the surface Se(Sh), see Eqs. (23,24)\nThis computed in the same way as Teby replacing\nthe label ein all the quantities by h. For equal masses\nm∗\ne=m∗\nh, we have clearly Te=Th. For example, the\norbits and amplitudes contributing to the mass me(1) =\n2m∗\ne+m∗\nhare\nˆPˆPˆQe∼ˆQeˆPˆP→(−p2)q,\nThis gives te(1) = (−p2)q. The 2 operators above cor-\nrespond to the same graph by a cyclic permutation of\ntheir elements ˆPˆPandˆQeand the sign ∼is understood\nas an equivalence between identical graphs. In the next\ncase, for the mass me(2) = 3m∗\ne+2m∗\nhwe have the pos-\nsibilities\nˆPˆQhˆPˆQ2\ne∼ˆQeˆPˆQhˆPˆQe∼ˆQ2\neˆPˆQhˆP→(−p2)q3,\nˆPˆPˆPˆPˆQe∼ˆPˆPˆQeˆPˆP∼ˆQeˆPˆPˆPˆP→(−p2)2q\nwe obtain te(2) = (−p2q3+p4q). In the general case,\nwe can construct all the possible orbits that contribute to\nthe mass me(n) by representing them as multiple prod-\nucts of elementary operatorsˆQn1\neˆPˆQm1\nhˆPˆQn2\neˆPˆQm2\nhˆP..ˆQnkeˆPˆQmk\nhˆP(B2)\nwhereni≥0 andmi≥0 are positive integers. The\nconstraints imposed by the mass on theses integers and\nkare\nk/summationdisplay\ni=1ni=n−k+1,k/summationdisplay\ni=1mi=n−k,1≤k≤n(B3)\nThere are also kcyclic permutations of the graph (B2)\nabove, corresponding to moving elements ˆQnieˆPˆQmi\nhˆPto\nthe right or to the left. For example, an equivalent graph\nwould be\nˆQn2\neˆPˆQm2\nhˆP..ˆQnkeˆPˆQmk\nhˆPˆQn1\neˆPˆQm1\nhˆP(B4)\nThe total amplitude te(n) for a given ncan be written\nas\nte(n) =n/summationdisplay\nk=1cn,k(−p2)kq2(n−k)+1(B5)\nwhere the coefficients cn,kenumerate the number of\nall nonequivalent graphs having the same mass, up to\ncyclic permutations. It corresponds to all possible sets of\nniandmirepresenting graphs (B2) with the constraint\n(B3), divided by the number of cyclic permutations k.\nThe total amplitude te(n) can be written more precisely\nas\nte(n) =n/summationdisplay\nk=1/summationdisplay\nni,mi≥0qPk\ni=1ni+Pk\ni=1mi(−p2)k×\n1\nkδPk\ni=1ni,n−k+1δPk\ni=1mi,n−k (B6)\nThe Kronecker functions δk,0can be represented by\nintegrals δk,0=/integraltext1\n0exp(2iπkx)dx=/contintegraltext\ndz/(2iπz)×zk,\nand the last expression becomes\nte(n) =n/summationdisplay\nk=1(−p2)k\nk/contintegraldisplaydz\n2iπz/contintegraldisplaydz′\n2iπz′(B7)\n/parenleftBiggk/productdisplay\ni=1∞/summationdisplay\nni=0qnizni/parenrightBigg\nzk−n−1/parenleftBiggk/productdisplay\ni=1∞/summationdisplay\nmi=0qmiz′mi/parenrightBigg\nz′k−n\nSumming up the series, we obtain\nte(n) =n/summationdisplay\nk=1(−p2)k\nk/contintegraldisplaydz\n2iπzk−n−2\n(1−qz)k/contintegraldisplaydz′\n2iπz′k−n−1\n(1−qz′)k\nApplying the residue theorem, it is easy to show that11\n/contintegraldisplaydz\n2iπzk−n−2\n(1−qz)k=/parenleftbign\nk−1/parenrightbig\nqn−k+1(B8)\nand\n/contintegraldisplaydz′\n2iπz′k−n−1\n(1−qz′)k=/parenleftbign−1\nk−1/parenrightbig\nqn−k(B9)\nwhere (n\nk) =n!/k!(n−k)! are the binomial coefficients.\nWe obtain finally\ncn,k=1\nk/parenleftbign\nk−1/parenrightbig/parenleftbign−1\nk−1/parenrightbig\n(B10)\nThe values ofthe coefficients cn,kup ton= 7 aregiven\nin Table 1.TABLE I: Values of the coefficients cn,kwhich represent\nthe number of non-equivalent orbits for a given mass ( n+\n1)m∗\ne(h)+nm∗\nh(e)with 2kbreakdowns, 1 ≤k≤n.\nn\\k1 2 3 4 5 6 7 Mass\n11 2m∗\ne(h)+m∗\nh(e)\n21 1 3m∗\ne(h)+2m∗\nh(e)\n31 3 1 4m∗\ne(h)+3m∗\nh(e)\n41 6 6 1 5m∗\ne(h)+4m∗\nh(e)\n51 10 20 10 1 6m∗\ne(h)+5m∗\nh(e)\n61 15 50 50 15 1 7m∗\ne(h)+6m∗\nh(e)\n71 21 105 175 105 21 1 8m∗\ne(h)+7m∗\nh(e)\nWe also observe by symmetry that th(n) is equal to\nte(n) since this geometric coefficient does not depend ex-\nplicitely on masses.\n∗Electronic address: fortin@lpt1.u-strasbg.fr\n†Electronic address: audouard@lncmp.org\n1A. 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Khomenko, Synth.\nMetals42, 1907 (1991).\n21D. Shoenberg, Magnetic Oscillations in Metals (Cam-\nbridge University Press, Cambridge, England, 1984).\n22A. Kosevich and I. Lifschitz, Sov. Phys. JETP 2, 646\n(1956).\n23The twoelectronic orbits considered inFig. 1havedifferent\nfrequencies, namely f 0/ f1= 2 / 3. In the case where f 0=\nf1, the chemical potential oscillations are even larger than\nreported in Fig. 1 since the field-dependent contributions\nof the two orbits are in phase.\n24V. M. Gvozdikov and M. Taut, Phys. Rev. B 75, 155436\n(2007).\n25R. Chambers, Proc. Phys. Soc. 65, 458 (1952).\n26A. Slutskin and A. Kadigrobov, Sov. Phys. Solid State 9,\n138 (1967)."    },    {        "title": "2209.01893v1.Generation_and_routing_of_nanoscale_droplet_solitons_without_compensation_of_magnetic_damping.pdf",        "content": "Generation and routing of nanoscale droplet solitons\nwithout compensation of magnetic damping\nAndrei I. Nikitchenko and Nikolay A. Pertsev\u0003\nIoffe Institute, 194021 St. Petersburg, Russia\n*pertsev.domain@mail.ioffe.ru\nMagnetic droplet soliton is a localized dynamic spin state which can serve as a nanoscale informa-\ntion carrier and nonlinear oscillator. The present opinion is that the formation of droplet solitons\nrequires the compensation of magnetic damping by a torque created by a spin-polarized electric cur-\nrent or pure spin current. Here we demonstrate theoretically that nanoscale droplet solitons can be\ngenerated and routed in ferromagnetic nanostructures with voltage-controlled magnetic anisotropy\nin the presence of uncompensated magnetic damping. Performing micromagnetic simulations for the\nMgO/Fe/MgO trilayer with almost perpendicular-to-plane magnetization, we reveal the formation\nof the droplet soliton under a nanoscale gate electrode subjected to a sub-nanosecond voltage pulse.\nThe soliton lives up to 50 ns at room temperature and can propagate over micrometer distances in a\nferromagnetic waveguide due to nonzero gradient of the demagnetizing field. Furthermore, we show\nthat an electrical routing of the soliton to different outputs of a spintronic device can be realized\nwith the aid of an additional semiconducting nanostripe electrode creating controllable gradient of\nthe perpendicular magnetic anisotropy.\nI. INTRODUCTION\nMagnetic droplet soliton is a type of strongly self-\nlocalized mode of magnetization oscillations. Early the-\noretical studies predicted that a conservative version of\nsuch a soliton could form in an ideal ferromagnet having\nno magnetic damping [1]. Later, dissipative droplet soli-\ntons were described theoretically [2, 3] and demonstrated\nexperimentally in spin transfer nanocontact oscillators\n(STNOs) [4, 5] and nanoconstriction-based spin Hall\ndevices [6, 7] with perpendicular magnetic anisotropy\n(PMA). In both nanostructures, the formation of soli-\ntons is due to the compensation of magnetic damping,\nwhich is provided either by the spin-transfer torque gen-\nerated by a spin-polarized electric current [4, 5] or by the\nspin-orbit torque created by a pure spin current injected\ninto the ferromagnet by an adjacent heavy metal [6, 7].\nIn STNOs, the soliton forms under the nanocontact\nowing to local magnetization reversal induced by the\nspin-transfer torque. The droplet size is governed by\nthe nanocontact size and can exceed the latter signif-\nicantly [8] due to the current-induced Zhang-Li torque\nacting on the droplet boundary [9, 10]. The dynamic na-\nture of droplet solitons manifests itself in a large-angle\nmagnetization precession at the droplet boundary. The\nprecession frequency is well below the FMR frequency\nand practically independent of the driving spin-polarized\ncurrent [2, 4]. However, it can be tuned by an applied\nelectric field in STNOs with the free layer possessing\nelectric-field-dependent PMA created by an adjacent di-\nelectric layer [11].\nIn this paper, we show theoretically that droplet soli-\ntons can be created and routed in ferromagnetic nanolay-\nerswithoutcompensationofmagneticdamping. Thisop-\nportunity appears in ferromagnet-dielectric heterostruc-\ntures having strong interfacial PMA, which can be re-\nduced significantly by an electric field created in thedielectric nanolayer. Such a voltage-controlled mag-\nnetic anisotropy (VCMA) represents an efficient tool\nfor the excitation of magnetic dynamics in ferromag-\nnetic nanolayers, including the precessional magneti-\nzation switching [12, 13], spin reorientation transition\n(SRT) [14], coherent magnetization precession [15–17],\nand spin waves [18–20]. Performing micromagnetic sim-\nulations for a perpendicularly magnetized MgO/Fe/MgO\ntrilayer subjected to a weak in-plane magnetic field, we\nreveal the formation of a droplet soliton induced by a\nsub-nanosecond voltage pulse locally applied to the MgO\nnanolayer via a gate electrode. The soliton forms under\nthe gate electrode and lives up to 100 ns at low temper-\natures, experiencing size oscillations with the period of\n0.1-0.5 ns. Furthermore, we demonstrate the propaga-\ntion of the generated nanoscale soliton over micrometer\ndistances from the nucleation region, which is achieved\nin a ferromagnetic waveguide owing to the demagnetizing\nfield accelerating the soliton. Finally, an electrical rout-\ning of the droplet soliton is realized in a nanostructure\nwith a controllable PMA gradient created by a semicon-\nducting nanostripe electrode. Since only electric fields\nare needed to generate and route droplet solitons, the\nproposed technique is distinguished by a low energy con-\nsumption, which is advantageous for device applications.\nII. RESULTS AND DISCUSSION\nA. Micromagnetic modeling\nWe model the dynamics of the magnetization M(r;t)\nin a (001)-oriented Fe nanolayer grown on MgO(001) and\ncapped with an ultrathin MgO overlayer (Fig. 1). The\nchosen Fe thickness tF= 0:75nm is smaller than the crit-\nical thickness tSRT\u00190:9nm (see Appendix for the calcu-\nlated critical thickness), below which the perpendicular-arXiv:2209.01893v1  [cond-mat.mtrl-sci]  5 Sep 20222\nFigure 1. Cylindrical MgO/Fe/MgO trilayer with a cir-\ncular gate electrode connected to a voltage source. The Fe\nnanolayer with a perpendicular magnetic anisotropy is sub-\njected to a weak in-plane magnetic field H, which deflects\nthe magnetization Mfrom the perpendicular-to-plane orien-\ntation.\nto-plane (PP) orientation of Mbecomes energetically fa-\nvorable in the MgO/Fe/MgO structure [21]. The fer-romagnetic layer is modeled by a two-dimensional en-\nsemble ofNnanoscale computational cells with the sizes\nlx=ly= 1:5nm andlz=tFsmaller than the exchange\nlength\u0015ex\u00193:3nm of Fe [22]. Regarding the saturation\nmagnetization Msas a constant quantity at a given tem-\nperature, we calculate the temporal evolution of M(r;t)\nby numerically solving a system of the Landau-Lifshitz-\nGilbert (LLG) equations for the unit vectors m(rn;t) =\nM(rn;t)=Msdefining the magnetization directions in the\ncomputational cells situated at the points rn(n=1, 2, 3,\n...,N). The effective field He\u000binvolved in the LLG equa-\ntioniswrittenas He\u000b=H+Hex+Hdip+Han, where His\nthe external magnetic field, HexandHdipare the contri-\nbutions of the exchange and dipolar interactions between\nspins in Fe, and Hanallows for the magnetocrystalline,\nmagnetoelastic, and interfacial anisotropies existing in\nthe MgO/Fe/MgO structure. The exchange and dipolar\ncontributions to the effective field He\u000bare calculated as\ndescribed in our preceding paper [20], and we use the\nrelation Han=\u0000(\u00160Ms)\u00001@Fan=@mto determine the\nanisotropy field Han(\u00160is the magnetic permeability of\nfree space). The effective volumetric energy density Fan\nof the magnetic anisotropy can be approximated as [23]\nFan\u0019K1(m2\nxm2\ny+m2\nxm2\nz+m2\nym2\nz) +K2m2\nxm2\nym2\nz+B1(uxxm2\nx+uyym2\ny)\n\u0000B1\u0014B1\n6c11+c12\nc11(uxx+uyy)\u0015\nm2\nz+Ksk+K0\nsk\ntFm2\nxm2\ny+Ks?+K0\ns?\ntF(m2\nx+m2\ny)m2\nz+Ks+K0\ns\ntFm2\nz;(1)\nwhereK1andK2are the coefficients of the fourth- and\nsixth-order terms defining the cubic magnetocrystalline\nanisotropy of bulk Fe at constant lattice strains u,B1\nis the magnetoelastic constant, c11andc12denote the\nelastic stiffnesses at fixed magnetization, uxxanduyy\nare the substrate-induced in-plane (IP) strains of the Fe\nnanolayer, while Ks,Ksk,Ks?andK0\ns,K0\nsk,K0\ns?are the\nparameters characterizing the magnetic anisotropy cre-\nated by the bottom and top Fe jMgO interfaces, respec-\ntively. The factor 1=tFin the last three terms reflects\nthe introduction of only one computational cell in the\nthickness direction z, which is justified by the condition\ntF<\u0015ex=2.\nIn our simulations, the numerical integration of the\nLLG equation is carried out using the projective Runge-\nKutta algorithm with the time step of 10 fs, which is\nmuch smaller than the duration \u001cV>0:1ns of the rect-\nangular voltage pulses applied to the gate electrode. To\nmake possible a nonparametric excitation of the mag-\nnetic dynamics by VCMA, we introduce an IP magnetic\nfieldHjj[110]creating an oblique orientation of the equi-\nlibrium magnetization [17].B. Electrical generation of magnetic solitons\nWe first consider the heterostructure of a circular\nshape, which includes a nanoscale gate electrode on top\nof the MgO overlayer (Fig. 1). The electric field Ez\ncreated in MgO by a voltage V(t)applied to the gate\nelectrode changes the specific energy associated with the\ntop FejMgO interface [24, 25]. Therefore, the coefficient\nK0\nsin Eq. (1) should be regarded as a voltage-dependent\nquantity for the computational cells beneath the gate.\nAsK0\ns?andK0\nskare much smaller than K0\ns[23], possible\nvoltage dependences of these parameters can be ignored.\nSince the dependence K0\ns(Ez)is practically linear at the\nfield intensities up to about 2 V nm\u00001[25], the voltage\ndependenceof K0\nscanbewrittenas K0\ns=K0\ns+ksV=tMgO,\nwhereK0\nsis the value of the anisotropy parameter K0\nsat\nEz= 0,ksis the electric-field sensitivity of K0\ns[26], and\ntMgOis the thickness of the MgO overlayer.\nIn accordance with the available experimental\ndata [21], the saturation magnetization of the 0.75-nm-\nthick Fe film is taken to be Ms= 1:71\u0002106A m\u00001.\nThe lattice strains induced in the Fe layer by a thick\nMgO substrate are set equal to uxx=uyy= 3:9% [23].\nIn the numerical calculations, we also use the exchange\nconstantAex= 20pJ m\u00001[22], Gilbert damping pa-\nrameter\u000b= 0:0025[27], anisotropy coefficients K1=3\n-0.5 0 0.5 1.0mz\n0 1 2 3 40.00.10.20.30.4\nTimet(ns)Soliton momentμ z(106μB)\nFigure 2. Time dependence of the magnetic moment \u0016zcar-\nried by electrically generated droplet soliton. Voltage pulse\nwith the height V= 4V and duration \u001cV= 0:3ns is ap-\nplied to the gate electrode with the radius RG= 50nm, and\nthe magnetic field strength equals 300 Oe. The inset shows\nthe spatial distribution of the magnetization direction cosine\nmz(x;y)under the gate electrode at t= 0:28ns.\n48kJ m\u00003[22] andK2= 15kJ m\u00003[28], PMA param-\netersK0\ns=\u00009\u000210\u00004J m\u00002[21] andKsk=Ks?=\nK0\nsk=K0\ns?=\u00004:5\u000210\u00005J m\u00002[23], VCMA coeffi-\ncientks= 100fJ V\u00001m\u00001[29], magnetoelastic constant\nB1=\u00003:3\u0002106J m\u00003[28], and elastic stiffnesses c11=\n2:42\u00021011N m\u00002andc12= 1:465\u00021011N m\u00002[30].\nThe MgO thickness is set equal to the value of 2 nm, at\nwhich the influence of the voltage-induced tunnel current\nthrough MgO on the magnetization dynamics can be ne-\nglected. The diameter of the MgO/Fe/MgO trilayer is\ntaken to be 450 nm, while the radius RGof the gate\nelectrode varies from 30 to 120 nm.\nThe simulations show that the initial magnetic state\nof the considered 0.75-nm-thick Fe disk is practically ho-\nmogeneous. Owing to strong PMA of such a nanolayer,\nthe deviation of the magnetization vector Mfrom the PP\norientation, which is induced by the external IP magnetic\nfieldH, appears to be small even at the strongest field\nH= 400Oe used in our simulations. At the chosen\nfield orientation along the [110]crystallographic direc-\ntion, which represents the easy axis of the nanolayer’s IP\nanisotropy due to the condition K1+(Ksk+K0\nsk)=tF<0,\nthe mean value of the magnetization polar angle \u0012is\nfound to be less than 8\u000e(see Appendix).\nIn the study of electrically induced magnetization dy-\nnamics, we consider the MgO/Fe/MgO heterostructures\nsubjected to rectangular voltage pulses with the dura-\ntion\u001cVranging from 0.1 to 1 ns. The pulse ampli-\ntude is set to -4 V, at which the electric field Ezin\nthe 2-nm-thick MgO nanolayer is below its breakdown\nfieldEb\u00192:4V nm\u00001[31].The micromagnetic simula-tions demonstrate that a voltage pulse providing local\nreduction of PMA may induce precessional magnetiza-\ntion switching by about 180\u000einside the Fe region be-\nneath the gate electrode. The switched region occupies\nan areaSsmaller than the gate area SGand may have\nnearly a disk shape (see the inset in Fig. 2). The switch-\ning creates a change \u0016in the magnetic moment of the\nFe film, which has a dominant out-of-plane component\n\u0016z(t) =\u0000MstFZ\nSGdxdy [mz(r;t)\u0000mz(r;t= 0)]. Since\n\u0016z(t=\u001cV)\u00182MstFS, the switched region represents a\nmagnetic droplet soliton. Figure 2 shows a representative\ntime dependence of the droplet moment \u0016z(t). We see\nthat\u0016zoscillates with a gradually increasing frequency\nf\u0016(t)and decreasing amplitude and becomes negligible\nafter a few nanoseconds. The analysis of the simulation\ndata reveals that such a behavior is mostly due to the\noscillations of the droplet area S. Importantly, the soli-\nton does not experience any significant drift towards the\nboundary of the Fe disk during the whole period of its\nexistence.\nThe soliton lifetime \u001cSdepends on the pulse dura-\ntion\u001cV, gate radius RG, and strength of external mag-\nnetic field H. By analyzing the results of simulations\nperformed at different values of \u001cV, we find that the op-\ntimal pulse duration \u001c\u0003\nV, which maximizes the soliton\nlifetime, corresponds to the minimum of the quantityZ\nSGdxdy\u0012\nHe\u000b\nz\u00002ksV\n\u00160MstMgOtFmz\u0013\n. At a fixed magnetic\nfield,theoptimalduration \u001c\u0003\nVfirstgrowswiththeincreas-\ninggateradius RG(seeAppendix). However, abovesome\nthreshold radius Rthtwo or three solitons form under\nthe gate instead of one. In what follows we present only\nthe results of simulations performed for the heterostruc-\ntures involving gates with RG<R thsubjected to voltage\npulses of the optimal duration \u001c\u0003\nV(RG;H).\nVariations of the soliton lifetime \u001cSwith the gate ra-\ndius and the field strength are presented in Fig. 3(a). At\na fixed field strength, \u001cSgrows with increasing gate size\nup to some optimal radius R\u0003\nG(H), at which\u001cSreaches\nmaximal value. Remarkably, the soliton lifetime attained\natH= 200Oe exceeds 100 ns when the gate radius is\nclose toR\u0003\nG(H= 200Oe)\u0019120nm. As may be ex-\npected, the growth of \u001cSatRG<R\u0003\nGcorrelates with the\ndependence of the maximal value of the soliton moment\n\u0016z(t)on the gate radius [see Fig. 3(b)]. The decrease of\n\u001cSatRG> R\u0003\nGmay be attributed to the enhancement\nof spin-wave radiation by larger droplets, which takes the\nmagnetic moment \u0016z(t)away from the soliton (see Ap-\npendix).\nTo evaluate the influence of thermal fluctuations on\nthe soliton lifetime, we carry out additional simulations\nwith the account of a stochastic Gaussian noise. In these\nsimulations, the effective field He\u000binvolved in the LLG\nequation includes a thermal random field Hthin the form\nemployedbytheMuMax3software[32]. Theresultsshow\nthat the introduction of Hthcorresponding to 300 K re-\nduces\u001cSapproximately by a factor of two. Hence the4\n(a)H=200 Oe\nH=300 Oe\nH=400 Oe\n40 60 80 100 120020406080100120\nGate electrode radius RG(nm)Soliton lifetime τS(ns)\n(b)\nH=200 Oe\nH=400 OeH=300 Oe\n40 60 80 100 12001234567\nGate electrode radius RG(nm)Maximal soliton moment μzmax(106μB)\nFigure 3. Soliton lifetime \u001cS(a) and maximal value of the\nsoliton magnetic moment \u0016z(b) plotted as a function of the\ngate electrode radius RG. The strengths Hof the applied\nmagnetic field are indicated on the plots.\nsoliton can live up to 50 ns at the room temperature.\nC. Propagation and routing of droplet solitons\nNext, it is important to determine how far the gener-\nated droplet soliton can propagate along a ferromagnetic\nwaveguide. To address this question, we carry out mi-\ncromagnetic simulations for the MgO/Fe/MgO trilayer\nwith a rectangular shape, representative IP dimensions\nLx= 1:5\u0016m andLy= 300nm, and a circular gate elec-\ntrodeplacednearthewaveguidebeginning[seeFig.4(a)].\nV а \nу н \nMgO/Fe/MgO \n1 2 3 4 \n....... cll ......... ..... �··· .................... � •••••• ••• �-•••••••• ,,..... •••••••••• • ► • • .............. 1 .,\nх \nH=280 Oe\nH=240 OeH=320 Oe (b)\n1234\n0 10 20 300.20.40.60.81.01.21.4\nTime t(ns)Droplet coordinate x(μm)Figure 4. Propagation of electrically generated droplet soli-\nton in the MgO/Fe/MgO waveguide. Panel (a) shows a rep-\nresentative trajectory of the soliton formed at H= 320Oe\nunder circular gate electrode placed near the waveguide edge.\nYellow regions depict the droplet area at four different po-\nsitions. Panel (b) presents the droplet position xin the Fe\nmicrostripe as a function of time at different field strengths H\nindicated on the plot. The radius RGof gate electrode equals\n70 nm. The curves break when the droplet magnetic moment\ngoes to zero.\nIn such a heterostructure, the generated soliton moves\naway from the nucleation region beneath the gate elec-\ntrode owing to the existence of a gradient @Hdip=@xof\nthe demagnetizing field Hdip[3], which is nonuniform in\nthe rectangular Fe microstripe. As a result, the soliton\npropagates along the waveguide, experiencing small devi-\nations from its central line [Fig. 4 (a)], which are caused\nby initial misalignment of the droplet velocity and restor-\ning forces created by the microstripe edges.\nFigure 4 (b) shows the droplet position in the waveg-\nuide as a function of time at different external magnetic\nfields. Itisseenthatthepropagationdistancegrowswith\nincreasing field strength H, exceeding one micrometer at\nH > 250Oe. Thisbehavioriscausedbythefield-induced\nincrease of the soliton lifetime in the Fe microstripe,\nwhich overcompensates the decrease in the average soli-\nton velocity ranging from 49.3 m s\u00001atH= 240Oe to\n38.5 m s\u00001atH= 320Oe. As the departure droplet size\ngrows with increasing field strength, we arrive at the con-5\nclusion that small solitons move faster than large ones.\nFinally, we describe an efficient method of electrical\nrouting of the droplet solitons to different outputs of a\nspintronic device. Such a routing can be realized in the\nMgO/Fe/MgO-based structure shown in Fig. 5, where\nadditional semiconducting nanostripe electrode is placed\non the upper MgO nanolayer near the circular gate elec-\ntrode. The application of dc voltages U=2and\u0000U=2\nto the ends of semiconducting nanostripe gives rise to an\nelectric current flowing along the electrode, which creates\na linear variation of the voltage applied to the underly-\ning MgO area. As a result, a voltage-controlled gradient\nSiU/2\n-U/2MgO/Fe/MgO\n0.0 V\n1.2 V0.8 V0.4 V-0.4 V-0.8 V-1.2 V\nMTJsV\nFigure 5. Routing of droplet solitons in the MgO/Fe/MgO-\nbased structure comprising circular gate electrode and semi-\nconducting nanostripe electrode placed on the upper MgO\nnanolayer. Dashed lines show the trajectories of the droplet\ncenter predicted by micromagnetic simulations at H=\n300Oe. Magnitude Uof dc voltages U=2and\u0000U=2applied\nto the ends of semiconducting nanostripe is indicated near\nthe corresponding path line. Gray arcs depict ferromagnetic\nelectrodes deposited on the upper MgO nanolayer at the dis-\ntanceD= 0:5\u0016m from the nanostripe center. Together with\nthe extended Fe interlayer, these electrodes form magnetic\ntunnel junctions enabling electrical detection of the soliton\ntrajectory.\nof PMA appears in the Fe region beneath the semicon-\nducting electrode. The simulations reveal that the PMA\ngradient strongly affects the soliton trajectory in the Fe\nfilm owing to additional acceleration of the droplet in\nthe direction antiparallel to the gradient vector. Fig-\nure 5 shows the trajectories of the soliton generated by\nthe gate electrode with the radius RG= 70nm and\nrouted by the Si nanostripe with the length \u000ey= 200nm,width\u000ex= 50nm, and resistivity \u001a= 83m\nm [33].\nWhen voltages U=2and\u0000U=2are applied to the nanos-\ntripe ends, the trajectory of the droplet center changes\nbeneath the nanostripe, deviating from approximately\nstraight path forming at U= 0. The direction of this\ndeviation depends on the sign of U, and the deviation\nmagnitude can be characterized by an angle \fbetween\nthe straight soliton trajectory and a line connecting the\nnanostripe center and the droplet position at a fixed dis-\ntanceDfrom this point. Figure 6 presents the volt-\n-4 -2 0 2 4-50050\nVoltage drop on Si U(V)Routing angle β(deg)\nFigure 6. Dependence of the soliton routing angle \fon\nthe magnitude Uof dc voltages U=2and\u0000U=2applied to\nthe ends of Si nanostripe with the length \u000ey= 200nm and\nwidth\u000ex= 50nm. Simulation data (points) are fitted by the\nfunction\f= 52\u000earctan(1:4U=V)(curve).\nage dependence of the routing angle \f(U)determined\natD= 0:5\u0016m. Remarkably, the routing angle reaches\nabout 50\u000eatU= 1V, varying almost linearly up to\nU= 0:8V with the mean rate d\f=dU\u001956\u000eV\u00001. At\nvoltagesU > 1V, the dependence \f(U)becomes non-\nlinear. This feature may be attributed to the arising\nsignificant influence of the sample edges on the soliton\npath, which is evidenced by the curved droplet trajecto-\nries obtained at U=\u00061:2V (see Fig. 5).\nThe demonstrated electrical control of the soliton tra-\njectory makes it possible to transfer the magnetic signal\nto one of several outputs of the device. The signal can be\nread electrically with the aid of a magnetic tunnel junc-\ntion (MTJ) formed by a nanoscale ferromagnetic elec-\ntrode deposited on the upper MgO nanolayer and the\nunderlying region of the Fe interlayer. Indeed, owing to\nthe phenomenon of spin-dependent tunneling, the MTJ\nresistance changes strongly after the magnetization re-\nversal in one of ferromagnetic electrodes [34]. Therefore,\nthe soliton appearance in the Fe region below the per-\npendicularly magnetized top ferromagnetic electrode will6\nmanifest itself in a resistance change, which can be easily\ndetected electrically.\nIII. CONCLUSIONS\nIn summary, we theoretically studied the electrically\ndriven magnetization dynamics in the MgO/Fe/MgO tri-\nlayer with the voltage-controlled magnetic anisotropy.\nThe micromagnetic simulations demonstrated that the\napplication of a sub-nanosecond voltage pulse to the\nnanoscale gate electrode placed on the MgO nanolayer\ngives rise to the formation of the magnetic droplet soliton\ndespite the presence of nonzero magnetic damping. The\nsoliton lifetime, which depends on the gate size and the\nstrength of in-plane external magnetic field, can reach\n50 ns at room temperature and 100 ns in the absence\nof thermal fluctuations. When generated near the edgeof the Fe microstripe, the soliton can propagate over a\ndistance exceeding one micrometer with the mean speed\nabout 40 m s\u00001owing to the existing gradient of the\ndemagnetizing field. By passing a small electric current\ndensity\u0018108A m\u00002along additional Si nanostripe elec-\ntrode, we also achieved an efficient electrical routing of\nthe soliton in the extended Fe interlayer.\nOur theoretical results provide guidelines for the de-\nvelopment of an energy-efficient information-processing\ndevice based on the electrical generation, propagation,\nand routing of magnetic solitons. The device converts\nthe input voltage signal into the magnetic information\ncarrier, which propagates to one of several outputs. The\ndesired output is selected by the voltage applied to the\nrouting electrode and involves the magnetic tunnel junc-\ntion, which provides electrical reading of the output sig-\nnal via the measurement of the junction’s resistance.\n[1] A.Kosevich, B.Ivanov, andA.Kovalev,PhysicsReports\n194, 117 (1990).\n[2] M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev.\nB82, 054432 (2010).\n[3] M. A. Hoefer, M. Sommacal, and T. J. Silva, Phys. Rev.\nB85, 214433 (2012).\n[4] S. Chung, S. M. Mohseni, S. R. Sani, et al., Journal of\nApplied Physics 115, 172612 (2014).\n[5] F. Maci` a, D. Backes, and A. D. Kent, Nat. Nanotechnol.\n9, 992 (2014).\n[6] B. Divinskiy, S. Urazhdin, V. E. Demidov, et al., Phys.\nRev. B 96, 224419 (2017).\n[7] M. Dvornik, A. A. Awad, and J. ˚Akerman, Phys. Rev.\nApplied 9, 014017 (2018).\n[8] S. Chung, Q. T. Le, M. Ahlberg, et al., Phys. Rev. Lett.\n120, 217204 (2018).\n[9] Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004).\n[10] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[11] C. Zheng, M. Dvornik, C. Wang, et al., Phys. Rev. Ap-\nplied14, 054001 (2020).\n[12] Y. Shiota, T. Nozaki, F. Bonell, et al., Nature Mater 11,\n39 (2012).\n[13] S. Kanai, M. Yamanouchi, S. Ikeda, et al., Applied\nPhysics Letters 101, 122403 (2012).\n[14] Y. Shiota, T. Maruyama, T. Nozaki, et al., Applied\nPhysics Express 2, 063001 (2009).\n[15] T. Nozaki, Y. Shiota, S. Miwa, et al., Nature Phys. 8,\n491 (2012).\n[16] J. Zhu, J. A. Katine, G. E. Rowlands, et al., Phys. Rev.\nLett.108, 197203 (2012).\n[17] G. Viaud and N. A. Pertsev, Phys. Rev. B 90, 064429\n(2014).\n[18] R. Verba, V. Tiberkevich, I. Krivorotov, and A. Slavin,\nPhys. Rev. Applied 1, 044006 (2014).\n[19] B. Rana, Y. Fukuma, K. Miura, et al., Applied Physics\nLetters 111, 052404 (2017).\n[20] A. I. Nikitchenko and N. A. Pertsev, Phys. Rev. B 104,\n134422 (2021).\n[21] A. Kozio  l-Rachwa  l, W. Skowro´ nski, T. ´Sle ˛zak,et al.,\nJournal of Applied Physics 114, 224307 (2013).[22] C. A. F. Vaz, J. A. C. Bland, and G. Lauhoff, Reports\non Progress in Physics 71, 056501 (2008).\n[23] N. A. Pertsev, Phys. Rev. B 92, 014416 (2015).\n[24] T. Maruyama, Y. Shiota, T. Nozaki, et al., Nature Nan-\notech. 4, 158 (2009).\n[25] M. K. Niranjan, C.-G. Duan, S. S. Jaswal, and E. Y.\nTsymbal, Applied Physics Letters 96, 222504 (2010).\n[26] N. Pertsev, Sci Rep. 3, 2757 (2013).\n[27] N. Kamiya, D. Oshima, S. Iwata, and T. Kato, Journal\nof the Magnetics Society of Japan 45, 96 (2021).\n[28] M. B. Stearns, “in Magnetic Properties of Metals ·3d,\n4d and 5d Elements, Alloys and Compounds,” Landolt-\nB¨ ornstein, New Series, Group III, Vol. 19a (Springer-\nVerlag, Berlin) (1986).\n[29] T. Nozaki, A. Kozio  l-Rachwa  l, W. Skowro´ nski, et al.,\nPhys. Rev. Applied 5, 044006 (2016).\n[30] J. P. Hirth and J. Lothe, “Theory of dislocations,” New\nYork ; London : McGraw-Hill (1968).\n[31] D. V. Dimitrov, Z. Gao, X. Wang, et al., Applied Physics\nLetters 94, 123110 (2009).\n[32] A. Vansteenkiste, J. Leliaert, M. Dvornik, et al., AIP\nAdvances 4, 107133 (2014).\n[33] M. Sharmin, S. Choudhury, and T. Begum, Dhaka Uni-\nversity Journal of Science 63, 37–41 (2015).\n[34] J. Faure-Vincent, C. Tiusan, E. Jouguelet, et al., Applied\nPhysics Letters 82, 4507 (2003).7\nAppendix\nTo confirm that the Fe film in the MgO/Fe/MgO tri-\nlayerretainsalmostperpendicular-to-planeorientationin\nall performed micromagnetic simulations, we calculated\nthe critical Fe thickness tSRTand the mean value h\u0012i\nof the magnetization polar angle \u0012as a function of the\nstrengthHof applied in-plane magnetic field. The re-\nsults presented in Fig. A1 show that tSRTexceeds the Fe\nthicknesstF= 0:75nm and the mean value h\u0012iis smaller\nthan 8\u000eeven at the highest field H= 400Oe used in the\nsimulations.\ntF\n0 0.1 0.2 0.3 0.4 0.57.57.77.98.18.38.5\n02468\nExternal fieldH(kOe)tSRT(Å)\nAngle〈θ〉 r(deg)\nFigureA1. ThecriticalFethickness tSRTandthemeanvalue\nh\u0012iofthemagnetizationpolarangle \u0012attF= 0:75nm(dashed\nline) plotted as a function of the magnetic field strength H.\nH=200 Oe\nH=300 Oe\nH=400 Oe\n40 60 80 100 1200.20.40.60.81.0\nGate electrode radiusR G(nm)Optimal pulse durationτ V(ns)\nFigure A2. Dependences of optimal pulse duration \u001c\u0003\nVon\ngate radius RGcalculated at different strengths Hof applied\nmagnetic field.The optimal duration \u001c\u0003\nVof the voltage pulse, which\nmaximizes the soliton lifetime \u001cS, depends on the size\nof the gate electrode and on the magnetic field strength.\nFigure A2 demonstrates variations of \u001c\u0003\nVwith the gate\nradiusRGcalculated at three different field strengths.\nIt is seen that the optimal pulse duration first increases\nwith the growing gate radius but begins to decrease when\nRGexceeds some threshold value. Such a change in the\ndependence \u001c\u0003\nV(RG)maybeattributedtotheappearance\nof inhomogeneous magnetization switching under larger\ngate electrodes.\nThe simulations also show that the gate radius RG\naffects both the soliton size and shape. Figure A3 com-\npares the temporal evolutions of the droplets formed un-\nderthegateelectrodeswithdifferentdiameters. Itcanbe\nseen that both solitons experience significant shape vari-\nations during the decay process. However, the droplet\ngenerated by the voltage pulse applied to the electrode\nwith the smaller radius RG= 90nm becomes nearly cir-\ncular when the time approaches 40 ns. In contrast, the\nsoliton formed under the larger gate with RG= 110nm\nhas a strongly anisotropic shape and fluctuating bound-\nary at this time. This feature gives rise to a high-power\nspin-wave radiation from the soliton, which causes rapid\nreduction of its area and magnetic moment.8\n1.0 ns 4.5 ns 8.0 ns 11.5 ns 15.0 ns 18.5 ns 22.0 ns 25.5 ns 29.0 ns 32.5 ns 36.0 ns 39.5 ns\n1.0 ns 4.5 ns 8.0 ns 11.5 ns 15.0 ns 18.5 ns 22.0 ns 25.5 ns 29.0 ns 32.5 ns 36.0 ns 39.5 ns\nFigure A3. Temporal evolutions of droplet solitons generated by optimal voltage pulses applied to gate electrodes with radius\nRG= 90nm (upper panel) and RG= 110nm (lower panel). The black line shows the soliton boundary, while the yellow area\nindicates the Fe region beneath the gate electrode. The magnetic field strength H= 200Oe."    },    {        "title": "1501.07861v4.A_large_scale_magnetic_shield_with_10_6_damping_at_mHz_frequencies.pdf",        "content": "A large-scale magnetic shield with 106damping at mHz frequencies\nI. Altarev,1M. Bales,1D. H. Beck,2T. Chupp,3K. Fierlinger,1P. Fierlinger,1F. Kuchler,1T. Lins,1, 4,a)M. G.\nMarino,1B. Niessen,1G. Petzoldt,1U. Schl apfer,5A. Schnabel,6J. T. Singh,1,b)R. Stoepler,1S. Stuiber,1M.\nSturm,1B. Taubenheim,1and J. Voigt6\n1)Physikdepartment, Technische Universit at M unchen, D-85748 Garching, Germany\n2)University of Illinois at Urbana-Champaign, Urbana, Il 61801, USA\n3)University of Michigan, Ann Arbor, MI 48109, USA\n4)FRM-II, Technische Universit at M unchen, D-85748 Garching, Germany\n5)IMEDCO AG, CH-4614 H agendorf, Switzerland\n6)Physikalisch-Technische Bundesanstalt, 10587 Berlin, Germany\n(Dated: 26 May 2015)\nWe present a magnetically shielded environment with a damping factor larger than one million at the mHz\nfrequency regime and an extremely low \feld and gradient over an extended volume. This extraordinary\nshielding performance represents an improvement of the state-of-the-art in the di\u000ecult regime of damping\nvery low-frequency distortions by more than an order of magnitude. This technology enables a new generation\nof high-precision measurements in fundamental physics and metrology, including searches for new physics far\nbeyond the reach of accelerator-based experiments. We discuss the technical realization of the shield with its\nimprovements in design.\nI. INTRODUCTION\nReduction of electromagnetic distortions and temporal\nvariations are crucial parameters for a variety of high-\nprecision measurements. High-quality magnetic shield-\ning is particularly important for fundamental physics\nmeasurements based on spin precession at extremely low\nmagnetic \felds. Prominent examples are next-generation\nmeasurements of electric dipole moments of fundamental\nparticles1, tests of CPT and Lorentz-invariance, and ap-\nplications of spin clocks2{4. Other disciplines, such as\nbiomagnetic signal measurements5or the investigation\nof magnetic nanoparticles for cancer therapy6, also use\nsimilar techniques.\nPrecision measurements target the detection of small\ne\u000bects from phenomena that were once dominant in the\nearly universe, thereby probing energies far beyond the\nreach of accelerator-based physics. With many unre-\nsolved questions in particle physics in the LHC era, these\ntechniques are getting increased attention.\nThe time-reversal-symmetry-violating electric dipole\nmoment of the neutron (nEDM) is considered to be one of\nthe most promising systems to discover physics beyond\nthe Standard Model1. An nEDM experiment7is cur-\nrently under development at the Technische Universit at\nM unchen (TUM), aiming to improve the current limit on\nthe nEDM (2.9\u000110\u000026ecm8) by two orders of magnitude.\nThis experiment will use two chambers with an overall\nvolume of\u001850 cm\u000250 cm\u000250 cm \flled with spin-\npolarized species and placed in the center of a shielded en-\nvironment in an applied magnetic \feld of 1-2.5 \u0016T. Ram-\nsey's technique9,10will be applied to the spin-polarized\na)Electronic mail: tobias.lins@ph.tum.de\nb)now at NCSL, Michigan State University, East Lansing, Michi-\ngan, USAspecies to search for e\u000bects from an nEDM.\nImproving the current limit on the nEDM demands\nboth stringent control on the magnitude of \feld gradi-\nents (for further discussion, see e.g.11) and temporal sta-\nbility of magnetic gradients better than 3 pTm\u00001over\n300 s. The latter requires a strong damping factor of\nthe magnetically shielded environment at extremely low\nfrequencies between 1 and 100 mHz.\nHere we describe a magnetic shield with a damping\nfactor exceeding 106for low-amplitude ( \u00181\u0016T) exter-\nnal distortions at the low frequency limit. This shield\ncan reduce distortions caused by typical external sources\n(crane, people, doors, cars or other machinery within\n\u001810 m distance from the shield) to below 1 pT. Typi-\ncal natural ambient magnetic \feld drifts of \u0018100 nT are\nreduced to\u0018100 fT inside the shield, limiting gradient\ndrifts to\u00181 pT=m over several hours.\nII. THE APPARATUS\nThe shield is comprised of a magnetically shielded\nroom (MSR) (see12for more details) forming the out-\nside part of the apparatus. It consists of two shells made\nof MagniferR\r13, a highly magnetizable alloy. Each shell\nconsists of 2\u00021 mm thick heat-treated MagniferR\rsheets\nin a crossed arrangement. A 8 mm thick aluminum\nshell is placed in between these shells for shielding of\nhigher frequencies. The inside dimensions of the MSR\nare 2.78 m length, 2.5 m width and 2.35 m height, with\n250 mm spacing between the MagniferR\rshells. The\nMSR has a door with 1.92 m width and 2 m height,\nplaced symmetrically in a side-wall of the room. On the\n\roor, a non-magnetic set of rails with 1.4 m spacing is\nmounted, which can carry any cuboid load of maximum\nlength 2.78 m\u0002width 1.92 m\u0002height 2 m and up to\n5.5 tonnes of weight. The rails are comprised of plas-\ntic wheels with 250 mm spacing, an aluminum frame toarXiv:1501.07861v4  [physics.ins-det]  23 May 20152\nhold the wheels and titanium bolts as bearings. After the\ndoor of the MSR is opened, a detachable set of rails may\nbe inserted between the inner rails and an outer stor-\nage area. A manually operated mechanical winch can be\nused to move heavy parts between the storage area and\nthe MSR. The detachable rails may be removed once they\nare no longer needed, allowing the door of the MSR to\nbe closed.\nAn additional, cube-shaped magnetic shield (`insert')\nwith 1.92 m width, 1.92 m height, 2.7 m length and\n4 tonnes weight is placed on the rails. An overview of\nthe assembly with MSR and insert is shown in Fig. 1.\nThe insert consists of three shells made from MagniferR\r\nwith 80 mm spacing between the shells. The outer shell\nhas a thickness of 2 \u00021 mm, the middle shell 4 \u00021 mm\nand the innermost shell 2 \u00021 mm. The dimensions inside\nof the insert are 2.2 m length \u00021.54 m width\u00021.54 m\nheight. Another rail system is installed in the space in-\nside the insert to allow the usage of another cylindrical\nMagniferR\rshield (not used in this work). The insert may\nbe moved manually along the rails using the winch sys-\ntem described above. When the insert is deployed inside\nFIG. 1. Cut-through view of the magnetic shield. The outer\nMSR is described in detail in Ref. [12]. Item (1) is the large\ndoor of the MSR, (2) the outer MagniferR\rshell, (3) the in-\nner MagniferR\rshell together with the aluminum shell for RF\nshielding. A rail system is mounted inside this second shell,\nallowing the deployment of the insert inside. Items (4)-(6)\nare the MagniferR\rshells of the insert. The end cap (7) of\nthe insert is rigidly mounted to the MSR. Item (8) is a \feld\ncoil for ultra-low \feld NMR and (9) the experimental cham-\nber. (2)-(7) have a cuboid geometry and (8) has a cylindrical\ngeometry.the MSR (see e.g. Fig. 1), the inner MagniferR\rshell of\nthe MSR and the outer shell of the insert are separated\nby 120 mm in the y direction, 250 mm in the x direction\nand 220 mm in the z direction. Movement of the insert\nusing the winch system in to, or out of, the MSR may\nbe performed in 10 minutes, enabling reasonably quick\naccess to the space inside. In addition, 84 holes with\n>?40 mm and 3 holes with ?130 mm provide access for\nprobes and other equipment while the shield is closed.\nThe alignment of these access holes between the MSR\nand insert is maintained within 1 mm. Figure 2 shows\nan annotated picture of the insert being deployed inside\nthe MSR.\nThe end cap of the insert is mounted on the inside\nwall of the MSR opposite the door, allowing the remain-\nder of the insert to be removed separately from the room.\nThis also ensures that neither the innermost shell nor any\nequipment inside is ever exposed to the full strength of\nthe ambient magnetic \felds at any time. Sixteen 2.8 m\nlong titanium bolts, placed between the outer and mid-\ndle shell and extending the entire length (in the y direc-\ntion) of the insert, are used to rigidly fasten the end cap\nmounted inside the MSR to the rest of the insert. These\nare tightened using a torque wrench while the door of the\nMSR is still open, and press the outer shell of the insert\nto the outer shell of the end cap. The two inner shells of\nthe end cap are then pressed pneumatically to the two\ninner shells of the insert. The mechanical force imparted\nby the pneumatic system is carried by the 16 bolts to\nthe main frame of the insert and is not transferred to the\nMSR.\nThe space in the gap between the MSR and the insert\nFIG. 2. A photograph showing the insert (1) halfway moved\ninto the MSR (2). The weight of the insert is carried by\nwheels (3) mounted to an aluminum frame (4). A winch (5)\nis used to move the insert in to, or out of, the MSR. Titanium\nbolts (6) of length 2.8 m are used to connect the insert to its\nend cap (item 7 in Fig. 1). After the insert is completely inside\nthe MSR, a part of the aluminum frame may be detached to\nallow the door (7) to be closed.3\nmay be used to deploy sensitive electronics. These elec-\ntronics can operate without in\ruencing the magnetic \feld\ninside the insert, but are still within the low-magnetic-\n\feld environment and RF shielding of the MSR.\nCoils are mounted along the walls of the insert. These\nare used for `magnetic equilibration'14, an improved\nprocedure15based on commonly used degaussing tech-\nniques12,16. The coils are wired to the external driving\nelectronics of the equilibration system through connec-\ntors inside the door of the MSR. No coils are installed\nalong the edges of the cuboid shells or on the end caps\nof the insert.\nIII. MEASUREMENTS\nA. Residual \feld of the insert\nAt the factory site, a magnetic equilibration of the in-\nsert was performed using procedures that were not yet\nfully optimized. Following this, a map of the residual\n\feld inside the insert ( without the MSR) was measured\nusing a Mag03-MCL7017low-noise \ruxgate probe. The\nresults from this illustrative measurement are shown in\nFig. 3, demonstrating that all \feld values within the in-\nner cubic meter stay below 1.5 nT.\nFollowing the arrival of the insert at the TUM site a\ntest consisting of two measurements was performed us-\ning the insert within the MSR. The purpose of these tests\nwas twofold: (1) to measure the residual \feld of the com-\nbined MSR+insert system, and (2) to determine how the\nmagnetic equilibration process was a\u000bected by the ac-\ncess holes in the shield and by the di\u000berent contact pres-\nsures (provided by the pneumatic system) between the\ninsert and its end cap. After magnetic equilibration of\nthe shielding, the \feld was measured with the \ruxgate\nprobe along the y-direction from the center to the end\ncap of the insert where the largest access hole ( ?130 mm)\nis located. Figure 4 shows the magnetic \feld (measured\nwith a Mag03-IEHV7017low-noise \ruxgate) as a func-\ntion of the distance from the center of the insert as an\nillustration for a typical magnitude of the residual \feld.\nWhen appropriate contact pressure is applied with the\npneumatic system (blue solid line, Fig. 4), the resid-\nual \feld following magnetic equilibration does not ex-\nceed 0.3 nT. Within the central cubic meter in the in-\nsert, we see no signi\fcant e\u000bect from the largest access\nhole. These results are comparable to a similar measure-\nment with only the MSR12. When little to no contact\npressure between the end cap and the insert is applied, a\nlarger residual \feld is observed (red dashed line, Fig. 4).\nIn other words, good contact between the end cap and\ninsert improves the e\u000bectiveness of the magnetic equili-\nbration.B. Temporal stability of the insert inside the MSR\nA 10-hour-long overnight measurement was performed\nto investigate the temporal stability of the \feld inside\nthe insert and MSR . Six SQUID magnetometers (W9L,\nB [nT]\n2.5\n2.0\n1.5\n0.5\n0.01.0x [mm]500\n-5000500\n0\n-500\n-5000500\ny [mm]0 -500 500(a)\n(b)\n(c)\nFIG. 3. Illustration of the absolute value of the magnetic\n\feld in the center of the insert measured with a \ruxgate probe\n(see text for more details). (a), (b) and (c) are 1 m \u00021 m\nplanes o\u000bset in the z-direction from the central plane by -\n0.25 m, 0 m and +0.25 m, respectively. Measurements were\nperformed at points on a 5 \u00025 grid. The coordinate center\n(0;0;0) corresponds to the center of the shield. The larger\ndistortions are observed opposite the end cap.4\nsee18) were arranged in a cube formation with a side\nlength of 3 cm, enabling the measurement of the magnetic\n\feldBiand the longitudinal gradient \u0001 Bi=@Bi=@riin\nall spatial directions ( Bi=Bx,By,Bzandri=x;y;z ).\nA representative 1000-second-long excerpt from the data\nis shown in Fig. 5 ((a) and (b)). Within the 1000 s time\ninterval, which is about 3-4 times longer than a typical\nEDM measurement cycle, corrections for internal \feld\ndrifts of the liquid-helium-cooled SQUID magnetometers\nare not necessary. Figure 5 (c) shows the average Allan\ndeviation of several of these 1000-second-long series. For\nintegration times of \u0018300 s all components of the mag-\nnetic \feld (gradient) are <100 fT (<2 pT/m). The mea-\nsurement is currently limited by the noise of the SQUID\nsystem, which is worse than the demonstrated intrinsic\nnoise of the SQUIDs18.\nC. Shielding performance\nAn important characterization of the performance of\na magnetic shield is the so-called shielding (damping)\nfactor `SF', de\fned as\nSF =max(B 0sin(!t))\nmax(B insidesin(!t +\u001e)): (1)\nwith\u001ethe relative phase of the signal inside the room\nto the applied signal, B 0the amplitude of the applied\n\feld measured at the position of the center of the shield\nbefore the shield is installed, Binside the amplitude of the\nmagnetic \feld measured in the center of the shield, and !\nthe frequency of the applied AC \feld. The SF was mea-\nsured in di\u000berent con\fgurations of the MSR and insert,\n0.00.20.40.60.81.00100200300400500600\ny@mDB@pTDwp\nwop\nwall\nFIG. 4. Magnetic \feld magnitude of whole 5-layer shield\nmeasured with a \ruxgate along a line concentric in the cen-\ntral 130 mm hole in the back wall of the shield. At distance\ny = 0 m the center of the shield is reached and y \u00191.1 m is\nthe position of the innermost shield layer. The magnetic \feld\nwas measured with the end cap closed with (blue) and without\n(red) pneumatic pressure. In both cases magnetic equilibra-\ntion was applied. The reproducibility of the measurement is\n<50 pT.\n0200 400 600 8001000-0.4-0.20.00.2\nt@sDB@pTDHaL\nHxLHyL\nHzL\n0200 400 600 8001000-50510\nt@sDDB@pTmDHbL\nHxLHyLHzL\n12510205020 1002005001035102050100\n12510\nt@sDB@fTD\nDB@pTmDHcL\nHxLHyLHzL\nHxLHyL\nHzLFIG. 5. Temporal stability of the magnetic \feld in x-, y-\nand z-direction (blue, red and black) at the center of the in-\nsert measured with SQUID magnetometers. (a) is a typical\ntime series of the magnetic \feld taken during a night under\ntypical conditions. (b) is the gradiometer measurement, si-\nmultaneously performed with (a). (c) is the Allan deviation\nof the magnetic \feld (solid) and the gradient (dashed) of sev-\neral 1000 s series.\nas well as for a few of the individual shielding layers. The\nmeasurements used a variety of di\u000berent sensors, includ-\ning a \ruxgate probe (FG), liquid-helium cooled SQUID\nmagnetometers (SQ),199Hg nuclear spin magnetometers\n(HG) and Cs atomic vapor magnetometers (CS).\nThe insert and its individual layers were characterized\nat the factory following fabrication and magnetic equili-\nbration. For these measurements, the insert was placed in\nthe earth \feld and a distortion B extof 2\u0016T peak-to-peak5\nTABLE I. Measured SF of di\u000berent shields as function of external excitation strengths B ext(peak-to-peak or root mean square)\nand frequency ffor very low frequencies. The observed variations between measurements with di\u000berent sensor types are\ndiscussed in the text. For comparison, the SFs of other highly shielded environments are listed. In the following, (t) indicates\nmeasurements in the transverse direction, (l) in the longitudinal direction.\nShield f[Hz] B ext[\u0016T] Sensor SF\nMSR outside layers 0.01 2 (ppa) FG 33\nMSR both layers (t) 0.01 2 (pp) FG 279\nMSR both layers (l) 0.01 2 (pp) FG 260\nMSR both layers (t) 0.01 32 (pp) FG \u0018400\nMSR both layers (l) 0.01 32 (pp) FG \u0018350\ninsert layer 1 (outs.) 0.01 2 (pp) FG 40\ninsert layer 1+2 0.01 2 (pp) FG 600\ninsert (l) 0.01 2 (pp) FG 4700\ninsert (t) 0.01 2 (pp) FG 6500\ninsert (l) 1 2 (pp) FG 4700\ninsert (t) 1 2 (pp) FG 6500\ninsert (l) 10 2 (pp) FG 56000\ninsert (t) 10 2 (pp) FG 100000\nMSR+insert (l) 0.001 16.25 (pp) HG 971000\nMSR+insert (l) 0.001 16.25 (pp) SQ 938000\nMSR+insert (t) 0.001 16.25 (pp) HG 1231060\nMSR+insert (t) 0.001 16.25 (pp) SQ 1173300\nMSR+insert (t) 0.001 16.25 (pp) CS 1390000\nMSR+insert (t) 0.003 16.25 (pp) CS 1510000\nMSR+insert (t) 0.02 16.25 (pp) CS 1580000\nMSR+insert (t) 0.05 16.25 (pp) CS 1660000\nMSR+insert (t) 0.003 3.2 (pp) SQ 1400000\nMSR+insert (t) 0.01 6.4 (pp) SQ 2000000\nMSR+insert (t) 0.05 6.4 (pp) SQ 2130000\nMSR+insert (t) 0.333 32 (pp) SQ \u00188000000\nMSR+insert (t) 1.25 32 (pp) SQ >1:7\u0002107\nBMSR-2190.01 1 (rmsb) 75000\nBMSR-2 1 1 (rms) 2000000\nBoston200.01 1 (rms) 1630\nBoston 1 1 (rms) 200000\napeak-to-peak value of a sinusoidal excitation\nbrms value of a sinusoidal excitation\namplitude at the position of the center of the insert with\nvariable frequency was applied via a 3D quasi-Helmholtz\ncoil system. The calibration of the distortion \feld was\nperformed before the insert was put in place. The SF\nat<0.1 Hz for the outer shell under these conditions is\nSF1= 40, for the outer two layers it is SF 2= 680. In\naddition, the residual \feld of only the outer layer of the\ninsert is<10 nT, a major improvement in performance\nof single-shell MSRs. It was clearly visible during tests\nthat the residual \feld of the 4 mm thick middle layer\ncould not be fully magnetically equilibrated, resulting in\na worse residual \feld. It is likely that a higher quality\nequilibration of the middle layer could have been achieved\nwith better equipment.\nThe SF of the combined MSR+insert system has been\nmeasured at the TUM. To provide an external distor-\ntion \feld, four combined coils of the external \feld com-\npensation system from Ref. [12] were operated as a\nsolenoid. External excitations from 3.2 { 32 \u0016T peak-\nto-peak amplitude at the center of the assembly wereapplied. The amplitude of the excitation at the center\nof the MSR+insert was calibrated before installation of\nthe shields. The SF measurements were performed using\n199Hg nuclear and Cs atomic magnetometers and com-\npared to measurements with a SQUID sensor.\nFor the Hg magnetometer system, a cylindrical quartz\ncell (\u00185 cm diameter, \u001810 cm length) \flled with\n10\u00005mbar Hg vapor is placed in the center of the insert.\nTwo frequency-stabilized 254 nm laser beams penetrate\nthe MSR and the cell along the x-direction, parallel to\nthe cylinder axis of the cell. The transmitted light is\ndetected on the opposite side outside the MSR. A reso-\nnant laser beam is used to transversely optically pump\nthe nuclear spins of the199Hg isotope along the quan-\ntization axis given by the propagation direction of the\nlaser beam. Pumping is performed for 3 s with a light\nmodulation frequency of \u00186.2 Hz, corresponding to the\nLarmor frequency of199Hg in the applied magnetic \feld\n(B0\u00180.8\u0016T). The B 0\feld was generated by portable\nHelmholtz coils. (The B 0coil described in Fig. 1 was not6\ninstalled at the time of the measurement.) The polariza-\ntion is then observed during free precession of the spins\nfor 100 s, an interval chosen to optimally resolve exter-\nnal sinusoidal distortions with 1 mHz amplitude. Typical\ntransverse spin life-times are >150 s for the magnetome-\nter cell, and the sensitivity in 100 s integration time is\n12 fT. The systematic e\u000bect in the amplitude determina-\ntion due to the long averaging time (100 s) for each mea-\nsured point is only a few percent and has been corrected\nfor. Due to the time-dependent variation, the frequency\nresolution is not as good as that normally achieved with\na quasi-static \feld measurement. A SQUID cryostat is\nmounted directly on top of the Hg cell (at a distance of\n\u001815 cm) and the signal is measured simultaneously for\ncomparison.\nCs atomic magnetometers7,21were also used for com-\nparison. (Measurements with these magnetometers were\nnottaken simultaneously with the SQUID and Hg mea-\nsurements.) These magnetometers use the electronic spin\nof Cs atoms in an evacuated cell. The spins are optically\npumped using alignment pumping and interrogated using\nlinearly polarized laser light with a operation frequency\nof\u00187 kHz/\u0016T. A typical performance of the sensors is\npT/p\nHzin the con\fguration used for the SF measure-\nments. The stability of the whole Cs setup | including\nthe Cs magnetometer, the \u00181\u0016T holding \feld required\nfor magnetometer operation, and the shield | was deter-\nmined by performing overnight measurements and found\nto be on the order of 2 pT.\nResults for the di\u000berent magnetometer systems are\nshown in Table I, together with a comparison of other\nshields, including the `BMSR-2' at the Physikalisch-\nTechnische Bundesanstalt in Berlin, Germany, (PTB\nBerlin), which has been considered until now the refer-\nence facility for magnetic shielding19. In the longitudinal\ndirection, the geometric aspect ratio of the inner volume,\nthe door and the placement of all large access holes along\nthe path of the magnetic \rux lead to a notable reduc-\ntion of the SF. Results from the di\u000berent magnetometers\nare within reasonable agreement and show a consistently\nlarge SF at very low frequencies. Variations in the ob-\nserved SF can be explained since not all measurements\ncould be performed in the exact same con\fguration, ei-\nther simultaneously or in the exact same position.\nThe performance of the Hg and Cs magnetometers for\nSF measurements at even lower frequencies than shown\nhere is currently limited by the lack of temperature stabi-\nlization of the shields, which in turn also a\u000bects the sta-\nbility of the applied magnetic \feld and the thermal stabil-\nity of the laser systems outside the shields. These probes\nare nevertheless better suited than the used SQUID sys-\ntem to measure long-term drifts, since this particular\nSQUID system has a known long-term stability problem.\nDirect comparison between SF measurements of dif-\nferent MSRs is di\u000ecult due to the sensitivity of the re-\nsults on the actual measurement conditions. In particu-\nlar, the geometry of the coils around the MSR and the\nexact strength of the applied excitation \feld can alterthe apparent SF value. In the presented measurements,\nthe coils | with dimension 6 m \u00029 m (6 m\u00026 m)\nin the transverse (longitudinal) direction | are signi\f-\ncantly larger than the MSR and produce a \feld with a\nhomogeneity of\u0018\u0006 20% over the volume of the MSR.\nFor the case of Helmholtz-like con\fgurations of small\nsize relative to the dimension of the shield, the SF ap-\npears higher owing to the intrinsically stronger damping\nof larger distortions. In the BMSR-2 apparatus, the coils\nare placed closer to the shield, which increases the ap-\nparent SF value. The coil con\fguration is unknown for\nthe Boston setup. It should also be noted that shielding\nbecomes signi\fcantly more di\u000ecult with increased size of\nthe shield. For these reasons, the quality of a shield is\nnot only determined by its SF, but also by its size and\nthe amount of material used for its construction.\nThe SF of the TUM MSR+insert has been measured\nusing an in-house-fabricated Cs atomic vapor magne-\ntometer and con\frmed using simultaneous measurements\nwith a Hg magnetometer and a mobile low-temperature\nSQUID system from PTB Berlin. The use of several inde-\npendent measurement systems ensures that the SF mea-\nsurement is not dominated by systematic issues. The\nmeasurement could be further improved by reducing the\nnoise \roor and drift of the sensors and applied internal\n\felds. However, such improvements would not a\u000bect the\nconclusions of this work. It should be noted, that the\ncontact pressure of the pneumatic system had no observ-\nable e\u000bect on the SF.\nIV. CONSIDERATIONS FOR MAGNETIC SHIELD\nDESIGN\nThe con\fguration (MSR+insert) uses only a 2 mm thin\nshell of magnetizable alloy for the innermost shield layer,\nwhich allows a high-quality magnetic equilibration to ob-\ntain very low residual \felds. This demonstrates that\nit is possible to sacri\fce some thickness of the inner-\nmost shield layer without experiencing excessive losses\nin damping capability. The performance of this shield\nwas expected to be high compared to the reference room\nBMSR-2 (cubic design with 3.2 m dimension of the inner\nMu-Metal layer22) due to its smaller size. However, it\nwas notexpected that its SF would vastly exceed that of\nthe BMSR-2, since it only has \fve layers of MagniferR\r\nwith (inside to outside) 2-4-2-2-8(aluminum)-2 mm thick-\nness. For comparison, the BMSR-2 has seven layers of\nMu-Metal with thicknesses (inside to outside) 4-7-6-3-\n10(aluminum)-3-2-2 mm. Major di\u000berences in the design\nconcept between the two shields are the thickness of the\nmetal sheets, as well as the width and length of individ-\nual sheets, which are connected many times to form one\nclosed shell of shielding material. We believe that the\nlower number of joints between the individual sheets due\nto their larger size (3000 mm \u0002750 mm) signi\fcantly\ncontributes to this improved performance.\nWe have demonstrated that the shielding performance7\nof the insert is also very high, although the spacing of\nshield layers is only 80 mm. This contradicts common\ndesign criteria, which suggest a signi\fcantly larger spac-\ning of the layers is necessary to obtain good shielding\nperformance.\nThe arrangement of the magnetic equilibration coils\nalso does not follow common criteria, which maintain\nsuch coils should be installed along the edges of the\nshield. We observe, however, no notable issues in the\nresidual \feld following magnetic equilibration. Indeed,\nthis is the \frst time such a small residual \feld has been\ndemonstrated over such a large volume.\nThe in\ruence of the access holes on the residual \feld\nis small for the 3-layer insert. The access holes have a\ndiameter up to 130 mm, which may be compared to the\n80 mm layer spacing along the path of the magnetic \rux\nfor longitudinally applied external distortions. Despite\nthis relatively large size, a surprisingly small distortion\nin the residual \feld due to the holes is observed. However,\nwe do \fnd that the shielding factor in the longitudinal\ndirection is lower. We attribute this to e\u000bects from the\nholes, the doors and the geometrical aspect ratio of the\ncuboid structure.\nThree di\u000berent types of mechanisms for connecting\nMu-Metal layers are built into this shield: (1) a pneu-\nmatically clamped door in the outer MSR, (2) a bolted\nconnection in the outer layer of the insert and (3) two\npneumatically pressed connections for the two inner lay-\ners in the insert. Due to the staged tests during con-\nstruction, the performance of each of these mechanisms\ncould be demonstrated independently. All three exhibit\ngood magnetic shielding properties and low residual \feld\ndistortions.\nAn additional advantage of the shielding system de-\nscribed above is that both the MSR and insert are\nportable and may be used independently as magnetic\nshields. Despite having demonstrated excellent shielding\nperformance with the MSR+insert, we believe that this\ncon\fguration can be further optimized. A detailed de-\nscription of the relevant aspects to achieve these further\nimprovements is in preparation.\nV. CONCLUSION\nAlthough an accurate comparison of magnetic shield-\ning installations is di\u000ecult, the shield described in this\nmanuscript is, to our knowledge, the by far strongest ex-\nisting large-scale magnetic shield in terms of magnetic\ndamping. In addition, its observed residual \feld, even\nwith environmental distortions and built-in experimen-\ntal hardware, is <1 nT. Many new design concepts\nhave been implemented, including reduced shielding-\nlayer spacing, modi\fed magnetic equilibration coil ar-\nrangements, optimized sheet metal arrangements and\ncomparably large access holes without large distortions\nin the observed residual \feld. With a low-frequency SF\nexceeding 106even for small amplitude distortions, thedesigns featured here have implications for the quality\nand complexity of shields for next-generation precision\nexperiments.\nAdditional improvements to the current setup are al-\nready planned. For example, a cylinder made from\nMagniferR\rwill be added into the insert to act as a mag-\nnetic yoke for the holding \feld of the nEDM experiment\nat the TUM. This is expected to increase the SF by at\nleast a factor of two.\nACKNOWLEDGMENTS\nWe acknowledge the loan of a sensor from PTB Berlin\nfor con\frmation of the shielding factors measurement, as\nwell as the support of the machine shops at the FRM-II\nand at the Physics Department of the TUM in Garching.\nThis work was supported by the DFG Priority Program\nSPP 1491 and the DFG Cluster of Excellence `Origin and\nStructure of the Universe'.\n1J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Progress in\nParticle and Nuclear Physics 71, 21 (2013).\n2D. Colladay and V. A. Kostelecky, Phys.Rev. D55, 6760 (1997).\n3I. Altarev, C. A. Baker, G. Ban, G. Bison, K. Bodek, M. Daum,\nP. Fierlinger, P. Geltenbort, K. Green, M. G. D. van der Grinten,\nE. Gutsmiedl, P. G. Harris, W. Heil, R. Henneck, M. Horras,\nP. Iaydjiev, S. N. Ivanov, N. Khomutov, K. Kirch, S. Kistryn,\nA. Knecht, P. Knowles, A. Kozela, F. Kuchler, M. Ku\u0013 zniak,\nT. Lauer, B. Lauss, T. Lefort, A. Mtchedlishvili, O. Naviliat-\nCuncic, A. Pazgalev, J. M. Pendlebury, G. Petzoldt, E. Pierre,\nG. Pignol, G. Qu\u0013 em\u0013 ener, M. Rebetez, D. Rebreyend, S. Roccia,\nG. Rogel, N. Severijns, D. Shiers, Y. Sobolev, A. Weis, J. Zejma,\nand G. Zsigmond, Phys. Rev. Lett. 103, 081602 (2009).\n4K. Tullney, F. Allmendinger, M. Burgho\u000b, W. Heil, S. Karpuk,\nW. Kilian, S. Knappe-Gr uneberg, W. M uller, U. Schmidt,\nA. Schnabel, F. Seifert, Y. Sobolev, and L. Trahms, Phys. Rev.\nLett.111, 100801 (2013).\n5L. Trahms and M. Burgho\u000b, Magnetic Resonance Imaging 28,\n1244 (2010).\n6T. Apih, B. Rameev, G. Mozzhukhin, and J. Barras, Mag-\nnetic Resonance Detection of Explosives and Illicit Materials\n(Springer Netherlands, 2014) pp. 99{110.\n7I. Altarev, D. H. Beck, S. Chesnevskaya, et al. , Il nuovo cimento\n4, 122 (2012).\n8C. A. Baker, D. D. Doyle, P. Geltenbort, et al. , Phys. Rev. Lett.\n97, 131801 (2006).\n9N. F. Ramsey, Phys. Rev. 76, 996 (1949).\n10N. F. Ramsey, Rev. Mod. Phys. 62, 541 (1990).\n11J. M. Pendlebury, W. Heil, Y. Sobolev, P. G. Harris, J. D.\nRichardson, R. J. Baskin, D. D. Doyle, P. Geltenbort, K. Green,\nM. G. D. van der Grinten, P. S. Iaydjiev, S. N. Ivanov, D. J. R.\nMay, and K. F. Smith, Phys. Rev. A 70, 032102 (2004).\n12I. Altarev, E. Babcock, D. Beck, M. Burgho\u000b, S. Ches-\nnevskaya, T. Chupp, S. Degenkolb, I. Fan, P. Fierlinger, A. Frei,\nE. Gutsmiedl, S. Knappe-Grneberg, F. Kuchler, T. Lauer,\nP. Link, T. Lins, M. Marino, J. McAndrew, B. Niessen, S. Paul,\nG. Petzoldt, U. Schlpfer, A. Schnabel, S. Sharma, J. Singh,\nR. Stoepler, S. Stuiber, M. Sturm, B. Taubenheim, L. Trahms,\nJ. Voigt, and T. Zechlau, Review of Scienti\fc Instruments 85,\n075106 (2014).\n13Krupp, \\MagniferR\r7904,\".\n14The commonly used term `degaussing' implies the removal of\nmagnetic \feld from within the material itself. Especially in the\ncase of Mu-metal magnetic shielding, which acts as a conduit for\nmagnetic \feld lines to produce a shielding e\u000bect, this is mislead-8\ning. We use `magnetic equilibration' because it appropriately de-\nscribes what occurs: the residual \feld from the shield is brought\ninto equilibrium with its surroundings.\n15I. Altarev, P. Fierlinger, T. Lins, M. Marino, B. Nieen, G. Pet-\nzoldt, M. Reisner, S. Stuiber, M. Sturm, J. Singh, B. Tauben-\nheim, H. Rohrer, and U. Schlpfer, (2015), to be published,\narXiv:1501.07408 [physics.ins-det].\n16J. Voigt, S. Knappe-Grneberg, A. Schnabel, R. Krber, and\nM. Burgho\u000b, Metrol. Meas. Syst. XX, 239 (2013).\n17Bartington Instruments, Witney, Oxon, England,.18D. Drung, Physica C: Superconductivity 368, 134 (2002).\n19O. Kosch, P. Meindl, U. Steinhoof, and L. Trahms, in\nBiomag2000, Proc. 12th Int. Conf. on Biomagnetism (2001) pp.\n553{556.\n20D. Cohen, U. Schlpfer, S. Ahlfors, E. Hamalainen, and E. Hal-\ngren, in Biomag2002, Proc. 13th Int. Conf. on Biomagnetism\n(2002) p. 919.\n21S. Pustelny, W. Gawlik, S. M. Rochester, D. F. J. Kimball, V. V.\nYashchuk, and D. Budker, Phys. Rev. A 74, 063420 (2006).\n22\\Mu-metal is a trademark of vacuumschmelze gmbh, hanau, ger-\nmany,\"."    },    {        "title": "1906.06255v2.Influence_of_External_Magnetic_Field_on_Dust___Acoustic_Waves_in_a_Capacitive_RF_Discharge.pdf",        "content": "In\ruence of External Magnetic Field on Dust \u0000Acoustic Waves in a Capacitive RF\nDischarge\nMangilal Choudhary,1,a)Roman Bergert, Slobodan Mitic, and Markus H. Thoma\nI. Physikalisches Institut, Justus{Liebig Universitt Giessen, Henrich{Bu\u000b{Ring 16,\nD 35392 Giessen, Germany\nThis paper reports experiments on self \u0000excited dust acoustic waves (DAWs) and\nits propagation characteristics in a magnetized rf discharge plasma. The DAWs are\nspontaneously excited in dusty plasma after adding more particles in the con\fning\npotential well and found to propagate in the direction of streaming ions. The spon-\ntaneous excitation of such low-frequency modes is possible due to the instabilities\nassociated with streaming ions through the dust grain medium. The background\nE-\feld and neutral pressure determine the stability of excited DAWs. The charac-\nteristics of DAWs strongly depend on the strength of external magnetic \feld. The\nmagnetic \feld of strength B <0.05 T only modi\fes the characteristics of propagating\nwaves in dusty plasma at moderate power and pressure, P = 3.5 W and p = 27 Pa\nrespectively. It is found that DAWs start to be damped with increasing the magnetic\n\feld beyond B >0.05 T and get completely damped at higher magnetic \feld B \u0018\n0.13 T. After lowering the power and pressure to 3 W and 23 Pa respectively, the\nexcited DAWs in the absence of B are slightly unstable. In this case, the magnetic\n\feld only stabilizes and modi\fes the propagation characteristics of DAWs while the\nstrength of B is increased up to 0.1 T or even higher. The modi\fcation of the sheath\nelectric \feld where particles are con\fned in the presence of the external magnetic \feld\nis the main cause of the modi\fcation and damping of the DAWs in a magnetized rf\ndischarge plasma.\na)Electronic mail: Mangilal.Choudhary@exp1.physik.uni-giessen.de\n1arXiv:1906.06255v2  [physics.plasm-ph]  29 Sep 2019I. INTRODUCTION\nThe presence of submicron to micron-sized particles in a plasma makes it more complex\nbecause these particles alter the dynamics of the plasma species (electrons and ions) as well\nas they exhibit their own dynamics. Such medium, which consists of three charged species\nnamely electrons, ions, and charged solid particles, is termed as a dusty plasma or complex\nplasma. In the background of a low-temperature plasma, energetic electrons impinge on the\nsurface of the solid particle and charge their surface negatively up to 103\u0000105times of an\nelectron charge1to balance the \ruxes of electrons and ions. After the density of negatively\ncharged dust particles crosses a critical value, then long-range coloumbic interaction among\nthe dust particles turns the dust grain medium to respond collectively similar to the plasma.\nThe results of the collective phenomena of dusty plasma are predicted theoretically and\nexperimentally in the form of linear and non-linear waves such as longitudinal dust acous-\ntic waves2{5dust acoustic transverse waves6{8, dust lattice waves9{11, dust acoustic solitary\nwaves12,13and dust acoustic shock waves14{16as well as dust vortices17{19\nThe study of dust acoustic waves (DAWs) has a more than 25 years old history20. In low-\ntemperature discharges (DC or RF), the DAWs are excited in the dust grain medium which\nis con\fned in a strong electric \feld (E) of the cathode sheath5,13or anodic plasma3or rf\nsheath21{23or di\u000bused plasma17,19,24. Such low frequency dust acoustic modes are either ex-\ncited by inherent instabilities such as ion-streaming instability and dust{dust instability25{28\nor externally forced triggered instabilities29,30. However, the ion-streaming instability which\narises due to the streaming of ions relative to the stationary dust particles is considered as a\nmain free energy source to excite DAWs in DC or RF discharge. The amount of free energy\ndepends on the velocity of the streaming ions ( vi=\u0016iE) through the dust grain medium,\nwhere\u0016iis the mobility of ion. In the laboratory, both of the variables ( Eand\u0016i) can be\nexternally controlled by changing the plasma or discharge conditions. Previous experimen-\ntal studies on the excitation and damping of DAWs were performed in the unmagnetized\nDC or RF plasma background20. Theoretical studies on DAWs for a magnetized plasma\nbackground have been reported in refs.31,32. In the plasma, an external magnetic \feld (B)\nis considered as a variable which modi\fes the dynamics of electrons and ions. Since the\ndust particles dynamics is associated with the plasma background, the propagation charac-\nteristics of DAWs is expected to be modi\fed in the magnetized plasma. There is a lack of\n2experimental work reported on the dust acoustic waves in a magnetized rf discharge. There-\nfore, the in\ruence of the external magnetic \feld on the longitudinal DAWs in a capacitively\ncoupled discharge has been the subject matter of present study.\nThe propagation characteristics of DAWs in the presence of an external magnetic \feld\nin a capacitively coupled discharge plasma are discussed. The dust grains are con\fned in\nthe sheath region of the lower electrode and form a stable dusty plasma. The dusty plasma\nbecomes unstable above a threshold dust number density at low background gas pressure\n(p<28 Pa). The inherent instabilities trigger the low-frequency acoustics modes, which\npropagate along the direction of streaming ions and gravity. It is found that the magnetic\n\feld quenches the dust{acoustic waves, which are excited at moderate rf power (P = 3.5\nW) and pressure (p = 27 Pa). However, waves excited at low pressure (23 Pa) and power\n(P = 3 W) remain with signi\fcant modi\fcations in their characteristics in the presence\nof an external magnetic \feld. The quenching and modi\fed characteristics of DAWs are\nexplained on the basis of the modi\fed sheath electric \feld and dust charge in the presence\nof an external magnetic \feld. The manuscript is organized as follows: Sec. II deals with the\ndetailed description of the experimental setup and diagnostics for the dusty plasma studies.\nSec. III describes the excitation and characteristics of DAWs without external magnetic \feld.\nThe propagation characteristics of DAWs in the presence of the external magnetic \feld is\ndiscussed in Sec. IV. The discussion on the observed results is provided in Sec. V. A brief\nsummary of the work is provided in Sec. VI.\nII. EXPERIMENTAL SETUP\nExperiments are performed in an aluminium vacuum chamber, which is placed in the\ncenter of a superconducting electromagnet ( Bmax\u00184 T) to introduce the homogeneous\nmagnetic \feld in the plasma volume. The magnetized dusty plasma device which is used in\nthe present study is shown in Fig.1(a). The schematic diagram of the experimental setup\nis presented in Fig.1 (b). More details about the superconducting electromagnet can be\nfound in ref.33. Before the experiments, vacuum chamber is evacuated to base pressure p\n<10\u00002Pa using a pumping system consisting of a rotary and turbo molecular pump. A\ngas mass \row controller (MFC) and gate valve controller are used to control the argon gas\npressure inside the vacuum chamber during the experiments. For a given argon pressure,\n3 \nSuper conducting \nelectromagnet  (a) \nVacuum \nchamber  \n \n(2) (1) \n(3) \n(10) \n(4) (9) \n   \n  \n(6) (7) \n(5) \n(8)  Y Z \nX B FIG. 1. (a) Magnetized dusty plasma device. (b) Schematic diagram of the experimental setup.\n(1) Full view of super conducting magnet, (2) Vacuum chamber, (3) ITO Coated (upper) electrode\n(4) aluminium (lower) electrode (5) levitated dust particles (6) 13.56 MHz RF power source, (8)\nLaser, (8) mirror, (9) CMOS camera for horizontal view, and (10) CCD camera for the vertical\nview.\nplasma is ignited between an aluminium electrode of 65 mm diameter (lower electrode) and\nan indium tin oxide (ITO) coated electrode of 65 mm diameter (upper electrode) using a\n13.56 MHz rf generator with a matching network. Both electrodes are separated by 30 mm\nand upper electrode is grounded along with the vacuum chamber. Special design of lower\nelectrode22, having a ring shaped periphery of height 2 mm and width of 5 mm, provides\na radial con\fnement to the negatively charged dust particles which are levitated over the\nlower electrode. The vacuum chamber has 8 side (or radial) ports with total volume of \u0018\n1000cm3. Two opposite side ports are used to diagnose the dusty plasma using a laser\nand CCD camera in a vertical plane. A CMOS camera is also installed to observe the dust\ndynamics in the horizontal plane through the transparent upper electrode. A dust dispenser,\nwhich is installed at one of the side ports of vacuum chamber, is used for injecting the dust\nparticles into the plasma.\n4III. DUST{ACOUSTIC WAVES AND ITS CHARACTERISTICS IN\nABSENCE OF MAGNETIC FIELD\nFor creating a dusty plasma in the laboratory, mono-dispersive Melamine Formaldehyde\n(MF) particles of radius, r= 1.025\u0016m are introduced using dust dispenser in a capacitively\ncoupled (RF) argon plasma. They get negatively charged in the plasma and con\fned at the\nsheath edge of the lower electrode where the dust particles achieve an equilibrium position\nunder the action of various forces34,35. To record the dynamics of the con\fned dust particles,\na red laser to illuminate the dust and a CCD camera to record the scattering light from\nthe grains are used. The CCD camera used in the present sets of the experiment is able\nto record 20 fps at the resolution of 1024 \u0002768 pixels. These stored images are further\nanalyzed using Matlab based code and ImageJ software36. It should be noted that the\nexcitation of waves in the dusty plasma is only possible when the dust cloud width exceeds\nthe wavelength of dust{acoustic wave at small dust{neutral friction or low gas pressure. In\nthe laboratory, this is achieved at low-pressure large volume dusty plasma. It is well known\nthat the gravitational force acting on massive particles mainly determines the volume of the\ndusty plasma. At ground-based experiments, a large volume dusty plasma is possible either\nin speci\fc discharge con\fguration17,37or discharges with nano to sub-micron sized particles38.\nThe computer simulation by Chutov and Goedheer39suggests an rf sheath expansion in the\npresence of a large number of particles. To keep this idea in mind, we have used particles\nwith sizer= 1.025\u0016m to create a volumetric dusty plasma at low pressure. Since the dust\ngrains exhibit a rotational motion near the outer edge of the lower electrode, the central\nregion of dusty plasma is considered for the study of wave phenomena. For such a study, a\n2D vertical plane (Y{Z plane) at the centre (X = 0 cm) is chosen to record the dynamics\nof particles at various discharge conditions. In this Y{Z plane, dusty plasma is observed to\nbe stable in the case of insu\u000ecient free energy to excite the instabilities. It is the case when\nfewer particles are con\fned in the potential well, as shown in Fig. 2(I). After adding more\nparticles into the con\fning potential well, the number of layers (or dusty plasma volume), as\nwell as dust density increases and the medium, starts to turn into wave-like motion, which\nis shown in Fig. 2(II) and Fig. 2(III). The instabilities \frst appear in the lower part (or\nbottom) of dust grain medium and after that in the upper part (or top). The instabilities\narise after adding more particles are a possible source of free energy for the excitation of\n5waves. The growth of instabilities depends on the E=p ratio25, which increases with the\nincrease of electric \feld ( E) or decrease of gas pressure ( p). Actually, this ratio determines\nthe velocity of streaming ions through the con\fned dust grain medium, which is considered\nas the main source of free energy25{28.\nThese self-excited waves propagate in the direction of streaming ions at given discharge\nconditions. The stability of the self-excited DAWs in discharges depends on the E=p ratio,\nwhich determines the streaming ion \rux through the con\fned dust grain medium25,26. At\nlow pressure p <20 Pa, the energy gain of ions is quite higher than dissipation arising from\nneutral pressure and hence, excited waves are highly unstable. At high pressure p >30 Pa,\nthe energy loss of ions with neutrals is higher than the energy gain in the sheath electric\n\feld and DAWs can not be excited. Moreover, the DAWs stability also depends on E-\feld,\nwhich is connected to input rf power. At higher power P >4 W, dust grains are con\fned\nin a strong E-\feld region with less number of layers (low volume) and DAWs are observed\nto be highly unstable. In view of this, we have chosen a pressure and power range of 23 Pa\nto 28 Pa and 3.5 W to 3 W, respectively for the present study of DAWs. For the detailed\ncharacterization of DAWs, the time evolution of the average intensity (or nd) of consecutive\nstill images are considered. Fig. 2(b) represents the time evolution of average intensity\npro\fle of DAWs in the absence of B. Maxima of the intensity corresponds to the wave crest\nand minima represents the wave trough. Distance between two consecutive maxima gives\nthe wavelength of DAW. The pink dashed line in Fig. 2(b) indicates the trajectory of a\nparticular wave crest and this estimates the phase velocity of DAWs. The characteristics of\npropagating DAWs at argon pressure, p = 27 Pa and rf power, P = 3.5 W are depicted in\nFig. 2(c). It is found that the wavelength ( \u0015) of DAW decreases from top to bottom and\nthe velocity ( Cda) increases from top to bottom. The frequency of the waves ( fd=Cda=\u0015)\nremains almost unchanged in the unmagnetized case (B = 0 T). The variation of particles\ndensity (nd) as well as of the E-\feld throughout the dust grain medium (from top to bottom)\ncauses the variation in the wavelength, \u0015\u00181.9 mm to 1.3 mm and phase velocity, Cda\u0018\n10.5 mm/s to 13.5 mm/s of DAWs, while they propagate towards the lower electrode. The\nfrequency of DAWs comes out to be \u00187 Hz. The average phase velocity of DAW is \u0018\n12 mm at the mentioned discharge parameters. For the theoretical estimation of Cda, the\nwave dispersion relation402in the case of cold dust ( Td= 0) and long wavelength limit\n((K2\u00152\nD<<1)) gives the phase velocity Cda=Zd(kBTind0=mdni0)1=2, whereK,\u0015D,kb,Ti,\n6 \n2\n \nmm\n \n(\nII\n)\n \n(\nIII\n)\n \n(\nI\n)\n \nY\n \nZ\n \ng\n \nThis is a watermark for trial version, register to get full one!\nBenefits for registered user:\n1. Can remove all trial watermark.\n2. No trial watermark on the output documents.\nRemove it Now\n02 4 6 8 1 00100200300Intensity (arb.)D\nistance z (mm) t = 0.13 S \nt = 0.065 S \nt = 0 Sλ TopB ottom(b)\n0 1 2 3 4 5 6 7 891011121314\n1.01.52.02.53.0\nWavelength (mm)\nTop to bottom\n Phase velocity\n Wavelength\nPhase velocity (mm/s)\nDust cloud length (mm)(c)FIG. 2. Dust grain dynamics in a vertical (Y{Z) plane at center (X = 0) with di\u000berent dust grain\nvolume. (b) Representation of the time evolution of intensity pro\fle of video frames taken at time\nstep of 65 ms in the Y{Z plane. (c) Wavelength and phase velocity of DAWs from top to bottom\nin the dusty plasma medium at B = 0 T. The argon pressure and rf power are set to 27 Pa and\n3.5 W respectively during the experiments\n.7md,nd0,ni0, andZd=Qd=eare the wave vector, the dust Debye length, the Boltzmann\nconstant, ion temperature, dust particle mass, equilibrium dust density, ion density and dust\ncharge number, respectively2,20,40The theoretical estimated phase velocity Cdaof the DAW\nis found to be \u001816mm=s for the parameters Te\u00183 eV,Ti= 0:025 eV,md\u00186\u000210\u000015\nkg,nd0\u00188\u0002105cm\u00003,ni0\u00189\u0002108cm\u00003, andZd\u00182\u0002103. Both the theoretically\nestimated and experimentally measured phase velocity of DAWs are in good agreement,\nwhich con\frms the acoustics nature of the self-excited waves in the dusty plasma.\n \nB = 0 T\n \nB = 0.02 T\n \nB = 0.06\n \nT\n \nB = 0.08\n \nT\n \nB = 0.1\n \nT\n \nB = 0.\n15\n \nT\n \n2 mm\n \nThis is a watermark for trial version, register to get full one!\nBenefits for registered user:\n1. Can remove all trial watermark.\n2. No trial watermark on the output documents.\nRemove it Now\nFIG. 3. Dust-acoustics waves in a vertical (Y{Z) plane at center (X = 0) in the presence of\ndi\u000berent strengths of external magnetic \feld (B). Experiments are performed at argon pressure p\n= 27 Pa and input rf power P = 3.5 W.\nIV. CHARACTERISTICS OF DAW IN THE PRESENCE OF MAGNETIC\nFIELD\nIn Section III, the excitation of DAWs and its propagation characteristics in the absence\nof an external magnetic \feld, B = 0 T have been discussed. The e\u000bect of an external\nmagnetic \feld on the propagating DAWs, which are excited at p = 27 Pa and P = 3.5 W,\nis shown in Fig. 3. At this discharge conditions, the excited waves are slightly unstable at\nB = 0 T and get stabilized at a low magnetic \feld (B <0.05 T). Only the propagation\n8 \nB = 0 \nT\n \nB = 0.06 T\n \nB = 0.\n15\n \nT\n \n2 mm\n \nThis is a watermark for trial version, register to get full one!\nBenefits for registered user:\n1. Can remove all trial watermark.\n2. No trial watermark on the output documents.\nRemove it NowFIG. 4. Dust-acoustics waves in a vertical (Y{Z) plane at center (X = 0) in the presence of\ndi\u000berent strengths of external magnetic \feld (B). The argon pressure and rf power are set to 26\nPa and 3 W respectively for the experiements.\ncharacteristics such as frequency, wavlength and velocity of the DAWs get modi\fed at low\nB range. Further increase of B (B >0.05 T) starts to damp the DAWs in the upper part\nof the con\fned dusty plasma medium. In the range of B, 0.05 T up to 0.13 T, the DAWs\ncontinuously damp from top to bottom of the dust grain medium. At higher B \u00180.13\nT, the DAWs are found to be completely damped and the dust particles do not show any\noscillatory motion around their equilibrium position. Further increase in B (B >0.13 T)\nturns the central region of the dusty plasma in a E \u0002B motion41,42which is not the focus of\nthe present study and therefore not considered. It should be noted that dust grains show\na rotational motion or E \u0002B motion due to the radial electric \feld. The magnitude of this\nE \feld decreases from the edge region to the central region of the lower electrode. Large\ndiameter of the lower electrode provides a weak radial E-\feld at the central region of the\ndust cloud, therefore, the oscillatory motion of dust grains (along B) dominate over the E \u0002B\nmotion (perpendicular to B). Hence, the e\u000bect of radial E-\feld during the wave motion is\nobserved to be insigni\fcant.\nIt has been discussed the excitation and stability of DAWs dependence on the E=p ratio.\nWe observe the characteristics of DAWs at a higher value of E=p, which is possible at\nslightly lower power (P = 3 W) and lower pressure (p = 26 Pa). Fig. 4 represents the\npropagation characteristics of DAWs at various strength of the magnetic \feld. At this\n9discharge conditions, the DAWs are slightly unstable compared to the previous discharge\ncase (Fig. 3) in unmagnetized plasma. In the range of magnetic \feld B <0.06 T, the dusty\nplasma medium gets stabilized and the wave parameters change. With increasing B from\n0.06 T to 0.15 T, we see the damping of DAWs in the upper part of dusty plasma. In this\ncase, the DAW are not completely damped in the lower part of dusty plasma even at the\nhigher magnetic \feld (B = 0.15 T). The role of B on the damping of the wave becomes\nnegligible after a certain value. Therefore, the bottom part of the dusty plasma medium\nexhibits a wave like motion even after increasing the magnetic \feld, B >0.15 T. It is also\nfound that higher magnetic \feld is required to damp the wave when the rf power is lowered.\nFurther lowering of the pressure to 23 Pa at P = 3 W, the DAWs are observed to be\nunstable and stabilize in the presence of a magnetic \feld. Fig. 5 represents the DAWs at\ndi\u000berent strength of B. It is found that DAWs do not damp even at the higher magnetic \feld\n(B>0.1 T) at this discharge conditions. The magnetic \feld only modi\fes the characteristics\nof the propagating DAWs or wave parameters. The wave parameters such as \u0015,Cdaandfd\nare measured for di\u000berent B. The wave parameters with various strength of B are listed in\ntable I. It is found that wavelength of DAW \frst decreases at lower B (B <0.05 T) and\nafter that its value slightly increases again towards higher B (B >0.1 T). The opposite\ntrend is seen in the case of the velocity and frequency of DAWs. The Cdaandfdincreases\nwith increasing the B (B <0.05 T) and after that, both values start to decrease towards\nhigher magnetic \feld. Similar kind of variation in wave parameters is also observed for other\ndischarge conditions (P = 3.5 and 3 W, p = 27 and 26 Pa). Thus, the results clearly show\nthe modi\fcation and damping of dust-acoustic waves in the magnetized rf discharge plasma.\nV. DISCUSSION\nIn the present sets of experiments, discharge is ignited between electrodes using a 13.56\nMHz rf source. In such high-frequency discharge, the ion frequency ( !pi) is lower than the\nrf frequency ( !rf). Therefore they are not able to react to the fast changing rf electric\n\feld. It means that ions only respond to the E-\feld averaged over an rf cycle, resulting in\na constant \rux of ions to the electrodes. However, the electron frequency ( !pe) is higher\nthan!rf. Hence electrons follow the E-\feld during the rf cycle and oscillate between the\n10 \n \nB = 0 T\n \nB = 0.1 T\n \nB = 0.\n04\n \nT\n \n2 mm\n \nThis is a watermark for trial version, register to get full one!\nBenefits for registered user:\n1. Can remove all trial watermark.\n2. No trial watermark on the output documents.\nRemove it NowFIG. 5. Dust-acoustics waves in a vertical (Y{Z) plane at center (X = 0) in the presence of\ndi\u000berent strengths of external magnetic \feld (B). The argon pressure and rf power are set to 23\nPa and 3 W, respectively.\nTABLE I. Dust acoustic wave characteristics with magnetic \feld. The experiments are performed\nat \fxed gas pressure p = 23 Pa and rf power P = 3 W .\nMagnetic \feld Wavelength Phase velocity Frequency\nB in T (\u0015) in mm (vph) in mm/sec (fd=vph/\u0015) in Hz\n0 1.90\u00060.5 10.2\u00060.3 5.5\u00060.4\n0.02 1.63\u00060.4 13.12\u00060.3 8.2\u00060.3\n0.04 1.55\u00060.4 13.45\u00060.3 8.7\u00060.3\n0.06 1.59\u00060.4 13.12\u00060.5 8.2\u00060.4\n0.1 1.66\u00060.4 12.34\u00060.3 7.4\u00060.3\nelectrodes on the background of the positive space charge of ions. The energetic electrons\ngain energy in the oscillating sheath E-\feld and lose energy via collisions with neutral gas\nresulting in a net energy loss of electrons sustaining the discharge through the ionization\nmechanism43,44. It should be noted that during the rf period, a constant ion \rux is lost from\nthe bulk plasma to the electrodes. At each electrode, the ion current ( Ii) is compensated\nby the electron current ( Ie) that leave the plasma only at the time of the sheath collapse in\na rf cycle at the respective electrode. When the plasma equilibrium is achieved, the rate of\nchange of plasma density ( n) which depends on the ionization rate ( RI) and loss rate ( RL)\nis zero, i.e, dn=dt =RI\u0000RL= 0. The ionization rate depends on the electron density,\nelectron energy, and ionization frequency. The plasma loss rate is mainly de\fned due to the\n110.000.020.040.060.080.10-0.5-0.4-0.3-0.2-0.10.0Upper dust edge (mm)M\nagnetic field (T) Sheath edge (p = 23 Pa, P = 3 W) \nSheath edge (p = 27 Pa, P = 3.5 W)FIG. 6. Relative variation of upper edge particles (or sheath edge particles) with the external\nmagnetic \feld for discharge conditions of Fig. 3 and Fig. 5. The negative value represents a\nsuppression of the dust cloud or shifting of upper edge to a lower position. The error in calculated\nvalues are within \u000610 %\nambipolar di\u000busion of plasma species to the chamber wall. The rate of change of electrons\nmainly depends on the input power, ionization rate, and the internal energy of electrons43{46.\nAs dust particles are introduced in the plasma volume, the energetic electrons are lost\nto their surface and make the dust negatively charged47. Since electrons are \frst lost to\nthe particle surface to reduce the electron \rux and enhance the ion \rux, the density of free\nelectrons is less than the ion density48,49. In other words, dust particles provide an extra\nsurface area in addition to the chamber wall to the plasma losses through ambipolar di\u000busion.\nMore dust particles into the con\fning potential well reduce more free electrons or enhance\nthe plasma loss rate. At this condition, an accelerating E-\feld is necessary26to compensate\nfor the electron loss rate or to keep the current constant. Thus, the time-averaged E-\feld\nof sheath region where dust particles are levitated will be higher (Fig. 2(III)) than for the\ncase of less particles (Fig. 2(I)). Since the ions are drifting through the dust cloud, the drift\n12velocity,vi=\u0016iE, also increases in the higher E-\feld. It has been discussed that the drifting\nions relative to the stationary dust particles cause the instabilities, which excites the low-\nfrequency acoustic modes in a dusty plasma. It has been observed that excitation of waves\nis possible when the ion drift velocity crosses a threshold value which is higher than the ion\nthermal speed25{28, i.e.,vi> vith. For a \fxed ion mobility, \u0016i, the drift velocity increases\nlinearly with the E-\feld. Therefore, the DAWs are excited after adding more dust particles\nin a potential well of the rf sheath at given power and pressure, as is shown in Fig. 2(III).\nIt is a challenge to diagnose the background plasma of the dust grain medium which is\nnecessary to understand the dynamics of a volumetric dusty plasma. It has been discussed\nthat dust grains absorb free electrons and reduce the density of free electrons in a plasma.\nA dimensionless parameter, which is called Havnes parameter50,51Ph=Zdnd=nidecides the\ndensity of free electrons in a dusty plasma. It has been assumed that plasma parameters\nare not strongly a\u000bected by the dust grains if Ph<1 and plasma parameters can be used to\nestimate the dusty plasma parameters. In the case of high dust density, Ph>1, density of\nfree electrons is very low or one can consider the electron depleted dusty plasma. Recently, a\nnew technique has been proposed to diagnose the dusty plasma using the dispersion relation\nof excited waves in a nanodusty plasma51,52, where the havnes parameter is high ( Ph>1).\nIn the present work, havnes parameter Phhas the value between 0.1 to 0.3 for nd\u00183\u00021010\nto 9\u00021010cm\u00003,ni\u00186\u00021014to 8\u00021014cm\u00003, andZd\u00182\u0002103. In such dusty plasma\n(Ph<1), density of free electrons play a dominate role to determine the dust charges, sheath\nE-\feld, sheath thickness etc. Therefore, we \fnd di\u000eculty to use this new proposed technique\nto estimate the dust charge and electric \feld in the presence of external magnetic \feld. It\nhas been noticed in earlier studies as well in present study that the electrostatic probes\nmodify the dynamics of dust grains in a local region5,53,54and can not be used to measure\nthe plasma parameters and E-\feld during the wave motion. Since DAWs are excited due\nto adding more dust particles, plasma parameters without dust grains are not su\u000ecient to\nexplain the observed results. Therefore, observed results are qualitatively understood on the\nbasis of dust dynamics in the magnetized rf plasma. As the magnetic \feld is applied, \frst it\nreduces the loss rate of the energetic electrons to the chamber wall due to the reduction of\ngyroradius. At this discharge conditions, electrons are magnetized even at low B (B <0.01\nT) and ions remain unmagnetized below B <0.15 T. A fraction of these free electrons are\nlost to the dust grains and the remaining electrons gain energy in the time varying E-\feld\n13of sheath and lose their energy through collisions with neutrals. Hence, the plasma density\nis expected to increase with increasing the magnetic \feld. There is an another possible\nmechanism to increase the free electron density ( ne) or plasma density ( n) by applying a\nB-\feld. The magnetic \feld can also reduce the electron \rux to the dust grain due to the\ncross-\feld di\u000busion55. Since the radius of particles is 1 \u0016m, the electron current \rowing\nalong B to dust particles will not be a\u000bected by B56. However, it is expected to decrease\nthe electron current perpendicular to B which reduces the electron loss to dust grains. A\nreduction of the electron \rux to the dust surface in a strong B has been con\frmed by the\nnumerical simulation57. These additional free electrons involve in the ionization process\nto increase the plasma density. Hence, both mechanisms continuously increase the plasma\ndensity or free electron density in the dusty plasma volume with increasing magnetic \feld. It\nis also expected that the \frst mechanism to increase nedominates over the later mechanism\nat lower B (B <0.05 T). This increment in nereduces the sheath thickness at lower B (B <\n0.05 T), resulting in a reduction of the volume occupied by the dust grains. In a 2D (Y{Z)\nplane, the shifting of upper edge particles to a lower position, as presented in Fig. 6 is an\nindication of changing the sheath thickness. It is assumed that upper edge of dust cloud\nrepresents the sheath edge. Previous studies report a higher electron temperature ( Te) in\nthe dusty plasma volume48,49, which is expected due to the more energy gain by the free\nelectrons in the time-varying sheath E-\feld. Therefore, a higher Teis also expected in our\nexperiments which can a cause of the sheath expansion. However, the role of Teremains\nless e\u000bective than that of nein the lower B range. It has been discussed that the cause\nof instability is due to the loss of free electrons in a dusty plasma. As the density of free\nelectrons start to increase in the dusty plasma, the accelerating E-\feld starts to decrease.\nSince the stability of DAWs depends on the E/p ratio, which increases with lowering the\nratio by reducing E-\feld at constant pressure (p = 27 Pa). Therefore, the stable DAWs\npropagate in the lower magnetic \feld range (B <0.05 T). Further increase in magnetic\n\feld, B>0.05 T, increases the plasma density and electron temperature via ionization\nprocess in the time-varying E-\feld of the sheath. The increase in neshould further suppress\nthe dust grain medium but we observe a slight expansion of the dust cloud (see Fig. 6).\nShifting of the upper edge particles to a higher position indicates either the expansion of rf\nsheath or expansion of dust cloud. Expansion of the sheath is possible due to the dominant\nrole ofTerather than nein the higher B range. This variation in Temay also increase\n14the charge Qdon the dust particles58. Thus, the shifting of particles to higher position is\nan indication of increase in Tein the range of B 0.05 \u0015B\u00140.13T. Since the increase in\nneandTewith B continuously lowers the sheath E-\feld, the DAWs start to be damped\nafter B>0.05 T and are completely damped at higher B \u00180.13 T (see Fig. 3). Damping\nof DAWs is possible due to the suppression of instabilities associated with streaming ions\nin the sheath electric \feld. Since external magnetic \feld lowers the sheath E-\feld, drift\nvelocity (vi=\u0016iE) decreases with lowering the E-\feld and hence, DAWs get damped at\nhigher B. There is a limited theoretical work which reports the DAWs in a magnetized dusty\nplasma31,32. Salimullah et al.31have investigated the role of magnetic \feld on the DAWs in\nthe plasma and observed the modi\fcation as well as damping of DAWs in the presence of\nmagnetic \feld. Hence, the theoretical studies also support our experimental investigations.\nIn the case of low power and pressure (P = 3 W and p = 23 Pa) dusty plasma, ultimately\nwe are reducing the ionization rate, therefore, relative higher E-\feld is required to keep the\ncurrent constant or to balance the plasma loss rates. This higher accelerating E-\feld and\nlower ion mobility at lower pressure increase the E/p ratio and the dusty plasma becomes\nmore unstable. After turning on the B, the reduction in electron loss rates to the wall as\nwell as dust particles enhances the ionization rate which lowers the E-\feld. Therefore, the\npropagating characteristics of DAW as well as its stability change with increasing B. Since\nthe ion mobility does not depend on B, its value remains unchanged in the presence of B-\n\feld. Therefore, the medium support the DAWs even at higher magnetic \feld (Fig. 4 and\nFig. 5). The modi\fcation of the wave parameters in the presence of B at P = 3 W and p\n= 23 Pa is shown in Table I. The phase velocity ( Cda) of DAWs depends on Qd,nd, andni.\nThend,niandQdincreases with magnetic and the combined e\u000bect of all these variables\nincreasesCdaat lower B (B <0.05 T). Towards higher B, the dust cloud expansion decreases\nndand we also expect a sight higher value of ni. These both variables may decide the value\nofCda. Similarly, the change in wavelength can also be understood on the basis of dust\ncloud suppression and expansion in the presence of B. Hence, the basic properties of an rf\ndischarge and dusty plasma help to understand the stability and damping of DAWs in the\npresence of an external magnetic \feld.\n15VI. CONCLUSION\nThis paper highlights the results of dust acoustics waves in an rf plasma and its prop-\nagation characteristics in the presence of an external magnetic \feld. The dusty plasma is\ncreated by introducing MF particles of 1.025 \u0016m radius into the electrostatic potential well\nover the lower electrode created by the rf sheath \feld. The dusty plasma medium supports\nlow-frequency acoustics waves after adding more particles in the potential well. The insta-\nbility due to the streaming ions through the dust grain medium provides su\u000ecient energy\nto grow such modes. These self-excited waves propagate in the direction of drifting ions\nand have a characteristics similar to the sound waves or acoustic waves. The characteristics\nof the propagating dust-acoustics waves (DAWs) get modi\fed while the external magnetic\n\feld is introduced. For moderate pressure and power (P = 3.5 W, p = 27 Pa), the lower\nmagnetic \feld (B <0.05 T) modi\fes the propagation characteristics of DAWs and the waves\nget damped at the higher magnetic \feld (B \u00180.13 T). On the other hand, the magnetic\n\feld only stabilizes and modi\fes the characteristics of propagating DAWs while the power\nand pressure are lowered to 3 W and 23 Pa, respectively. The stability and damping of\nDAWs are qualitatively explained on the basis of the change of E=p ratio in the presence of\na magnetic \feld. The magnetic \feld lowers the E-\feld of sheath where particles are con\fned\nat given discharge conditions. At weak E-\feld streaming ions do not cross a threshold drift\nvelocity to excite the oscillatory motion of dust particles through the instabilities. But in\nlower pressure, the ion mobility is higher which increases the ion drift velocity even in a\nweak E-\feld. Therefore, the dust grain medium exhibits a wave like motion. In the dusty\nplasma, one of the challenges is to diagnose the background plasma in the presence of dust\nparticles. Probe techniques are not reliable to use in a dusty plasma because probes either\nperturb the dusty plasma or modify its characteristics. The use of spectroscopy also be-\ncomes complicated in the case of dusty plasma with an external magnetic \feld. Therefore,\nnumerical simulations of dusty plasmas in the presence of a magnetic \feld is required for\ndetailed study of DAWs in the magnetized rf plasma, which will be the scope of our future\nwork.\n16ACKNOWLEDGEMENT\nThis work is supported by the Deutsche Forschungsgemeinschaft (DFG). The authors are\ngrateful to Christopher Dietz and Benjamin Steinmueller for experimental assistance.\nREFERENCES\n1A. Barkan, N. D'Angelo, and R. L. Merlino, \\Charging of dust grains in a plasma,\" Phys.\nRev. Lett. 73, 3093{3096 (1994).\n2N. N. Rao, P. K. Shukla, and M. Y. Yu, \\Dust-acoustic waves in dusty plasmas,\" Planet.\nSpace Sci. 38, 543{546 (1990).\n3A. Barkan, R. L. Merlino, and N. 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Lett. 73, 3093{3096 (1994).\n21"    },    {        "title": "1112.5562v1.Temperature_gradient_assisted_magnetodynamics_in_a_ferromagnetic_nanowire.pdf",        "content": "Temperature gradient assisted magnetodynamics in a\nferromagnetic nanowire\nThomas Bose and Ste\u000ben Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany\u0003\n(Dated: June 1, 2022)\nAbstract\nThe dynamics of the low energy excitations in a ferromagnet is studied in case a temperature\ngradient is coupled to the local magnetization. Due to the di\u000berent time scales of changing temper-\nature and magnetization it is argued that only the coupling between the spatially varying part of\nthe temperature \feld and the magnetization is relevant. Using variational principles the evolution\nequation for the magnetic system is found which is strongly in\ruenced by the local temperature\npro\fle. The system o\u000bers damped spin wave excitations where the strength of damping is de-\ntermined by the magneto-thermal coupling. Applying the model to nanowires it is demonstrated\nthat the energy spectrum is signi\fcantly a\u000bected by the boundary conditions as well as the initial\ntemperature distribution. In particular, the coupling between temperature and magnetization is\nexpected to be several orders stronger for the open as for the isolated wire.\nPACS numbers: 75.78.-n; 75.30.Ds; 75.10.Hk; 05.70.Ln\n\u0003thomas.bose@physik.uni-halle.de; ste\u000ben.trimper@physik.uni-halle.de\n1arXiv:1112.5562v1  [cond-mat.mes-hall]  23 Dec 2011I. INTRODUCTION\nSpin-based electronics is a topical and challenging subject in solid-state physics includ-\ning the manipulation of spin degrees of freedom, in particular the spin dynamics and the\nspin-polarized transport [1]. In this regard it is desirable to \fnd proper means for the\ncreation and control of spin currents. As it was \frstly observed in [2] a spin current can\nbe thermally generated by placing a magnetic metal in a temperature gradient. This phe-\nnomenon is called spin Seebeck e\u000bect (SSE) and can be detected by attaching a Pt wire on\ntop of the magnetic metal and measuring a transverse electrical voltage in the Pt wire via\nthe inverse Spin Hall e\u000bect, the conversion of a spin current into an electric current [3, 4].\nNow a new sub\feld has emerged called 'spin caloritronics' [5] including the SSE as a local\nthermal transport e\u000bect [6]. In particular, the SSE combines spin degrees of freedom with\ncaloric properties. Further, the SSE was also observed in ferromagnetic semiconductors [7]\nand insulators [8]. Despite the absence of conduction electrons, a magnetic insulator can\nconvert heat \row into a spin voltage. The occurrence of a temperature-gradient induced\nspin current along the magnetization direction in a ferromagnet is called longitudinal SSE\n[9]. Here, the magnon-induced spin current is injected parallel to the temperature gradient\nfrom a ferromagnet into an attached paramagnetic metal. Theoretically the interplay be-\ntween temperature gradient and spin current within the SSE is investigated intensively. So,\nthe SSE is attributed to a temperature di\u000berence between the magnon temperature in the\nferromagnet and the electron temperature in the attached Pt contact [10]. Whereas in the\nconventional charge assisted Seebeck e\u000bect the Seebeck coe\u000ecient can be expressed by the\nelectronic conductivity, a microscopic model for spin transport driven by the temperature\ngradient was proposed in [11]. The authors \fnd thermally driven contributions to the spin\ncurrent in ferromagnetic metals. In case of ferromagnetic insulators a linear-response theory\nwas formulated [12] where the collective spin-wave excitations are taken into account. A\ndetailed numerical study of the SSE based on a modi\fed Landau-Lifshitz-Gilbert equation\nis presented in [13]. In addition, the scattering of a spin current generated by a temperature\ngradient is analyzed in [14] where the setup initiates a spin current torque acting on the\nmagnetic scatterer. As pointed out in [15] the domain wall motion can be also originated\nin magnonic spin currents. Likewise it is stressed in [16] a spin-transfer torque in a multi-\nlayer structure can be initiated by thermal transport from magnons. Recently [17] reported\n2that the SSE is considerably enhanced by nonequilibrium phonons that drag the low-lying\nspin excitations. Due to a phonon-magnon drag mechanism one \fnds a phonon driven spin\ndistribution [18] which in\ruences the SSE. The ampli\fcation of magnons by thermal spin-\ntransfer torque is observed in [19] where the system is subjected to a transverse temperature\ngradient. Another aspect of such collective magnetic excitations in a magnetic insulator\nis the transmission of electrical signals [20]. Measurements of the Seebeck coe\u000ecient and\nthe electrical resistivity of polycrystalline magnetic \flms are presented in [21]. The results\nprovide important information for further studies of the SSE in nanomagnets. Whereas\nmost studies of spin caloric e\u000bects are obtained for thin \flms, patterned ferromagnetic \flms\ndemonstrate the profound e\u000bect of substrate on the spin-assisted thermal transport [22].\nMotivated by recent experimental and theoretical \fndings related to SSE we study the\nmagnetization dynamics under the in\ruence of a temperature gradient. Because the relevant\ntime scale of the magnetization seems to be totally di\u000berent from the time scale of the tem-\nperature \feld our analysis is focused on the case that the temperature gradient is preferably\ncoupled to spatial variation of the magnetization. Due to spin-wave excitations those spatial\nvariations of the local magnetization have an impact on the spin dynamics. To that aim\nwe propose an action functional including both \felds, the local magnetization \feld m(x;t)\nas well as the local temperature \feld T(x;t). The functional includes a symmetry allowed\ncoupling between the \felds. Using variational principles we derive the relevant Bloch-type\nequations of motion. The evolution equations are discussed in detail for one-dimensional\nstructures under distinct boundary conditions. In detail we study a ferromagnetic nanowire.\nII. DERIVATION OF THE EQUATION OF MOTION\nLet us consider a ferromagnet below its Curie temperature under the in\ruence of a\ntemperature \feld varying in space and time. Then the action for the total system is composed\nof three parts\nS=Sm+ST+SI: (1)\nThe magnetic part Smgiving rise to the local precession of the magnetic moments m(x;t)\nreads [23]\nSm=Z\ndtZ\nddx\u0012J\u000b\f\n2@m\r\n@x\u000b@m\r\n@x\f+A\u000b(m)@m\u000b\n@t\u0013\n: (2)\n3Here, theJ\u000b\fcharacterize the exchange interaction between the magnetic moments while\nthe vector \feld Ais a functional of the magnetic moments mas indicated in Eq. (2). It\nwill be speci\fed below. Because our system is con\fned in a \fnite volume it is appropriate\nto separate the scalar temperature \feld T(x;t) = \b( x)\u0002(t). Following [24] the temperature\npart is written as\nST=Z\ndtZ\nddx\u0012@2\n@x2\n\u000b+1\n\u0014@\n@t+k2\u0013\nT(x;t)\n=Z\ndtZ\nddx\u0012\n\u0002@2\b\n@x2\n\u000b+\b\n\u0014@\u0002\n@t+k2\b\u0002\u0013\n:(3)\nThe constant \u0014is the heat conductivity or temperature di\u000busivity while kis the separation\nparameter with the dimension of a wave vector. Its physical meaning will be discussed\nbelow. The separation of the temperature \feld is likewise suggested by the well separated\ntime scales of the magnetization and the heat conduction. For that reason we argue that the\ninteraction part SIincludes only the coupling of the magnetization to the spatial varying\npart of the temperature \feld:\nSI=Z\ndtZ\nddx\u0012\n\u0000\u000b\f\r@m\u000b\n@x\f@\b\n@x\r\u0013\n: (4)\nWith other words the coupling re\rects the assumption that the spatial variation of the\nmagnetization is directly coupled to the gradient of the x-depending part \b( x). The tensor\n\u0000\u000b\f\rre\rects the coupling strength. As noticed above the separation parameter kin Eq. (3) is\nproportional to an inverse length scale k'l\u00001\nT, describing the range where the temperature\nis signi\fcantly changed in space. Having a nanowire or thin \flms in mind we assume\n\u0016m\u0014lT\u0014mm de\fned by the length extensions of the sample. The characteristic time\n\u001cTfor pure heat conductance in a con\fned medium is \u001cT'(\u0014k2)\u00001'\u0014\u00001l2\nT. Taking the\nthermal di\u000busivity \u0014= 2:51 mm2s\u00001for Y 3Fe5O12(YIG crystal) [25] one estimates the time\nscale\u001cTin the range \u0016s\u0014\u001cT\u0014s. In contrast magnetodynamics occurs a considerably\nshorter time scale within the nanosecond range: \u001cm\u0014ns, i.e.\u001cTis supposed to di\u000ber from\n\u001cmabout a few orders of magnitude, \u001cm\u001c\u001cT. From Eqs. (1)-(4) one \fnds the equations of\n4motion by variational principles:\n@2\b\n@x2\n\u000b+k2\b = 0; (5a)\n1\n\u0014@\u0002\n@t+k2\u0002 = \u0000\u000b\f\r@2m\u000b\n@x\f@x\r; (5b)\n\u0012@A\f\n@m\u000b\u0000@A\u000b\n@m\f\u0013@m\f\n@t=J\f\r@2m\u000b\n@x\f@\r+ \u0000\u000b\f\r@2\b\n@x\f@x\r: (5c)\nThis set of equations describes the in\ruence of a temperature pro\fle on the magnetization,\nEq. (5c), and the feedback of the magnetization on the temperature, Eq. (5b). In a \frst\napproximation we neglect the latter one. Consequently, Eqs. (5a) and (5b) represent the\nconventional heat equation after the separation of the variables. In the following we consider\nan isotropic magnet characterized by J\u000b\f=J\u000e\u000b\f, see Eq. (5c). Multiplying Eq. (5c) with\nmvectorial it results\n_m\u000b(m\u0001curlmA) =J\u0000\nm\u0002r2m\u0001\n\u000b+ \u0003\u000b\f\r\u000em\f@2\b\n@x\r@x\u000e: (6)\nHere there appears a tensor \u0003 \u000b\f\r\u000e, which includes the coupling between magnetization and\ntemperature pro\fle according to Eq. (4). Since \u0003is symmetric with respect to the indices\n\rand\u000eone gets for isotropic systems \u0003 \u000b\f\r\u000e =C1\u000e\u000b\f\u000e\r\u000e+C2(\u000e\u000b\r\u000e\f\u000e+\u000e\u000b\u000e\u000e\f\r). If\u0003= 0\nEq. (6) should conserve the spin length m2= 1 because of the absence of dissipative forces.\nFrom here we conclude that a pure reversible equation of motion can be obtained by setting\ncurlmA=\u0000\r\u00001mwhere\ris the gyromagnetic ratio. As the result one gets the Landau-\nLifshitz equation without relaxation. Using the same ansatz also for nonzero \u0003and setting\nin leading order C1=\u0000CandC2= 0 one \fnds from Eq. (6) the following equation\n@m(x;t)\n@t=\u0000\rJm(x;t)\u0002r2m(x;t) +\rC\u0002\nr2\b(x)\u0003\nm(x;t): (7)\nThe sign of the coupling parameter Cbetween temperature and magnetization will be dis-\ncussed in detail below. The \fnal evolution equation Eq. (7) does not conserve the spin\nlength anymore and has the form of a Bloch-Bloembergen-like equation [26, 27]. Therefore\nit should be applicable for ferromagnetic systems on a mesoscopic scale. To a\u000erm the rea-\nsonability of the result in Eq. (7) we brie\ry outline an alternative way of its derivation. Now\nthe starting point is the energy functional of the total system\nF=Fm+FT+FI; (8)\n5where it is likewise decomposed into a magnetic part Fm, a temperature part FTand an\ninteraction partFI. Within the Ginzburg-Landau theory the functional is given by\nF=Z\nddx\u0012J\u000b\f\n2@m\r\n@x\u000b@m\r\n@x\f+\u001f\u000b\f\n2@T\n@x\u000b@T\n@x\f+ \u0000\u000b\f\r@m\u000b\n@x\f@\b\n@x\r\u0013\n; (9)\nwhere the coupling tensors J\u000b\fand \u0000\u000b\f\rare the same as before in Eqs. (2) and (4). Moreover,\nwe have chosen a quadratic coupling of the gradient of the temperature characterized by\n\u001f\u000b\f. The function \b = \b( x) is again the space-dependent part of the scalar temperature\n\feldT(x;t) = \b( x)\u0002(t). In order to derive the desired equations of motion we follow the\ncommon approach [28]\n@T\n@t=\u0000\u001aT\u000eF\n\u000eT; (10a)\n@m\u000b\n@t=\u0000\u001amZ\nddyfm\u000b(x);m\f(y)g\u000eF\n\u000em\f(y)=\u001am\u000f\u000b\f\rm\f\u000eF\n\u000em\r: (10b)\nHere, the Poisson brackets fm\u000b(x);m\f(y)g=\u000e(x\u0000y)\u000f\u000b\f\rm\r(x) for classical magneti-\nzation densities are introduced, see [28, 29]. Now we use the separation ansatz for the\ntemperature in Eq. (10a) and restrict the calculation to isotropic situations; J\u000b\f=J\u000e\u000b\f,\n\u0003\u000b\f\r\u000e=\u0000C\u000e\u000b\f\u000e\r\u000eand additionally \u001f\u000b\f=\u001f\u000e\u000b\f. Identifying further \u001f\u001aT=\u0014and\u001am=\u0000\r\nthe resulting evolution equations coincide exactly with Eqs. (5a), (5b) and (7) for the tem-\nperature and the magnetization, respectively, as derived from the expressions for the action\nin Eqs. (1)-(4). The advantage of the second method is that the \frstly unknown quan-\ntityA(m), introduced in Eq. (2), is not necessary. On the other hand, the coincidence of\nthe obtained results for the equations of motion based on di\u000berent approaches justi\fes the\nassumptions made for the vector \feld A.\nIII. APPLICATION TO FERROMAGNETIC NANOWIRES\nIn this section Eq. (7) is analyzed for a ferromagnetic nanowire as schematically de-\npicted in Fig. 1. Because the model is focused on the ferromagnetic phase and the related\ncollective low energetic excitations, the magnetization-\feld is decomposed into m(x;t) =\nm0+ (x;t)\u0011 1(x;t)e1+ 2(x;t)e2+m0e3. Herem0is the constant magnetization\nwhereas the spin-wave amplitudes  1; 2obey the coupled equations\n@ 1\n@t=\rJm 0@2 2\n@x2+\rC@2\b\n@x2 1;\n@ 2\n@t=\u0000\rJm 0@2 1\n@x2+\rC@2\b\n@x2 2:(11)\n6y\nxz\n∇TT1 T2m(x1)m(x2)m(x3)FIG. 1. (Color online) Nanowire with locally varying temperature.\nThe magnetization dynamics is accordingly in\ruenced only by the spatially varying temper-\nature \feld \b( x), the complete dynamics of which are given by Eqs. (5a) and Eq. (5b). In\none dimension it results\n@2\b\n@x2=\u0000k2\b;@\u0002\n@t=\u0000\u0014k2\u0002: (12)\nThe solution for \b( x) in Eq. (12) can be written as a superposition \b( x) =P\nn\bn(x). Hence\n@2\b=@x2=\u0000P\nnk2\nn\bncan be substituted in Eq. (11). In terms of the Fourier transformed\n~\bn(q;t) =FTf \b(x;t)gand'i(q;t) =FTf i(x;t)gEq. (11) can then be transformed into\n@\n@t'1(q;t) =\u0000a2q2'2(q;t)\u0000ra2X\nn\u0012\nk2\nnZ\ndq0'1(q\u0000q0;t)~\bn(q0)\u0013\n;\n@\n@t'2(q;t) =a2q2'1(q;t)\u0000ra2X\nn\u0012\nk2\nnZ\ndq0'2(q\u0000q0;t)~\bn(q0)\u0013\n:(13)\nHere the time tis measured in terms of \u0016t=\r~Jm 0t, where for convenience \u0016tis already\ncalledtin Eq. (13). The other dimensionless quantities are de\fned by\nJ=~Ja2; C=Km 0a2; r=K=~J; (14)\nwhereais the lattice constant. The separation parameters knand the ~\bn(q) are determined\nas the solution of Eq. (12) under consideration of the relevant boundary conditions.\n7A. Zero heat \row at the boundaries\nLet us \frstly study the case when there is no heat current \rowing through the edges of\nthe wires. The initial and boundary conditions for the heat equation Eq. (12) read\nT(x;t= 0) = \u0002 0X\nn\bn(x);@T(x;t)\n@x\f\f\f\f\nx=0;L= 0; (15)\nwhereLis the length of the nanowire. The solution of Eq. (12) ful\flling the boundary\ncondition Eq. (15) is\n\bn(x) =uncos(knx); kn=n\u0019\nL; n = 0;1;2;:::: (16)\nPerforming the Fourier transformation we obtain\n~\bn(q) =un\u0019[\u000e(q+kn) +\u000e(q\u0000kn)]: (17)\nInserting this relation into Eq. (13) leads to terms of the form 'i(q\u0006kn) which can be\nexpanded with respect to the small parameters kn:\n'i(q\u0006kn)''i(q)\u0006@'i\n@qkn+O\u0000\nk2\nn\u0001\n: (18)\nIs\u0015the wave length of the low energetic excitation in the ferromagnet with the corresponding\nwave vector q\u00181=\u0015and setting kn\u00181=Lthe expansion made in Eq. (18) is valid for\nkn\u001cq\u001c1=aorL\u001d\u0015\u001da, respectively. Introducing \u0011='1+ i'2and combining\nEqs. (17)-(18) with Eq. (13) one obtains the following evolution equation for \u0011:\n@\n@\u0016t\u0011(q;\u0016t) =\u0000\nia2q2\u0000\u0016\u000b\u0001\n\u0011(q;\u0016t);\u0016\u000b= 2\u0019ra2X\nnk2\nnun:\nThe solution for the '= ('1;'2) with the initial conditions '(t= 0) ='0= ('01;'02) is\nwritten in terms of the real time tas\n'(q;t) =0\n@cos [!(q)t]\u0000sin [!(q)t]\nsin [!(q)t] cos [!(q)t]1\nA'0exp[\u0000\u000bt]; ! (q) =\rJm 0a2q2: (19)\nHere the spin wave energy !(q) and the damping constant \u000bare de\fned by\n!(q) =\rJm 0q2; \u000b = 2\u0019rJ\rm 0X\nnk2\nnun: (20)\nThe expression for !(q) re\rects the common quadratic spin wave dispersion relation. The\ntemperature gradient leads to damped spin waves whereas the damping parameter \u000bis\n8q108m−1\nt(a)\nq108m−1\nt\n(b)\nFIG. 2. (Color online) Spin waves '1(q;t)='0(color scale) for a= 1\u0017A, L = 1\u0016m andr= 100\nfor the cases: (a) vanishing heat current at the edges of the nanowire according to the initial heat\npro\fle I in the inset in Fig. 3(a); (b) \fxed temperatures at the edges referring to the initial heat\npro\fle III in the inset in Fig. 3(b).\nde\fned in Eq. (20). The strength of the attenuation \u000bcan be estimated from Eq. (20)\nand Eq. (14) using \u000b/CP\nnk2\nnun. In order that the damping is physically realized we\nhave to discuss the sign of the parameter C/rintroduced in (7). It characterizes the\ncoupling between the local magnetization and the local temperature. Moreover, the sign of\nP\nnk2\nnunis determined by the sign of the coe\u000ecients unwhich is given in terms of the initial\ntemperature distribution, compare Eqs. (15)-(16). A positive damping parameter \u000b >0 is\nrealized by assigning\nX\nnk2\nnun70 =)r70: (21)\nThet- andq-dependent ratio '1='0is plotted in Fig. 2(a). As expected the frequency is\nincreased when qis enlarged. Moreover the damping should be observed within a picoseconds\n9ϕ1/ϕ0\ntr= 100\nr=−100T0/Θ0\nx/LI II(a)\nϕ1/ϕ0\ntr= 100\nr=−100˜T0/Θ0\nx/LIII IV\n(b)\nFIG. 3. (Color online) Spin wave solutions for a= 1\u0017A, L = 1\u0016m andq= 4\u0002108m\u00001for the cases:\n(a) vanishing heat current at the edges of the nanowire with the initial heat pro\fles I ( un= 1=n2,\nu0= 10) and II ( un=u4=\u00001,u0= 10) mentioned in the text; (b) \fxed temperatures at the\nedges with the initial heat pro\fles III ( vn=\u00001=n2,T1=\u00020= 12,T2=\u00020= 8:5) and IV ( vn=v3,\nT1=\u00020= 10 =T2=\u00020) in the inset mentioned in the text. Here only the envelope of the solution\nreferring to heat pro\fle III is plotted.\ntime-scale. To illustrate that we present the dynamic solutions for di\u000berent heat pro\fles in\nFig. 3(a). The initial temperature distribution is chosen by T0=\u00020=u0+P\nnuncos[knx]\nand is depicted in the inset of Fig. 3(a). To reach a damping of the spin wave in the ps-\nrange we setjrj=K=~J= 100, where ris a measure for the relation of the coupling of the\ntemperature and magnetization pro\fles (i.e. r\brm) and the magnetization-magnetization\n(i.e.rmrm) coupling, see Eq. (14). In that case one observes a strong coupling between\ntemperature and magnetization.\n10B. Fixed temperatures at the boundaries\nAs the second realization we consider the case that the edges of the wire are held at \fxed\ntemperatures:\nT(x;t= 0) = \u0002 0X\nn\bn(x);\b(x= 0) = \b(x=L) = 0: (22)\nTherefore, Eq. (22) refers to the transformed temperature \feld T(x;t) = ~T(x;t)\u0000(1\u0000\nx=L)T1\u0000x=LT 2, whereT1=~T(x= 0;t) andT2=~T(x=L;t) are \fxed and ~T(x;t) is the\noriginal temperature pro\fle in the wire. The spatial temperature pro\fle \b( x) is obtained\nfrom Eq. (12) combined with the boundary conditions in Eq. (22). The solution is the\nsuperposition of\n\bn(x) =vnsin (knx); kn=n\u0019\nL; n = 1;2;:::: (23)\nThe Fourier transform of that \b n(x) reads\n~\bn(q) = ivn\u0019[\u000e(q+kn)\u0000\u000e(q\u0000kn)]: (24)\nUtilizing again the expansion in Eq. (18) and the transformation \u0011='1+ i'2we \fnd by\ninserting Eq. (24) in Eq. (13)\n@\n@\u0016t\u0011(q;\u0016t) + i \u0016\f@\n@q\u0011(q;\u0016t) = ia2q2\u0011(q;\u0016t); \u0016\f= 2\u0019ra2X\nnk3\nnvn: (25)\nThis partial di\u000berential equation of \frst order can be solved by using the method of char-\nacteristics. The solution for '= ('1;'2) with the initial conditions '(t= 0) ='0is given\nby\n'(q;t) =0\n@cos [\n(q;t)t]\u0000sin [\n(q;t)t]\nsin [\n(q;t)t] cos [\n(q;t)t]1\nA'0exp[J\rm 0\f(t)qt]: (26)\nThe quadratic spin wave dispersion relation \n( q;t) and the damping parameter \f(t) are\nde\fned by\n\n(q;t) =J\rm 0\u0012\nq2\u0000\f2(t)\n3\u0013\n; \f (t) = 2\u0019rJ\rm 0tX\nnk3\nnvn: (27)\nObviously the quadratic spin wave dispersion relation is now modi\fed by a time depending\ncorrection term J\rm 0\f2(t)=3, i.e. the spin wave dispersion is changed by the damping\n11parameter\f(t) which occurs as a factor in the exponential function in Eq. (26). The decay\nof this function is guaranteed for negative values \f(t)<0. In the same manner as before,\nusing Eq. (27) and Eq. (14), we estimate \f/CP\nnk3\nnvn. Thus, one has to distinguish\nX\nnk3\nnvn70 =)r?0: (28)\nReferring to the initial heat distribution ( ~Twas introduced below in Eq. (22)) we assume\n~T0=\u00020= (1\u0000x=L)T1=\u00020+xT2=L\u00020+Pvnsin[knx], see the inset in Fig. 3(b). The\nsolution'1(q;t)='0is shown in Fig. 2(b). Contrary to the previous case of an absent heat\n\row through the ends of the wire now we observe an enhancement of the lifetime of the spin\nwaves, cf. the time scales of Figs. 2(a) and 2(b). The same e\u000bect of increased spin wave\nlifetimes is noticeable in Fig. 3 where the solution for q= 4\u0002108m\u00001is illustrated for di\u000berent\nheat pro\fles but identical values of the coupling relation rfor both types of boundary\nconditions. In Fig. 3(b) we only plot the envelope of the solution for one of both cases.\nHowever, the curve shown is featured by a decrease of the frequency up to a time of \u0018430ps\nwhich is determined by the maximum of the argument \n( q;t)tof the periodic functions.\nThereafter the frequency increases continuously with time. A quantitative comparison of\nthe lifetimes for comparable temperature distributions in Fig. 3 o\u000bers that the characteristic\ntime scale is 40\u000090 times larger for the case when heat \rows through the edges than for\nthe case the nanowire is isolated. The reason for that consists of the expansion made in\nEq. (18). Because in Eq. (25) all even terms cancel mutually in the combination 'i(q+\nkn)\u0000'i(q\u0000kn) and hence only the odd terms remain. This leads to an alteration of\nthe damping constant in Eqs. (26)-(27) compared to \u000bin Eqs. (19)-(20). The physical\npicture behind might be as follows: As discussed the type of the temperature pro\fle \b( x)\ndetermines the strength of coupling. In our model the solution for \b( x) is selected by the\nboundary conditions either in Eq. (15) or in Eq. (22). The complete isolation of the nanowire\nimposes the restriction on the thermo-magnetic system that it is not able to exchange energy\nwith the environment. This leads to very limited possibilities for the self-organization of the\nmagneto-caloric behavior. Otherwise, if heat is allowed to \row through the edges of the wire\nthe thermo-magnetic system is in contact with its surrounding. Thus the self-organization\nprocess can be optimized due to a broader variety of its realization.\n12IV. CONCLUSIONS\nIn the present work the low energetic excitations in a ferromagnetic nanowire had been\nanalyzed when the spatial variations of the magnetization is coupled to the correspond-\ning temperature gradient. The resulting spin-waves are damped due to the temperature\ngradient. This result contributes to the recent e\u000bort in understanding spin caloritronics\nphenomena. Whereas the spin Seebeck e\u000bect is focused on the creation of a spin current\ndue to a temperature gradient the goal of our study is to elucidate the behavior of the per-\nmanent e\u000bect of that temperature gradient on the excitation of the magnetic system. Once\nthe magnetic system is excited the shape of the temperature pro\fle in\ruences the damping,\ntoo. Further, the external conditions under which an experiment is realized should a\u000bect\nthe dynamics as well. In this regard we found that the system is strongly controlled by the\nboundary conditions which modify the damping parameter. While the complete isolated\nnanowire o\u000bers conventional spin wave excitations which are damped due to the coupling to\na spatial changing temperature \feld the situation is more complex in case the ends of the\nnanowire are held at \fxed temperatures. Referring to the latter realization it results that\nthe damping parameter is time-dependent and modi\fes the spin wave dispersion relation.\nHowever, if one considers systems which are subjected to identical boundary conditions the\nactual initial temperature distribution also a\u000bects the spin dynamics. In principle it should\nbe possible to validate our theoretical \fndings experimentally by spin Seebeck e\u000bect mea-\nsurements or ferromagnetic resonance techniques in low dimensional samples. Our studies\ncan be accordingly extended to higher dimensions. Further, the model can be re\fned by\ntaking into account higher order terms in the expansion made in Eq. (18). The inclusion of\nstochastic forces should also improve the model slightly.\nOne of us (T.B.) is grateful to the Research Network 'Nanostructured Materials' , which\nis supported by the Saxony-Anhalt State, Germany.\n13[1] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004)\n[2] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh,\nNature 455, 778 (2008)\n[3] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006)\n[4] S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006)\n[5] G. E. Bauer, A. H. MacDonald, and S. 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China, and\nE-mail: shaozhenli@yahoo.com\n\u0003To whom correspondence should be addressed\n†Materials Science Division, Argonne National Laboratory\n‡School of Physics and Institute for Quantum Materials, Hubei Polytechnic University\n¶S.L., W.Z., and J.D. contributed equally to this work.\n1arXiv:1512.00286v1  [cond-mat.mes-hall]  1 Dec 2015Abstract\nMagnetic insulators such as yttrium iron garnet, Y 3Fe5O12, with extremely low magnetic\ndamping have opened the door for low power spin-orbitronics due to their low energy dissi-\npation and efficient spin current generation and transmission. We demonstrate reliable and\nefficient epitaxial growth and nanopatterning of Y 3Fe5O12thin-film based nanostructures on\ninsulating Gd 3Ga5O12substrates. In particular, our fabrication process is compatible with\nconventional sputtering and liftoff, and does not require aggressive ion milling which may be\ndetrimental to the oxide thin films. Structural and magnetic properties indicate good quali-\nties, in particular low magnetic damping of both films and patterned structures. The dynamic\nmagnetic properties of the nanostructures are systematically investigated as a function of the\nlateral dimension. By comparing to ferromagnetic nanowire structures, a distinct edge mode\nin addition to the main mode is identified by both experiments and simulations, which also\nexhbits cross-over with the main mode upon varying the width of the wires. The non-linear\nevolution of dynamic modes over nanostructural dimensions highlights the important role of\nsize confinement to their material properties in magnetic devices where Y 3Fe5O12nanostruc-\ntures serve as the key functional component.\nKeywords: nanopatterning, ferromagnetic resonance, magnetic oxides, spin current\n2Introduction\nMagnetic insulators such as yttrium iron garnet, Y 3Fe5O12(YIG) with extremely low magnetic\ndamping have been indispensable in almost every aspect of contemporary spin-orbitronics re-\nsearch.1,2For many decades, the growth of single crystal YIG films has been dominated by liquid\nphase epitaxy (LPE),3which yields films in the thickness range from several hundreds of nanome-\nters to millimeters. On the other hand, the exotic magnetic properties revealed recenlty from var-\nious YIG-based magnetic heterostructures4–8call for urgent need for their nanostructured forms\nin order to realize practical devices for applications, including spin transfer torque devices,9–11\nmagnetic logic devices,12,13auto-oscilators,9,14and skyrmion memories.15However, the con-\nventinal LPE method is incompatible with current industrial top-down nanofabrication technolo-\ngies. Recently, great advances have been achieved in the growth of high quality YIG films using\npulsed laser deposition (PLD) and magnetron sputtering at elevated temperatures,16–22yielding\nnanometer-thick films with low magnetic damping similar to single-crystal YIG bulk materials.\nFor example, a Gilbert damping constant of a= 0.00023 has been achieved by Sun et al16and\nby d’Allivy Kelly et al17using PLD, and a\u00180.00009 has been achieved by Chang et al19using\nmagnetron sputtering. These demonstrations, more suited for commercial production, are of great\ntechnological significance.\nDespite the intensive investigations on YIG continuous films, an efficient yet reliable method\nfor epitaxial patterning of YIG nanostructures is still missing. Microstructured YIG films have\nbeen prepared in the past using aggressive ion etching of sputtered YIG films while using resist\nmask to define the morphological structures.9,23,24However, the injected, highly energized Ar ions\nare also detrimental to the films, particularly harmful to oxides with higher stiffness as opposed to\nmetals.25Mechanical defects and even cracks, as well as non-trival modifications of the magnetic\nproperties such as saturation magnetization and damping constant have been observed after the ion\nmilling process.24In this work, we report high-quality growth of YIG films using combined room\ntemperature (RT) magnetron sputtering and ex-situ post annealing. In particular, our approach\nis also compatible with modern nano-lithgraphy techniques, taking advantage of the RT deposi-\n3tion. We demonstrate epitaixal patterned YIG nanostructures using electron beam lithography and\nliftoff. A Gilbert damping constant comparable to the extended thin-films is achieved even for the\npatterned structures.\nResults and discussion\nThe YIG films are deposited on (111)-oriented gadolinium gallium garnet (Gd 3Ga5O12, GGG)\nsingle crystal substrates at room temperature (RT) from a commercial YIG sputter target. The Ar\ngas flow, chamber pressure, and sputtering power are kept at 16 sccm, 10 mT, and 75 W, respec-\ntively, which is the optimal deposition enviroment for RT growth. It has been shown previously\nthat these films can display different surface morphologies26depending on the growth conditions.\nHere, we used very low sputtering rate and deposited our films from a stoichiometric YIG targets\nunder ultrahigh vacuum. In addition, our sputtering system has an on-axis geometry which does\nnot induce any substrate misorientation. The as-grown films have a dark gray color implying that\nthe stoichiometry has changed during the RT deposition. The films are subsequently subjected\nto an ex-situ post annealing at 800\u000eC for 2 hours in a tube furnance with continuous air flow.\nThe temperature ramping rate is \u0018120\u000eC / hour. The key attributes for obtaining a good flim\nin our process is to ensure an oxygen-rich environment during annealing. Nevertheless, we found\nthat flowing just air at ambient pressure is sufficient enough instead of using pure oxygen. Alter-\nnatively, such annealing can also be done in-situ after the film growth;18however, much longer\nannealing time ( >10 hours) is required primarily due to the limitation of the maximum achieve-\nable oxygen partial pressure in the commercial sputtering chamber. The annealed films have a light\nyellow color (nanometer-thick YIG films).\nNanostructured YIG films are fabricated by RT deposition onto PMMA/PMGI bilayer resisits\ndefined via electron beam lithography (Raith 150). Since the GGG substrate is highly insulating,\nwe sputtered 5-nm-thick Au after spin coating the bilayer resists to allow efficient electron charge\ndissipation during the lithography. The Au layer is subsequently removed by gold etcher, followed\n4by wet development of PMMA (MicroChem MIBK) and PMGI (Shipley CD-26), respectively.\nThe PMMA/PMGI bilayer resists can form an undercut cross-section profile which is suitable for\nmagnetron sputtering deposition. After deposition of 40 nm YIG film using the same recipe as\nbefore, we remove the resists and surplus materials on top by a resist remover (Shipley 1165).\nFinally, the nanostructured samples are annealed using the same recipe as for continuous films.\nAlthough we choose to use electron beam lithography for our demonstration, the general process\nis also applicable to other unconventional lithography techniques.27\n20 30 40 50 60 70 80 90100102104106\nGGG (666)GGG (444)YIG (444)\n  Intensity (a.u.)\n2 (o)YIG (222)\n50.0 50.5 51.0 51.5 52.0102104106 K2K1GGG (444)YIG (444)\n  Intensity (a.u.)\n2 (o)a b \n-40 -20 0 20 40-150-100-50050100150\n   Magnetization (emu/cm3)\nH (Oe) c \nFigure 1: (a) X-ray diffraction pattern of a 40 nm annealed YIG film, and (b) the same spectrum in\nan expanded scale showing the YIG(444) peak. (c) Magnetic hysteresis loop of the same sample\nmeasured with an in-plane magnetic field, yielding a saturation magnetization value of \u0018130\nemu/cm3.\nWe first characterized our YIG films structurally and magnetically. Figure 1(a) and (b) show\nthe x-ray diffraction pattern our YIG films. The data confirm the (111)-oriented YIG phase in the\nsamples and show no evidence for the existence of any additional phases. Figure 1(c) shows the\nmagnetic hysteresis loop measured by a vibrating sample magnetometry with an in-plane magnetic\nfield. The data indicates very small coercivity, below 1 Oe, and a saturation magnetization, Ms=\n130\u000620 emu/cm3. This value is in equivalent to a 4 pMs=1633\u0006251 G, which is only 6%\nsmaller than the literature value for bulk YIG crystals.1A more precise value of 4 pMswill be\ndetermined via dynamic measurements below.\nDynamic magnetic properties are investigated by a vector-network-analyzer ferromagnetic-\nresonance method (VNA-FMR) in a probe station using low-loss rf probes. We make 18 mm\n\u00022 mm large YIG bars using the previously described patterning method. Considering that the\nthickness of the YIG film is only 40 nm, these structures are large enough to be considered as\n5-3000 -2500 -2000 -1500 -10004681012\n  \n Fm1 (1827 G ) \n Fm2 (1817 G )\n Fm3 (1856 G )f (GHz)\nH (Oe)\n0 2 4 6 8 10 12 1403691215\n  H (Oe)\nf (GHz) Fm1 ( = 3.53 x 10-4)\n Fm2 ( = 2.93 x 10-4)\n Fm3 ( = 7.56 x 10-4)b c \nhrf \nH G S G a Figure 2: (a) VNA-FMR measurement configuration showing the directions of static and dynamic\nmagnetic fields with respect to the samples. Ferromagnetic resonance data of three YIG large\nbars (Fm 1;2;3) showing (b) resonance frequency versus field and corresponding fitting with Kittel\nequation, and (c) resonance linewidth versus frequency and corresponding linear fitting.\n6continuous films, which also serve as the reference samples for our YIG nanostructures as will be\ndiscussed later. On chip coplanar-waveguides made from Ti(5 nm)/Au(150 nm) are subsequently\nfabricated on top of the YIG bars by photolithography and liftoff. The measurement configuration\nis illustrated in Figure 2(a). We show in Figure 2(b) and (c) the ferromagnetic resonance data\nof three different YIG samples (Fm 1;2;3, where Fm denotes ‘Film’). The resonance field, HFMR,\nversus frequency, f, can be fitted by the Kittel equation:\nf=jgjp\nHFMR(HFMR +4pMs); (1)\nwhere gis the gyromagnetic ratio. Our fitting (Fig. 2(b)) yields 4 pMsvalues of 1827, 1817, and\n1856 G for Fm 1, Fm 2, and Fm 3, respectively. On the other hand, the resonance linewidth, DH,\nversus fcan be linearly fitted with:\nDH=2a\njgjf+DH0; (2)\nwhere DH0denotes the inhomogenous linewidth broadening. Our fitting28[Fig. 2(c)] yields mag-\nnetic damping values of 3.53, 2.93, and 7.56 ( \u000210\u00004), respectively. The similar 4 pMsas well as\nthe low magnetic damping values for these different samples demonstrate the reproducibility of our\nfabrication process. These values are also comparable to that of similar YIG thin-films fabricated\nby other approaches.16–19\nWe next move onto the discussions of nanostructured YIG. We make arrays of YIG nanowires\n(NW), and nanodots (ND) with varying dimensions using the previously described method. Owing\nto the top-down lithography used here, all nanostructures exhibit quite uniform size and spacing.29\nAs a demonstration, we fabricated NW samples with different widths, denoted as NW 300;450;600;750;1800,\nwhere the subscript indicates the wire width in nanometer. We also make circular and elliptical dots\nof similar dimensions. Figure 3 shows the morphology of our nanostructured YIG samples by us-\ning scanning electron microscopy after 5 nm Au coating. Clean edges and faithful pattern transfer\nare achieved thanks to the well-defined undercut resist bilayer. The morphology of these patterned\n7films is also similar to their continuous-film counterparts due to the identical growth and annealing\nconditions.\n2mm \n 200 nm  \n2mm 200 nm  \n2mm 200 nm  \n2mm 200 nm  a \nb \nc \nd e \nf \ng \nh \nFigure 3: Scanning electron microsopy images of epitaxial patterned YIG nanostructures on GGG\nsubstrates showing (a) 300-nm-wide wires, (b) 750-nm-wide wires, (c) 600-nm-radius circular\ndots, (d) 410-nm\u0002670-nm elliptical dots, and their corresponding zoomed-in images (e-h).\nWe fabricate on-chip coplanar waveguides on top of each patterned arrays in order to study the\ndynamic properties of nanostructured YIG. We focus primarily on the NW samples with varying\nwidths so that the effects of geometrical confinement to the magnetization dynamics can be sys-\ntematically investigated. The gray scale mapping in Figure 4(a) and (b) compare the FMR property\nof Fm 1and NW 300. The bright color indicates a low microwave absorption, while the dark color\ncorresponds to a high microwave absorption. The FMR spectrum of the continuous film is sym-\nmetric, while the spectrum of NW 300exhbits a clear asymmetry with respect to H= 0 due to the\nshape anisotropy originated from the confined edges of the nanostructures. In addition to the main\nFMR mode, we clearly identify an appreciable edge mode for the NW 300sample. Both the main\n8and edge modes are accurately reproduced by our micromagnetic simulations using MuMax Sim-\nulator, see Fig. 4(c) and (d). The parameters used for the simulation are: Ms=147:77 emu/cm3,\nAex=4\u000210\u000013J/m, a=7:561\u000210\u00004,g=0:00284 GHz/Oe. All the values are based on the\nfitting results of the reference sample. At H= -1000 Oe, the simulated edge mode sits at \u00185.5\nGHz [Fig. 4(d)], which agrees very well with the experiment.\n4.04.24.44.64.85.05.25.45.65.86.0FMR absorption (a.u.)\n  \nf (GHz)Fm1 \nH = -1000 Oe  \n4.04.24.44.64.85.05.25.45.65.86.0\n  FMR absorption (a.u.)\nf (GHz)NW300 \n5.4 5.5 5.6\n  \nFm1 NW300 a b \nc d \nFigure 4: FMR 2D-spectrum measured for samples (a) Fm 1and (b) NW 300, and the corresponding\nmicromagnetic simulations at H= -1000 Oe (c-d). Both main and edge modes are identified for\nNW 300. Inset pictures of (d) show simulated spatial distribution of magnetization dynamics from\nthe two modes, in which the red color indicates a high spin precession amplitude, and the blue\ncolor corresponds to low amplitude.\nEdge modes can be expected for generic nanopatterned magnetic films. However, earlier work\nhas shown that such modes are missing in high damping ferromagnetic metals such as Permalloy\n(Py, Ni 80Fe20) nanowires with similar structural dimensions.30The distinct edge modes of YIG\nnanostructures observed here is owing to the intrinsic low magnetic damping and weak magnon\nexchange interactions of YIG, so that they can be well separated from the main FMR mode.\nFigure 5 further demonstrates the evolution of such edge mode with the width of the nanowires.\nStarting from NW 300, the edge mode moves closer for NW 450as the width increases, and finally\nmerges with the main FMR mode for NW 600. Notably, this mode starts to reoccur for NW 750and\n9NW300 \nNW450 \nNW600 \nNW750 \nNW1800  \nFm1 Figure 5: FMR 2D-spectrum of Fm 1and NW series of samples, showing evoluation of the coer-\ncivity (dashed line) and the magnetization dynamic modes over the width of the wires. Arrows\nindicate the existence of edge modes. Only the ascending field branch of the hysteresis is shown\nfor simplicity.\n10NW 1800as the width further increases. This cross-over between main and edge modes indicates\na higher sensitiviy of the edge mode than the main mode to the change of local demagnetization\nfield since they reside primarily at the edges of the nanowires where the pinning is much stronger\nas opposed to the center [Fig. 4(d)]. The strong demagnetization field is also evidenced by the fact\nthat even the main mode shows clear evolution with width at H= 0 Oe, as shown in Fig. 6(a) and\n(b). However, when the external field is sufficiently high, no appreciable evolution with width can\nbe observed for the main mode, Fig. 6(b).\n0246810121416182001020304050\n  H (Oe)\nf (GHz)Fm1\nNW300\nNW450\nNW600\nNW750\nNW1800\n02468101214161820012345\n   (x10-3)\nWire Width (x 102 nm)c d b \n0 2 4 6 810 12 14 16 18 200.00.20.40.60.81.01.21.4\n  f = (fNW - fFm1) / fFm1\nWire Width (x 102 nm)H (Oe)\n 0  -500 \n -1000  -2000\n0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0FMR Absorption (a.u.)H = 0 OeNW1800\nNW750\nNW600\nNW450\nNW300 \nf (GHz)Fm1 a \nFigure 6: (a) FMR 1D-spectrum of Fm 1and NW series samples measured at zero external field\nshowing the evolution of the main FMR mode. (b) Percent change in the resonance frequency\nof the main FMR mode at different external applied field, H= 0, -500, -1000, and -2000 Oe for\ndifferent width of the nanowire. (c) Resonance linewidth versus frequency of the different samples.\n(d) Extracted magnetic damping values for nanowires with different width. Dashed line indicates\nthe damping value for the continuous film.\nThe ‘appear-merge-reoccur’ behavior of the edge mode to the main mode (Fig. 5) is an inter-\nesting observation, which also finds good agreement with the evolution of the damping constant.\nWe study the frequency dependent resonance linewidth for all our NW samples [Fig. 6(c)] and ex-\ntract the damping parameters via fitting with Eq.(2), as shown in Fig. 6(d). Starting from NW 300,\nthe damping constant increases as width increases, and peaks at NW 600, corresponding to exactly\nthe merge point of the edge and main modes in Fig. 5. This is explained by the fact that when\n11the two modes degenerate, they are coupled due to magnon-magnon scattering leading to efficient\nenergy transfer between the modes,31which as a result has their effective damping increased. The\nloss of the edge mode at this point is compensated by the large damping enhancement for the main\nmode. After all, it is the total linewidth that truly quantifies the losses of magnetic energy regard-\nless of the nature and number of microscopic mechanisms involved. In fact, such degeneracy is\nsomething to be avoided for device applications such as for the magnetic auto-oscilators, because it\ncould lead to selflimiting damping and prevent the onset of auto-oscilations. Therefore, our results\nhere highlight the significance and general guidance of proper device engineering for YIG based\nspintronics.\nConclusions\nMagnetic insulators with low magnetic damping such as Y 3Fe5O12have been widely proposed and\ninvestigated as good candidates for pure spin current spintronics concepts. Fundamental studies\nhave reached several milestones in many sub-fields such as spin pumping, spin-Seebeck effect,\nspin transfer torque, and auto-oscilation. However, successful patterning of nanostructures from\nsuch materials for practical device applications has only recently become feasible.\nHere, we have demonstrated reliable and efficient epitaixal growth and nanopatterning of\nY3Fe5O12thin-film based nanostructures on insulating GGG substrates. Strucutral and magnetic\nproperties indicate good qualities, in particular low magnetic damping of both films and patterned\nstructures. We systematically studied the evolution of the dynamic magnetization parameters over\nthe change of the lateral dimensions. A distinct edge mode in addition to the main mode is iden-\ntified by both experiments and simulations, which also exhbit cross-over with the main mode over\nchanging the width of the wires. The cross-over leads to also a significantly enhanced magnetic\ndamping. The non-linear evolution of dynamic modes over nanostructural dimensions highlights\nthe important role of size confinement to their material properties in magnetic devices where YIG\nnanostructures serve as the key functional component.\n12Acknowledgement\nWe thank Dr. Jennifer Zheng and Dr. Junjie Zhang for technical help for using the tube furnace.\nWork at Argonne was supported by the U.S. Department of Energy, Office of Science, Materials\nScience and Engineering Division. Use of the Center for Nanoscale Materials was supported by\nthe U. S. Department of Energy, Office of Science, Basic Energy Sciences, under Contract No.\nDE-AC02-06CH11357. S.L. acknowledge the National Natural Science Foundation of China (No.\n51302074 and 11374147), the Natural Science Foundation of Hubei Province (No. 2012FFB010),\nthe Creative team of Hubei Polytechnic University of China (Project No. 13xtz05), and the Edu-\ncation Foundation of Hubei Province (D20144402).\nConflict of Interest : The authors declare no competing financial interest.\nReferences\n(1) Edited by Wu, M. and Hoffmann, A. Recent Advances in Magnetic Insulators – From Spin-\ntronics to Microwave Applications Solid State Physics 64, (Academic Press, 2013).\n(2) Hoffmann, A. and Bader, S. D. ‘Opportunities at the frontier of spintronics’, Phys. Rev. Ap-\nplied (2015) 4, 047001.\n(3) Linares, R. C. McGraw, R. B. and Schroeder, J. B. ’Growth and Properties of Yttrium Iron\nGarnet Single Crystal Films’ , J. Appl. Phys. (1965) 36, 2884.\n(4) Kajiwara, Y . Harii, K. Takahashi, S. Ohe, J. Uchida, K. Mizuguchi, M. Umezawa, H. Kawai,\nH. Ando, K. Takanashi, K. 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V . ‘Surface morphological changes and mag-\nnetic properties of Sc-substituted Y3Fe5O12 epitaxial films deposited on the GGG substrate’ ,\nJ. Magn. Magn. Mater. (2010) 322, 3314.\n(27) Zhang, W. et al, J. Appl. Phys. (2010) 107, 09D724; ibid(2013) 113, 17B502; J. Micromech.\nMicroeng. (2011) 21, 045024.\n(28) Kalarickal, S. S. Krivosik, P. Wu, M. Patton, C. E. Schneider, M. L. Kabos, P. Silva, T. J.\nand Nibarger, J. P. ’Ferromagnetic resonance linewidth in metallic thin films: Comparison of\nmeasurement methods’ , J. Appl. Phys. (2006) 99, 093909.\n(29) Adeyeye, A. O. and Singh, N. ‘Large area patterned magnetic nanostructures’ , J. Phys. D:\nAppl. Phys. (2008) 41, 153001.\n(30) Ding, J. Kostylev, M. and Adeyeye, A. O. ’Magnetic hysteresis of dynamic response of one-\ndimensional magnonic crystals consisting of homogenous width nanowires observed with\nbroadband ferromagnetic resonance’ , Phys. Rev. B. (2011) 84, 054425.\n(31) Adur, R. Du, C. Wang, H. Manuilov, S. A. Bhallamudi, V . P. Zhang, C. Pelekhov, D. V . Yang,\nF. and Hammel, C. P. ’Damping of Confined Modes in a Ferromagnetic Thin Insulating Film:\nAngular Momentum Transfer across a Nanoscale Field-Defined Interface’ , Phys. Rev. Lett.\n(2014) 113, 176601.\n15Graphical TOC Entry\nSome journals require a graphical entry for the Table of Contents. This should be\nlaid out “print ready” so that the sizing of the text is correct.\nInside the tocentry environment, the font used is Helvetica 8 pt, as required\nbyJournal of the American Chemical Society .\nThe surrounding frame is 9 cm by 3.5 cm, which is the maximum permitted for\nJournal of the American Chemical Society graphical table of content entries. The\nbox will not resize if the content is too big: instead it will overflow the edge of the\nbox.\nThis box and the associated title will always be printed on a separate page at the\nend of the document.\n16"    },    {        "title": "0710.0986v2.Activation_of_additional_energy_dissipation_processes_in_the_magnetization_dynamics_of_epitaxial_chromium_dioxide_films.pdf",        "content": "arXiv:0710.0986v2  [cond-mat.other]  8 Oct 2007Activation of additional energy dissipation processes in t he magnetization dynamics of\nepitaxial chromium dioxide films\nG. M. M¨ uller∗and M. M¨ unzenberg\nIV. Physikalisches Institut, Universit¨ at G¨ ottingen, D- 37077 G¨ ottingen, Germany\nG.-X. Miao and A. Gupta\nMINT Center, Department of Chemistry, Chemical and Biologi cal Engineering, University of Alabama, Tuscaloosa, AL 354 87\n(Dated: October 25, 2018)\nThe precessional magnetization dynamics of a chromium diox ide(100) film is examined in an all-\noptical pump-probe setup. The frequency dependence on the e xternal field is used to extract the\nuniaxial in-plane anisotropy constant. The damping shows a strong dependence on the frequency,\nbut also on the laser pump fluency, which is revealed as an impo rtant experiment parameter in\nthis work: above a certain threshold further channels of ene rgy dissipation open and the damping\nincreases discontinuously. This behavior might stem from s pin-wave instabilities.\nPACS numbers: 75.30.Gw, 76.90.+d\nAs a predicted half-metallic ferromagnet1, Chromium\ndioxide (CrO 2) shows a spin polarization at the Fermi\nlevel that comes close to a full polarization2,3,4, which is\nthe defining property of a half-metal. Therefore, CrO 2\nhas attracted a lot of interest as a possible material for\nfuture spintronic devices.5,6To achieve high processing\nspeed in such devices, a fundamental insight into the\nmagnetization dynamics of CrO 2is needed.\nAn all-optical pump-probe setup utilizing fs laser\npulses7,8(time-resolved magneto-optical Kerr effect,\nTRMOKE) allows investigation of the magnetization dy-\nnamics of ferromagneticfilms in the time domain. In this\nsetup, thelaserinduced demagnetization9,10andthe sub-\nsequent remagnetization11is accompanied by a changeof\nthe equilibrium direction of magnetization, which can be\nunderstood as a ps field pulse12that leads to precessional\nmotion according to the Landau-Lifschitz-Gilbert (LLG)\nequation13,14\nd\ndtM=−γ0M×Heff−α\nMM×d\ndtM,(1)\nwhereγ0=µ0|γ|and the dimensionless parameter\nαaccounts for the damping of the magnetic motion.\nThis damping term is derived by introducing isotropic\nRayleigh-like energy dissipation. In general, the mi-\ncroscopic mechanism for damping does not obey these\nassumptions. Nevertheless, the precessional motion\ncan still be described by an effective and possibly fre-\nquency dependent damping parameter αeff. It has been\nshown that this parameter can be extracted from the\nprecessional motion traced in an all-optical TRMOKE\nsetup.8,15In these experiments, the equilibrium magne-\ntization is canted out of the film plane so that the laser\ninduced demagnetization comes along with a change of\nthe direction of the shape anisotropy field of the sam-\nple. Zhang et al. have demonstrated16that the in-plane\nanisotropy of a CrO 2(100) film can be utilized in an op-\ntical pump-probe setup for the generation of an in-plane\nanisotropy field pulse that induces precessional motion.\nRecently, several TRMOKE experiments with a similarconfiguration were reported.17,18,19,20Here, we present\na systematic all-optical measurement of the precessional\nfrequency and damping of a 300nm CrO 2(100) film. The\nexamined pump fluency dependence of the sample shows\ntheopeningofanadditionalchannelofenergydissipation\nat a sufficiently high perturbation from the equilibrium\nconfiguration.\nHaniso\n(i) (ii) (iii)(a) (b)\n(c)/c113zM\nHext /c102easy axishard□axis\nFIG.1: (Color online)(a)Configuration ofthesample system :\nThe external field is applied in hard axis direction; the angl es\nθandφare chosen according to Eq. (2). (b) φcalculated\nfrom Eq. (2) with constants as given in the text. (c) In-plane\nanisotropy field pulse: the impinging pump pulse changes the\nequilibrium configuration (i) due to lattice heating and ad-\njustment of the magnetic anisotropy on the timescale of 1 ps\n(ii); the slow recovery of the intial configuration is accomp a-\nnied by precessional motion (iii).\nThe CrO 2film is grown epitaxially by CVD on a\nTiO2(100) (rutile) substrate.21The 300 nm thick film,\nexamined in detail for this work, is expected to show\nuniaxial magnetic in-plane anisotropy with the c-axis\n([001]) being the in-plane easy axis and an effective\nfirst order anisotropy constant of 15 ·103J/m3at room\ntemperature.21This anisotropy originates from an in-\nterplay between crystalline and strain induced magnetic\nanisotropy. With the external field applied in the in-\nplane hard axis direction, the free energy density of the2\n/s48 /s51/s48/s48 /s54/s48/s48 /s57/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s48 /s51/s48/s48 /s54/s48/s48 /s57/s48/s48/s32/s32/s32/s47/s32/s109 /s86\n/s116/s32/s47/s32/s112/s115\n/s32/s48/s72\n/s101/s120/s116/s32/s61/s32/s45/s32/s57/s48/s109/s84/s32\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s45/s32/s55/s48/s109/s84\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s45/s32/s53/s48/s109/s84/s32\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s45/s32/s51/s48/s109/s84/s32\n/s32\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s45/s32/s49/s48/s109/s84/s108/s111/s110/s103/s105/s116/s117/s100/s105/s110/s97/s108/s32/s99/s111/s109 /s112/s111/s110/s101/s110/s116/s32/s100/s111/s109 /s105/s110/s97/s110/s116/s32\n/s32/s32\n/s116/s32/s47/s32/s112/s115/s48/s72\n/s101/s120/s116/s32/s61/s32/s57/s48/s109/s84/s32\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s55/s48/s109/s84\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s53/s48/s109/s84/s32\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s51/s48/s109/s84/s32\n/s32/s32\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s49/s48/s109/s84/s116/s114/s97/s110/s115/s118/s101/s114/s115/s101/s32\n/s99/s111/s109 /s112/s111/s110/s101/s110/s116\n/s100/s111/s109 /s105/s110/s97/s110/s116\nFIG. 2: (Color online) Changes in the reflectivity and in\nthe real part of the Kerr angle become manifest in the mea-\nsured transients; the fits are given by the dashed lines. For\nlower fields, the transverse component of magnetization pre -\ncesses dominantly where, for higher fields, the longitudi-\nnal component is the dominant one. The pump fluency is\nF= 10mJ/cm2. The curves are shifted for clarity.\nsystem can be written as\nF=−µ0MsHextsinφsinθ+1\n2µ0M2\nscos2θ+K1,effsin2φ,\n(2)\nwhere the first term represents the Zeeman energy, con-\ntributions fromthe shapeand the in-plane anisotropyare\ntaken into account by the second and the third term, re-\nspectively. The angles φandθare named according to\nFig. 1 (a).\nIn our experiment, probe and pump pulse are gener-\nated by a Titanium:sapphire fs laser together with a re-\ngenerativeamplifier(repetition rate250kHz, pulsewidth\n60 fs, spot size of probe and pump pulse 40 and 60 µm,\nrespectively). We utilize a double modulation scheme so\nthatthepolarizationoftheprobepulseismodulatedwith\na photo-elastic modulator and the intensity of the pump\npulse with a mechanicalchopper. The sample ismounted\nat room temperature with the external magnetic field in\nthe plane of incidence parallel to the hard axis [cf. Fig. 1\n(a)]. It has been shown that the hysteresis curve for this\nsamplein this configurationcan be explainedbycoherent\nrotation of a single domain.22Therefore, the equilibrium\nconfiguration, i.e., the azimuthal angle φ, is given by the\nminimum of the free energy in Eq. (2). In Fig. 1 (b), φ\nis depicted as a function of the external field where the\nroom temperature material parameters are chosen to be\nK1,eff= 15·103J/m3andµ0Ms= 603mT.23\nThe laser induced demagnetization in CrO 2reaches its\nmaximum after 200 to 300 ps – in contrast to, e.g., nickel\nwhere the demagnetization occurs on a timescale of hun-\ndreds of fs. This behavior was attributed to the half-\nmetallic character of chromium dioxide and a resulting(a)\n(b)\nFIG. 3: (Color online) (a) Measured precessional frequency\nas a function of the external field for pump fluencies F=\n10mJ/cm2andF= 20mJ/cm2and fits according to Eq. (3).\n(b) Free energy density as a function of the azimuthal angle\nfor different external field values.\ndecouplingbetweenthespinandtheelectronsystem.24,25\nNevertheless, the relevant time scale for electron-lattice\nequilibration is in the order of 1 ps in CrO 2as well as in\nnickel. Therefore, the magnetic in-plane anisotropy con-\nstant undergoes a fast change in magnitude on the same\ntimescale, which results in an anisotropy field pulse. The\noriginal configuration recovers accompanied by preces-\nsional motion on a timescale of 100 ps [cf. Fig. 1 (c)].\nThe measured transients for pump fluency of F=\n10mJ/cm2are depicted in Fig. 2; it should be noted\nthat these transients portray temporal changes in the\nreal part of the Kerr angle as well as in the reflectiv-\nity. In the field range where the transverse component\nprecesses dominantly (40mT < µ0Hext<40mT, cf.\nFig. 2), the traced magnetic motion changes its phase\nbyπunder switching the external field due to the the\ntwofold degeneracy of the equilibrium configuration and\nthe pecularities of the exciting field pulse. At higher\nfields where the longitudinal component is the dominant\none, almost no change of phase is observed. Thus, a sep-\naration of the magnetic signal from the temporal change\nof the reflectivity would be complicated. These tran-\nsients can be perfectly fitted with functions of the type\nA·exp(−t/T) +B·exp(−t/τ)·sin(ωt) where the first\nsummand portrays the change in reflectivity and a small\nchange in the length of the Kerr vector. The second\none represents a solution of the linearized LLG equation\n[Eq. (1)]. In the framework of this linearization, which\ncorresponds to a parabolic approximation of the free en-\nergy, the precessional frequency ω= 2πfand the damp-\ning time τcan be expressed by the sample parameters\nMs,K1,eff,αeff, and the Land´ e factor gas well as the3\nexternal field Hext(neglecting quadratic terms in αeff):\nω=γ0\nµ0Mssinθ0/radicalBig\nFθθFφφ−F2\nθφ,\nτ=2µ0Ms\nαγ0/parenleftBig\nFθθ+1\nsin2θ0Fφφ/parenrightBig, (3)\nwhereFijdenotes the coefficients of the parabolic ap-\nproximation of the free energy around the minimum. In\nFig. 3(a), the extracted precessional frequency is plot-\nted as a function of the external field for pump flu-\nencies of F= 10mJ /cm2and 20mJ /cm2. The over-\nall reduced frequency for the higher fluency can be ex-\nplained by the higher average heating, which also man-\nifests in a reduction of the Kerr signal at negative\ndelay times. The measured frequencies are fitted by\nEq. (3), where, to avoid ambiguities, the Land´ e fac-\ntor and the magnetization are held fixed at g= 226\nandµ0Ms= 603mT33. The determined values are\nK1,eff= 15,990(75)·103J/m3(15,350(150) ·103J/m3)\nforF= 10mJ/cm2(20mJ/cm2).34The drop of the fre-\nquency at about µ0Hext= 65mT can be understood\neasily: the precessional motion of magnetization means\n(in first order) a harmonic oscillation around the mini-\nmum of the free energy landscape of the sample system\n[cf. Fig. 3(b)]. At external fields below the anisotropy\nfieldµ0Haniso=2K1,eff\nMs,therearetwodegenerateminima,\nwhich move towards each other with increasing field. At\nexternal fields above the anisotropy field, the magneti-\nzation is, of course, aligned with the external field. This\ntransition from one to two minima at Hext=Hanisomust\nbe accompanied by vanishing second order derivatives\nat the minimum position and, thus, vanishing restoring\nforces for the precessional motion as well.\n/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48 /s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48\n/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s49/s53/s48/s51/s48/s48/s52/s53/s48/s54/s48/s48/s55/s53/s48\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s49/s53/s48/s51/s48/s48/s52/s53/s48/s54/s48/s48/s55/s53/s48/s32/s70/s32/s61/s32/s49/s48/s32/s109/s74/s47/s99/s109/s50/s48/s72\n/s101/s120/s116/s32/s47/s32/s109/s84\n/s32/s32/s101/s102/s102\n/s32/s32\n/s101/s102/s102/s48/s72\n/s101/s120/s116/s32/s47/s32/s109/s84\n/s32/s32\n/s70/s32/s61/s32/s50/s48/s32/s109/s74/s47/s99/s109/s50/s32/s32/s32/s47/s32/s112/s115\n/s48/s72\n/s101/s120/s116/s32/s47/s32/s109/s84\n/s32/s47/s32/s112/s115/s32/s32\n/s32\n/s48/s72\n/s101/s120/s116/s32/s47/s32/s109/s84/s32\nFIG. 4: (Color online) Field depencene of the damping time\nand the effective Gilbert damping parameter for fluencies F=\n10mJ/cm2andF= 20mJ/cm2; the lines are guides for the\neyes.\nAlso, the damping time τis extracted from the mea-\nsured Kerr transients and plotted for the two pump flu-\nencies in Fig. 4. The damping parameter αeffis cal-/s52/s44/s56/s53/s44/s48/s53/s44/s50/s53/s44/s52/s53/s44/s54/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48/s51/s44/s53/s52/s44/s48/s70/s32/s47/s32/s109/s74/s47/s99/s109/s50\n/s32/s32/s102/s40\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s50/s48/s32/s109 /s84/s41/s32/s47/s32/s71 /s72/s122\n/s32/s32/s49/s32/s47/s32 /s40\n/s48/s72\n/s101/s120/s116/s32/s61/s32/s50/s48/s32/s109 /s84/s41/s32/s47/s32/s110/s115/s45/s49\n/s70/s32/s47/s32/s109/s74/s47/s99/s109/s50\nFIG. 5: (Color online) Pump fluency depencene of the preces-\nsional frequency and the inverse damping time for µ0Hext=\n20mT; the lines are guides for the eyes. The energy dis-\nsipation rate increases above the threshold value of F=\n20mJ/cm2\nculated from τand the above determined sample pa-\nrameters according to Eq. 3 and is included in Fig. 4.\nThe damping shows a strong increase with decreasing\nprecessional frequency as it was also observed in sim-\nilar experiments17,19,20so that genuine Gilbert damp-\ning cannot be the dominant process of energy dissipa-\ntion operative in this sample. For CrO 2films, an in-\ncrease of the damping for lower precession frequencies\nwas likewise found in ferromagnetic resonance (FMR).26\nIn addition, these FMR measurements revealed that the\nlinewidth in the in-plane hard axis direction is broadened\ncompared to the easy axis direction. Thus, the damping\nin this sample might not only be frequency dependent,\nbut also direction dependent. Woltersdorf and Heinrich\ncould demonstrate that the magnetic damping of an iron\nthin film on GaAs due to two-magnon scattering is en-\nhancedindirectionsofhighermisfitdislocationdensity.27\nIn the CrO 2film, there might be a higher misfit dislo-\ncation density in the b-axis direction compared to the\nc-axis direction due to the lattice mismatch anisotropy\nwith the TiO 2substrate.28Qualititatively, transmission\nelectron microscopy reveals that the strain is relieved by\nformation of misfit dislocations that stretch to the sur-\nface of the film; a quantitative study of the supposed\ndislocation density anisotropy is not possible. Another\nstriking feature of the curves in Fig. 4 is the qualita-\ntively different field dependence of the damping in the\nfield range above µ0Hext= 60mT for the two pump\nfluencies. Thus, the pump fluency is an important pa-\nrameter in this all-optical measurement. In Fig. 5, the\nfrequency and the inverse damping time τ−1is plotted\nas a function of the laser fluency for an applied external\nfield ofµ0Hext= 20mT. The frequency decreases with4\nthe fluency as the equilibrium temperature rises due to\nthe increased average heating. In the room temperature\nregime up to 350 K, according to FMR results26, the\nstrength of damping is expected to decrease slightly with\ntemperature. This behavior is observed in our experi-\nment up to pump fluencies of F= 20mJ /cm2; above\nthis threshold, there is a discontinuous increase of the\ndamping. By measuring the length of the Kerr vector in\ncomparisonwith SQUIDmeasurements,it isverifiedthat\nthe spin temperature remains below 330 K for all fluen-\ncies and time delays. Therefore, the increase of damping\ncannot be attributed to the increase of the spin temper-\nature consistently with the FMR results. We suggest\nthat this additional damping stems from spin wave in-\nstabilities that are long known to occur threshold-like inhigh-field-pumping FMR.29There, because of the high\nangle of excitation, the uniform precession becomes un-\nstable due to spin wave disturbances and energy of the\nuniform mode is transferred to non-uniform ones. In\ntime-resolved FMR where the precession is excited by\na single field pulse, these instabilities have only been ob-\nserved in very certain configurations.30,31The question,\nwhich pecularities of the optically induced field pulse or\nthe CrO 2sample become manifest in the spin waveinsta-\nbilities, suggested in this report, remains open for future\nresearch.\nThe authors gratefully acknowledge support by the\nDeutsche Forschungsgemeinschaft within the priority\nprogram SPP 1133.\n∗Electronic address: mueller@ph4.physik.uni-goettingen .de\n1K. Schwarz, J. Phys. F 16, L211 (1986).\n2Y. S. Dedkov, M. Fonine, C. K¨ onig, U. R¨ udiger,\nG. 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B. Kos, and T. J. Silva,\nPhys. Rev. B 73, 094454 (2006).\n32H. B. Callen and E. Callen, J. Phys. Chem. Solids 27, 1271\n(1966).\n33Therewith, the total temperature dependence of the sam-\nple due to the average heating is included in the anisotropy\nconstant which should exhibit the strongest temperature\ndependence.32\n34The good agreement between measured and calculated fre-\nquenciesalso justifies thenon-considerationofthefact th at\nthe length of the magnetization vector is not conserved in\nthe experiment due to the slow laser-induced demagneti-\nzation. For all pump fluencies used in this work, the length\nof the magnetization vector (compared to negative pump\nprobe delay times) is reduced by less than 4%."    },    {        "title": "1810.10595v4.Nearly_isotropic_spin_pumping_related_Gilbert_damping_in_Pt_Ni___81__Fe___19___Pt.pdf",        "content": "Nearly isotropic spin-pumping related Gilbert damping in Pt/Ni 81Fe19/Pt\nW. Cao,1,\u0003L. Yang,1S. Au\u000bret,2and W.E. Bailey1, 2,y\n1Materials Science and Engineering, Department of Applied Physics and Applied Mathematics,\nColumbia University, New York, New York 10027, USA\n2SPINTEC, Universit \u0013eGrenoble Alpes/CEA/CNRS, F-38000 Grenoble, France\n(Dated: July 12, 2021)\nA recent theory by Chen and Zhang [Phys. Rev. Lett. 114, 126602 (2015)] predicts strongly\nanisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic \flms. Interfacial\nGilbert-type relaxation, due to the spin pumping e\u000bect, is predicted to be signi\fcantly larger for\nmagnetization oriented parallel to compared with perpendicular to the \flm plane. Here, we have\nmeasured the anisotropy in the Pt/Ni 81Fe19/Pt system via variable-frequency, swept-\feld ferromag-\nnetic resonance (FMR). We \fnd a very small anisotropy of enhanced Gilbert damping with sign\nopposite to the prediction from the Rashba e\u000bect at the FM/Pt interface. The results are contrary\nto the predicted anisotropy and suggest that a mechanism separate from Rashba spin-orbit coupling\ncauses the rapid onset of spin-current absorption in Pt.\nINTRODUCTION\nThe spin-transport properties of Pt have been studied\nintensively. Pt exhibits e\u000ecient, reciprocal conversion\nof charge to spin currents through the spin Hall e\u000bect\n(SHE)[1{4]. It is typically used as detection layer for\nspin current evaluated in novel con\fgurations[5{7]. Even\nso, consensus has not yet been reached on the experi-\nmental parameters which characterize its spin transport.\nThe spin Hall angle of Pt, the spin di\u000busion length of Pt,\nand the spin mixing conductance of Pt at di\u000berent inter-\nfaces di\u000ber by as much as an order of magnitude when\nevaluated by di\u000berent techniques[2, 3, 8{12].\nRecently, Chen and Zhang [13, 14] (hereafter CZ) have\nproposed that interfacial spin-orbit coupling (SOC) is\na missing ingredient which can bring the measurements\ninto greater agreement with each other. Measurements of\nspin-pumping-related damping, particularly, report spin\ndi\u000busion lengths which are much shorter than those es-\ntimated through other techniques[15, 16]. The introduc-\ntion of Rashba SOC at the FM/Pt interface leads to\ninterfacial spin-memory loss, with discontinuous loss of\nspin current incident to the FM/Pt interface. The model\nsuggests that the small saturation length of damping en-\nhancement re\rects an interfacial discontinuity, while the\ninverse spin Hall e\u000bect (ISHE) measurements re\rect the\nbulk absorption in the Pt layer[15, 16].\nThe CZ model predicts a strong anisotropy of the en-\nhanced damping due to spin pumping, as measured in\nferromagnetic resonance (FMR). The damping enhance-\nment for time-averaged magnetization lying in the \flm\nplane ( pc-FMR, or parallel condition) is predicted to be\nsigni\fcantly larger than that for magnetization oriented\nnormal to the \flm plane ( nc-FMR, or normal condition).\nThe predicted anisotropy can be as large as 30%, with\npc-FMR damping exceeding nc-FMR damping, as will be\nshown shortly.\nIn this paper, we have measured the anisotropy of the\nenhanced damping due to the addition of Pt in symmet-ric Pt/Ni 81Fe19(Py)/Pt structures. We \fnd that the\nanisotropy is very weak, less than 5%, and with the op-\nposite sign from that predicted in [13].\nTHEORY\nWe \frst quantify the CZ-model prediction for\nanisotropic damping due to the Rashba e\u000bect at the\nFM/Pt interface. In the theory, the spin-memory loss\nfor spin current polarized perpendicular to the interfa-\ncial plane is always larger than that for spin current po-\nlarized in the interfacial plane. The pumped spin po-\nlarization\u001b=m\u0002_mis always perpendicular to the\ntime-averaged or static magnetization hmit'm. For\nnc-FMR, the polarization \u001bof pumped spin current is\nalways in the interfacial plane, but for pc-FMR, is nearly\nequally in-plane and out-of-plane. A greater damping\nenhancement is predicted in the pccondition than in the\nnccondition, \u0001 \u000bpc>\u0001\u000bnc:\n\u0001\u000bnc=Kh1 + 4\u0011\u0018(tPt)\n1 +\u0018(tPt)i\n(1)\n\u0001\u000bpc=Kh1 + 6\u0011\u0018(tPt)\n1 +\u0018(tPt)+\u0011\n2[1 +\u0018(tPt)]2i\n(2)\n\u0018(tPt) =\u0018(1)\u0002coth(tPt=\u0015sd) (3)\nwhere the constant of proportionality K is the same for\nboth conditions and the dimensionless parameters, \u0011and\n\u0018, are always real and positive. The Rashba parameter\n\u0011= (\u000bRkF=EF)2(4)\nis proportional to the square of the Rashba coe\u000ecient\n\u000bR, de\fned as the strength of the Rashba potential,arXiv:1810.10595v4  [cond-mat.mtrl-sci]  22 Feb 20192\nFIG. 1. Frequency-dependent half-power FMR linewidth\n\u0001H1=2(!) of the reference sample Py(5 nm) (black) and sym-\nmetric trilayer samples Pt(t)/Py(5 nm)/Pt(t) (colored). (a)\npc-FMR measurements. (b) nc-FMR measurements. Solid\nlines are linear \fts to extract Gilbert damping \u000b. (Inset):\ninhomogeneous broadening \u0001 H0inpc-FMR (blue) and nc-\nFMR (red).\nV(r) =\u000bR\u000e(z)(^k\u0002^z)\u0001\u001b, where\u000e(z) is a delta function\nlocalizing the e\u000bect to the interface at z= 0 (\flm plane\nisxy),kFis the Fermi wavenumber, and EFis the Fermi\nenergy. The back\row factor \u0018is a function of Pt layer\nthickness, where the back\row fraction at in\fnitely large\nPt thickness de\fned as \u000f=\u0018(1)=[1 +\u0018(1)].\u000f= 0 (1)\nrefers to zero (complete) back\row of spin current across\nthe interface. \u0015sdis the spin di\u000busion length in the Pt\nlayer.\nTo quantify the anisotropy of the damping, we de\fne\nQ:\nQ\u0011(\u0001\u000bpc\u0000\u0001\u000bnc)=\u0001\u000bnc (5)\nas an anisotropy factor , the fractional di\u000berence be-\ntween the enhanced damping in pc and nc conditions.\nPositive Q (Q >0) is predicted by the CZ model. A\nspin-memory loss \u000efactor of 0.9\u00060.1, corresponding\nto nearly complete relaxation of spin current at the in-\nterface with Pt, was measured through current perpen-\ndicular to plane-magnetoresistance (CPP-GMR)[8] Ac-\ncording to the theory[13, 14], the spin-memory loss can\nbe related to the Rashba parameter by \u000e= 2\u0011, so we\ntake\u0011\u00180:45. The e\u000bect of variable \u0011 < 0:45 will be\nshown in Figure 3. To evaluate the thickness dependent\nback\row\u0018(tPt), we assume \u0015Pt\nsd= 14 nm, which is asso-\nciated with the absorption of the spin current in the bulk\nof Pt layer, as found from CPP-GMR measurements[8]\nand cited in [13]. Note that this \u0015Pt\nsdis longer than that\nused sometimes to \ft FMR data[15, 16]; Rashba interfa-\ncial coupling in the CZ model brings the onset thickness\ndown. The calculated anisotropy factor Q should then\nFIG. 2. Pt thickness dependence of Gilbert damping \u000b=\n\u000b(tPt) inpc-FMR (blue) and nc-FMR (red). \u000b0refers to the\nreference sample ( tPt= 0). (Inset): Damping enhancement\n\u0001\u000b(tPt) =\u000b(tPt)\u0000\u000b0due to the addition of Pt layers in\npc-FMR (blue) and nc-FMR (red). Dashed lines refer to cal-\nculated \u0001\u000bncusing Equation 1 by assuming \u0015Pt\nsd= 14 nm\nand\u000f= 10%. The red dashed line ( \u0011= 0:15) shows a similar\ncurvature with experiments; The black dashed line ( \u0011\u00150:25)\nshows a curvature with the opposite sign.\nbe as large as 0.3, indicating that \u0001 \u000bpcis 30% greater\nthan \u0001\u000bnc(see Results for details).\nEXPERIMENT\nIn this paper, we present measurements of the\nanisotropy of damping in the symmetric Pt( tPt)/Py(5\nnm)/Pt(tPt) system, where \\Py\"=Ni 81Fe19. Because\nthe Py thickness is much thicker than its spin coher-\nence length[17], we expect that spin-pumping-related\ndamping at the two Py/Pt interfaces will sum. The\nfull deposited stack is Ta(5 nm)/Cu(5 nm)/Pt( tPt)/Py(5\nnm)/Pt(tPt)/Al 2O3(3 nm),tPt= 1{10 nm, deposited\nvia DC magnetron sputtering under computer control on\nion-cleaned Si/SiO 2substrates at ambient temperature.\nThe deposition rates were 0.14 nm/s for Py and 0.07\nnm/s for Pt. Heterostructures deposited identically, in\nthe same deposition chamber, have been shown to exhibit\nboth robust spin pumping e\u000bects, as measured through\nFMR linewidth[18, 19], and robust Rashba e\u000bects (in\nCo/Pt), as measured through Kerr microscopy[20, 21].\nThe stack without Pt layers was also deposited as the ref-\nerence sample. The \flms were characterized using vari-\nable frequency FMR on a coplanar waveguide (CPW)\nwith center conductor width of 300 \u0016m. The bias mag-\nnetic \feld was applied both in the \flm plane ( pc) and\nperpendicular to the plane ( nc), as previously shown in\n[22]. The nc-FMR measurements require precise align-\nment of the \feld with respect to the \flm normal. Here,3\nFIG. 3. Anisotropy factor Q for spin-pumping enhanced damping, de\fned in Equation 5. Solid lines are calculations using the\nCZ theory[13], Equations 1{3, for variable Rashba parameter 0 :01\u0014\u0011\u00140:45.\u0015Pt\nsdis set to be 14 nm. Back\row fraction \u000fis\nset to be 10% in (a) and 40% in (b). Black triangles, duplicate in (a) and (b), show the experimental values from Figure 2.\nsamples were aligned by rotation on two axes to maxi-\nmize the resonance \feld at 3 GHz.\nRESULTS AND ANALYSIS\nFigure 1 shows frequency-dependent half-power\nlinewidth \u0001 H1=2(!) in pc- and nc-FMR. The measure-\nments were taken at frequencies from 3 GHz to a cut-o\u000b\nfrequency above which the signal-to-noise ratio becomes\ntoo small for reliable measurement of linewidth. The\ncuto\u000b ranged from 12{14 GHz for the samples with Pt\n(linewidth\u0018200{300 G) to above 20 GHz for tPt= 0.\nSolid lines stand for linear regression of the variable-\nfrequency FMR linewidth \u0001 H1=2= \u0001H0+2\u000b!=\r , where\n\u0001H1=2is the full-width at half-maximum, \u0001 H0is the in-\nhomogeneous broadening, \u000bis the Gilbert damping, !\nis the resonance frequency and \ris the gyromagnetic ra-\ntio. The \fts show good linearity with frequency !=2\u0019for\nall experimental linewidths \u0001 H1=2(!). The inset sum-\nmarizes inhomogeneous broadening \u0001 H0inpc- and nc-\nFMR; its errorbar is \u00182 Oe.\nIn Figure 2, we plot Pt thickness dependence of damp-\ning parameters \u000b(tPt) extracted from the linear \fts in\nFigure 1, for both pc-FMR and nc-FMR measurements.\nStandard deviation errors in the \fts for \u000bare\u00183\u000210\u00004.\nThe Gilbert damping \u000bsaturates quickly as a function\noftPtin both pc and nc conditions, with 90% of the ef-\nfect realized with Pt(3 nm). The inset shows the damp-\ning enhancement \u0001 \u000bdue to the addition of Pt layers\u0001\u000b=\u000b\u0000\u000b0, normalized to the Gilbert damping \u000b0of\nthe reference sample without Pt layers. The Pt thickness\ndependence of \u0001 \u000bmatches our previous study on Py/Pt\nheterostructures[19] reasonably; the saturation value of\n\u0001\u000bPt=Py=Pt is 1.7x larger than that measured for the\nsingle interface \u0001 \u000bPy=Pt [19] (2x expected). The dashed\nlines in the inset refer to calculated \u0001 \u000bncusing Equation\n1 (assuming \u0015Pt\nsd= 14 nm and \u000f= 10%).\u0011= 0:25 shows\na threshold of Pt thickness dependence. When \u0011>0:25,\nthe curvature of \u0001 \u000b(tPt) will have the opposite sign to\nthat observed in experiments, so \u0011= 0:25 is the maxi-\nmum which can qualitatively reproduce the Pt thickness\ndependence of the damping.\nAs shown in Figure 2 inset, the damping enhancement\ndue to the addition of Pt layers is slightly larger in the\nncgeometry than in the pcgeometry: \u0001 \u000bnc>\u0001\u000bpc.\nThis is opposite to the prediction of the model in [13].\nThe anisotropy factor Q\u0011(\u0001\u000bpc\u0000\u0001\u000bnc)=\u0001\u000bncfor the\nmodel (Q>0) and the experiment (Q <0) are shown to-\ngether in Figure 3 (a) and (b). The magnitude of Q\nfor the experiment is also quite small, with -0.05 <Q<0.\nThis very weak anisotropy, or near isotropy, of the spin-\npumping damping is contrary to the prediction in [13],\nand is the central result of our paper.\nThe two panels (a) and (b), which present the same\nexperimental data (triangles), consider di\u000berent model\nparameters, corresponding to negligible back\row ( \u000f=\n0:1, panel a) and moderate back\row ( \u000f= 0:4, panel b)\nfor a range of Rashba couplings 0 :01\u0014\u0011\u00140:45. A spin\ndi\u000busion length \u0015sd= 14 nm for Pt[8] was assumed in all4\ncases.\nThe choice of back\row fraction \u000f= 0:1 or 0:4 and the\nchoice of spin di\u000busion length of Pt \u0015sd= 14 nm follow\nthe CZ paper[13] for better evaluation of their theory.\nFor good spin sinks like Pt, the back\row fraction is usu-\nally quite small. If \u000f= 0, then there will be no spin\nback\row. In this limit, \u0001 \u000bpc, \u0001\u000bncand the Q factor\nwill be independent of Pt thickness.\nIn the case of a short spin di\u000busion length of Pt, e.g.,\n\u0015sd= 3 nm, the anisotropy Q as a function of Pt thick-\nness decreases more quickly for ultrathin Pt, closer to\nour experimental observations. However, we note that\nthe CZ theory requires a long spin di\u000busion length in or-\nder to reconcile di\u000berent experiments, particularly CPP-\nGMR with spin pumping, and is not relevant to evaluate\nthe theory in this limit.\nLeaving apart the question of the sign of Q, we can see\nthat the observed absolute magnitude is lower than that\npredicted for \u0011= 0:05 for small back\row and 0.01 for\nmoderate back\row. According to ref [13], a minimum\nlevel for the theory to describe the system with strong\ninterfacial SOC is \u0011= 0:3.\nDISCUSSION\nHere, we discuss extrinsic e\u000bects which may result in\na discrepancy between the CZ model (Q \u0018+0.3) and our\nexperimental result (-0.05 <Q<0). A possible role of two-\nmagnon scattering[23, 24], known to be an anisotropic\ncontribution to linewidth \u0001 H1=2, must be considered.\nTwo-magnon scattering is present for pc-FMR and nearly\nabsent for nc-FMR. This mechanism does not seem to\nplay an important role in the results presented. It is\ndi\u000ecult to locate a two-magnon scattering contribution\nto linewidth in the pure Py \flm: Figure 1 shows highly\nlinear \u0001H1=2(!), without o\u000bset, over the full range to\n!=2\u0019= 20 GHz, thereby re\recting Gilbert-type damp-\ning. The damping for this \flm is much smaller than\nthat added by the Pt layers. If the introduction of Pt\nadds some two-magnon linewidth, eventually mistaken\nfor intrinsic Gilbert damping \u000b, this could only produce\na measurement of Q >0, which was not observed.\nOne may also ask whether the samples are appropriate\nto test the theory. The \frst question regards sample qual-\nity. The Rashba Hamiltonian models a very abrupt inter-\nface. Samples deposited identically, in the same deposi-\ntion chamber, have exhibited strong Rashba e\u000bects, so we\nexpect the samples to be generally appropriate in terms\nof quality. Intermixing of Pt in Ni 81Fe19(Py)/Pt[25] may\nplay a greater role than it does in Co/Pt[26], although\ndefocused TEM images have shown fairly well-de\fned in-\nterfaces for our samples[27].\nA second question might be about the magnitude of\nthe Rashba parameter \u0011in the materials systems of in-\nterest. Our observation of nearly isotropic damping isconsistent with the theory, within experimental error and\napart from the opposite sign, if the Rashba parameter \u0011is\nvery low and the back\row fraction \u000fis very low. Ab-initio\ncalculations for (epitaxial) Co/Pt in the ref[28] have in-\ndicated\u0011= 0.02{0.03, lower than the values of \u0011\u00180.45\nassumed in [13, 14] to treat interfacial spin-memory loss.\nThe origin of the small, negative Q observed here is un-\nclear. A recent paper has reported that \u0001 \u000bpcis smaller\nthan \u0001\u000bncin the YIG/Pt system via single-frequency,\nvariable-angle measurements[7], which is contrary to the\nCZ model prediction as well. It is also possible that a\nfew monolayers of Pt next to the Py/Pt interfaces are\nmagnetized in the samples[19], and this may have an un-\nknown e\u000bect on the sign, not taken into account in the\ntheory.\nCONCLUSIONS\nIn summary, we have experimentally demonstrated\nthat in Pt/Py/Pt trilayers the interfacial damping at-\ntributed to spin pumping is nearly isotropic, with an\nanisotropy between \flm-parallel and \flm-normal mea-\nsurements of <5%. The nearly isotropic character of the\ne\u000bect is more compatible with conventional descriptions\nof spin pumping than with the Rashba spin-memory loss\nmodel predicted in [13].\nACKNOWLEDGEMENTS\nWe acknowledge support from the US NSF-DMR-\n1411160 and the Nanosciences Foundation, Grenoble.\n\u0003wc2476@columbia.edu\nyweb54@columbia.edu\n[1] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Ap-\nplied Physics Letters 88, 182509 (2006).\n[2] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. 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Sinclair, Journal of Magnetism and\nMagnetic Materials 134, 173 (1994).\n[27] W. E. Bailey, A. Ghosh, S. Au\u000bret, E. Gautier, U. Ebels,\nF. Wilhelm, and A. Rogalev, Phys. Rev. B 86, 144403\n(2012).\n[28] S. Grytsyuk, A. Belabbes, P. M. Haney, H.-W. Lee, K.-J.\nLee, M. D. Stiles, U. Schwingenschl ogl, and A. Manchon,\nPhys. Rev. B 93, 174421 (2016)."    },    {        "title": "2209.12302v1.Formation_of_the_cosmic_ray_halo__The_role_of_nonlinear_Landau_damping.pdf",        "content": "arXiv:2209.12302v1  [astro-ph.GA]  25 Sep 2022Draft version September 27, 2022\nTypeset using L ATEXtwocolumn style in AASTeX63\nFormation of the cosmic-ray halo: The role of nonlinear Land au damping\nD. O. Chernyshov ,1V. A. Dogiel,1A. V. Ivlev,2A. D. Erlykin,1and A. M. Kiselev1\n1I. E. Tamm Theoretical Physics Division of P. N. Lebedev Inst itute of Physics, 119991 Moscow, Russia\n2Max-Planck-Institut f¨ ur extraterrestrische Physik, 857 48 Garching, Germany\nABSTRACT\nWe present a nonlinear model of self-consistent Galactic halo, wher e the processes of cosmic ray\n(CR) propagation and excitation/damping of MHD waves are included . The MHD-turbulence, which\nprevents CR escape from the Galaxy, is entirely generated by the r esonant streaming instability. The\nkey mechanism controlling the halo size is the nonlinear Landau (NL) da mping, which suppresses the\namplitude of MHD fluctuations and, thus, makes the halo larger. The equilibrium turbulence spectrum\nis determined by a balance of CR excitation and NL damping, which sets the regions of diffusive and\nadvective propagation of CRs. The boundary zcr(E) between the two regions is the halo size, which\nslowly increases with the energy. For the vertical magnetic field of ∼1µG, we estimate zcr∼1 kpc\nfor GeV protons. The derived proton spectrum is in a good agreeme nt with observational data.\nKeywords: cosmic rays – Galaxy: halo – MHD-turbulence\n1.INTRODUCTION\nThe problem of the Galactic halo is being discussed\nfrom the beginning of the 1950s. Before that time,\na sharp transition was assumed between the Galactic\ndisk of the thickness of ∼100 pc and the extragalactic\nmedium. Later, Ginzburg (1953) developed conceptions\nof the physical cosmic ray (CR) halo with the size of\nabout 15 kpc, where CRs are trapped by scattering (i.e.,\npropagate diffusively). The characteristic CR age in the\nGalaxy was estimated as ∼108yr, which is confirmed\nby radio data and by the information on CR chemi-\ncal composition (e.g., about the abundance of unsta-\nble isotope10Be) (see, e.g., Ginzburg & Ptuskin 1976 ;\nSzabelski et al. 1980 , etc.).\nThe static halo model of a fixed height was presented\ninGinzburg & Syrovatskii (1964). It assumes that the\nCR density at a certain distance from the Galacticplane\nbecomes negligible. This model is currently broadly im-\nplemented in advanced numerical codes, such as GAL-\nPROP (Moskalenko & Strong 1998 ).\nThe downside of the model is that it depends on two\narbitrary parameters, namely the diffusion coefficient\nand the halo size, whose values are ambiguously de-\nfined. Therefore, it is necessary to describe the pro-\ncesses of generation and damping of MHD turbulence\nin the halo and their connections to the CR transport\nself-consistently.\nCorresponding author: D. O. Chernyshov\nchernyshov@lpi.ruInDogiel et al. (2020), we have suggested a model of\nCRself-confinementintheGalaxy,wheretheturbulence\ngenerated in the Galactic disk was amplified by stream-\ning CRs. However,the turbulence excitation rateis very\nhigh in that model, and hence the size of the halo is\ntoo small at GeV energies. To resolve this issue, in the\npresent paper we also take into account the nonlinear\nLandau (NL) damping, which was neglected in the orig-\ninal work. We show that the inclusion of the damping\nterm leads to a significantly larger halo size. We also\nshow that the MHD turbulence which confines CRs in\nthe halo can be entirely self-generated by CRs.\n2.SELF-CONSISTENT NONLINEAR MODEL OF\nCR HALO\nUnlike models with a pre-defined halo size, self-\nconsistent halo models include a mechanism of MHD-\nwave excitation. In these models, CR propagation is de-\nscribed by a system of nonlinear equations (see, e.g., in\nDogiel et al. 1994 ;Evoli et al. 2018 ;Dogiel et al. 2020 ,\nand references therein).\nA general system of simplified one-dimensional non-\nlinear equations for the CR spectrum N(p,z) and the\nenergy density of MHD fluctuations W(k,z) can be pre-\nsented in the following form:\n∂\n∂z/parenleftbigg\nuadvN−D∂N\n∂z/parenrightbigg\n−∂\n∂p/parenleftbigg1\n3duadv\ndzpN−˙pN/parenrightbigg\n=Q,\n∂uAW\n∂z−duA\ndz∂(kW)\n∂k+∂\n∂k/parenleftbiggkW\nτcas/parenrightbigg\n= (ΓCR−ν)W ,\n(1)2\nwhereQ(p,z) is the source term of CRs, uadv(z) is the\nCR advection velocity, which depends on the difference\nbetween outward- and inward-propagating MHD waves,\nuA(z) =B(z)//radicalbig\n4πρ(z) is the Alfven velocity, ˙ p <0\nis the rate of momentum loss due to interaction with\ngas,ρ=mpnis the mass density of ionized hydro-\ngen (mpis proton mass), and Bis the strength of the\nlongitudinal large-scale magnetic field. Furthermore,\nΓCR(k,z) is the rate of resonant wave excitation, νis\nthe wave damping rate, and τcas(W) is the characteris-\ntic timescale of turbulent cascade to larger k; the latter\ndepends on the particular process of MHD-generation\n(see, e.g., Ptuskin et al. 2006 , this process is discussed\ninSection 2.1). Thespectrum N(p,z)isnormalizedsuch\nthat/integraltext\nN(p)dpis the total number density of CRs.\nThe wavenumber kof MHD fluctuations is related\nto the CR momentum pvia the resonance condition\n(Skilling 1975 ),\nkp≈mpΩ∗, (2)\nwhere Ω ∗=eB/m pcis the gyrofrequency of non-\nrelativistic CR protons. The resulting CR diffusion co-\nefficient is ( Skilling 1975 )\nD(p,z)≈vB2\n6π2k2W. (3)\nIn this approximation, the excitation rate is propor-\ntional to the CR diffusion flux,\nΓCR(k,z)≈ −2π2euAp\nBcD∂N\n∂z. (4)\nThere are very few known parameters and processes\nthat can govern the density of MHD-fluctuations in the\nhalo (and thus the CR diffusion). These are the spatial\ndependencies of the magnetic field and the gas density,\nthe magnitude and the spectrum of CR source in the\ndisk, and nonlinear processes involving MHD waves. In\nthisrespect,thevarietyofmodelsforthewaveexcitation\nin the halo is very restricted.\n2.1.Development of Dogiel et al. (2020)\nEvoli et al. (2018) andDogiel et al. (2020) presented\none-dimensional models of CR propagation along the\nmagnetic field lines, with MHD-fluctuations excited by\nthe resonant CR-streaming instability.\nEvoli et al. (2018) developed a model of MHD-\nturbulence with nonlinear cascading to larger k. They\nconsidered three sources of waves responsible for CR\nscattering in the halo: (i) self-generatedMHD-waves ex-\ncited by CRs through the streaming instability, (ii) pro-\ncessesmimicking wavegenerationby, e.g., supernovaex-\nplosionsinthediskwhichejectwavesatlargescales,and\n(iii) cascading process which is determined by an initial\narbitrary source of background turbulence distributed\nover the halo. In case the cascading is responsible for\nthe damping of MHD-fluctuations in the halo, the CRhalo can be about of few kpc, which is compatible with\nthe estimations of GALPROP.\nOn the contrary, Dogiel et al. (2020) showed that the\ncascading process in the halo is negligible for relevant\nvalues of k, i.e., the term containing τcasin the sec-\nond Equation ( 1) can be omitted. We considered two\nsources of waves responsible for CR scattering in the\nhalo, namely, (i) self-generated MHD waves excited by\nCRs through the streaminginstability, and (ii) the spec-\ntrum of MHD-fluctuations generated by sources in the\nGalactic disk. In that model, magnetic fluctuations are\nonly excited in the direction away from the disk. How-\never, the resulting CR flux excites waves too efficiently,\nwhich yields the halo size of only ∼100 pc at low ener-\ngies.\nInDogiel et al. (2020), we have not considered the\npossibility that outgoing waves may be reflected by a\nnonuniformmedium(see Ferraro1954 ;Kulsrud2005 ,for\ndetails). In fact, that happens if the approximation of\ngeometrical optics is no longer applicable. According to\nGinzburg (1970), ifthewavephasevelocitychangesfrom\numintoumaxwithin a layer of thickness ℓ, the reflection\ncoefficient Rof the outgoing waves from the layer is\nR2∼exp/parenleftbigg\n−4πkℓumin\numax/parenrightbigg\n. (5)\nApplying this expression to the halo with ℓ= 1 kpc, we\nsee that even for very long waves, resonant with PeV\nprotons, only ∼0.1% of the total energy is reflected.\nTherefore, we indeed can safely assume that there are\nno backward-moving waves present in the halo.\nAnother physical process neglected in Dogiel et al.\n(2020) is nonlinear Landau (NL) damping. A two-\ndimensional halo model including this process has\nalready been developed by Dogel’ et al. (1993) and\nDogiel et al. (1994). The authors used the equation for\nCR propagation complemented with the equation for\nMHD-fluctuations which are excited by the CR flux and\nattenuated by the NL damping, cascading, and adia-\nbatic losses. It was shown that CR distribution is quasi-\nisotropic near the Galactic plane, but becomes more fo-\ncused along the radial coordinate as particles propagate\nfurther away. At some point the scattering becomes un-\nabletoreflectparticlesback,whichsetstheouterbound-\nary of the halo. The halo size was estimated to be about\n10 kpc. Howeverthe advectivetransport ofCRs wasnot\ntaken into account in this model.\nAccording to V¨ olk & Cesarsky (1982) andMiller\n(1991), the rate of NL damping is given by\nνNL(k)≈g(n,T)8πuA\nB2kk/integraldisplay\nkminW(k1)dk1,(6)\nwhere the lower integration limit kminis unimportant\nfor our self-consistent model (see Section 4.2). Follow-3\nFigure 1. Schematic representation of the model considered\nin the paper.\ningMiller(1991), the dimensionless factor g(β) in Equa-\ntion (6) can be approximated by\ng(β)≈√π\n4β1/2/parenleftbigg\ne−β−1+1\n2ǫ1/2e−ǫβ−1/parenrightbigg\n.(7)\nHere,ǫ=me/mpis electron-to-proton mass ratio and\nβ=nkT\nB2/8π≡u2\nth\nu2\nA, (8)\nis the plasma- βparameter expressed via the thermal\nvelocity of protons uth. We assume that temperatures\nTof both protons and electrons are equal.\nThe idea that the excitation of MHD turbulence in a\nhalo can be balanced by NL damping has been previ-\nously discussed by Ptuskin et al. (1997). In the present\nwork, we use a different expression for NL damping,\nwhich takes into account contributions of both thermal\nprotons and electrons. Furthermore, in our model βsig-\nnificantly drops with the height, which results in a much\nweaker damping for waves excited by CRs with energies\nabove100GeVand, thus, helpstoconfinesuchparticles.\n3.THE HALO MODEL WITH NL DAMPING\nThe idealized structure of our model is sketched in\nFigure1. We consider two characteristic regions along\nthez-axis: the Galactic disk, where MHD-turbulence is\nassumed to be generated by sources distributed over the\nGalactic plane, and the CR halo, where the turbulence\nis self-generated by the outgoing CR flux. We assume\nthat the magnetic field is practically vertical in the halo\n(while its geometry can be arbitrary in the disk), and\nthat CRs do not diffuse across the magnetic field lines.This model allowsus to reduce the set ofequations ( 1)\nto\n∂\n∂z/parenleftbigg\nuadvN−D∂N\n∂z/parenrightbigg\n−∂\n∂p/parenleftbigg1\n3duadv\ndzpN−˙pN/parenrightbigg\n(9)\n= 2Q(p)δ(z),\n∂vAW\n∂z−duA\ndz∂kW\n∂k= (ΓCR−νNL)W , (10)\nwhereQ(p) is the source of CRs above/belowthe Galac-\ntic plane, ˙ p <0 is the momentum loss rate due to\nionization or proton-proton collisions within the disk\n(0< z < z d), while in the halo ( z≥zd) the CR losses\naresolelydue toadiabaticcooling. Hereandbelow, zdis\nthe characteristic height of the disk. The CR advection\nvelocity changes discontinuously at the disk boundary,\nuadv(z) =uAθ(z−zd), (11)\nwhereθ(z) is the Heaviside function. We assume that\nsources of the turbulence in the disk do not contribute\nto the turbulence in the halo, i.e.\nW(k,zd) = 0, (12)\ni.e., the halo turbulence is entirely self-generated by\nCRs. However, the existence of turbulence within the\ndisk is essential ( Evoli et al. 2018 ;Dogiel et al. 2020 ),\nand we take this into account in Section 4.\nAccording to Equation ( 7), MHD-waves are damped\non plasma electrons in a low- βplasma, and on protons\nin a high- βplasma. Most of the thermal electrons or\nprotons contribute to the damping if, respectively,\n0.01≤β≤0.2, (13)\nor\nβ≥10. (14)\nFor these values of β, the respective dominant expo-\nnential factor in Equation ( 7) can be set to unity, and\nνNLin Equation ( 6) becomes independent of the plasma\ndensity.\nBelow in Section 4we obtain a simplified analytical\nsolution of Equations ( 9) and (10), while in Section 5\nwe present and discuss the exact numerical solution.\n4.ANALYTIC APPROXIMATION\nIn this section we derive an analytical solution which\nqualitatively explainsthe roleof NL damping in the self-\nconsistent halo model.\n4.1.CR spectrum in the disk, and CR flux from the\ndisk to the halo\nTo simplify CR propagation in the disk (see, e.g.\nBerezinskii et al. 1990 ), we consider the following two\nkey parameters: the outgoing CR flux and CR distribu-\ntion function at the boundary between the disk and the\nhalo. They both should be continuous at boundary.4\nA general equation for the outgoing flux S0(p) =\nS(p,zd) at the boundary is obtained by integrating\nEquation ( 9),\nS0(p) =Q(p)+d\ndp\n1\n3uA0pN0(p)−zd/integraldisplay\n0˙pN(z,p)dz\n,\n(15)\nwhereuA0=uA(zd) andN0(p) =N(p,zd). The flux\nS0derived from Equation ( 15) can be considered as\nthe boundary condition for Equation ( 9) atz=zd.\nSince the halo size should be much larger than zd, we\nassume that the CR spectrum does not change sig-\nnificantly across the disk. Then we can approximate\nN(p,z)≈N0(p) for 0≤z≤zd.\nAs pointed out in Dogiel et al. (2020), energy losses\nin the halo are unimportant and thus the CR flux S0(p)\nis conserved. Then we obtain the following solution of\nEquation ( 9) forz≥zd:\nN(p,z) =S0(p)\nu(p,z), (16)\nwhereu(p,z) is the outflow velocity of CRs,\nu=\nη∞/integraldisplay\nηeη−η1dη1\nuA(η1)\n−1\n, (17)\nandη(p,z) is a dimensionless variable\nη=z/integraldisplay\nzduA\nDdz1. (18)\nThevalueof η∞generallydepends ontheboundarycon-\ndition at z→ ∞. Substituting N0(p) =N(p,zd) from\nEquation ( 16) in Equation ( 15), we derive the flux,\nS0(p) =ud(p)\nE(p)∞/integraldisplay\npQ(p1)exp\n−p1/integraldisplay\npud(p2)\nE(p2)dp2\ndp1.\n(19)\nHere,ud(p) =u(p,zd) andE(p) =1\n3puA0−/integraltextzd\n0˙pdz=\n1\n3puA0+1\n2NHL(p), where NHis the vertical column den-\nsity of hydrogen atoms in the disk and L(p) =−˙p/nH\nis energy loss function (per unit column density) due to\ninteraction with the disk gas. We note that udin fact\ndepends on S0, and, therefore, Equation ( 19) is an inte-\ngral equation for S0(p). If the dependence udversusS0\nis weak, the equation can be solved iteratively.\nWe can obtain a simple approximation for S0(p) as-\nsuming that ES0/udis a power-law function ∝p−α.\nAccording to experimental data, S0/ud≡N(p) has a\nnegative spectral index smaller than that of S0(p) (both\naresmallerthan −2),while Ecannotincreasefasterthan∝p. Therefore, α >0 and we readily obtain from Equa-\ntion (15),\nS0(p) =Q(p)\n1+αE/(pud). (20)\nWe notice that both the energy loss rate E/pand the in-\nverse outflow velocity u−1\nd=N/S0decrease with p, and\ntherefore S0(p)≈Q(p) for sufficiently high energies. To\nevaluate a critical energy where E/(pud) = 1, we assume\nNH≈6×1020cm−2andud∼uA∼106cm/s. This\nyields the proton energy about 0.5 GeV, above which we\ncan setS0=Q.\nAs discussed in Dogiel et al. (2020), the value of η\nis the key parameter characterizing CR propagation in\nthe halo. The entire halo can be approximately split\ninto two regions: one is called the halo sheath , where\nη(z)≪1 and the diffusion term −D∂N/∂z dominates\nin the CR flux S0; the other is where η(z)≫1 and\nthe advection term uadvNdominates. The critical point\nzcr(p) separating these two regions can be determined\nfrom the condition η(p,zcr)≈1. Since the dominant\nadvection irreversibly carries CRs away from the disk,\nthe halo size can be set equal to zcr. Therefore, the\nboundary condition at z→ ∞becomes unimportant as\nlong asη∞≫1.\nInordertoderive ηfromEquation( 18), weneedtoob-\ntain the diffusion coefficient Dfrom Equation ( 3), which\nrequires the solution of Equation ( 10).\n4.2.Excitation-damping balance\nThe numerical solution of Equations ( 9) and (10) (see\nSection5) suggests that W(k,z) in the diffusion region\ncan be estimated from the excitation-damping balance,\nΓCR=νNL. (21)\nWe rewrite it using Equations ( 4) and (6),\n4g(z)c2\nπe2B2k2k/integraldisplay\nkminW(k1)dk1=S0(p)−uAN .(22)\nIn the halosheath ( η <1) the last termofEquation( 22)\ncan be neglected. In this case, S0(p)∝Q(p) decreases\nwithpfaster than p−2, and thus the integral on the left-\nhand size of Equation ( 22) is dominated by the upper\nlimitk. Therefore,\nW(k,z) =π\n4g(z)∂\n∂k/bracketleftbig\np2S0(p)/bracketrightbig\n, (23)\nand\nη(p,z) =−3π3e\n2vc∂\n∂p/bracketleftbig\np2S0(p)/bracketrightbigz/integraldisplay\nzduA\nBg(z1)dz1.(24)5\nFrom Equation ( 7) we obtain\nη=−6π5/2e\nvc∂\n∂p/bracketleftbig\np2S0(p)/bracketrightbigz/integraldisplay\nzduth(T)β−1dz1\nB(e−β−1+1\n2ǫ1/2e−ǫβ−1).\n(25)\nFor simplicity, below we assume n(z) =n0exp(−z/zn),\nB(z) =B0exp(−z/zB), andT(z) =T0exp(z/zT).\n4.3.Spectrum of CRs in the halo sheath\nIfβis within the ranges defined in Equations ( 13)\nand (14), the expression for g(β) simplifies significantly.\nIn this regime, previously considered by Ptuskin et al.\n(1997), wecanneglecttheexponentialdependenceinthe\ndenominator of Equation ( 25) and rewrite the equation\nas\nη(p,z) =A(p)/parenleftBig\nez/zη−ezd/zη/parenrightBig\n,(26)\nwherez−1\nη=z−1\nn−z−1\nB−1\n2z−1\nT. Forthesakeofsimplicity,\nbelow we assume zd≈0.\nThe magnitude of the dimensionless factor A(p) de-\npends on the dominant mechanism of NL damping. In\na low-βplasma with 0 .01≤β≤0.2 the damping on\nthermal electrons dominates, and\nA(p)≈Ae(p) =−12π5/2eu2\nA0zη\nvcB0uth0ǫ1/2∂\n∂p/bracketleftbig\np2S0(p)/bracketrightbig\n,(27)\nwhereuth0=uth(zd), while in a high- βplasma with\nβ≥10 the damping is due to thermal protons, and\nA(p)≈Ap(p) =1\n2ǫ1/2Ae(p). (28)\nThecriticalpoint zcrisderivedfromthecondition η≈1.\nThus, the halo size is estimated from Equation ( 26) as\nzcr(p) =zηln[1+1/A(p)]. (29)\nFor low energies, where A(p)≫1, the halo size in-\ncreases with paszcr(p)≈zη/A(p); for high energies,\nthe halo size zcr(p)≈ −zηlnA(p) is almost indepen-\ndent of p. We point out that the model is not vi-\nable in the former regime, normally corresponding to\nthe electron-dominated damping, because the resulting\nhalo size becomes too small. On the other hand, for the\nproton-dominated damping with β >10, the function\nAp(p)∼1 for the following halo parameters: B≤1µG,\nn≥10−2cm−3, andT≥100 eV. The halo size in this\ncase exceeds 1 kpc at energies above 1 GeV.\nThe CR spectrum is given by Equation ( 16),\nN(p,z) =S0(p)\nuA0∞/integraldisplay\nηeη−η1dη1\n[1+η1/A(p)]zη/zA,(30)\nwherez−1\nA=1\n2z−1\nn−z−1\nBcharacterizes the spatial scale\nof variation of vA(z). This result can be expressed interms of the incomplete gamma-function Γ(a,z),\nN(p,z) =S0(p)\nuA0eη+A(p)A(p)zη/zA(31)\n×Γ(1−zη/zA,A(p)+η).\nIfA(p)+η≫1, the solution corresponds to the advec-\ntion flux with the Alfven velocity at the halo periphery.\nThis represents low-energy CRs at large distances from\nthe disk,\nN(p,z)≈S0(p)\nuA0[1+η/A(p)]−zη/zA(32)\n=S0(p)\nuA0e−z/zA≡S0(p)\nuA(z).\nIfA(p)+η≪1, the CR spectrum tends to\nN(p,z)≈S0(p)\nuA0A(p)zη/zAΓ(1−zη/zA) (33)\n∝S0(p)[pS0(p)]zη/zA.\nThe resulting spectrum, corresponding to the diffusion-\ndominated flux, does not practically depend on zup\nto the critical point zcr. We note that Equation ( 33)\nresembles Equation (34) from Ptuskin et al. (1997).\nThe derived approximate solution has important im-\nplications. We conclude that CRs escape from the halo\natz= 1−10 kpc, and for energetic CRs the halo size\nweakly depends on their energy. Given S0(p)∝p−2.4\nandN(p)∝p−2.7, our solution suggests that zη/zA≈\n0.3/1.4 orzn/zB−0.15zn/zT= 0.35.\n5.NUMERICAL SOLUTION AND DISCUSSION\nEquations ( 32) and (33) provide sufficiently good ap-\nproximations for the CR spectrum as long as βis within\nthe ranges defined in Equations ( 13) and (14). How-\never, the magnitude of βvaries strongly with zand,\ntherefore, the exponential terms in the denominator of\nEquation ( 25) cannot be generally ignored. As a result,\nthe expressions for ηandNbecome complicated and\ncan only be obtained numerically.\nTo reduce the number of free parameters, we con-\nsider a simple isothermal model ( z−1\nT= 0) with a con-\nstant magnetic field ( z−1\nB= 0). We use the follow-\ning set of parameters: B= 1µG,n0= 0.1 cm−3,\nzn= 1 kpc, and T= 10 eV. For the CR source func-\ntion, we use Q(p)≃Q∗(p/mpc)−2.4withQ∗mpc=\n9.4×10−4cm−2s−1, which is similar to the value given\nbyStrong et al. (2010). In this case, β= 40 at z= 0.\nThe total power of CR sources in the Galaxy can be\nroughly estimated as W= 2πR2\nGal/integraltext\nEkin(p)Q(p)dp=\n4.5×10−3erg×2πR2\nGalQ∗mpc. Assuming RGal= 20\nkpc for the Galactic disk radius, we obtain W ≈1041\nerg/s.\nEquations ( 9) and (10) are solved numerically by em-\nploying the procedure described in Dogiel et al. (2020).6\nFigure 2. Halo size zcr(Ekin) obtained from the numerical\nsolution of our model. The dashed line shows the case of\na single-component gas with B= 1µG,n0= 0.1 cm−3,\nzn= 1 kpc, and T= 10 eV ( β= 40), the solid line represents\nthe case of a two-component gas (see Section 5).\nTo account for CR species heavier than protons, the ex-\ncitation rate Γ CRis multiplied by a factor of 1.5 (see,\ne.g.,Dogiel et al. 2018 ). For the initial CR density we\nuseN(p,z) = 0, to avoid appearance of a sharp dis-\ncontinuity at the upper halo boundary. Both the initial\nMHD spectrum and the boundary condition at z= 0\nare equal to a small non-zero function W0(k,z), as it\nis necessary for the waves excitation. To ensure a weak\n(logarithmic) dependence of the integral in Equation ( 6)\non the integration limits, we use W0(k,z)∝k−1.\nThe resulting halo size, zcr, and the differential CR\nspectrum, N/4π, are plotted versus the proton ki-\nnetic energy Ekinby the dashed lines in Figures 2\nand3, respectively. To account for the solar modula-\ntion, we use the force-field approximation with poten-\ntialφ= 0.5 GV (Gleeson & Axford 1968 ). Observa-\ntional data are taken from Aguilar et al. (2015) (AMS-\n02),Adriani et al. (2019) (CALET), Grebenyuk et al.\n(2019) (NUCLEON), Yoon et al. (2011) (CREAM-\nI),Yoon et al. (2017) (CREAM-I+III), and An et al.\n(2019) (DAMPE). The data are collected using Cosmic-\nRay DataBase (CRDB v4.0) by Maurin et al. (2020).\nWe stress that the CR spectra strongly depend on\na particular model of NL damping. In our case, the\ndamping is described by Equations ( 6) and (7). Since\nβrapidly drops with the height, so does the damping\nand, hence, the CR diffusion coefficient. Therefore the\nCR spectra plotted in Figure 3can be interpreted as\nfollows:\n•Ekin<10 GeV: At such energies, A(p) is suffi-\nciently large and, thus, the halo size is small in\naccordance with Equation ( 29). For this reason,\nβ(zcr)≈β(0)>10andNLdampingisduetother-Figure 3. Energy spectra of CR protons obtained from the\nnumerical solution of our model (lines) and the observation al\ndata (symbols). All parameters are the same as in Figure 2.\nmal protons. Equation ( 32) is applicable, which\ngivesN(p)∝Q(p)∝p−2.4.\n•10 GeV< Ekin<1 TeV:N(p) starts approaching\nasofterspectrumdescribedbyEquation( 33). The\nhalo size increases with energy as zcr(p)∝1/A(p),\nand thus β(zcr) rapidly decreases, so that even-\ntually a mixed damping both on thermal protons\nand electrons operates.\n•100 GeV < Ekin<10 TeV: In the mixed-damping\nregime, a smooth transition from Ap(p) to much\nlargerAe(p) occurs. According to Equation ( 33),\nthat makes N(p) harder (NL damping rapidly re-\nduces with CR energy as the proton contribution\nbecomes negligible, and therefore the CR confine-\nmentincreases). InFigure 3, thetransitionisman-\nifested by the increase seen at 1 TeV < Ekin<\n10 TeV.\n•Ekin>10 TeV: Finally, at very high energies\nβ(zcr) decreases below 0 .1, where the damping\nis due to thermal electrons. Equation ( 33) be-\ncomes applicable; since zη/zA= 1/2 in our case,\nN(p)∝p−3.1.\nFigure3shows that the theoretical curve and the\nexperimental data are in good qualitative agreement.\nHowever, we should also keep in mind that gas in the\nhalo consists of several components. In particular, the\nwarm ionized gas (WIM) dominates at lower altitudes,\nwhile at higher zit is mostly hot coronal gas ( Ferri` ere\n1998;Gaensler et al. 2008 ). To account for multiple gas\ncomponents, we assume that the total gas density in\nour model is determined by a sum of the two phases:\nn(z) =nhot(z) +nWIM(z). The same principle ap-\nplies to the magnitude of NL damping in Equation ( 6):7\ng(z) =g(βhot)+g(βWIM). Note that the factor kuAin\nEquation ( 6) is the wave frequency, and therefore is the\nsame in both phases. Assuming B= 1µG, we use the\nfollowing set of parameters:\n•Warm phase ( β= 4):n0= 0.1 cm−3,T= 1 eV,\nzn= 0.4 kpc.\n•Hot phase ( β= 4):n0= 10−3cm−3,T= 100 eV,\nzn= 2 kpc.\nThe source function is Q(p)≃Q∗(p/mpc)−2.32with\nQ∗mpc= 9×10−4cm−2s−1.\nThe results for the two-phase model are depicted in\nFigures2and3by the solid lines. We see that the\ntheoretical curve show a much better agreement with\nthe observational data in this case, which is due to a\nmuch weaker dependence of βonz.\nWhile the two-phase model provides a remarkably\ngood overall agreement with the experimental data in a\nwide energy range, the discrepancy below 10 GeV is up\nto 20%. We believe that this is because the effect of disk\nturbulence on the vertical profile of the spectrum can no\nlongerbe ignoredat such lowenergies. Indeed, by deriv-\ning Equation ( 19) we assume that N(p,zd) =N(p,0).\nWhile this assumption is certainly reasonable for high\nenergies, the low-energy part of the spectrum should be\nstronger affected by the fact that the diffusion coeffi-\ncient in the disk decreases with energy, which inevitably\nleads to an increasing vertical gradient of N(z). There-\nfore, the low-energy spectrum should be more inhomo-\ngeneous at 0 < z < z d, and the actual spectrum at z= 0\nshould go somewhat above the theoretical curves plot-\nted in Figure 3. Furthermore, the diffusion coefficient\nin the Galactic disk is likely not affected by the CR\nstreaming (e.g., due to heavy damping on neutrals), but\nrather depends on external sources of turbulence (such\nas supernova explosions and stellar winds).\nApart from the halo size, another important parame-\nter characterizingpropagationofCRs is their grammage\nX, i.e., the average surface density traversed by CRs\nduring their lifetime in the Galaxy. The grammage de-\ntermines the ratio of secondary-to-primary nuclei, and\nthus can be derived from experimental data. For our\nmodel, it can be roughly estimated as\nX≈ NHmpc\nud, (34)\nwhich gives X(10 GeV) ≈12 g/cm2for our parameters\natEkin= 10 GeV. This value is close to that obtained\nby, e.g., Engelmann et al. (1990). We stress, however,\nthat such estimates are very approximate: to properly\ntest the model, we need to accuratelycalculate the spec-\ntra of secondary and primary nuclei, and compare them\nto the experimental data. This work will be reported in\na separate paper.\n6.CONCLUSIONSWe present a development of the self-consistent model\nof the Galactic CR halo, extending the model by\nDogiel et al. (2020). Our earlier model by Dogiel et al.\n(2020) predicts a small size of the halo at low ener-\ngies, which does not agree with experimental data. To\novercome this discrepancy, we include nonlinear Landau\n(NL) damping in the present model.\nThe key input parameters of the proposed model are\nthe CR source Q(p) as well as the spatial profiles of\nthe vertical magnetic field B, ionized gas density n, and\ntemperature T. We show that all these parameters may\nsignificantly affect the size of the halo, in particular at\nrelatively low CR energies. The MHD-turbulence in the\nhalo, which controls the vertical escape of CRs, is en-\ntirely generated by the resonant CR-streaming instabil-\nity. The equilibrium spectrum of MHD waves in our\npresent model is reached when the CR excitation rate is\nbalanced by NL damping. This significantly suppresses\nthe amplitude of MHD waves compared to the model of\nDogiel et al. (2020), thus making the halo size substan-\ntially larger.\nWe consider two alternative models of gas distribu-\ntions in the halo: a single-component isothermal model\nand a two-phase model composed of hot coronal gas and\nwarmionizedgas. Weshowedthatthesingle-component\nmodel requires very dense and hot gas with β≈40 at\nlow altitudes to be able to reproduce the experimental\ndata. For the two-phase model, the required gas param-\neters are much closer to those reported in the literature\n(e.g.,Ferri` ere 1998 ).\nOur model is able to reproduce the spectrum of CR\nprotons in a wide range of energies, including the spec-\ntral features observed between ∼10 GeV and ∼10 TeV\n(see Fig. 3). Despite some 20% discrepancy with ex-\nperimental data below 10 GeV, our model predicts a\nreasonable halo size of about 1 kpc at 1 GeV. We argue\nthat such a discrepancy may be due to increasing influ-\nence of the Galactic disk at lower energies, which is still\nneglected in our model.\nACKNOWLEDGMENTS\nThe authors are grateful to an anonymous referee for\nconstructive suggestions, and to Andrey Bykov for use-\nful discussions and comments. The work of DOC, VAD,\nADE, and AMK is supported by the Russian Science\nFoundation via the Project 20-12-00047.\nNOTE ADDED IN PROOF\nNew data on CR proton spectrum reported by\nCALET ( Adriani et al. 2022 ) confirms the existence of\nthe second spectral break at 10 TeV. The break position\nsuggests that the scale height of hot gas should be about\n2 kpc or less.8\nREFERENCES\nAdriani, O., Akaike, Y., Asano, K., et al. 2019, PhRvL,\n122, 181102\n—. 2022, PhRvL, 129, 101102\nAguilar, M., Aisa, D., Alpat, B., et al. 2015, PhRvL, 114,\n171103\nAn, Q., Asfandiyarov, R., Azzarello, P., et al. 2019, Scienc e\nAdvances, 5, eaax3793\nBerezinskii, V. S., Bulanov, S. V., Dogiel, V. A., Ginzburg,\nV. L., & Ptuskin, V. S. 1990, Astrophysics of cosmic rays\n(Amsterdam: North Holland)\nDogel’, V. A., Gurevich, A. V., & Zybin, K. P. 1993, A&A,\n268, 356\nDogiel, V. A., Chernyshov, D. O., Ivlev, A. V., et al. 2018,\nApJ, 868, 114\nDogiel, V. A., Gurevich, A. V., & Zybin, K. 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S., Anderson, T., Barrau, A., et al. 2017, ApJ,\n839, 5"    },    {        "title": "1809.01042v1.Separation_of_the_two_magnon_scattering_contribution_to_damping_for_the_determination_of_the_spin_mixing_conductance.pdf",        "content": "arXiv:1809.01042v1  [cond-mat.mtrl-sci]  4 Sep 2018Separation of the two-magnon scattering contribution to da mping for the\ndetermination of the spin mixing conductance\nA. Conca,1,∗S. Keller,1M. R. Schweizer,1E. Th. Papaioannou,1and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n(Dated: September 5, 2018)\nWe present angle dependent measurements of the damping prop erties of epitaxial Fe layers with\nMgO, Al and Pt capping layers. Based on the preferential dist ribution of lattice defects following the\ncrystal symmetry, we make use of a model of the defect density to separate the contribution of two-\nmagnon scattering to the damping from the isotropic contrib ution originating in the spin pumping\neffect, the viscous Gilbert damping and the magnetic proximi ty effect. The separation of the two-\nmagnon contribution, which depends strongly on the defect d ensity, allows for the measurement of\na value of the effective spin mixing conductance which is clos er to the value exclusively due to spin\npumping. The influence of the defect density for bilayers sys tems due to the different capping layers\nand to the unavoidable spread in defect density from sample t o sample is thus removed. This shows\nthe potential of studying spin pumping phenomena in fully or dered systems in which this separation\nis possible, contrary to polycrystalline or amorphous meta llic thin films.\nINTRODUCTION\nIn bilayers systems formed by a ferromagnetic (FM)\nlayer in contact with a metallic non-magnetic (NM) one,\na pure spin current can be generated and injected in the\nlatterwhen the ferromagneticresonanceisexcited. Typi-\ncally, a microwavemagnetic field is used for this purpose.\nThe whole processis commonly referredto as spin pump-\ning [1, 2]. If the non-magnetic layer is formed by a heavy\nmetal with large spin-orbit coupling (Pt, Ta or similar),\nthe spin current can be detected by using the inversespin\nHall effect (ISHE) for conversion into a charge current.\nSince the spin current leaving the magnetic layer car-\nries away angular momentum from the magnetization\nprecession,it representsanadditionallosschannelforthe\nmagnetic system and consequently causes an increase in\nthe measured Gilbert damping parameter α[1]:\n∆αsp=γ/planckover2pi1\n4πMsdFMg↑↓(1)\nwhereg↑↓is the real part of the spin mixing conductance\nwhich is controlling the magnitude of the generated spin\ncurrent and γis the gyromagnetic ratio.\nThis expression is only valid for sufficiently thick NM\nlayers where no reflection of the spin current takes place\nat the film surface or interface with other materials, i.e.\nno spin current is flowing back into the magnetic layer.\nIn principle, it allows the estimation of g↑↓by measur-\ning the increase in damping compared to the intrinsic\nvalue. However, to perform this measurement is not\nstraightforward. If the estimation of g↑↓for a FM/Pt\nsystem is needed, ideally one should measure the effec-\ntive Gilbert damping parameter α0for a single stand-\ning magnetic layer acting as a reference sample with no\nlosses due to spin pumping and repeat the same after de-\npositing a thick Pt layer. However, most of the commonferromagnetic materials, with exception of the magnetic\ninsulators like YIG, will change its properties due to ox-\nidation processes. Therefore, a capping layer is required\nand one has to find an appropriate one, in the sense that\nits introduction must not modify the damping properties\nof the magnetic layer. Examples in the literature show\nthat this is far to be a trivial task [3–5]. In addition to\nthis, the emergence of a finite magnetic polarization in\nPt in contact with a ferromagnetic layers has an impact\non damping which further hinders the estimation of g↑↓\n[5–12].\nFor the reference layers, the most convenient candi-\ndates as capping material are oxides like MgO, for which\nit has been proven that they are able to block the flow\nof spin current and therefore to deactivate spin pumping\n[13–15], or metals with weak spin-orbit interaction like\nAl or Ru. But even for these cases, it has been shown\nthat an increase of damping not related to spin pumping\nis possible. Ruiz et al.show for instance that a MgO\ncapping layer increases strongly the damping in permal-\nloy while this is not the case for Al capping layer [5].\nThe reason has nothing to do with the metallic char-\nacter of the capping layer since the increase for Ru is\neven larger than with MgO. The same work [5] already\nprovides a hint for a possible reason since the increase\nof damping roughly scales with the value of the inter-\nface perpendicular anisotropy constant K⊥\nS. Theoretical\nworks [16] show that the counterplay between the de-\nmagnetizing field responsible for the in-plane orientation\nof the magnetization and the perpendicular anisotropy\nfield can induce inhomogeneous magnetization states for\ncertain field strengths combinations which are responsi-\nble for an increased damping. In this sense, this effect\nhas been also adduced to explain the damping thickness\ndependence in Co 2FeAl/MgO systems [3].\nHere we present angle dependent measurements of the\ndamping properties of epitaxial Fe layers with MgO, Al2\nFIG. 1. (Color online) Dependence of the FMR linewidth on the frequency for different orientations φHof the external magnetic\nfield with respect to the [100] crystallographic axis of Fe fo r (a) Fe/Al and (b) Fe/Pt systems. The lines correspond to a li near\nfit to extract the effective damping parameter αeff. Forφ= 30◦a strong non-linearity due to magnetic dragging is observed .\nFor visibility reasons, each data set is shifted vertically by 1.25 mT with respect the previous one.\nand Pt capping layers. Fully epitaxial systems consti-\ntute a perfect ordered model with almost ideal and well\ndefined interfaces. Here, we will show that the angle de-\npendence of damping allows for a measurement of the\nstrength of the two-magnon scattering and of its contri-\nbution to the effective damping parameter. With the\nseparation of this contribution we access the increase\nof damping caused only by spin pumping and magnetic\nproximity effect and to an estimation of g↑↓without the\ncontamination of defects effects.\nEXPERIMENTAL DETAILS\nThe samples were deposited by e-beam evaporation\non MgO(100) substrates in a molecular beam epitaxy\n(MBE) chamber with a base pressure P b= 5×\n10−10mbar. A set of Fe/Pt bilayers with fixed Fe thick-\nness (12 nm) and varying Pt thickness were prepared.\nAdditional reference samples, where Pt is substituted by\nMgO or Al, have also been prepared. The Fe and Pt\nfilms were grown with a deposition rate of 0.05 ˚A/s. The\nsamples were deposited with a substrate temperature of\n300◦C and subsequently annealed at the same tempera-\nture.\nThe characterization by X-ray diffractometry (XRD)\n(presented elsewhere [17]) shows that the Fe/Pt bilayers\nare fully epitaxial with the Fe unit cell rotated by 45◦\nwith respect to the MgO substrate unit cell and with\nPt rotated again 45◦with respect to Fe. In the case of\nFe/Al, epitaxial growth of the upper layer could not be\nachieved.\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). Forthis, the samples were placed facing the strip-line and\nthe˜S12transmission parameter was recorded.\nRESULTS AND DISCUSSION\nFigures 1 shows the dependence of the measured FMR\nline width ∆ Hon the frequency for the reference layer\nwith Al capping (a) and a Fe/Pt system (b). The data\nis shown for different orientation of the external static\nmagnetic field varying from φH= 0◦([100], easy axis) to\nφH= 45◦([110], hard axis). For visibility reasons, each\ndata set is shifted vertically by 1.25 mT with respect to\nthe previous one.\nAs commented before, the choice of capping layer can\nhave a large influence on the linewidth and effective\ndampingofthe magneticlayer,evenforlightmetals. The\nmagnetic proximity effect (MPE) in the case of Pt also\ncontributes to an increase on damping, [5, 9–12] which\nadditionally challenges the measurement of the contri-\nbution from the spin pumping. Taking into account all\nthese considerations, the effective increase on damping\nwhen comparing a reference system and a system with a\nheavy metal can be separated as follows:\nαeff=α0+αmpe+αsp+αi. (2)\nHereα0is the intrinsic damping parameter which can\nbe defined as characteristic of the material under in-\nvestigation (growth conditions however may influence it\nstrongly) and it is the sum of the losses by two-magnon\nscattering and by energy transfer to the phonon system.\nαmpeis the contribution due to the dynamic coupling be-\ntween the ordered spins in Pt due to the MPE and the3\nFIG. 2. (Color online) (a) Dependence of the FMR resonance\nfieldHFMRon the in-plane direction of the static magnetic\nfield for two values of the resonant frequency. (b) Dependenc e\nof the in-plane angle of the magnetization vector φMon the\nexternal field direction φH. Both angles are measured relative\ntothe[100] axis. Thedottedline represents thecase of perf ect\ncollinearity between magnetization and external field.\nmagnetization in the magnetic layer. αspis the result\nof the losses by the spin current generated in the fer-\nromagnetic layer by the precession of the magnetization\nand that flows into the Pt layer (spin pumping). The last\ntermαisummarizes the increase of damping due to other\ninterfacial effects such as interface PMA as commented\nabove, spin memory loss [18] or isotropic scattering at\ninterface defects [19].\nSeveral efforts have been made in order to separate\nsome of the contributions to αeff. In a recent work with\nCoFeB/Pt [9] we were able to separate αmpedue to the\ndependence on the Pt thickness. As already reported by\nCaminale et al.[11], a linear Pt thickness dependence of\nthe spin-current absorption in spin-sink layers exhibiting\nMPE and of αmpeis expected [12]. A detailed vector\nnetwork analyzer FMR study has also been recently re-\nported to separate the different contributions in NiFe/Pt\nsystems [20].\nTheterm α0isaresultoftwocontributions[22]. Oneis\nthe pure Gilbert damping, which is of viscous nature and\ngenerates a dissipation of energy and angular momentum\ntothe lattice. The secondoneisthe transfertospin-wavemodes with k/negationslash= 0 from the FMR mode via two-magnon\nscattering. For a pure Gilbert-like viscous damping the\nlinewidth dependence on the frequency is purely linear:\nµ0∆H=µ0∆H0+4παf\nγ. (3)\nHere, ∆H0is the inhomogeneous broadening and is re-\nlated to film quality.\nThe lines in Figs. 1 (a) and (b) are a fit to this ex-\npression. It has to be mentioned that although a viscous\ndamping generates a linear dependence, on the contrary\nit is not possible to assume that the observation of a lin-\nearbehavior provesthat only viscous damping is present.\nThe reason for that is that two-magnon scattering can\nmimic also a linear dependence [21–23]. For both sam-\nples, and for the MgO capped sample not shown here,\nforφ= 30◦a strongly non linear behavior with a large\nincrease in linewidth values for smaller frequencies is ob-\nserved. For this reason, the hollow points in Fig. 1 have\nbeen excluded from the fit. The non-linearity at low fre-\nquencies cannot be explained by viscous damping and it\nis caused by magnetic dragging. The magnetic dragging\neffect describesthe increaseofthe linewidth of precessing\nmagneticlayerswithlargemagneticcrystalineanisotropy\ndue to the non-collinearity of the magnetization and the\nexternal magnetic field. In Fig. 2 (a), the dependence\nof the resonance field HFMRon the in-plane direction of\nthe external magnetic field is shown for two fixed fre-\nquency values. As a result of the four-fold anisotropy ex-\npectedfromthecubiclatticeofFeandassumingaperfect\ncollinearity between magnetization vector and external\nfield,HFMRcan be modeled as: [10, 24]\nµ0HFMR=µ0˜HFMR+2K1\nMscos(4φ),(4)\nwhereK1is the cubic anisotropyconstant, φthe in-plane\nazimuthal angle and ˜HFMRis the averaged resonance\nfield value. The fraction2K1\nMsis directly the anisotropy\nfield H B. In Fig. 2(a) a deviation from this model is ob-\nserved for angles between the hard and easy axis and it\nis due to magnetic dragging, i.e., the magnetization is\nnot aligned to the external field due to the effect of the\nanisotropy field. The fact that the deviation from the\nmodel in Eq.4 is smallerfor largerfrequencies(i.e. larger\napplied field) alsosupportsthis interpretation. The same\nbehavior observed for φ= 30◦has been also been re-\nported for ultrathin Fe films [25] or for insulating LSMO\nfilms [28] and attributed to magnetic dragging. The de-\ngree of non-collinearity can be estimated by solving the\nequilibrium condition for the angle defining the orienta-\ntion of the magnetization φMfor each value of φH:\nHsin(φM−φH)+HB\n4sin(4φM) = 0,(5)4\nwhere the value for the cubic anisotropy field was taken\nfrom [10]. Fig. 2(b) shows the obtained value of φMfor\nthe data shown in Fig. 2(a). The angle between mag-\nnetization and magnetic field can be as large as 10◦for\n13 GHz and it is decreased to a maximum around 4.5◦\nfor 18 GHz. The magnetic dragging effect is largest for\nφHbetween the easy and hard axis and vanishes along\nthe main crystallographic axes.\nFigure 3 shows the value of the effective damping pa-\nrameter αeffas obtained from the fits in Fig. 1 for the\nthree capping layers. In all of them, an eight-fold sym-\nmetry on the in-plane angle φHis observed with maxima\nalong the easy and hard axis of the Fe layers and min-\nima in between. For the Fe/Al and Fe/MgO samples,\nwhere spin pumping has no influence, αeff=α0+αi\nwhile for the Fe/Pt sample, where both losses through\nspin pumping and due to the MPE are active, we obtain\nthe situation shown in Eq. 2. It is remarkable that the\ndifferent origins of the damping do not change the overall\nsymmetry of the angular dependence. It has though an\nimpact on the absolute values, which are larger for the\nFe/Pt sample.\nIn the literature concerning epitaxial layers, it is possi-\nble to find different symmetries for the dependence of the\nFMRlinewidthorthedampingparameteronthein-plane\nfield direction. For the Heusler alloy Co 2FeAl both four-\nand eight-fold symmetries for the linewidth have been\nreported. The situation differs depending on the thick-\nnessofthe film [23] andalsobetween different groups[30]\npointing out to a role of the growth conditions. For Fe 3Si\nfilms and Fe/V multilayer systems a four-fold symmetry\nis reported [22, 26] and for ultrathin Fe layers, where the\nrole of the interface is strong, a two-fold symmetry of\nαeffhas been measured [25]. Eight-fold symmetry has\nbeen also observed in epitaxial FeSi systems [26, 29]. In\na different work on Fe layers, a decrease on the obtained\nαvalue along the intermediate orientation between the\ntwomainaxisrelativetothe onemeasuredalongtheeasy\nand hard axis was reported [27], pointing to an angular\ndependence very similar to ours. Concerning insulating\nsystems, two- and four-fold symmetries have been ob-\nserved in LSMO films [28].\nTwo-magnon scattering can only occur if scattering\ncenters in form of defects are present. If, as expected,\nthese are present as point lattice defects or dislocation\nlinesalongthemaincrystallographicdirections, itisclear\nthat the scattering intensity should reflect the symmetry\nof the lattice. This fact would for certain explain a four-\nor eight-fold anisotropy in damping observed in some on\nthe reports mentioned above and the maxima in αefffor\nour samples for φ= 0◦,45◦,90◦,135◦.\nFollowingZakeri et al. andAria et al., the contribution\nto damping due to two-magnonscattering can be written\nas [21, 26]:α2M=/summationdisplay\n/angbracketleftxi/angbracketrightΓ/angbracketleftxi/angbracketrightf(φH−φ/angbracketleftxi/angbracketright), (6)\nwhere Γ /angbracketleftxi/angbracketrightrepresents the strength of the two-magnon\nscattering contribution along the in-plane crystallo-\ngraphic direction /angbracketleftxi/angbracketright. The function f(φH−φ/angbracketleftxi/angbracketright) al-\nlows for an angle dependent two-magnon contribution to\ndamping with respect to the orientation of the external\nfieldHrelative to the crystallographic directions /angbracketleftxi/angbracketright.\nThe physical interpretation of the function f(φH−φ/angbracketleftxi/angbracketright)\nlays in the Fourier transform of the defects in the film\n[26, 34]. By using the ansatz f(φH−φ/angbracketleftxi/angbracketright) = cos2(4φH−\nφ/angbracketleftxi/angbracketright) we can fit the damping dependence using a simpli-\nfied version:\nαeff=αiso+α2M=αiso+Γ2Mcos2(4φH−φ[100]) (7)\nwhereαisoincludes now all the isotropic contributions to\ndamping, i.e. αmpe,αsp, pure Gilbert damping and po-\ntentially isotropic interface contributions from the term\nαi, mainly spin memory loss and interface PMA related\neffects.\nThe red lines in Fig. 3 show the fit to this model. The\nobtained parameters are summarized in Table. I. A very\nlow value below 1 ×10−3is obtained for αisofor the Fe/Al\nsample. Since αsp,MPE= 0 is expected and due to the\nlow value we consider that the obtained αisomust be\nvery close to the value corresponding only to pure vis-\ncous Gilbert damping corresponding to high quality Fe.\nHowever, strictly speaking, the obtained value is only\nan upper limit since still other effects might contribute.\nConcerning 3d metals with no half-metallic character, a\nvery low damping value of 0.7 ×10−3has been reported\nby Leeet al.for CoFe [35]. This value is comparable\nto theαisomeasured here for Fe/Al. The fact that the\nCoFe samples in which the low value was obtained are\nalso fully epitaxial with an exceptionally high crystalline\nquality explains the similarity in values. The low defect\ndensity in CoFe almost suppresses two-magnon scatter-\ning in the CoFe samplesand thereforeis comparablewith\nourαisowhere that contribution is already separated.\nFor the Fe/MgO sample the value for αisoincreases by\na factor larger than 2 although also here αsp,MPE= 0.\nαiso Γ2M\n(10−3) (10−3)\nFe/Al 0.8 ±0.3 3.6 ±0.4\nFe/Pt 3.4 ±0.3 2.4 ±0.4\nFe/MgO 1.9 ±0.1 1.3 ±0.1\nTABLE I. Isotropic contribution αisoand two-magnon scat-\ntering contribution Γ 2Mto the total effective damping param-\neterαeff.5\nFIG. 3. (Color online) Angular dependence of the effective da mping parameter αeffin the in-plane direction of the static\nmagnetic field φHfor (a) Fe/Al, (b) Fe/Pt and (c) Fe/MgO. The red lines are a fit t o Eq. 7.\nThe main differences between Fe/Al and Fe/MgO are\nthat the MgO is single crystalline while Al is polycrys-\ntalline and the contrast between the metallic character\nof Al with the insulating oxide. The lattice mismatch\nbetween MgO and Fe is around 4% and introduces there-\nfore a certain degree of stress in the Fe layer which is\nnot present when the capping is polycrystalline Al and\nwhichcanhaveanimpactondamping. Atthesametime,\nsince the Gilbert damping is sensitive to the density of\nstates and this one is modified at the interface by the\nkind of bonds between the Fe atom and the atoms from\nthe cappinglayer, the simple materialdifferencemay also\nexplain the difference. In this sense it is remarkable that\nthe low damping value by Lee et al.commented before is\nonly observed for CoFe with a MgO capping layer and a\nlargervalue is measuredwhen MgAl 2O4is used [35]. Our\ndata confirms the important role of the capping layer on\ndamping observed in other works [5].\nA further increase in the value of αisois observed for\nthe Fe/Pt sample where additional losses through spin\npumping and MPE are present. Unfortunately the data\npresented in this paper does not allow to disentangle\nthese two contributions. For this reason, when using\nEq. 1 for the calculation of spin mixing conductance, it\nmakes sense to refer to an effective value g↑↓\neffwhich is at\nthe same time an upper limit for the corresponding value\nfor spin pumping alone. Using the Fe/Al sample as a\nreference we obtain a value for the spin mixing conduc-\ntance of 3 .7±0.9×1019m−2. This value is lower than\nthe one presented in our previous report [10] and shows\nthat the value of g↑↓\neffcan be easily overestimated if the\neffect of two-magnon scattering on damping is not sepa-\nrated, with the consequent overestimation of the injected\nspin current and underestimation of the spin Hall angle\nfrom the ISHE voltage [17]. The advantage of using epi-\ntaxial magnetic layers is that they allow the separation\nof the contribution of the two-magnon scattering due to\nthe strongangulardependenceand welldefinedcrystallo-\ngraphicdirections. Thesameisnotpossibleincommonly\nused material as CoFeB or NiFe where the amorphous or\npolycrystalline nature of the layers blends the scatteringdependence on the in-plane angle.\nThe parameter Γ 2Mprovides further insight into the\norigin of total damping in the samples. This parameter\nis larger for the Fe/Al sample in comparison to the fully\nepitaxial bilayers being almost three times larger than\nfor Fe/MgO. As a result, the total damping in the Fe/Al\nsample is dominated by the two-magnon scattering due\nalsoto the low αisowhile the sameis not true in the other\ntwo systems. It has to be taken into account that, since\nas scattering centers for magnon scattering the defects at\nthe interfaces play a role, they can be dominant in thin\nfilms. From TEM images (presented for instance in [10]),\nwe can prove the existence of a highly ordered interface\nin the fully epitaxial samples. Of course, the same is not\ntrue for the case with polycrystalline Al capping. We\nbelieve that the dominant role of the interface here is\npossible, also due to the overall low defect density in the\nbulk of the Fe layer.\nFor completeness we want to discuss two additional\neffects potentially affecting the linewidth and damping.\nDue to the spread of internal and anisotropy field due to\nmosaicity in the film, there is a contribution to the line\nbroadening which has the following form [26, 33]:\n∆Hmosaic=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂HFMR\n∂φH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆φH, (8)\nwhere ∆φHis the average spread of the direction of the\neasy axes in the film plane. From Fig. 1(c) it is clear\nthat this contribution should increase the linewidth in\ntheregion φ= 15−30◦andequivalentonesbutthis isnot\nobserved pointing to a weak impact of mosaicity. In any\ncase, the mosaicity term is frequency independent and\nwill be only visible in the inhomogeneous linebroadening\n∆H0and will not affect the determination of αeff.\nThe discussion followingthe introduction ofEqs.6 and\n7 was focused on crystalline lattice defects as the origin\nof two-magnon scattering. However any kind on inhomo-\ngeneity in the magnetic state of the sample may play the\nsame role. The presence of magnetic dragging, visible for\ninstance for φ= 30◦in Fig. 1 can create a slight inhomo-\ngeneity in the magnetization state for field orientations6\nclose to the hard axis direction and an increase of damp-\ning around the hard axis orientation. In any case, this\ncontribution follows also the symmetry of the lattice and\nit is accounted in the Γ 2Mparameter.\nAlthough certain theoretical works point to an\nanisotropic Gilbert damping in fully epitaxial systems\ndue to its dependence on the density of states at the\nFermi energy [31, 32], experimentally this has been only\nseen in ultrathin Fe films [22] due to the modification of\nthe electronic structure induced by the interfacial spin-\norbit coupling. The anisotropy in αeffpresented here can\nbe fully explained by two-magnon scattering, and there-\nfore an isotropic Gilbert damping can be assumed.\nCONCLUSIONS\nMaking use of the well defined dependence of the two-\nmagnon scattering mechanism on the in-plane field di-\nrection, we have been able to separate this contribution\nto damping from the isotropic contributions originating\nfrom the viscous Gilbert damping mechanism, from spin\npumping and from the magnetic proximity effect in Pt.\nThe method can be implemented thanks to the pref-\nerential ordering of crystalline defects with respect to\nthe crystallographic directions in epitaxial systems and\ntherefore cannot be extended to amorphous or polycrys-\ntalline magnetic films. This shows the potential of the\nstudy of spin pumping related phenomena in ordered\nsystems. Without the contribution of the two-magnon\nscattering, which depends strongly on the chosen cap-\nping layer and defect density, a value of the effective spin\nmixing conductance g↑↓\neffis obtained which is closer to\ntheg↑↓associated only to spin pumping. This approach\nallows for a better estimation of the spin Hall angle in\nmetals.\nACKNOWLEDGEMENTS\nFinancial support by M-era.Net through the\nHEUMEM project and by the Carl Zeiss Stiftung\nis gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and\nB. I. Halperin, Rev. Mod. Phys. 77, No. 4, 1375 (2005).\n[2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[3] A. Conca, A. Niesen, G. Reiss, and B. Hillebrands,\nJ. Phys. D: Appl. Phys. 51, 165303 (2018).\n[4] A. Natarajarathinam, Z. R. Tadisina, T. Mewes,\nS. Watts, E. Chen, and S. Gupta, J. Appl. Phys. 112,\n053909 (2012)[5] A. Ruiz-Calaforra, T. Br¨ acher, V. Lauer, P. Pirro,\nB. Heinz, M. 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P. White,\nW. T. Ruane, B. D. Esser, D. W. McComb, P. C. Ham-\nmel, andF. Yang, NatureCommunications 8, 234(2017)."    },    {        "title": "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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Stiles1\n1Center for Nanoscale Science and Technology\nNational Institute of Standards and Technology,\nGaithersburg, MD 20899-6202\n2Maryland Nanocenter, University of Maryland,\nCollege Park, MD 20742-3511\n(Dated: December 4, 2018.)\nThe Landau-Lifshitz-Gilbert damping parameter is typical ly assumed to be a local quantity, in-\ndependent of magnetic configuration. To test the validity of this assumption we calculate the\nprecession damping rate of small amplitude non-uniform mod e magnons in iron, cobalt, and nickel.\nAt scattering rates expected near and above room temperatur e, little change in the damping rate is\nfound as the magnon wavelength is decreased from infinity to a length shorter than features probed\nin recent experiments. This result indicates that non-loca l effects due to the presence of weakly\nnon-uniform modes, expected in real devices, should not app reciably affect the dynamic response of\nthe element at typical operating temperatures. Conversely , at scattering rates expected in very pure\nsamples around cryogenic temperatures, non-local effects r esult in an order of magnitude decrease\nin damping rates for magnons with wavelengths commensurate with domain wall widths. While this\nlow temperature result is likely of little practical import ance, it provides an experimentally testable\nprediction of the non-local contribution of the spin-orbit torque-correlation model of precession\ndamping. None of these results exhibit strong dependence on the magnon propagation direction.\nMagnetization dynamics continues to be a techno-\nlogically important, but incompletely understood topic.\nHistorically, field induced magnetization dynamics have\nbeen described adequately by the phenomenological\nLandau-Lifshitz (LL) equation [1]\n˙m=−|γM|m×H+λˆm×(m×H),(1)\nor the mathematically equivalent Gilbert form [2, 3].\nEquation 1 accounts for the near equilibrium dynamics\nof systems in the absence of an electrical current. γM\nis the gyromagnetic ratio and λis the phenomenological\ndamping parameter, which quantifies the decay of the\nexcited system back to equilibrium. The LL equation\nis a rather simple approximation to very intricate dy-\nnamic processes. The limitations of the approximations\nentering into the LL equation are likely to be tested by\nthe next generation of magnetodynamic devices. While\nmanygeneralizationsforthe LLequationarepossible, we\nfocus on investigatingthe importance of non-local contri-\nbutions to damping. It is generally assumed in both ana-\nlyzing experimental results and in performing micromag-\nnetic simulations that damping is a local phenomenon.\nWhile no clearexperimental evidence exists to contradict\nthis assumption, the possibility that the damping is non-\nlocal – that it depends, for example, on the local gradient\nofthemagnetization–wouldhaveparticularimplications\nfor experiments that quantify spin-current polarization\n[4], for storage [5] and logic [6] devices based on using\nthis spin-current to move domain-walls, quantifying vor-\ntex [7] and mode [8] dynamics in patterned samples, and\nthe behavior of nano-contact oscillators [9, 10].\nWhile several viable mechanisms have been proposed\nto explain the damping process in different systems\n[11, 12, 13, 14, 15, 16, 17], werestrictthescopeofthispa-\nper to investigating the degree to which the assumptionof local damping is violated for small amplitude dynam-\nics within pure bulk transition metal systems where the\ndominant source of damping is the intrinsic spin-orbit in-\nteraction. For such systems, Kambersk´ y’s [14] spin-orbit\ntorque-correlationmodel, which predicts a decay rate for\nthe uniform precession mode of\nλ0=π¯hγ2\nM\nµ0/summationdisplay\nnm/integraldisplay\ndk/vextendsingle/vextendsingleΓ−\nnm(k)/vextendsingle/vextendsingle2Wnm(k),(2)\nhasrecentlybeen demonstratedtoaccountforthe major-\nity of damping [18, 19]. The matrix elements |Γ−\nnm(k)|2\nrepresent a scattering event in which a quantum of the\nuniformmodedecaysintoasinglequasi-particleelectron-\nhole excitation. This annihilation of a magnon raises the\nangularmomentum of the system, orienting the magneti-\nzation closer to equilibrium. The excited electron, which\nhas wavevector kand band index m, and the hole, with\nwavevector kand band index n, carry off the energy and\nangular momentum of the magnon. This electron-hole\npair is rapidly quenched through lattice scattering. The\nweighting function Wnm(k) measures the rate at which\nthe scattering event occurs. The very short lifetime of\nthe electron-hole pair quasiparticle (on the order of fs at\nroom temperature) introduces significant energy broad-\nening (several hundred meV). The weighting function,\nwhich is a generalization of the delta function appearing\nin a simple Fermi’s golden rule expression, quantifies the\nenergy overlap of the broadened electron and hole states\nwith each other and with the Fermi level.\nEquation 2, which has been discussed extensively\n[14, 18, 19, 20], considers only local contributions to the\ndamping rate. Non-local contributions to damping may\nbe studied through the decay of non-uniform spin-waves.\nAlthough recent efforts have approached the problem of2\nnon-localcontributionsto the dissipationofnon-collinear\nexcited states [21, 22] the simple step of generalizing\nKambersk´ y’s theory to non-uniform mode magnons has\nnot yet been taken. We fill this obvious gap, obtaining a\ndamping rate of\nλq=π¯hγ2\nM\nµ0/summationdisplay\nnm/integraldisplay\ndk/vextendsingle/vextendsingleΓ−\nnm(k,k+q)/vextendsingle/vextendsingle2Wnm(k,k+q)\n(3)\nfor a magnon with wavevector q. We test the impor-\ntance of non-local effects by quantifying this expression\nfor varying degrees of magnetic non-collinearity (magnon\nwavevector magnitude). The numerical evaluation of\nEq. 3 for the damping rate of finite wavelength magnons\nin transition metal systems, presented in Fig. 1, and the\nensuing physical discussion form the primary contribu-\ntion of this paper. We find that the damping rate ex-\npected inverypuresamplesatlowtemperatureisrapidly\nreduced as the magnon wavevector |q|grows, but the\ndamping rate anticipated outside of this ideal limit is\nbarely affected. We provide a simple band structure ar-\ngument to explain these observations. The results are\nrelevant to systems for which the non-collinear excita-\ntion may be expanded in long wavelength spin-waves,\nprovided the amplitude of these waves is small enough to\nneglect magnon-magnon scattering.\nCalculations for the single-mode damping constant\n(Eq. 3) as a function of electron scattering rate are pre-\nsented in Fig. 1 for iron, cobalt, and nickel. The Gilbert\ndamping parameter α=λ/γMis also given. Damp-\ning rates are given for magnons with wavevectors along\nthe bulk equilibrium directions, which are /angbracketleft100/angbracketrightfor Fe,\n/angbracketleft0001/angbracketrightfor Co, and /angbracketleft111/angbracketrightfor Ni. Qualitatively and quan-\ntitatively similar results were obtained for other magnon\nwavevector directions for each metal. The magnons re-\nportedoninFig.1constitutesmalldeviationsofthemag-\nnetization transverse to the equilibrium direction with\nwavevectormagnitudes between zero and 1 % of the Bril-\nlouin zone edge. This wavevector range corresponds to\nmagnon half-wavelengths between infinity and 100 lat-\ntice spacings, which is 28.7 nm for Fe, 40.7 nm for Co,\nand 35.2 nm for Ni. This range includes the wavelengths\nreported by Vlaminck and Bailleul in their recent mea-\nsurement of spin polarization [4].\nResults for the three metals are qualitatively similar.\nThemoststrikingtrendisadramatic,orderofmagnitude\ndecreaseofthe damping rate at the lowestscatteringrate\ntested as the wavevector magnitude increases from zero\nto 1 % of the Brillouin zone edge. This observation holds\nin each metal for every magnon propagation direction\ninvestigated. For the higher scattering rates expected in\ndevices at room temperature there is almost no change\nin the damping rate as the magnon wavevector increases\nfrom zero to 1 % of the Brillouin zone edge in any of the\ndirections investigated for any of the metals.\nTo understand the different dependences of the damp-\ning rate on the magnon wavevector at low versus high\nscattering rates we first note that the damping rate10121013101410151091010\n0.010.1λ (s-1)\nγ (s-1)\n α 1081090.01λ (s-1) \n α1090.01 1.0 0.1\n0.01\n λ (s-1)\n αhγ (eV)\n10-310-3\n10-30.0\n0.001\n0.003\n0.005\n0.01\n0.0\n0.001\n0.003\n0.005\n0.01\n0.0\n0.001\n0.003\n0.005\n0.01Fe\nCo\nNiq\nq\nq\nFIG. 1: Damping rates versus scattering rate. The preces-\nsion damping rates for magnons in iron, cobalt, and nickel\nare plotted versus electron scattering rate for several mag non\nwavevectors. A dramatic reduction in damping rate is ob-\nserved at the lowest scattering rates. The Landau-Lifshitz λ\n(Gilbert α) damping parameter is given on the left (right)\naxes. Electron scattering rate is given in eV on the top axis.\nMagnon wavevector magnitudes are given in units of the Bril-\nlouin zone edge and directions are as indicated in the text.\n(Eqs. 2 & 3) is a convolution of two factors: the torque\nmatrix elements and the weighting function. The ma-\ntrix elements do not change significantly as the magnon\nwavevector increases, however, the weighting function\ncan change substantially. The weighting function\nWnm(k,k+q)≈An,k(ǫF)Am,k+q(ǫF) (4)3\n789101112\n789101112\n Energy  (eV)\nH F P\nFIG. 2: Partial band structure of bcc iron. The horizontal\nblack line indicates the Fermi level and the shaded region\nrepresents the degree of spectral broadening. The solid dot is\na hypothetical initial electron state while the open circle is a\npotential final scattering state. (Initial and final state wa ve-\nvector separations are exaggerated for clarity of illustra tion.)\nThe intraband magnon decay rate diminishes as the energy\nseparation of the states exceeds the spectral broadening.\ncontains a product of the initial and final state electron\nspectral functions\nAn,k(ǫ) =1\nπ¯hγ\n(ǫ−ǫn,k)2+(¯hγ)2, (5)\nwhichareLorentziansinenergyspace. Thespectralfunc-\ntion for state |n,k/angbracketright, which has nominal band energy ǫn,k,\nis evaluated within a verynarrowrangeofthe Fermi level\nǫF. The width of the spectral function ¯ hγis given by\nthe electron scattering rate γ= 1/2τwhereτis the\norbital lifetime. (The lifetimes of all orbital states are\ntaken to be equal for these calculations and no specific\nscattering mechanism is implied.) The weighting func-\ntion restricts the electron-hole pair generated during the\nmagnon decay to states close in energy to each other and\nnear the Fermi level. For high scattering rates, the elec-\ntron spectral functions are significantly broadened and\nthe weighting function incorporates states within an ap-\npreciablerange(severalhundredmeV) ofthe Fermi level.\nFor low scattering rates, the spectral functions are quite\nnarrow (only a few meV) and both the electron and hole\nstate must be very close to the Fermi level.\nThe second consideration useful for understanding the\nresults of Fig. 1 is that the sum in Eqs. 2 & 3 can be\ndivided into intraband ( n=m) and interband ( n/negationslash=m)\nterms. For the uniform mode, these two contributions\ncorrespond to different physical processes with the intra-\nband contributiondominatingatlowscatteringratesand\nthe interband terms dominating at high scattering rates\n[14, 18, 19, 20].101210131014101510-3\n10-3\n10-4\n1071081090.01 0. 11\n0.01\n 0.0\n 0.001\n 0.003\n 0.005\n 0.01λintra,inte r (s-1)\nγ (s-1)q in ( π/a)interband\nintraband\n αintra,inter hγ (eV)Increasing q\nFIG. 3: Intraband and interband damping contributions in\niron. Theintrabandandinterbandcontributionstothedamp -\ning rate of magnons in the /angbracketleft100/angbracketrightdirection in iron are plot-\nted versus scattering rate for several magnitudes of magnon\nwavevector. Magnitudes are given in units of the Brillouin\nzone edge.\nFor intraband scattering, the electron and hole occupy\nthe sameband and must haveessentiallythe sameenergy\n(within ¯hγ). The energy difference between the electron\nand hole states may be approximated as ǫn,k+q−ǫn,k≈\nq·∂ǫn,k/∂k. The generation of intraband electron-hole\npairs responsible for intraband damping gets suppressed\nasq·∂ǫn,k/∂kbecomes largecomparedto ¯ hγ. Unless the\nbands are very flat at the Fermi level there will be few lo-\ncations on the Fermi surface that maintain the condition\nq·∂ǫn,k/∂k<¯hγfor low scattering rates as the magnon\nwavevectorgrows. (See Fig. 2). Indeed, at low scattering\nrates when ¯ hγis only a few meV, Fig. 3 shows that the\nintraband contribution to damping decreases markedly\nwith only modest increase of the magnon wavevector.\nSince the intraband contribution dominates the inter-\nband term in this limit the total damping rate also de-\ncreases sharply as the magnon wavevector is increased\nfor low scattering rates. For higher scattering rates, the\nelectronspectralfunctionsaresufficientlybroadenedthat\nthe overlap of intraband states does not decrease appre-\nciably as the states are separated by finite wavevector\n(q·∂ǫn,k/∂k<¯hγgenerally holds over the Fermi sur-\nface). Therefore, the intraband contribution is largelyin-\ndependentofmagnonwavevectorathighscatteringrates.\nThe interband contribution to damping involves scat-\ntering between states in different bands, separated by the\nmagnon wavevector q. Isolating the interband damping\ncontribution reveals that these contributions are insensi-\ntive to the magnon wavevector at higher scattering rates\nwhere they form the dominant contribution to damp-\ning (see Fig. 3). To understand these observations we\nagain compare the spectral broadening ¯ hγto the quasi-\nparticle energy difference ∆m,k+q\nn,k=ǫm,k+q−ǫn,k. The\nquasiparticle energy difference may be approximated as4\n∆m,k\nn,k+q·∂∆m,k\nn,k/∂k. The interband energy spacings\nare effectively modulated by the product of the magnon\nwavevector and the slopes of the bands. At high scatter-\ning rates, when the spectral broadening exceeds the ver-\ntical band spacings, this energy modulation is unimpor-\ntant and the damping rate is independent of the magnon\nwavevector. At low scattering rates, when the spec-\ntral broadening is less than many of the band spacings,\nthis modulation can alter the interband energy spacings\nenough to allow or forbid generation of these electron-\nhole pairs. For Fe, Co, and Ni, this produces a modest\nincrease in the interband damping rate at low scattering\nrates as the magnon wavevector increases. However, this\neffect is unimportant to the total damping rate, which\nremains dominated by the intraband terms at low scat-\ntering rates.\nLastly, we describe the numerical methods employed\nin this study. Converged ground state electron densities\nwere first obtained via the linear-augmented-plane-wave\nmethod. The Perdew-Wang functional for exchange-\ncorrelation within the local spin density approximation\nwas implemented. Many details of the ground state den-\nsity convergence process are given in [23]. Densities were\nthen expanded into Kohn-Sham orbitals using a scalar-\nrelativistic spin-orbit interaction with the magnetiza-\ntion aligned along the experimentally determined mag-\nnetocrystalline anisotropy easy axis. The Kohn-Sham\nenergies were artificially broadened through the ad hoc\nintroduction of an electron lifetime. Matrix elements of\nthe torque operator Γ−= [σ−,Hso] were evaluated sim-\nilarly to the spin-orbit matrix elements [24]. ( σ−is the\nspin lowering operator and Hsois the spin-orbit Hamil-\ntonian.) The product of the matrix elements and the\nweightingfunction wereintegratedover k-spaceusingthe\nspecial points method with a fictitious smearing of the\nFermi surface for numerical stability. Convergence wasobtained by sampling the full Brillouin zone with 1603\nk-points for Fe and Ni, and 1602x 91 points for Co.\nIn summary, we have investigated the importance of\nnon-local damping effects by calculating the intrinsic\nspin-orbit contribution to precession damping in bulk\ntransition metal ferromagnets for small amplitude spin-\nwaveswith finite wavelengths. Results ofthe calculations\ndo not contradict the common-practice assumption that\ndamping is a local phenomenon. For transition metals,\nat scattering rates corresponding to room temperature,\nwe find that the single-mode damping rate is essentially\nindependent of magnon wavevector for wavevectors be-\ntween zero and 1 % of the Brillouin zone edge. It is not\nuntil low temperatures in the most pure samples that\nnon-local effects become significant. At these scatter-\ning rates, damping rates decrease by as much as an or-\nder of magnitude as the magnon wavevector is increased.\nThe insensitivity of damping rate to magnon wavevector\nat high scattering rates versus the strong sensitivity at\nlow scattering rates can be explained in terms of band\nstructure effects. Due to electron spectral broadening at\nhigh scattering rates the energy conservation constraint\nduring magnon decay is effectively relaxed, making the\ndamping rate independent of magnon wavevector. The\nminimal spectral broadening at low scattering rates –\nseenonlyinverypureandcoldsamples–greatlyrestricts\nthe possible intraband scattering processes, lowering the\ndamping rate. The prediction of reduced damping at low\nscattering rates and non-zero magnon wavevectors is of\nlittle practical importance, but could provide an accessi-\nble test of the torque-correlationmodel. Specifically, this\nmight be testable in ferromagnetic semiconductors such\nas (Ga,Mn)As forwhich manyspin-waveresonanceshave\nbeen experimentally observed at low temperatures [25].\nThis work has been supported in part through NIST-\nCNST / UMD-Nanocenter cooperative agreement.\n[1] L.LandauandE. Lifshitz, Phys.Z.Sowjet. 8, 153 (1935).\n[2] T. L. Gilbert, Armour research foundation project No.\nA059, supplementary report, unpublished (1956).\n[3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[4] V. Vlaminck and M. Bailleul, Science 322, 410 (2008).\n[5] S.S.P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n[6] M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and\nS.S.P. Parkin, Science 320, 209 (2008).\n[7] B.E. Argyle, E. Terrenzio, and J.C. Slonczewski,\nPhys. Rev. Lett. 53, 190 (1984).\n[8] R.D. McMichael, C.A. Ross, and V.P. Chuang,\nJ. Appl. Phys. 103, 07C505 (2008).\n[9] S. Kaka, M.R. Pufall, W.H. Rippard, T.J. Silva,\nS.E. Russek, and J.A. Katine, Nature 437, 389 (2005).\n[10] F.B. Mancoff, N.D. Rizzo, B.N. Engel, and S. Tehrani,\nNature437, 393 (2005).\n[11] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y,\nPhys. Stat. Sol. 23, 501 (1967).\n[12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).[13] V. Korenman and R.E. Prange, Phys. Rev. B 6, 2769\n(1972).\n[14] V. Kambersk´ y, Czech. J. Phys. B 26, 1366 (1976).\n[15] J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna ,\nW.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n[16] Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[17] M. Zwierzycki, Y. Tserkovnyak, P.J. Kelly, A. Brataas,\nand G.E.W. Bauer, Phys. Rev. B 71, 064420 (2005).\n[18] K. Gilmore, Y.U. Idzerda, and M.D. Stiles,\nPhys. Rev. Lett. 99, 027204 (2007).\n[19] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n[20] K.Gilmore, Y.U.Idzerda, andM.D.Stiles, J.Appl.Phys .\n103, 07D303 (2008).\n[21] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[22] J. Foros, A.Brataas, Y.Tserkovnyak,andG.E.W. Bauer,\nPhys. Rev. B 78, 140402(R) (2008).\n[23] L.F. Mattheiss and D.R. Hamann, Phys. Rev. B 33, 8235\n(1986).\n[24] M.D. Stiles, S.V. Halilov, R.A. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[25] S.T.B. Goennenwein, T. Graf, T. Wassner, M.S. Brandt,M. Stutzmann, A. Koeder, S. Frank, W. Schoch, and\nA. Waag, Journal of Superconductivity 16, 75 (2003)."    },    {        "title": "1903.04171v1.The_effect_of_magnetic_twist_on_resonant_absorption_of_slow_sausage_waves_in_magnetic_flux_tubes.pdf",        "content": "arXiv:1903.04171v1  [astro-ph.SR]  11 Mar 2019The effect of magnetic twist on resonant absorption\nof slow sausage waves in magnetic flux tubes\nMohammad Sadeghi1∗, Kayoomars Karami1,2†\n1Department of Physics, University of Kurdistan, Pasdaran Stree t, P.O. Box 66177-15175, Sanandaj, Iran\n2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran\nMarch 12, 2019\nAbstract\nObservations show that twisted magnetic flux tubes are present t hroughout the sun’s\natmosphere. The main aim of this work is to obtain the damping rate of sausage modes\nin the presence of magnetic twist. Using the connection formulae ob tained by Sakurai et\nal. (1991), we investigate resonant absorption of the sausage mo des in the slow continuum\nunder photosphere conditions. We derive the dispersion relation an d solve it numerically and\nconsequently obtain the frequencies and damping rates of the slow surface sausage modes.\nWe conclude that the magnetic twist can result in strong damping in co mparison with the\nuntwisted case.\n1 Introduction\nOne of important questions regarding the sun is that how the s olar corona reaches temperatures\nexceeding 1 MK. There are different theories to justify this pr oblem. One of these is propagation\nof magnetohydrodynamic (MHD) waves and their damping by the resonant absorbtion proposed\nfor the first time by Ionson [1]. At the observational point of view, there are some evidences for\npropagating and damping of MHD waves (see e.g. Yu et al. [2]). Based on propagating speed\nof MHD waves, they are classified into fast, slow and Alfv` en w aves. Here, we focus on the slow\nsausage MHD waves which has recently been observed by Dunn So lar Telescope [3].\nDorotoviˆ c et al. [4] observed the linear slow sausage waves in magnetic pore with the period\nfrom 20 minutes to 70 minutes in a photospheric pore size by th e Swedish Solar Telescope. Also,\nMorton et al. [5] observed sausage modes in magnetic pores wi th the period range from as short\nas 30 s up to 450 s. Grant et al. [3] reported the observational periodof sausage mode as 181 s to\n412 s. Moreover, Keys et al. [6] observationally found that i n solar magnetic pores the number\nof surface modes are more than body modes. They also pointed o ut that surface modes appear\nto carry more energy compared to body modes. The slow waves ar e usually generated by the\nmotions of sunspots, magnetic pores, and granules and they p lay significant role in heating of\nthe lower part of the sun’s atmosphere [2]. Yu et al. (2017) sh owed that the resonant absorption\nplays an important role in the wave damping for the slow surfa ce sausage modes in the slow\ncontinuum [7].\n∗m.sadeghi@uok.ac.ir\n†kkarami@uok.ac.ir\n1Besides, there is ample evidence on existence of magnetic tw ist in the solar atmosphere and\nbelow. For instance, it has been suggested that magnetic flux tubes are twisted while rising\nthrough the convection zone (e.g., Murray & Hood [8]; Hood et al. [9]; Luoni et al. [10]). Also,\nthe observations show that magnetic twist is one of importan t mechanisms in energy transport\nfrom the photosphere to the corona, see Wedemeyer-Bhm et al. [11].\nSo far, many studies have been done about the effect of twisted m agnetic fields on the\nkink and sausage MHD waves in magnetic flux tubes. For instanc e, the connection formulae\n(or conservation laws) were first used for studying surface w aves in cylindrical plasmas in the\npresence of magnetic twist by Goossens et al. [12]. Bennett e t al. [13] studied the sausage\nmodes in a magnetic flux tube containing a uniform magnetic tw ist. They pointed out that in\nthe presence of magnetic twist, an infinite set of body modes i s generated.\nErd´ elyi & Carter [14] considered magnetic twist just in the annulus for surface and hybrid\nmodes. They found that when the magnetic twist increases the hybrid modes include a wide\nrange of phase speeds for the sausage modes.\nErd´ elyi&Fedun[15]investigated thepropagationofMHDwa ves inanincompressibletwisted\nmagnetic flux tube. They showed that an increase in the twiste d magnetic field from 0 to 0.3\ncould lead to an increase of 1 to 2 percent of the period of the s ausage waves. Erd´ elyi & Fedun\n[16] extended their previous work [15] to the compressibili ty condition and concluded that the\nperiod of the sausage waves increases 3 to 5 percent.\nKarami & Bahari [17] considered the effect of twisted magnetic field on the resonant ab-\nsorption of MHD waves in coronal loops. They showed that the p eriod ratio P1/P2of the\nfundamental and its first-overtone surface waves for kink ( m= 1) and fluting ( m= 2,3) modes\nis lower than 2 in the presence of twisted magnetic field. Ebra himi & Karami [18] investigated\nresonant absorption of kink MHD waves by a magnetic twist in c oronal loops. They concluded\nthat the resonant absorption by the magnetic twist can justi fy the rapid damping of kink MHD\nwaves observed in coronal loops.\nGiagkiozis et al. [19] studied resonant absorption of axisy mmetric modes ( m= 0) in twisted\nmagnetic flux tube. They showed that in the presence of twiste d magnetic field, both the\nlongitudinal magnetic field and the density have crucial rol es in the wave damping. Giagkiozis\net al. [20] also elaborated that the magnetic twist can remov e the cut-off of fast body sausage\nmodes. Besides the works mentioned above, there are some fur ther studies on the effect of\nmagnetic twist on the MHD waves in the literature, see e.g. Er d´ elyi & Fedun [21]; Carter &\nErd´ elyi [22], [23]; Ruderman [24],[25]; Karami & Barin [26 ]; Terradas & Goossens [27]; Karami\n& Bahari [28].\nIn the present work, our main goal is to study the effect of magne tic twist on resonant\nabsorption of the slow surface sausage modes in the sun’s atm osphere. To do so, we apply the\nmagnetic twist to the model of Yu et al. [7]. To achieve this ai m, in Section 2 we introduce the\nmodel and solve the equations of motion governing the slow su rface sausage modes. In Section\n3, we obtain the dispersion relation under magnetic pore con ditions. In Section 4, using the\nconnection formulae, we derive the damping rate for the slow surface waves. The numerical\nresults are shown in Section 5. Finally, we conclude the pape r in Section 6.\n2 Equations of motion and model\nThe linearized ideal MHD equations are as follows [29]\n(1a) ρ∂2ξ\n∂t2=−∇δp−1\nµ0/parenleftBig\nδB×(∇×B)+B×(∇×δB)/parenrightBig\n,\n2(1b) δp=−ξ·∇p−γp∇·ξ,\n(1c) δB=−∇×(B×ξ),\nwhereρ,pandBare the background density, kinetic pressure and magnetic fi eld, respectively.\nAlsoξis the Lagrangian displacement vector, δpandδBare the Eulerian perturbations of the\npressure and magnetic field, respectively. Here, γis the ratio of specific heats (taken to be 5 /3\nin this work), and µ0is the permeability of free space.\nIn our model, we consider a magnetic field as follows\n(2) B=/parenleftBig\n0,Bφ(r),Bz(r)/parenrightBig\n.\nNow, the plasma pressure and magnetic field must satisfy the f ollowing magnetohydrostatic\nequation in the r-direction as\n(3)d\ndr/parenleftBigg\np+B2\nφ+B2\nz\n2µ0/parenrightBigg\n+B2\nφ\nµ0r= 0.\nHere, following Yu et al. [7] we consider the following profil es for the density and z-component\nof the background magnetic field as\nρ(r) =\n\nρi, r /lessorequalslantri,\nρi+(ρe−ρi)/parenleftBig\nr−ri\nre−ri/parenrightBig\n, ri< r < r e,\nρe, r /greaterorequalslantre,(4)\nB2\nz(r) =\n\nB2\nzi, r /lessorequalslantri,\nB2\nzi+/parenleftbig\nB2\nze−B2\nzi/parenrightbig/parenleftBig\nr−ri\nre−ri/parenrightBig\n, ri< r < r e,\nB2\nze, r /greaterorequalslantre,(5)\nwhereρiandρeare the constant densities of the interior and exterior regi ons of the flux tube,\nrespectively. Also BziandBzeare theinterior andexterior constant longitudinal magnet ic fields,\nrespectively. We further assume the φ-component of the background magnetic field takes the\nform\nB2\nφ(r) =\n\nB2\nφir2\nr2\ni, r /lessorequalslantri,\nB2\nφi+/parenleftBig\nB2\nφe−B2\nφi/parenrightBig/parenleftBig\nr−ri\nre−ri/parenrightBig\n, ri< r < r e,\nB2\nφe, r /greaterorequalslantre,(6)\nwhereBφiandBφeare constant. Putting Eqs. (5) and (6) into the magnetohydro static equation\n(3), we obtain the background gas pressure as follows\np(r) =\n\npi−B2\nφi\nµ0(r/ri)2, r /lessorequalslantri,\nA1+A2r+A3ln(r/ri), ri< r < r e,\npe−B2\nφe\nµ0ln(r/re), r /greaterorequalslantre,(7)\n3where\nA1=/parenleftBigg\npi−B2\nφi\nµ0/parenrightBigg\n−A2ri,\nA2≡3/parenleftBig\nB2\nφi−B2\nφe/parenrightBig\n+/parenleftbig\nB2\nzi−B2\nze/parenrightbig\n2µ0(re−ri),\nA3≡riB2\nφe−reB2\nφi\nµ0(re−ri),\npe=/parenleftBigg\npi−B2\nφi\nµ0/parenrightBigg\n+A2(re−ri)+A3ln(re/ri), (8)\nandpiis an arbitrary constant. Also the constants A1andpehave been obtained from the\ncontinuity of the gas pressure across the boundaries r=riandr=re. Note that Eq. (7) in the\nabsence of twist, i.e. Bφi=Bφe= 0 recovers the gas pressure profile given by Yu et al. [7].\nIn addition, we define the following quantities\nv2\nAi≡B2(ri)\nµ0ρi, (9)\nv2\nAe≡B2(re)\nµ0ρe, (10)\nv2\nsi≡γp(ri)\nρi=γ/parenleftBigg\npi−B2\nφi\nµ0/parenrightBigg\n/ρi, (11)\nv2\nse≡γp(re)\nρe=γpe/ρe, (12)\nv2\nc(i,e)≡v2\ns(i,e)v2\nA(i,e)\nv2\ns(i,e)+v2\nA(i,e), (13)\nwhereB2=B2\nφ+B2\nz. AlsovA(i,e),vs(i,e)andvc(i,e)are the interior/exterior Alfv´ en, sound\nand cusp velocities, respectively. Besides, we define the pa rameterβas the ratio of the plasma\npressure to the magnetic field pressure, inside the flux tube a s\nβ≡p(ri)\nB2(ri)/(2µ0)= 2v2\nsi/(γv2\nAi). (14)\nWith the help of the parameter β, Eq. (14), the arbitrary constant pican be determined as\npi\nB2\ni/(2µ0)=β+2/parenleftbiggBφi\nBi/parenrightbigg2\n, (15)\nwhereBi=B(ri). Using Eqs. (11), (12) and (14), one can find the density rati oρe/ρias\nρe\nρi=/parenleftbiggvsi\nvse/parenrightbigg2pe/parenleftbigg\npi−B2\nφi\nµ0/parenrightbigg,\n=2\nβ/parenleftbiggvsi\nvse/parenrightbigg2/bracketleftBigg\nβ/2+(/tildewideB2\ni−/tildewideB2\ne)/2+/parenleftbigg\n1−reln(re/ri)\nre−ri/parenrightbigg\n/tildewideB2\nφi−/parenleftbigg\n1−riln(re/ri)\nre−ri/parenrightbigg\n/tildewideB2\nφe/bracketrightBigg\n,\n(16)\n40 0.8 1 1.2 1.511.21.3(r)/i\n0 0.8 1 1.2 1.500.51B2z(r)/B2i\n0 0.8 1 1.2 1.500.050.1B2(r)/B2i\n0 0.8 1 1.2 1.5\nr/R0.41.4p(r)/[B2i/2\n0]\nFigure 1: Variations of the background quantities of the flux tube including the density, the\nmagnetic field components and the plasma pressureversus the fractional radius r/Rfor thetwist\nparameter Bφi/Bzi= 0.3 andri/R= 0.8 andre/R= 1.2. For the photospheric conditions, the\nauxiliaryparametersaretakenfrom[3]as vAi= 12km s−1,vAe= 0km s−1(i.e.Bφe=Bze= 0),\nvsi= 7 km s−1,vse= 11.5 km s−1. With the help of these values and using γ= 5/3, from Eqs.\n(14) and (16) we find β= 0.41 andρe/ρi= 1.242.\nwhere/tildewideB=B/Biand/tildewideB2\ni= 1. Figure 1 shows the background quantities containing the density\n(4), the magnetic field components, Eqs. (5) and (6), and the p lasma pressure (7) for the twist\nparameter Bφi/Bzi= 0.3 under the magnetic pore conditions.\nThe set of Eqs. (1a)-(1c) in cylindrical coordinates can be s olved by Fourier decomposition\nof the perturbed quantities as follows\n(17) (ξ,δPT)∝ei(mφ+kzz−ωt),\nwhereωis the angular frequency, mis the azimuthal wavenumber for which only integer values\nareallowed and, kz, is thelongitudinal wavenumber inthe zdirection. Also δPT=δp+B.δB/µ0\nis the Eulerian perturbation of total (gas and magnetic) pre ssure. Putting Eq. (17) into (1a)-\n(1c), we obtain the two coupled first order differential equati ons\n(18a) Drdξ\ndr=C1(rξ)−rC2δPT,\n(18b) DrdδPT\ndr=1\nrC3(rξ)−C1δPT.\nThe above equations derived earlier by Appert et al. [30] and later by Hain & Lust, Goedbloed\nand Sakurai et al. [31, 32, 33]. Here, the multiplicative fac tors are defined as\n(19a) D≡ρ/parenleftbig\nω2−ω2\nA/parenrightbig\nC4,\n(19b) C1≡2Bφ\nµ0r/parenleftBig\nω4Bφ−m\nrfBC4/parenrightBig\n,\n5(19c) C2≡n2−k2\nz,\n(19d) C3≡ρD/bracketleftbigg\nω2−ω2\nA+2Bφ\nµ0ρd\ndr/parenleftbiggBφ\nr/parenrightbigg/bracketrightbigg\n+4ω4B2\nφ\nµ2\n0r2−ρC44B2\nφω2\nA\nµ0r2,\n(19e) C4≡/parenleftbig\nv2\ns+v2\nA/parenrightbig/parenleftbig\nω2−ω2\nc/parenrightbig\n,\nwhere\nfB≡m\nrBφ+kzBz,\nω2\nA≡f2\nB\nµ0ρ,\nω2\nc≡/parenleftbiggv2\ns\nv2\nA+v2s/parenrightbigg\nω2\nA,\nand\nn2≡ω4\n(v2\nA+v2s)(ω2−ω2c). (20)\nHereωA(=kzvA) is the Alfv´ en angular frequency and ωc(=kzvc) is the cusp angular frequency.\nAlsovA=|B|/√µ0ρis the Alfv´ en speed, vs=/radicalbig\nγp/ρis the sound speed, and vc=vsvA\n(v2s+v2\nA)1/2\nis the cusp velocity.\nCombiningEqs. (18a)and(18b), onecanobtainasecond-orde rordinarydifferentialequation\nfor radial component of the Lagrangian displacement ξas [16, 19]\n(21)d\ndr/bracketleftbiggD\nrC2d\ndr(rξr)/bracketrightbigg\n+/bracketleftbigg1\nD/parenleftbigg\nC3−C2\n1\nC2/parenrightbigg\n−rd\ndr/parenleftbiggC1\nrC2/parenrightbigg/bracketrightbigg\nξr= 0.\nFor the sausage modes ( m= 0), solutions of Eq. (21) in the interior ( r/lessorequalslantri) and exterior\n(r/greaterorequalslantre) regions are given by [20]\n(22a) ξri(s) =Ais1/2\nE1/4e−s/2M(a,b;s),\n(22b) ξre(r) =AeKν(krer),\nwhereAiandAeare constant. Also M(.) is the Kummer function, and K(.) is the modified\nBessel function of the second kind [34]. Replacing the solut ions (22a) and (22b) into Eq. (18b)\nread\n(23a) δPTi(s) =Aie−s/2/parenleftbiggkaDi\nn2\ni−k2z/parenrightbigg/bracketleftbigg/parenleftbiggni+kz\nkz/parenrightbigg\nsM(a,b;s)−2M(a,b−1;s)/bracketrightbigg\n,\n(23b) δPTe(r) =Ae/bracketleftBigg/parenleftBigg\nµ0(1−ν)De−2B2\nφen2\ne\nµ0r(k2z−n2e)/parenrightBigg\nKν(krer)−De\nkreKν−1(krer)/bracketrightBigg\n.\nThe parameters appeared in Eqs. (22a)-(23b) are defined as\ns≡k2\naE1/2r2, E≡4B4\nφin2\ni\nµ2\n0r4\niD2\nik2z(1−α2)2, (24)\nka≡kz(1−α2)1/2, α2≡4B2\nφiω2\nAi\nµ0r2\niρi(ω2−ω2\nAi)2, (25)\na≡1+k2\nri\n4k2zE1/2, b= 2, (26)\n6k2\nr≡(ω2\ns−ω2)(ω2\nA−ω2)\n(v2\nA+v2s)(ω2c−ω2), (27)\nDi≡ρi/parenleftbig\nω2−ω2\nAi/parenrightbig\n, De≡ρe/parenleftbig\nω2−ω2\nAe/parenrightbig\n, (28)\n(29) ν2(0,r)≡1+2B2\nφe\nµ2\n0D2e/parenleftBig\n2B2\nφen2\nek2\nz+µ0ρe/bracketleftbig\nω2\nAe/parenleftbig\n3n2\ne−k2\nz/parenrightbig\n−ω2/parenleftbig\nn2\ne+k2\nz/parenrightbig/bracketrightbig/parenrightBig\n.\nNote that in the annulus region ( ri< r < r e), we don’t solve the MHD equations. Instead,\nwe relate the interior solutions to exterior ones by using th e connection formula introduced in\nsection 4.\n3 Dispersion relation for the case of no inhomogeneous layer\nHere, we are interested in obtaining the dispersion relatio n for the sausage mode in the case of\nno annulus region. The solutions (22a)-(23b) for inside and outside of the flux tube must satisfy\nthe following boundary conditions\n(30a) ξri/vextendsingle/vextendsingle/vextendsingle\nr=R=ξre/vextendsingle/vextendsingle/vextendsingle\nr=R,\n(30b)/parenleftBig\nδPTi−B2\nφi\nµ0rξri/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nr=R=/parenleftBig\nδPTe−B2\nφe\nµ0rξre/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nr=R,\nwhereR=ri=reis the tube radius. The above relations show continuity cond itions for the\nLagrangian displacement andLagrangian changes of thetota l pressureacross thetubeboundary,\nrespectively. Inserting the solutions (22a)-(23b) into th e boundary conditions (30a) and (30b),\nafter some algebra one can find the following dispersion rela tion\n(31)−µ0Di\nk2\nri/bracketleftbigg/parenleftbiggni+kz\nkz/parenrightbigg\ns−2M(a,b−1,s)\nM(a,b,s)/bracketrightbigg\n=µ0De\nk2re/parenleftbigg\n1−ν−kreR Kν−1(kreR)\nKν(kreR)/parenrightbigg\n−/parenleftbigg\n1+2n2\ne\nk2re/parenrightbigg\nB2\nφe+B2\nφi,\nwheres=k2\naE1/2R2.\nNow, we are interested in investigating the dispersion rela tion (31) in the limit of no twist\ninside and outside the tube, i.e. Bφi=Bφe= 0. For the small twist, from Eq. (24) we have\nE≪1 and then from the first relation of Eq. (26) we get E1/2≃k2\nri/(4ak2\nz). Also from Eq.\n(25) we obtain α2≪1 and then ka≃kz. Consequently, from the first relation of Eq. (24)\nwe finds≃k2\nriR2/(4a). Using these approximations for the small twist, the Kumme r functions\nappeared in the dispersion relation (31) behave as\nlim\na→∞M(a,b−1,s) = lim\na→∞M/parenleftbigg\na,1,k2\nriR2\n4a/parenrightbigg\n= Γ(1)I0/parenleftBigg\n2/radicalbigg\nk2\nriR2\n4/parenrightBigg\n=I0(kriR),(32)\nlim\na→∞M(a,b,s) = lim\na→∞M/parenleftbigg\na,2,k2\nriR2\n4a/parenrightbigg\n= Γ(2)/parenleftbiggk2\nriR2\n4/parenrightbigg−1/2\nI1/parenleftBigg\n2/radicalbigg\nk2\nriR2\n4/parenrightBigg\n=2\nkriRI1(kriR),\n(33)\n7where we have used the following relation [34]\nlim\na→∞M(a,b,z/a) = Γ(b)z(1−b)/2Ib−1(2√z), (34)\nin which Γ( b) is the Gamma function and I(.) is the modified Bessel function of the first kind.\nNow, putting Eqs. (32) and (33) in the dispersion relation (3 1) and using Bφi→0,Bφe→0,\ns→0, andν→1 (see Eq. (29)), one can find\nDi\nkriI0(kriR)\nI1(kriR)=−De\nkreK0(kreR)\nK1(kreR). (35)\nReplacing DiandDefrom Eq. (28) into the above relation, we get\nρi/parenleftbig\nω2−ω2\nAi/parenrightbig\n+kri\nkreρe/parenleftbig\nω2−ω2\nAe/parenrightbig\nQ0= 0, (36)\nwhereQ0=−I1(kriR)K0(kreR)\nI0(kriR)K1(kreR). Finally, Eq. (36) can be recast as\nω2=ρiω2\nAi−ρekri\nkreω2\nAeQ0\nρi−ρekri\nkreQ0, (37)\nwhich recovers exactly the same result obtained by Edwin & Ro bertes [35] and Yu et al. [7] in\nthe absence of magnetic twist.\nIn addition, we turn to solve the dispersion relation (31), n umerically, in the presence of\ntwist. To do this, we use its dimensionless form (see Eq. (133 ) in Appendix D). In Fig. 2, we\nplot the results obtained for the phase speed v/vsi=ω/ωsiof the slow surface sausage mode\n(m= 0) versus kzRfor different twist parameters Bφi/Bzi. The figure shows that (i) for a given\nkzR, when the twist increases the phase speed decreases. (ii) Fo r a given twist Bφi/Bzi, the\nphase speed decreases for larger kzRvalues. (iii) For the case of no twist, the result of [7] is\nrecovered.\nInthenextsection, weturnto obtain thedispersionrelatio n of sausagemodesin thepresence\nof annulus region.\n4 Dispersion relation in the presence of inhomogeneous laye r\nand resonant absorption\nConsidering an inhomogeneous layer, where the background d ensity, pressure and magnetic\nfield change continuously from inside to outside the flux tube , see Eqs. (4)-(6), the differential\nequation (21) may become singular at ω=ωc(r) andω=ωA(r). The resonant absorption\nthat may occur in these singular points causes damping of the wave amplitude. Because the\npresence of the inhomogeneous layer, the values of ωc(r) andωA(r) change continuously from\ninside to outside the flux tube. These processes are called sl ow (cusp) and Alfv´ en continua,\nrespectively. Note that following Yu et al. [7] under the mag netic pore conditions, only the cusp\nsingularity occurs where the phase speed of the slow surface sausage (sss) mode lies in the range\nofvce< vsss< vci.\nFollowing Sakurai et al. [33], in the resonant layer where th e singularity occurs because of\nmagnetic twist one does not need to solve Eq. (21). Instead, t he solutions inside and outside of\nthe flux tube are related to each other via the following conne ction formula given by [33]\n80 2 4 6 8 10\nkzR0.8350.840.8450.850.8550.86vciv/vsiBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.28\nBi/Bzi=0.3\nFigure 2: The phase speed v/vsi(=ω/ωsi), Eq. (31), of the slow surface sausage modes ( m= 0)\nversuskzRfordifferenttwistparameters Bφi/Bzi. Underthemagneticporeconditions, following\n[3] the auxiliary parameters are taken as vAi= 12 km s−1,vAe= 0 km s−1(i.e.Bφe=Bze= 0),\nvsi= 7 km s−1,vse= 11.5 km s−1,vci= 6.0464 km s−1(≃0.8638vsi) andvce= 0 km s−1. Here,\nthe results for Bφi/Bzi= 0 and 10−3overlap with each other.\n(38a)[ξr]≡ξre(re)−ξri(ri),\n=−iπµω4\nc\n|∆c|rB2ω2\nA/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBig\nδPTi−2B2\nφiξri\nµ0r/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nr=ri,\n(38b)[δPT]≡δPTe(re)−δPTi(ri),\n=−i2πω4\ncB2\nφ\n|∆c|rB2ω2\nA/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBig\nδPTi−2B2\nφiξri\nµ0r/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nr=ri,\nwhere [ξr] and [δPT] are the jump conditions in the Lagrangian radial displacem ent and total\npressure perturbation, respectively, across the inhomoge neous (resonant) layer. The subscript c\ndenotes the position of the slow resonance ( r=rc) and|∆c|≡/vextendsingle/vextendsingle/vextendsingled(ω2−ω2\nc)\ndr/vextendsingle/vextendsingle/vextendsingle\nr=rc. Note that we will\ndetermine the cusp resonance point rclater, see Eq. (52).\nSubstituting the solutions (22a)-(23b) into the connectio n formula (38a) and (38b), one can\nfindthe dispersionrelation governing the slow surface saus age modes in thepresence of magnetic\ntwist as\n(39)Di\nk2\nri/bracketleftbiggni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightbigg\n+ri/parenleftBigg\nµ0(1−v)De−2B2\nφen2\ne\nµ0rek2re−De\nkreKv−1(krere)\nKv(krere)/parenrightBigg\n+iπµ0ω4\nc\n|∆c|B2ω2\nA/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBigg\nDi\nk2\nri/bracketleftbiggni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightbigg\n+2B2\nφi\nµ0/parenrightBigg\n×/parenleftBigg\n−2B2\nφ\nµ0r/vextendsingle/vextendsingle/vextendsingle\nr=rc+µ0(1−v)De−2B2\nφen2\ne\nµ0rek2re−De\nkreKv−1(krere)\nKv(krere)/parenrightBigg\n= 0.\n9For the case of no twist, i.e. Bφi=Bφe= 0, the dispersion relation (39) following the same\napproach that was used in the previous section takes the form\nρi/parenleftbig\nω2−ω2\nAi/parenrightbig\n−ρe/parenleftbig\nω2−ω2\nAe/parenrightbigki\nkeQ0+iπk2\nz\nρ|∆c|/parenleftbiggv2\ns\nv2s+v2\nA/parenrightbigg2\nρiρe/parenleftbig\nω2−ω2\nAi/parenrightbig/parenleftbig\nω2−ω2\nAe/parenrightbigG0\nke= 0,\n(40)\nwhereG0=−K0(krere)\nK1(krere). The above relation is same as that obtained by Yu et al. [7] in the\nabsence of magnetic twist.\nUsing Eqs. (4) to (12), one can obtain the quantities vs=/radicalbig\nγp/ρ,vA=|B|/√µ0ρand the\ncusp velocity vc≡vsvA\n(v2s+v2\nA)1/2in the inhomogeneous layer ( ri< r < r e) as\n(41) v2\ns=v2\nsi/bracketleftbigg1+δ(χv2\nsei−1)+ζ\n1+δ(χ−1)/bracketrightbigg\n,\n(42) v2\nA=v2\nAi/bracketleftbigg1+δ(χv2\nAei−1)\n1+δ(χ−1)/bracketrightbigg\n,\n(43) v2\nc=v2\nsiv2\nAi/bracketleftBig\n1+δ(χv2\nsei−1)+ζ/bracketrightBig/bracketleftBig\n1+δ(χv2\nAei−1)/bracketrightBig\n/bracketleftBig\n1+δ(χ−1)/bracketrightBig/bracketleftBig\nv2\nsi/parenleftBig\n1+δ(χv2\nsei−1)+ζ/parenrightBig\n+v2\nAi/parenleftBig\n1+δ(χv2\nAei−1)/parenrightBig/bracketrightBig,\nwhereδ≡r−ri\nre−ri,χ≡ρe/ρi,vsei≡vse/vsi,vAei≡vAe/vAiand\nζ≡γv2\nAi\nv2\nsi/bracketleftBigg\nriB2\nφe−reB2\nφi\n(re−ri)B2\ni/bracketrightBigg/parenleftBig\nln(r/ri)−δln(re/ri)/parenrightBig\n. (44)\nNotice that in the absence of twist, i.e. Bφi=Bφe= 0, Eqs. (41), (42) and (43) transform to\nthe corresponding relations in Yu et al. [7].\nIn Fig. 3, using Eqs. (41), (42) and (43) we plot the sound, Alf v´ en and cusp velocities for\nthe twist parameters Bφi/Bzi= 0.3 under magnetic pore conditions. Figure 3 shows that for\nvc< vciandvci< vc< vcmax, respectively, the surface and body sausage modes can reson antly\ndamp in the slow continuum. Here, vcmaxis the maximum value of the cusp velocity.\nNote that according to Yu et al. [7], the position of the cusp r esonance point rcis obtained\nby setting ω2=ω2\nc/vextendsingle/vextendsingle/vextendsingle\nr=rc≡k2\nzv2\nc/vextendsingle/vextendsingle/vextendsingle\nr=rcin Eq. (43). Consequently, the resulting equation in terms\nof the variable δc≡δ/vextendsingle/vextendsingle/vextendsingle\nr=rc=rc−ri\nre−riyields the following second order equation\nAδ2\nc+Bδc+C= 0, (45)\n100 0.2m0.4 0.6 0.8 100.20.40.6vcivsi1.21.4vsevAi1.8v/vsi\nvs\nvA\nvcB i/Bzi=0.3\nFigure 3: Variations of the dimensionless velocity v/vsi, Eqs. (41), (42) and (43), as a function\nofδ≡r−ri\nre−riin the nonuniform (transitional) layer for the twist parame terBφi/Bzi= 0.3 under\nmagnetic pore conditions. The auxiliary parameters are vAi= 12 km s−1,vAe= 0 km s−1(i.e.\nBφe=Bze= 0),vsi= 7 km s−1,vse= 11.5 km s−1,vci= 6.0464 km s−1(≃0.8638vsi) and\nvce= 0 km s−1[3].\nwith\nA≡1−v2\nc\nv2\nci−γv2\nAi\nv2\nsi/parenleftBigg\nriB2\nφe−reB2\nφi\n(re−ri)B2\ni/parenrightBigg/parenleftbigg\n1−v2\nc\nv2\nAi/parenrightbigg/bracketleftBigg\n1\n2/parenleftbiggre−ri\nri/parenrightbigg2\n+re−ri\nri+ln(re/ri)/bracketrightBigg\n+χ/bracketleftBigg\n2v2\nc\nv2\nci−(v2\nsei+v2\nAei)−γv2\nAi\nv2\nsi/parenleftBigg\nriB2\nφe−reB2\nφi\n(re−ri)B2\ni/parenrightBigg/parenleftbigg\nv2\nAei−v2\nc\nv2\nAi/parenrightbigg/parenleftbiggre−ri\nri−ln(re/ri)/parenrightbigg/bracketrightBigg\n−χ2/parenleftbiggv2\nc\nv2\nci−v2\nseiv2\nAei/parenrightbigg\n, (46)\nB≡2/parenleftbiggv2\nc\nv2\nci−1/parenrightbigg\n+γv2\nAi\nv2\nsi/parenleftBigg\nriB2\nφe−reB2\nφi\n(re−ri)B2\ni/parenrightBigg/parenleftbigg\n1−v2\nc\nv2\nAi/parenrightbigg/parenleftbiggre−ri\nri+ln(re/ri)/parenrightbigg\n−χ/bracketleftbiggv2\nc\nv2\nci/parenleftbigg\n1+v2\nse+v2\nAe\nv2\nsi+v2\nAi/parenrightbigg\n−(v2\nsei+v2\nAei)/bracketrightbigg\n, (47)\nC≡1−v2\nc\nv2\nci, (48)\nwhere we have used the following approximation\nln(r/ri) = ln/bracketleftbigg\n1+/parenleftbiggre−ri\nri/parenrightbigg\nδc/bracketrightbigg\n≃/parenleftbiggre−ri\nri/parenrightbigg\nδc−1\n2/parenleftbiggre−ri\nri/parenrightbigg2\nδ2\nc, (49)\nfor/parenleftBig\nre−ri\nri/parenrightBig\nδc<1. We checked that keeping the higher order terms O(δ3\nc) does not affect the\nresults. Equation (45) has two roots for δcas\nδc1=−B\n2A+√\nB2−4AC\n2A,0< δ/lessorequalslantδm, (50)\n11δc2=−B\n2A−√\nB2−4AC\n2A, δm/lessorequalslantδ/lessorequalslant1, (51)\nwhereδmis the value of δwhenvc=vcmax. For instance, in Fig. 3 for Bφi/Bzi= 0.3 we have\nvcmax= 0.93vsiandδm= 0.27.\nUnder the magnetic pore conditions, for slow surface mode we have only one root denoted\nbyδc2. Consequently, the cusp resonance position rcreads\nrc=ri/bracketleftbigg/parenleftbiggre\nri−1/parenrightbigg\nδc2+1/bracketrightbigg\n. (52)\nNext, we turn to calculate the parameter ∆ cappeared in the dispersion relation (39). To this\naim, using Eq. (43) and ω2\nc(rc) =k2\nzv2\nc/vextendsingle/vextendsingle/vextendsingle\nr=rcwe obtain\n∆c≡/bracketleftbiggd\ndr(ω2−ω2\nc)/bracketrightbigg\nr=rc=−2/parenleftbigg\nωcdωc\ndr/parenrightbigg\nr=rc,\n=−/parenleftBigg\nω2\nc(rc)\nl/parenrightBigg/braceleftBigg/parenleftbig\nχv2\nsei−1/parenrightbig\n+lζ′\n1+δ/parenleftbig\nχv2\nsei−1/parenrightbig\n+ζ−(χ−1)\n1+δ(χ−1)\n+(χv2\nAei−1)\n1+δ/parenleftbig\nχv2\nAei−1/parenrightbig (53)\n−v2\nsi/parenleftBig\nχv2\nsei−1+lζ′/parenrightBig\n+v2\nAi/parenleftBig\nχv2\nAei−1/parenrightBig\nv2\nsi/bracketleftBig\n1+δ/parenleftBig\nχv2\nsei−1/parenrightBig\n+ζ/bracketrightBig\n+v2\nAi/bracketleftBig\n1+δ/parenleftbig\nχv2\nAei−1/parenrightbig/bracketrightBig/bracerightBigg\nr=rc,\nwhere\nζ′≡dζ\ndr=γv2\nAi\nv2\nsi/bracketleftBigg\nriB2\nφe−reB2\nφi\nl2B2\ni/bracketrightBigg/parenleftBig\nl/r−ln(re/ri)/parenrightBig\n. (54)\nNote that in both Eqs. (39) and (53) due to having the cusp reso nance,δ(r=rc) should be\nreplaced by δc2, Eq. (51).\n4.1 Weak damping limit - slow continuum\nHere, we are interested in investigating the dispersionrel ation (39) in the limit of weak damping.\nTo this aim, we first rewrite Eq. (39) as follows\n(55)Di+k2\nri/parenleftbigg\nµ0De(1−v)−2B2\nφen2\ne\nµ0rek2re−De\nkreKv−1(krere)\nKv(krere)/parenrightbigg\n/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig\n+iπµ0ω4\nc\n|∆c|B2ω2\nA/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftbigg\nDi/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig\n+2B2\nφi\nµ0k2\nri/parenrightbigg\n/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig\n×/parenleftBigg\n−2B2\nφ\nµ0r/vextendsingle/vextendsingle/vextendsingle\nr=rc+µ0(1−v)De−2B2\nφen2\ne\nµ0rek2re−De\nkreKv−1(krere)\nKv(krere)/parenrightBigg\n= 0.\n12This can also be recast in the following compact form\nDAR+iDAI= 0, (56)\nwhereDARandDAI, respectively, are the real and imaginary parts of Eq. (55) g iven by\n(57) DAR=ρi(ω2−ω2\nAi)−riρe(ω2−ω2\nAe)kri\nkreQ,\n(58) DAI=πρiρek2\nz\nkreρc|∆c|/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBigv2\nsc\nv2\nAc+v2sc/parenrightBig2/parenleftbig\n(ω2−ω2\nAi)+Z/parenrightbig\n(ω2−ω2\nAe)G,\nwhere\n(59) Q≡ −kriri/parenleftbigg\nµ0De(1−v)−2B2\nφen2\ne\nDeµ0rekre−Kv−1(krere)\nKv(krere)/parenrightbigg\n/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig,\n(60) G≡/parenleftBigg\n−2kreB2\nφ\nDeµ0r/vextendsingle/vextendsingle/vextendsingle\nr=rc+µ0(1−v)De−2B2\nφen2\ne\nµ0rekreDe−Kv−1(krere)\nKv(krere)/parenrightBigg\n,\n(61) Z≡2B2\nφik2\nri\nµ0ρi/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig.\nNote that in Eqs. (57) and (58) we have the complex frequency ω=ωr+iγ, in which ωrandγ\nare the cusp (slow) frequency and the damping rate, respecti vely. In the limit of weak damping,\ni.e.γ≪ωr, the damping rate γis given by Goossens et al. [12]\nγ=−DAI(ωr)/parenleftbigg∂DAR\n∂ω/vextendsingle/vextendsingle/vextendsingle\nωr/parenrightbigg−1\n. (62)\nWith the help of Eq. (62), one can obtain an analytical expres sion forγ(see Appendix A) in\nthe slow (cusp) continuum as follows\nγ=−πρek2\nz\nkreρc|∆c|/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBig\nv2\ns\nv2\nA+v2s/parenrightBig2/parenleftbig\n(ω2\nr−ω2\nAi)+Z/parenrightbig\n(ω2\nr−ω2\nAe)G\n2ωr/parenleftBig\n1−χkri\nkreQ/parenrightBig\n−ωrχT, (63)\nwhere the quantity Tis given by Eq. (85). Note that in the limit of no twist, i.e. Bφi=Bφe= 0,\nEqs. (59), (60), (61), (78) and (79) reduce to\nQ=Q0≡I′\n0(x)K0(y)\nI0(x)K′\n0(y),\nG=G0≡K0(y)\nK′\n0(y),\nZ= 0,\nP=P0≡/parenleftBigg\nI′′\n0(x)\nI0(x)−I′\n0(x)2\nI0(x)2/parenrightBigg\nK0(y)\nK′\n0(y),\nS=S0≡/parenleftBigg\n1−K′′\n0(y)K0(y)\nK′\n0(y)2/parenrightBigg\nI′\n0(x)\nI0(x), (64)\n13wherex=kririandy=krereand we have also used the relation\nlim\na→∞M(a,b−1,s)\nM(a,b,s)=x\n2I0(x)\nI1(x). (65)\nReplacing the relations (64) into Eq. (63), the damping rate γin the absence of twist takes the\nform\nγ=−πρek2\nz\nkreρc|∆c|/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBig\nv2\ns\nv2\nA+v2s/parenrightBig2\n(ω2\nr−ω2\nAi)(ω2\nr−ω2\nAe)G0\n2ωr/parenleftBig\n1−χkri\nkreQ0/parenrightBig\n−ωrχT0, (66)\nwhere\nT0=ω2\nr/parenleftbig\nω2\nr−ω2\nAe/parenrightbigkri\nkre/parenleftbigg(Q0+xP0)(ω2\nr−2ω2\nci)\n(ω2\nsi−ω2c)(ω2\nAi−ω2r)(ω2r−ω2\nci)−(Q0−yS0)(ω2\nr−2ω2\nce)\n(ω2se−ω2r)(ω2\nAe−ω2r)(ω2r−ω2ce)/parenrightbigg\n.\n(67)\nNotice that Eq. (66) is the same as the dispersion relation (2 8) in Yu et al. [7] for the slow\nsurface sausage modes when the twist is absent.\n4.2 Weak damping rate in long wavelength limit - slow continu um\nHere, we try to examine the damping rate (63) in the long wavel ength limit, i.e. kzR≪1\n(ωr≈ωci=kzvci). In this limit, one can show that Eq. (63) takes the form (see Eq. (88) in\nAppendix B)\nγ=−πρekz\n|∆c|ω4\nci\nω4\nAi/parenleftbig\n(ω2\nci−ω2\nAi)+Z/parenrightbig\n(ω2\nci−ω2\nAe)G\n2ωci/parenleftBig\nρi−ρekri\nkreQ/parenrightBig\n−ωciρeT, (68)\nwhere the quantities T,Q,GandZ, are given by Eqs. (89), (90), (91) and (92), respectively.\nUnder photospheric (magnetic pore) conditions, i.e. vAe=vAφe=vce≃0, one can show that\nEq. (68) reduces to (see Eq. (107) in Appendix B)\nγ=−πρekzR\n|∆c|ω5\nci\nω4\nAi/parenleftbig\n(ω2\nci−ω2\nAi)+Z/parenrightbig\nG\n2/parenleftBig\nρi−ρekri\nkreQ/parenrightBig\n−ρeT, (69)\nwhere now the quantities T,Q,GandZare given by Eqs. (108), (109), (110) and (111),\nrespectively.\nNote that in the absence of twist (i.e. Bφi=Bφe= 0), the weak damping rate γin long\nwavelength limit, Eq. (68), reduces to (see Eq. (129) in Appe ndix C)\nγ=2πχ3\n|∆c|R/bracketleftBigg\nω7\nciω2\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig3\n3ω10\nAiω2\nci+8χω8\nAiω2\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig\nln(kzR)/bracketrightBigg\n(kzR)4ln3(kzR). (70)\nFor the photosphere conditions (i.e. ωAe=ωce= 0), Eq. (70) reads\nγ=2πχ3\n|∆c|R/bracketleftbiggω11\nciω2\nsi\n3ω10\nAi+8χω8\nAiω2\nsiln(kzR)/bracketrightbigg\n(kzR)4ln3(kzR). (71)\nIt should be noted that Eqs. (70) and (71) without the terms 8 χω8\nAiω2\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig\nln(kzR) and\n8χω8\nAiω2\nsiln(kzR) appeared in their denominator are same as Eqs. (36) and (37) in [7]. This\ndifference is because of wrong minus sign appeared in Eq. (A.7) in [2].\n145 Numerical results\nHere, we solve numerically the dispersion relation (39) to o btain the frequencies and damping\nrates of the slow surface sausage modes. To this aim, it is con venient to recast Eq. (39)\nin dimensionless form (see Eq. (130) in Appendix D). Under th e magnetic pore conditions,\nfollowing [3] we set again the model parameters as vAi= 12 km s−1,vAe= 0 km s−1(i.e.\nBφe=Bze= 0),vsi= 7 km s−1,vse= 11.5 km s−1,vci= 6.0464 km s−1(≃0.8638vsi) and\nvce= 0 km s−1. Our numerical results are shown in Figs. 4 to 14.\nFigures 4, 5, 6 and 7 present variations of the phase speed (or normalized frequency) v/vsi≡\nωr/ωsi, the dampingrate to frequency ratio |γ|/ωrand the dampingtime to periodratio τD/T=\nωr/(2π|γ|) oftheslow surfacesausagemodesversus kzRfordifferenttwist parameters Bφi/Bzi=\n(0,10−3,0.1,0.2,0.3) and different thickness of the inhomogeneous layer l/R= (0.1,0.2,0.3,0.4).\nThe figures clear that i) the minimum value of the phase speed v/vsidecreases and shifts to\nsmallerkzRwith increasing the twist parameter Bφi/Bzi. ii) For a given l/R, in the small\nwavelength limit ( kzR≫1), we have asymptotically v/vsi→vci/vsi= 0.8638 and |γ|/ωr→0.\nThis shows that the effect of magnetic twist for larger kzRis negligible. iii) The maximum\nvalue of|γ|/ωrincreases and its position moves to smaller kzRwhenBφi/Bziincreases. iv) In\nthe absence/presnt of twist, the maximum value of |γ|/ωrdecreases and its position moves to\nsmallerkzRwhenl/Rincreases (see also Fig. 8). v) For a given l/R, the minimum value of\nτD/Tdecreases with increasing Bφi/Bzi. For instance, for the case of l/R= 0.1, the minimum\nvalue ofτD/TforBφi/Bzi= 0.3 changes ∼38% less than the case of no twist. vi) The dashed-\nline curves in Figs. 4, 5, 6 and 7 present the analytical resul ts of the damping rate to frequency\nratio|γ|/ωrevaluated by Eq. (63). These curves show that for the weak dam ping (i.e. γ≪ωr)\nand in the long wavelength limit (i.e. kzR≪1), our numerical results are in good agreement\nwith analytical ones.\nIn Figs. 9, 10, 11, 12 and 13, we plot variations of v/vsi,|γ|/ωrandτD/Tversus the\nthickness of the inhomogeneous layer l/Rfor different Bφi/Bzi= (0,10−3,0.1,0.2,0.3) and\nkzR= (0.5,1,2,4,8). Figures show that i) the maximum value of |γ|/ωrincreases and moves\nto smaller l/RwhenBφi/Bziincreases. ii) In the absence/presnt of twist, the maximum v alue\nof|γ|/ωrincreases and its position moves to smaller l/RwhenkzRincreases (see also Fig. 14).\niii) For a given kzR, the phase speed and the damping rate to frequency ratio, res pectively,\napproach v/vsi→vci/vsi= 0.8638 and |γ|/ωr→0 in the limit of larger values of l/R. iv)\nFor a given kzR, the minimum value of τD/Tdecreases with increasing Bφi/Bzi. For instance,\nfor the case of kzR= 1, the minimum value of τD/TforBφi/Bzi= 0.3 decreases ∼63.5% in\ncomparison to the case of no twist.\nIt is worth to mention that for the case of no twist Bφi/Bzi= 0, the results of Figs. 4 to 14\nrecover those obtained in Yu et al. [7]. Note that the results forBφi/Bzi= 0 and 10−3overlap\nwith each other.\n6 Conclusions\nHere, weinvestigated theeffect ofmagnetic twist onresonant absorptionofslow sausagewaves in\nmagnetic flux tubes under the solar photospheric (or magneti c poor) conditions. We considered\na straight cylindrical flux tube with different magnetic twist profiles in the interior, annulus\nand exterior regions. Besides, we assumed the density and lo ngitudinal magnetic field to be\nconstant inside and outside of the flux tube, but to be inhomog eneous in the annulus layer.\nWe presented the solutions of ideal MHD equations for the int erior and exterior regions of the\nflux tube. In the case of no inhomogeneous (annulus) layer, we derived the dispersion relation\n15which recovers the result obtained by Edwin & Robertes [35] a nd Yu et al. [7] in the absence\nof magnetic twist. In the presence of inhomogeneous layer, w ith the help of the appropriate\nconnection formula of resonant absorption introduced by Sa kurai et al. [33], we obtained the\njump conditions governing the solutions inside and outside of the flux tube. Consequently, we\nderived the dispersion relation of the slow surface sausage modes in the presence of magnetic\ntwist. Using this, we first obtained an analytical relation f or damping rate of the slow surface\nsausage modes in the limit of weak damping and long wavelengt h limit. Then, we showed that\nour analytical expression for the damping rate in the absenc e of twist recover the result obtained\nby Yu et al. [7]. In addition, we solved the dispersion relati on, numerically, and obtain the phase\nspeed (or normalized frequency) v/vsi≡ωr/ωsi, the damping rate to frequency ratio |γ|/ωrand\nthe damping time to period ratio τD/T=ωr/(2π|γ|) of the slow surface sausage modes under\nthe photospheric (magnetic poor) conditions. Our results s how the following:\n•For a given thickness of the inhomogeneous layer l/R, with increasing the twist parameter\nBφi/Bzi(i) the minimum values of both the phase speed v/vsiand the damping time to\nperiod ratio τD/T=ωr/(2π|γ|) decrease and shift to smaller kzR; (ii) the maximum value\nof|γ|/ωrincreases and moves to smaller kzR.\n•For a given l/R, the phase speed and the damping rate to frequency ratio, app roach\nv/vsi→vci/vsi= 0.8638 and |γ|/ωr→0, respectively, in the small wavelength limit\n(kzR≫1). This asymptotic behaviours also hold for a given kzRin the limit of larger\nvalues of l/R.\n•For a given kzR, the maximum value of |γ|/ωr(or minimum value of τD/T) increases (or\ndecreases) and moves to smaller l/Rwhen the twist parameter increases.\n•For the case of l/R= 0.1, the minimum value of τD/TforBφi/Bzi= 0.3, for instance,\nchanges ∼38% less than the case of no twist. Also for kzR= 1, the minimum value of\nτD/TforBφi/Bzi= 0.3, for example, decreases ∼63.5% in comparison to the case of\nno twist. These results show that the magnetic twist can cons iderably affect the resonant\nabsorptionoftheslowsurfacesausagemodesinmagneticflux tubesunderthephotospheric\nconditions.\nAcknowledgements\nThe authors thank Tom Van Doorsselaere and Marcel Goossens f or reading the manuscript and\nuseful discussions. The work of K. Karami has been supported financially by Research Institute\nfor Astronomy and Astrophysics of Maragha (RIAAM) under res earch project No. 1/5440-61.\n16Appendix\nA Weak damping rate for the surface sausage mode\nHere, with the help of Eq. (62) we obtain the damping rate of su rface sausage modes in the\nweak damping limit, i.e. γ≪ωr. To this aim, we first calculate∂DAR\n∂ωfrom Eq. (57) as follows\n∂DAR\n∂ω= 2ρiω−2ρeωkri\nkreQ−ρe/parenleftbig\nω−ω2\nAe/parenrightbig/parenleftbigg1\nkredkri\ndω−kri\nk2redkre\ndω/parenrightbigg\nQ−ρe/parenleftbig\nω−ω2\nAe/parenrightbigkri\nkredQ\ndω.\n(72)\nNow from Eq. (27), one can obtain\ndkri\ndw=−ω3(ω2−2ω2\nci)\n(v2\nsi+v2\nAi)(ω2−ω2\nci)2kri, (73)\ndkre\ndw=−ω3(ω2−2ω2\nce)\n(v2se+v2\nAe)(ω2−ω2ce)2kre. (74)\nAlso from Eq. (59) fordQ\ndω, we get\ndQ\ndω=Q\nxdx\ndw−xd\ndω/parenleftbigg\nµ0De(1−v)−2B2\nφen2\ne\nµ0Dey−Kv−1(y)\nKv(y)/parenrightbigg\n/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig\n+x/parenleftbigg\nµ0De(1−v)−2B2\nφen2\ne\nµ0Dey−Kv−1(y)\nKv(y)/parenrightbigg\nd\ndω/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig\n/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig2. (75)\nAfter some algebra, we obtain\ndQ\ndω=\nQ\nxdx\ndw+x/parenleftbigg\n1−ν\ny2+2v2\nAφen2\ne\n(ω2−ω2\nAe)y2+/parenleftbigg\nK′\nv−1\nKv−K′\nvKv−1\nK2v/parenrightbigg/parenrightbigg\ndy\ndω+1\nydν\ndω−4v2\nAφe\n(ω2−ω2\nAe)y/parenleftbigg\nnedne\ndω−ωn2\ne\n(ω2−ω2\nAe)/parenrightbigg\n/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig\n+x/bracketleftBig\ns\nkzdni\ndω+ni+kz\nkzds−2d\ndω/parenleftBig\nM(a,b−1;s)\nM(a,b;s)/parenrightBig/bracketrightBig/parenleftbigg\nµ0De(1−v)−2B2\nφen2\ne\nµ0Dey−Kv−1(y)\nKv(y)/parenrightbigg\n/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig2, (76)\nwherex=kririandy=krere. This can be rewritten as\ndQ\ndω=Pdx\ndw+Sdy\ndω, (77)\nwhere\nP=Q\nx+x/bracketleftBig\ns\nkzdni\ndω+ni+kz\nkzds−2d\ndω/parenleftBig\nM(a,b−1;s)\nM(a,b;s)/parenrightBig/bracketrightBig/parenleftbigg\nµ0De(1−v)−2B2\nφen2\ne\nµ0Dey−Kv−1(y)\nKv(y)/parenrightbigg\ndx\ndw/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig2,(78)\n17S=x/parenleftbigg\n1−ν\ny2+2v2\nAφen2\ne\n(ω2−ω2\nAe)y2+/parenleftbigg\nK′\nv−1\nKv−K′\nvKv−1\nK2v/parenrightbigg/parenrightbigg\ndy\ndω+1\nydν\ndω−4v2\nAφe\n(ω2−ω2\nAe)y/parenleftbigg\nnedne\ndω−ωn2\ne\n(ω2−ω2\nAe)/parenrightbigg\ndy\ndω/bracketleftBig\nni+kz\nkzs−2M(a,b−1;s)\nM(a,b;s)/bracketrightBig .\n(79)\nIn addition, from Eq. (20) we have\ndni\ndw=ω3(ω2−2ω2\nci)\n(v2\nsi+v2\nAi)(ω2−ω2\nci)2ni, (80)\ndne\ndw=ω3(ω2−2ω2\nce)\n(v2se+v2\nAe)(ω2−ω2ce)2ne. (81)\nWith the help of Eqs. (73) and (74), Eq. (77) takes the form\ndQ\ndω=xPω3(ω2−2ω2\nci)\n(ω2\nsi−ω2)(ω2\nAi−ω2)(ω2−ω2\nci)+ySω3(ω2−2ω2\nce)\n(ω2se−ω2)(ω2\nAe−ω2)(ω2−ω2ce).(82)\nReplacing this into Eq. (72) yields\n∂DAR\n∂ω= 2ρiω−2ρeωkri\nkreQ−ρeω3/parenleftbig\nω−ω2\nAe/parenrightbigkri\nkre/parenleftbigg(Q+xP)(ω2−2ω2\nci)\n(ω2\nsi−ω2)(ω2\nAi−ω2)(ω2−ω2\nci)−(Q−yS)(ω2−2ω2\nce)\n(ω2se−ω2)(ω2\nAe−ω2)(ω2−ω2ce)/parenrightbigg\n. (83)\nFinally, substituting Eqs. (58) and (83) into Eq. (62) one ca n get the damping rate γin the\nlimit of weak damping for the surface sausage modes in the slo w continuum as\nγ/vextendsingle/vextendsingle/vextendsingle\nω=ωr=−πρek2\nz\nkreρc|∆c|/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBig\nv2\ns\nv2\nA+v2s/parenrightBig2/parenleftbig\n(ω2−ω2\nAi)+Z/parenrightbig\n(ω2−ω2\nAe)G\n2ω/parenleftBig\n1−χkri\nkreQ/parenrightBig\n−ωχT, (84)\nwhere\nT=ω2\nr/parenleftbig\nω2\nr−ω2\nAe/parenrightbigkri\nkre/parenleftbigg(Q+xP)(ω2\nr−2ω2\nci)\n(ω2\nsi−ω2r)(ω2\nAi−ω2r)(ω2r−ω2\nci)−(Q−yS)(ω2\nr−2ω2\nce)\n(ω2se−ω2r)(ω2\nAe−ω2r)(ω2r−ω2ce)/parenrightbigg\n.\n(85)\nB Weak damping rate in long wavelength limit\nHere, we turn to examine the dispersion relation (84) in the l ong wavelength limit, i.e. kzR≪1.\nIn this limit, Eq. (29) yields ν2= 1+O(k2\nzR2)≃1. Also following [34] we have\nlim\nkzR→0M(a,b−1;s)\nM(a,b;s)= 1+a\nbs+O(s2), (86)\nlim\nkzR→0K0(y)\nK1(y)=−yln(y). (87)\nIn the limit kzR≪1 (ωr≈ωci), one should note that the damping rate (84) at ω=ωr≈ωci\nbecomes singular. To avoid of this singularity, we follow th e approach of [7] in which one can\n18assumeω2\nr=ω2\nci−α, whereα≪ω2\nci. Substituting ω2\nci−ω2\nr=αinto Eq. (84) and using Eqs.\n(86) and (87), one can obtain\nγ=−πρekz\n|∆c|ω4\nci\nω4\nAi/parenleftbig\n(ω2\nci−ω2\nAi)+Z/parenrightbig\n(ω2\nci−ω2\nAe)G\n2ωci/parenleftBig\nρi−ρekri\nkreQ/parenrightBig\n−ωciρeT, (88)\nwhere we have used ri≈Randkre=kz. Also\nT=ω2\nci/parenleftbig\nω2\nci−ω2\nAe/parenrightbigkri\nkre/parenleftbigg(Q+xP)ω2\nci\n(ω2\nsi−ω2\nci)(ω2\nAi−ω2\nci)α−(Q−yS)(ω2\nci−2ω2\nce)\n(ω2se−ω2\nci)(ω2\nAe−ω2\nci)(ω2\nci−ω2ce)/parenrightbigg\n,(89)\n(90) Q=−x/parenleftbigg\n−2B2\nφen2\ne\nµ0Dey+yln(y)/parenrightbigg\n/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig,\n(91) G=/parenleftBigg\n−2kreB2\nφ\nDeµ0r/vextendsingle/vextendsingle/vextendsingle\nr=rc−2B2\nφen2\ne\nµ0yDe+yln(y)/parenrightBigg\n,\n(92) Z=2B2\nφik2\nri\nµ0ρi/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig,\nP=Q\nx+x/bracketleftBig\ns\nkzdni\ndω+ni\nkzds−2d\ndω/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig/parenleftbigg\n−2B2\nφen2\ne\nµ0Dey+yln(y)/parenrightbigg\ndx\ndw/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig2, (93)\nS=x/parenleftbigg\n2v2\nAφen2\ne\n(ω2\nci−ω2\nAe)y2−x\n2(1+ln(y))/parenrightbigg\ndy\ndω−4v2\nAφe\n(ω2\nci−ω2\nAe)y/parenleftbigg\nnedne\ndω−ωcin2\ne\n(ω2\nci−ω2\nAe)/parenrightbigg\ndy\ndω/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\nbs/parenrightbig/bracketrightBig .(94)\nNow from Eqs. (24), (26) and using (25), one can obtain\ns= 2v2\nAφinikz/parenleftbig\nω2\nci−ω2\nAi/parenrightbig, (95)\na= 1+x2/bracketleftBig\nr2\ni/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2−4v2\nAφiω2\nAi/bracketrightBig\n8v2\nAφinikzr2\ni/parenleftbig\nω2\nci−ω2\nAi/parenrightbig, (96)\nwherev2\nAφ≡B2\nφ\nµ0ρ. With the help of Eqs. (95) and (96), one can get\nas=x2\n4+v2\nAφi/parenleftBigg\n2nikz/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg\n, (97)\n19da\ndω=1\n8v2\nAφinikz/bracketleftBigg\n2x/parenleftbig\nω2\nci−ω2\nAi/parenrightbigdx\ndω+x2/parenleftBigg\n2ωci−/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\nnidni\ndω/parenrightBigg\n+4v2\nAφiω2\nAix2r−2\ni/parenleftBigg\n−2/parenleftbig\nω2\nci−ω2\nAi/parenrightbigdx\ndω+2ωci/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2+1\nni/parenleftbig\nω2\nci−ω2\nAi/parenrightbigdni\ndω/parenrightBigg/bracketrightBigg\n,(98)\nds\ndω= 2v2\nAφinikz/parenleftBigg\n1\nni/parenleftbig\nω2\nci−ω2\nAi/parenrightbigdni\ndω−2ωci/parenleftbig\nω2−ω2\nAi/parenrightbig2/parenrightBigg\n, (99)\nsda\ndω=1\n4/bracketleftBigg\n2xdx\ndω+x2\n/parenleftbig\nω2\nci−ω2\nAi/parenrightbig/parenleftBigg\n2ωci−/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\nnidni\ndω/parenrightBigg\n+4v2\nAφiω2\nAix2r−2\ni/parenleftBigg\n−2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2dx\ndω+2ωci/parenleftbig\nω2\nci−ω2\nAi/parenrightbig3+1\nni/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2dni\ndω/parenrightBigg/bracketrightBigg\n,(100)\nads\ndω=x2/parenleftBigg\n1\n4nidni\ndω−ωci\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig/parenrightBigg\n+2v2\nAφinikz/parenleftBigg\n1\nni/parenleftbig\nω2\nci−ω2\nAi/parenrightbigdni\ndω−2ωci/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg\n−v2\nAφiω2\nAix2r−2\ni/parenleftBigg\n1\nni/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2dni\ndω−2ωci/parenleftbig\nω2\nci−ω2\nAi/parenrightbig3/parenrightBigg\n. (101)\nCombining Eqs. (100) and (101) yields\nsda\ndω+ads\ndω=x\n2dx\ndω+4v2\nAφiω2\nAix2r−2\ni/parenleftBigg\n−2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2dx\ndω+9ωci\n4/parenleftbig\nω2\nci−ω2\nAi/parenrightbig3+3\n4ni/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2dni\ndω/parenrightBigg\n+2v2\nAφinikz/parenleftBigg\n1\nni/parenleftbig\nω2\nci−ω2\nAi/parenrightbigdni\ndω−2ωci/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg\n. (102)\nIn addition, we need to evaluate the quantity α. To this aim, following [7] we first replace\nω2=ω2\nci−αinto Eq. (27) and get\nk2\nri≃k2\nz\nα/parenleftbig\nω2\nci−ω2\nsi/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n/parenleftbig\nω2\nAi+ω2\nsi/parenrightbig=k2\nz\nαω6\nci\nω2\nsiω2\nAi, (103)\nwhere we have used the definition ω2\nc≡ω2\nsω2\nA\nω2s+ω2\nAin obtaining the second equality of the above\nrelation. In the next, the dispersion relation (31) in long w avelength limit ( kzR≪1) reads\n(104)(ω2\nci−ω2\nAi)\nk2\nri/bracketleftbigg/parenleftbiggni\nkz/parenrightbigg\ns−2/parenleftBig\n1+a\nbs/parenrightBig/bracketrightbigg\n=−χ(ω2\nci−ω2\nAe)R2ln(y)+/parenleftbigg\n1+2n2\ne\nk2re/parenrightbiggB2\nφe\nµ0ρi−B2\nφi\nµ0ρi.\nNow, replacing k2\nrifrom Eq. (103) into (104), the quantity αcan be obtained as follows\n(105) α=−k2\nzω4\nci\nω4\nAi\nχ(ω2−ω2\nAe)R2ln(y)−/parenleftBig\n1+2n2\ne\nk2re/parenrightBig\nv2\nAφe+v2\nAφi/bracketleftBig/parenleftBig\nni\nkz/parenrightBig\ns−2/parenleftbig\n1+a\nbs/parenrightbig/bracketrightBig\n.\n20Substituting this into Eq. (103) yields\n(106) k2\nri=ω2\nciω2\nAi\nω2\nsi/bracketleftBig/parenleftBig\nni\nkz/parenrightBig\ns−2/parenleftbig\n1+a\nbs/parenrightbig/bracketrightBig\n/parenleftBig\nχ(ω2−ω2\nAe)R2ln(y)−/parenleftBig\n1+2n2e\nk2re/parenrightBig\nv2\nAφe+v2\nAφi/parenrightBig.\nUnder photospheric (magnetic pore) conditions, i.e. vAe=vAφe=vce≃0, the weak damping\nrateγ, Eq. (88), in long wavelength limit reduces to\nγ=−πρekzR\n|∆c|ω5\nci\nω4\nAi/parenleftbig\n(ω2\nci−ω2\nAi)+Z/parenrightbig\nG\n2/parenleftBig\nρi−ρekri\nkreQ/parenrightBig\n−ρeT. (107)\nBesides, Eqs. (89), (90), (91), (92), (93), (94), (105) and ( 106) take the forms\nT=ω2\ncix\ny/parenleftbigg(Q+xP)\nα−(Q−yS)\n(ω2\nci−ω2se)/parenrightbigg\n, (108)\n(109)Q=−xyln(y)/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig\n=xyln(y)\n2/parenleftBigg\n1−a\n2s+v2\nAφin2\ni\nω2\nci−ω2\nAi/parenrightBigg\n=xyln(y)\n2/parenleftBigg\n1−v2\nAφi/parenleftBigg\n2/parenleftbig\nnikz−n2\ni/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg/parenrightBigg\n,\n(110) G=/parenleftBigg\n−2krev2\nAφ\nχω2\nciR+yln(y)/parenrightBigg\n,\n(111)Z=2v2\nAφik2\nri/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig\n=−v2\nAφik2\nri/parenleftBigg\n1−a\n2s+v2\nAφin2\ni\nω2\nci−ω2\nAi/parenrightBigg\n=−v2\nAφik2\nri/parenleftBigg\n1−v2\nAφi/parenleftBigg\n2/parenleftbig\nnikz−n2\ni/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg/parenrightBigg\n,\n21P=Q\nx+x/bracketleftBig\ns\nkzdni\ndω+ni\nkzds−2d\ndω/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig\nyln(y)\ndx\ndw/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig2\n=yln(y)\n2/parenleftBigg\n1−a\n2s+v2\nAφin2\ni\n(ω2\nci−ω2\nAi)/parenrightBigg\n+xyln(y)\n2/parenleftBigg\n1−as+2v2\nAφin2\ni\n(ω2\nci−ω2\nAi)/parenrightBigg\nd\ndω/parenleftBigg\nv2\nAφin2\ni\n(ω2\nci−ω2\nAi)−a\n2s/parenrightBigg\n=yln(y)\n2/parenleftBigg\n1−x2\n8−v2\nAφi/parenleftBigg\n2/parenleftbig\nnikz−n2\ni/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg/parenrightBigg\n+xyln(y)\n2/parenleftBigg\n1−v2\nAφi/parenleftBigg\n2/parenleftbig\nnikz−n2\ni/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg/parenrightBigg\n/parenleftBigg\n2v2\nAφini\n(ω2\nci−ωAi)dni\ndω−2v2\nAφiωcin2\ni\n(ω2\nci−ω2\nAi)−sda+ads\n2/parenrightBigg\n,\n(112)\nS=−x1+ln(y)/bracketleftBig\nni\nkzs−2/parenleftbig\n1+a\n2s/parenrightbig/bracketrightBig\n=x(1+ln(y))/parenleftBigg\n1\n2−a\n4s+v2\nAφin2\ni\n2(ω2\nci−ω2\nAi)/parenrightBigg\n=x(1+ln(y))\n2/parenleftBigg\n1−v2\nAφi/parenleftBigg\n2/parenleftbig\nnikz−n2\ni/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg/parenrightBigg\n,(113)\nα=−k2\nzω4\nci\nω4\nAi\nχω2\nciR2ln(y)+v2\nAφi/bracketleftBig/parenleftBig\nni\nkz/parenrightBig\ns−2/parenleftbig\n1+a\nbs/parenrightbig/bracketrightBig\n\n=k2\nzω4\nci\nω4\nAi/parenleftbig\nχω2\nciR2ln(y)+v2\nAφi/parenrightbig/parenleftBigg\n1\n2−a\n4s+v2\nAφin2\ni\n2(ω2\nci−ω2\nAi)/parenrightBigg\n=k2\nzω4\nci\nω4\nAi/parenleftbig\nχω2\nciR2ln(y)+v2\nAφi/parenrightbig/parenleftBigg\n1−v2\nAφi/parenleftBigg\n2/parenleftbig\nnikz−n2\ni/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg/parenrightBigg\n,(114)\nk2\nri=ω2\nciω2\nAi\nω2\nsi/bracketleftBig/parenleftBig\nni\nkz/parenrightBig\ns−2/parenleftbig\n1+a\nbs/parenrightbig/bracketrightBig\n/parenleftBig\nχω2\nciR2ln(y)+v2\nAφi/parenrightBig\n=−2ω2\nciω2\nAi\nω2\nsi1/parenleftBig\nχω2\nciR2ln(y)+v2\nAφi/parenrightBig/parenleftBigg\n1+a\n2s−v2\nAφin2\ni\n(ω2\nci−ω2\nAi)/parenrightBigg\n=−2ω2\nciω2\nAi\nω2\nsi1/parenleftBig\nχω2\nciR2ln(y)+v2\nAφi/parenrightBig/parenleftBigg\n1+v2\nAφi/parenleftBigg\n2/parenleftbig\nnikz−n2\ni/parenrightbig/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n−x2r−2\niω2\nAi\n2/parenleftbig\nω2\nci−ω2\nAi/parenrightbig2/parenrightBigg/parenrightBigg\n.\n(115)\nFinally, substituting Eqs. (108) to (115) into (107) gives a long analytical expression for the\nweak damping γin long wavelength limit for the photospheric conditions.\n22C Weak damping rate in long wavelength limit with no twist\nIn the limit of no twist, i.e. Bφi=Bφe= 0, Eqs. (97) and (102) read\nas=x2\n4, (116)\nsda\ndω+ads\ndω=x\n2dx\ndω. (117)\nSubstituting the above relations into Eqs. (90) to (94), (10 5) and (106) one can get\n(118) Q=xy\n2ln(y),\n(119) G=yln(y),\n(120) Z= 0,\nP=/parenleftbigg1\n2−3x2\n16/parenrightbigg\nyln(y), (121)\nS=x\n2(1+ln(y)). (122)\n(123) α=χω4\nci\n2ω4\nAi(ω2\nci−ω2\nAe)k2\nzR2ln(y),\n(124) k2\nri=−2ω2\nciω2\nAi\nχω2\nsi(ω2\nci−ω2\nAe)R2ln(y).\nInserting Eqs. (118), (119), (120) into (88) gives\nγ=−πρek2\nz\n|∆c|ω3\nci\nω4\nAi/parenleftbig\n(ω2\nci−ω2\nAi)/parenrightbig\n(ω2\nci−ω2\nAe)Rln(y)\n2/parenleftBig\nρi−ρex2\n2ln(y)/parenrightBig\n−ρeT. (125)\nPutting Eqs. (121) and (122) into (89), one can get\nT=ω2\nci/parenleftbig\nω2\nci−ω2\nAe/parenrightbig/parenleftBigg\nx2ln(y)ω2\nci\n(ω2\nsi−ω2\nci)(ω2\nAi−ω2\nci)α−3x4ln(y)ω2\nci\n16(ω2\nsi−ω2\nci)(ω2\nAi−ω2\nci)α\n+x2(ω2\nci−2ω2\nce)\n2(ω2se−ω2\nci)(ω2\nAe−ω2\nci)(ω2\nci−ω2ce)/parenrightBigg\n. (126)\nReplacing Eqs. (123) and (124) into (126) yields\nT=−4ω6\nAi\nχ2ω2\nciω2\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig\nk2zR2ln(kzR)−3ω8\nAi\n2χ3ω4\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig2k2zR2ln2(kzR)\n−ω4\nciω2\nAi(ω2\nci−2ω2\nce)\nχω2\nsi(ω2se−ω2\nci)(ω2\nAe−ω2\nci)(ω2\nci−ω2ce)ln(kzR). (127)\n23Note that in long wavelength limit ( kzR≪1), the third term appeared in Eq. (127) is small in\ncomparison to the first two ones. Hence, Eq. (127) reduces to\nT=−4ω6\nAi\nχ2ω2\nciω2\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig\nk2zR2ln(kzR)−3ω8\nAi\n2χ3ω4\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig2k2zR2ln2(kzR).(128)\nFinally, substituting Eq. (128) into (125) gives the weak da mping rate in long wavelength limit\nwith no twist as\nγ=2πχ3\n|∆c|R/bracketleftBigg\nω7\nciω2\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig3\n3ω10\nAiω2\nci+8χω8\nAiω2\nsi/parenleftbig\nω2\nci−ω2\nAe/parenrightbig\nln(kzR)/bracketrightBigg\n(kzR)4ln3(kzR).(129)\nD Dimensionless dispersion relation\nIn order to numerical solving the dispersion relation (39), we recast it in the following dimen-\nsionless form\nv2\nF−v2\nAI\nk2\nrI/bracketleftbigg\n(NI+1)s−2M(a,b−1;s)\nM(a,b;s)/bracketrightbigg\n+ri/bracketleftBigg\n(1−ν)χ/parenleftbig\nv2\nF−v2\nAE/parenrightbig\n−2χv2\nAφEN2\nE\nrek2\nrE−χkz/parenleftbig\nv2\nF−v2\nAE/parenrightbig\nKν−1(krEkzre)\nkrEKν(krEkzre)/bracketrightBigg\n+iπk2\nzv4\ncv2\nSi\n|∆c|v4\nA/parenleftbiggv2\nF−v2\nAI\nk2\nrI/bracketleftbigg\n(NI+1)s−2M(a,b−1;s)\nM(a,b;s)/bracketrightbigg\n+2v2\nAφi/parenrightbigg\n×/parenleftBigg\n2(1+δ(χ−1))v2\nAφc\nrc+(1−ν)χ/parenleftbig\nv2\nF−v2\nAE/parenrightbig\n−2χv2\nAφEN2\nE\nrek2\nrE−χkz/parenleftbig\nv2\nF−v2\nAE/parenrightbig\nKν−1(krEkzre)\nkrEKν(krEkzre)/parenrightBigg\n= 0,\n(130)\nwhere\na= 1+k2\nrI/bracketleftBig\nk2\nz/parenleftbig\nv2\nF−v2\nAI/parenrightbig2−4v2\nAφIv2\nAI/bracketrightBig\n8v2\nAφINI/parenleftbig\nv2\nF−v2\nAI/parenrightbig, s= 2v2\nAφINI/parenleftbig\nv2\nF−v2\nAI/parenrightbig,\nvF=ωr\nωsi=v\nvsi, vAE=vAe\nvsi, vAI=vAi\nvsi, b= 2, χ=ρe\nρi,\nvSE=vse\nvsi, vAφE=vAφe\nvsi, vAφI=vAφi\nvsi, vSI= 1, v2\nAφ=B2\nφ\nµ0ρ,\nk2\nre=k2\nzk2\nrE, k2\nri=k2\nzk2\nrI, n2\ne=k2\nzN2\nE, n2\ni=k2\nzN2\nI, (131)\nand\nN2\nI=v4\nF\nv2\nFv2\nAI+/parenleftbig\nv2\nF−1/parenrightbig, N2\nE=v4\nF\nv2\nFv2\nAE+v2\nSE/parenleftbig\nv2\nF−1/parenrightbig,\nk2\nrI=/parenleftbig\n1−v2\nF/parenrightbig/parenleftbig\nv2\nAI−v2\nF/parenrightbig\n/parenleftbig\n1+v2\nAI/parenrightbig/parenleftbig\nv2\ncI−v2vF/parenrightbig, k2\nrE=/parenleftbig\nv2\nSE−v2\nF/parenrightbig/parenleftbig\nv2\nAE−v2\nF/parenrightbig\n/parenleftbig\nv2\nSE+v2\nAE/parenrightbig/parenleftbig\nv2\ncE−v2vF/parenrightbig,\nvcE=vce\nvsi, vcI=vci\nvsi,\nν2= 1+2v2\nAφE/parenleftbig\nv2\nF−v2\nAE/parenrightbig/bracketleftbig\n2v2\nAφEN2\nE+/parenleftbig\nv2\nAE/parenleftbig\n3N2\nE−1/parenrightbig\n−v2\nF/parenleftbig\nN2\nE+1/parenrightbig/parenrightbig/bracketrightbig\n.(132)\n24Note that in the absence of inhomogeneous layer, Eq. (130) re duces to\nv2\nF−v2\nAI\nk2\nrI/bracketleftbigg\n(NI+1)s−2M(a,b−1;s)\nM(a,b;s)/bracketrightbigg\n=(1−ν)χ/parenleftbig\nv2\nF−v2\nAE/parenrightbig\n−2χv2\nAφiN2\nE\nrek2\nrE−χkz/parenleftbig\nv2\nF−v2\nAE/parenrightbig\nRKν−1(krEkzR)\nkrEKν(krEkzR)+v2\nAφI+v2\nAφE.\n(133)\nReferences\n[1] Ionson, J. A. 1978, ApJ, 226, 650\n[2] Yu, D. J., Van Doorsselaere, T., & Goossens, M. 2017a, A&A , 602, A108\n[3] Grant, S. D. T., Jess, D. B., Moreels, M. G., et al. 2015, Ap J, 806, 132\n[4] Dorotoviˆ c ,I., Erd´ elyi, R., &Karlovsky, V.2008, inIA U Symp.247, Waves &Oscillations in\nthe Solar Atmosphere: Heating and Magneto-Seismology, ed. R. Erd´ elyi & C. A. Mendoza-\nBriceno (Cambridge: Cambridge Univ. Press), 351\n[5] Morton, R. J., Erd´ elyi, R., Jess, D. B., & Mathioudakis, M. 2011, ApJL, 729, L18\n[6] Keys, P. H., Morton, R. J., Jess, D. B., et al. 2018, ApJ, 85 7, 28\n[7] Yu, D. J., Van Doorsselaere, T., & Goossens, M. 2017b, ApJ , 850, 44\n[8] Murray, M. J., & Hood, A. 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M., & Roberts, B. 1983, SoPh, 88, 179\n260 2 4 6 8 10\nkzR0.840.8450.850.8550.86vci0.868v/vsiBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.1\n(a)\n0 2 4 6 8 10\nkzR00.0050.010.0150.02||/r\nBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.1\n(b)\n0 2 4 6 8 10\nkzR5101001000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.1\n(c)\nFigure 4: (a) The phase speed v/vsi≡ωr/ωsi, (b) the damping rate to frequency ratio |γ|/ωr,\nand (c) the damping time to period ratio τD/T=ωr/(2π|γ|) of the slow surface sausage modes\nversuskzRforl/R= 0.1 and different twist parameters Bφi/Bzi= (0,10−3,0.1,0.2,0.3). For\ncomparison, the analytical results obtained by Eq. (63) are shown by the dashed-line curves.\nAuxiliary parameters are as in Fig. 2. The results for Bφi/Bzi= 0 and 10−3overlap with each\nother. 270 2 4 6 8 10\nkzR0.840.8450.850.8550.86vci0.868v/vsi Bi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.2\n(a)\n0 2 4 6 8 10\nkzR00.0050.010.0150.018||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.2\n(b)\n0 2 4 6 8 10\nkzR5101001000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.2\n(c)\nFigure 5: Same as Fig. 4, but for l/R= 0.2.\n280 2 4 6 8 10\nkzR0.840.8450.850.8550.86vci0.868v/vsi Bi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.3\n(a)\n0 2 4 6 8 10\nkzR00.0050.010.0150.018||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.3\n(b)\n0 2 4 6 8 10\nkzR5101001000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.3\n(c)\nFigure 6: Same as Fig. 4, but for l/R= 0.3.\n290 2 4 6 8 10\nkzR0.8450.850.8550.86vci0.868v/vsiBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.4\n(a)\n0 2 4 6 8 10\nkzR00.0050.010.0150.018||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.4\n(b)\n0 2 4 6 8 10\nkzR5101001000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3l/R=0.4\n(c)\nFigure 7: Same as Fig. 4, but for l/R= 0.4.\n300 2 4 6 8 10\nkz R0.850.8520.8540.8560.8580.860.862vci0.8660.868v/vsi\nl/R=0.1\nl/R=0.2\nl/R=0.3\nl/R=0.4\n(a)\n0 2 4 6 8 10\nkz R00.0020.0040.0060.0080.010.0120.014||/rl/R=0.1\nl/R=0.2\nl/R=0.3\nl/R=0.4\n(b)\n0 2 4 6 8 10\nkz R5101001000D/Tl/R=0.1\nl/R=0.2\nl/R=0.3\nl/R=0.4\n(c)\nFigure 8: (a) The phase speed v/vsi≡ωr/ωsi, (b) the damping rate to frequency ratio |γ|/ωr,\nand (c) the damping time to period ratio τD/T=ωr/(2π|γ|) of the slow surface sausage modes\nversuskzRfor different thickness of the inhomogeneous layer l/R= (0.1,0.2,0.3,0.4). Here,\nthe dashed and solid line curves, respectively, are related toBφi/Bzi= 0 and Bφi/Bzi= 0.2.\nAuxiliary parameters are as in Fig. 2.\n310 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R0.840.8450.850.8550.86vciv/vsi\nBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=0.5\n(a)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R00.0020.0040.0060.0080.010.0120.014||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1Bi/Bzi=0.2\nBi/Bzi=0.3\nkzR=0.5\n(b)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R101001000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=0.5\n(c)\nFigure 9: (a) The phase speed v/vsi≡ωr/ωsi, (b) the damping rate to frequency ratio |γ|/ωr,\nand (c) the damping time to period ratio τD/T=ωr/(2π|γ|) of the slow surface sausage modes\nversusl/RforkzR= 0.5 and different twist parameters Bφi/Bzi= (0,10−3,0.1,0.2,0.3). Aux-\niliary parameters are as in Fig. 2. The results for Bφi/Bzi= 0 and 10−3overlap with each\nother.\n320 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R0.840.8450.850.8550.86vci0.868v/vsi Bi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=1\n(a)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R00.0020.0040.0060.0080.010.0120.0140.016||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=1\n(b)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R5101001000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=1\n(c)\nFigure 10: Same as Fig. 9, but for kzR= 1.\n330 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R0.840.8450.850.8550.86vci0.868v/vsi Bi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=2\n(a)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R00.0050.010.0150.018||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=2\n(b)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R5101001000D/T\nBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=2\n(c)\nFigure 11: Same as Fig. 9, but for kzR= 2.\n340 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R0.840.8450.850.8550.86vci0.868v/vsi Bi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=4\n(a)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R00.0050.010.0150.018||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=4\n(b)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R510100100010000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=4\n(c)\nFigure 12: Same as Fig. 9, but for kzR= 4.\n350 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R0.840.8450.850.8550.86vci0.868v/vsi Bi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=8\n(a)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R00.0050.010.0150.02||/rBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=8\n(b)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R51010010001000010000D/TBi/Bzi=0\nBi/Bzi=10-3\nBi/Bzi=0.1\nBi/Bzi=0.2\nBi/Bzi=0.3kzR=8\n(c)\nFigure 13: Same as Fig. 9, but for kzR= 8.\n360 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R0.840.8450.850.8550.86vci0.868v/vsi\nkzR=0.5\nkzR=1\nkzR=2\nkzR=4\nkzR=8\n(a)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R00.0020.0040.0060.0080.010.0120.0140.016||/rkzR=0.5\nkzR=1\nkzR=2\nkzR=4\nkzR=8\n(b)\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nl/R5101001000d/T\nkzR=0.5\nkzR=1kzR=2\nkzR=4kzR=8\n(c)\nFigure 14: (a) The phase speed v/vsi≡ωr/ωsi, (b) the damping rate to frequency ratio |γ|/ωr,\nand (c) the damping time to period ratio τD/T=ωr/(2π|γ|) of the slow surface sausage modes\nversusl/Rfordifferent kzR= (0.5,1,2,4,8). Here, thedashedandsolidlinecurves, respectively,\nare related to Bφi/Bzi= 0 and Bφi/Bzi= 0.2. Auxiliary parameters are as in Fig. 2.\n37"    },    {        "title": "2103.15665v1.Nonequilibrium_Dynamics_of_the_Chiral_Quark_Condensate_under_a_Strong_Magnetic_Field.pdf",        "content": "symmetrySS\nArticle\nNonequilibrium Dynamics of the Chiral Quark\nCondensate under a Strong Magnetic Field\nGastão Krein1,‡\nand Carlisson Miller1,‡\n1Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II,\n01140-070 São Paulo, SP , Brazil\n*Correspondence: gastao.krein@unesp.br\n‡ These authors contributed equally to this work.\nReceived: date; Accepted: date; Published: date\nAbstract: Strong magnetic fields impact quantum-chromodynamics (QCD) properties in several\nsituations; examples include the early universe, magnetars, and heavy-ion collisions. These examples\nshare a common trait: time evolution. A prominent QCD property impacted by a strong magnetic field is\nthe quark condensate, an approximate order parameter of the QCD transition between a high-temperature\nquark-gluon phase and a low-temperature hadronic phase. We use the linear sigma model with quarks\nto address the quark condensate time evolution under a strong magnetic field. We use the closed time\npath formalism of nonequilibrium quantum field theory to integrate out the quarks and obtain a mean-\nfield Langevin equation for the condensate. The Langevin equation features dissipation and noise\nkernels controlled by a damping coefficient. We compute the damping coefficient for magnetic field\nand temperature values achieved in peripheral relativistic heavy-ion collisions and solve the Langevin\nequation for a temperature quench scenario. The magnetic field changes the dissipation and noise pattern\nby increasing the damping coefficient compared to the zero-field case. An increased damping coefficient\nincreases fluctuations and time scales controlling condensate’s short-time evolution, a feature that can\nimpact hadron formation at the QCD transition. The formalism developed here can be extended to\ninclude other order parameters, hydrodynamic modes, and system’s expansion to address magnetic field\neffects in complex settings as heavy-ion collisions, the early universe, and magnetars.\nKeywords: Quantum chromodynamics; Chiral symmetry; Quark Condensate; Quark-gluon plasma;\nNonequilibrium dynamics\n1. Introduction\nStrong magnetic fields impact prominent quantum-chromodynamics (QCD) phenomena, notably\nthose associated with QCD’s approximate chiral symmetry in the light-quark sector. Special in this\nrespect is the impact on the chiral condensate, as revealed by recent lattice QCD calculations [ 1–3].\nThe chiral condensate is an approximate order parameter for the finite temperature QCD transition between\na high-temperature quark-gluon phase (QGP) and a low-temperature hadronic phase. The transition likely\nqualifies as a crossover (not a phase transition), in that the chiral condensate is nearly zero in the QGP\nphase, and nonzero in the hadronic phase, with a rapid change (not a jump) around the pseudocritical\ntemperature Tpc'150MeV [ 4]. Such a rapid change in the condensate’s value is key to our understanding\nof how protons and neutrons (and other light-flavor hadrons) acquire their masses from almost massless\nquarks and gluons [ 5,6]. Phenomenologically, QCD matter under strong magnetic fields occurs in different\nsettings, to name three of great current interest: the early universe [7,8], magnetars [9,10], and relativistic\nSymmetry 2021 ,xx, 5; doi:10.3390/symxx010005 www.mdpi.com/journal/symmetryarXiv:2103.15665v1  [hep-ph]  29 Mar 2021Symmetry 2021 ,xx, 5 2 of 24\nheavy-ion collisions [ 11,12]. Magnetized QCD matter in those settings evolves in time, albeit under\nvery different time scales. A QGP to hadron transition occurring under such circumstances typifies a\nnonequilibrium phase change problem. In this paper, we present a first study of such a dynamical transition\nin magnetized QCD matter; to wit: we study the nonequilibrium dynamics of the chiral condensate under\na strong magnetic field.\nMagnetic field strengths and space-time scales in this study concern the phenomenology related to\nhigh-energy heavy-ion collision experiments. Relativistic heavy-ion collisions produce QCD matter of\ndeconfined quarks and gluons, the quark-gluon plasma (QGP). Noncentral collisions produce the QGP\nunder strong magnetic fields [ 11,12]; for example, noncentral Pb-Pb collisions at the Large Hadron Collider\n(LHC) can produce fields of strengths as large as [ 13]eB=15m2\np. We conduct our study within the\nperspective of a standard three-stage scenario of QGP’s time evolution [ 14–17]: (1) quarks and gluons\nare freed from the protons and neutrons of the colliding ions and form (2) hot matter that expands\nhydrodynamically until it (3) cools up to a temperature T\u0018150MeV , and finally disassembles into\nhadrons. More specifically, we work within the perspective that local thermodynamic properties as\ntemperature and order parameter acquire physical meaning.\nWe address magnetic field effects on the chiral condensate dynamics with Langevin field equations,\nequations widely used in field theory treatments of dynamical phase transitions [ 18,19]. A prototype\ndynamical transition addressed by these equations is the one of a temperature quench in a spin system, in\nthat a sudden drop in the system’s temperature takes the system out from a spin-disordered phase and\ndrives it irreversibly toward a spin-ordered phase. The quench-induced transition just described resembles,\nalbeit with differences, the heavy-ion collision evolution across the crossover from a quark-gluon phase, in\nwhich the condensate is very small, toward a hadron-dominated phase, in which the condensate ultimately\nreaches its vacuum value. Indeed, for zero magnetic field, there is a vast literature on the use of Langevin\nfield equations in this context—Refs. [ 20–39] are a sample of this literature. The Langevin equations\nfeatured in that literature are either postulated on phenomenological grounds [ 23,26,33–35,39], or derived\nfrom a microscopic model through a coarse-graining procedure [ 20–22,24,25,27–32,36–38]. We follow the\nlatter approach.\nWe extend the semiclassical approach of Ref. [ 29] to include magnetic field effects on the chiral\ncondensate dynamics. In that approach, the condensate dynamics is governed by a Langevin field\nequation derived from a semiclassical two-particle irreducible (2PI) effective action. The effective action,\ncomputed with the time path formalism of nonequilibrium quantum field theory [ 40,41], refers to the\nGell-Mann–Levy linear sigma model [ 42] with quarks (LSMq). The LSMq features degrees of freedom\nassociated with the long-wavelength QCD chiral physics: constituent quarks, pseudoscalar-isoscalar\nmesons (pions, pseudo-Goldstone bosons), and a scalar-isoscalar meson (the quark condensate). The\nmodel does not describe quark confinement. Despite of this limitation, the model describes many of the\nequilibrium, time-independent magnetic field effects on the QCD equation of state, phase structure and\nchiral condensate [ 43–55] brought out by lattice calculations. We direct the reader to Refs. [ 56–59] for\nreviews with additional references on works employing the LSMq and also other models.\nThis first study aims primarily to get insight into how a strong magnetic field affects condensate\ndynamics. To fulfil this aim, we simplify the analysis by omitting physical effects peculiar to a heavy-ion\ncollision. We address the omissions and ensuing consequences in the course of the presentation of our\nwork. Besides, we seek an analytical understanding and avoid, whenever possible numerical calculations.\nNotwithstanding the simplifications, our study brings new insight into a complex problem that offers\nenormous opportunities to learn about QCD matter.\nWe organize the presentation of the paper as follows. In the next section, we define the chiral quark\nmodel upon which we base our study and summarize its main features. In Section 3 we define the effective\naction and use the closed time path formalism to derive an equation of motion for the condensate, aSymmetry 2021 ,xx, 5 3 of 24\nLangevin equation featuring dissipation and noise kernels. The latter require the magnetized thermal\nquark propagator in the real time formalism. We derive the propagator in Section 4. We complete the\ncalculation of the the damping and noise kernels in Section 5. We present explicit numerical results in\nSection 6 and conclude in Section 7.\n2. The model\nWe present the main ingredients of the model upon which we base our study of magnetic field\neffects on the chiral condensate dynamics. The condensate dynamics is governed by a Langevin field\nequation derived from a semiclassical two-particle irreducible (2PI) effective action [ 29]. The effective\naction builds on effective degrees of freedom associated with the long wavelength chiral physics described\nby a Lagrangian featuring the approximate SU(2)L\u0002SU(2)Rsymmetry of QCD. The Lagrangian is that\nof the Gell-Mann–Levy linear sigma model [ 42], in which quarks replace the nucleons of the original\nmodel. As in the Lagrangian of the original model, a fermion isodoublet field, q= (u,d)T, representing\nthe light uand dquarks, Yukawa-couples to pseudoscalar-isotriplet pion pfield and a scalar-isoscalar\nsfield. The Lagrangian density of the linear sigma model with quarks (LSMq) is given by\nL=¯q[i/¶\u0000g(s+ig5t\u0001p)]q+1\n2\u0002\n¶ms¶ms+¶mp\u0001¶mp\u0003\u0000U(s,p), (1)\nwhere U(s,p)is the potential\nU(s,p) =l\n4(s2+p2\u0000v2)2\u0000hqs\u0000U0, (2)\nwhere U0is an arbitrary constant setting the zero of U(s,p). We use the metric signature gmn=\n(1,\u00001,\u00001,\u00001)and the Bjorken-Drell [ 60] conventions for the Dirac gmmatrices, for which fgm,gng=2gmn.\nForhq=0, the Lagrangian density is invariant under chiral SU(2)L\u0002SU(2)Rtransformations. This\nsymmetry can break spontaneously, in that sacquires a nonzero vacuum expectation value hsi=v6=0,\nwhereashpi=0due to parity. For hq6=0, the termLLSM\nSB=hqsbreaks the symmetry explicitly and plays\nthe role of the symmetry-breaking quark mass term in the QCD Lagrangian, LQCD\nm=\u0000m¯qq. Equality\nbetween the (vacuum or thermal) expectation values of LLSM\nSBandLQCD\nm implies mh¯qqiQCD=\u0000hqhsi,\nand establishes the physical correspondence between hsiand the quark condensate in QCD—Ref. [ 61]\npresents a didactic review on this and other topics relating the LSM and QCD. One can fit the parameters\nof the model to chiral physics observables—a fit at the classical level, for example, sets the parameters\nas:hq=fpm2\np,v2=f2\np\u0000m2\np/l2,m2\ns=2l2f2\np+m2\np, and mq=ghsi. Here fpand mpare the pion\nweak-decay constant and mass, msthes-meson mass, and mqthe constituent quark mass. We chose U0\nsuch that U(0, 0) =0 (Ref. [29] chooses U0such that U(fp, 0) =0).\nThe parameter gplays a very important role in the model’s equilibrium thermodynamics. For example,\nwhen solving the model in the mean field approximation for zero baryon chemical potential, one obtains a\nfirst order transition at a temperature T'123MeV with g=5.5, a second order transition at T'140MeV\nwith g=3.63, and a crossover at T'150MeV with g=3.3. We restrict our study of the condensate\ndynamics to the situation of a crossover, the situation seemingly relevant for QCD. The model the has\nalso been used to study equilibrium, time-independent magnetic field effects on the QCD equation of\nstate, phase structure and chiral condensate—for references, we direct the reader to Refs. [ 43–55] and the\nreviews in Refs. [56–59].\nWe derive the LSMq effective action within the semiclassical framework developed for zero magnetic\nfield in Ref. [29]. In that framework, the long wavelength (soft) modes control the sfield dynamics, with\nthe quarks providing a heat bath. In the present case, this means that the quarks are in equilibrium at\nsome local temperature and local magnetic field. The magnetic field enters the LSMq Lagrangian bySymmetry 2021 ,xx, 5 4 of 24\nreplacing in Eq. (1) ¶mbyDm=¶m+iqA m, where qstands for the (quark or pion) electric charge and Am\nthe electromagnetic vector field. We neglect pion fields in this first study but discuss in Section 6 their\npossible implications on our results. In this semiclassical framework, the magnetic field is a background\nfield, not a dynamical degree of freedom. The effective action is then a functional of the s(x)mean field\nand of the magnetic-field dependent quark propagator S(x,y). We denote the effective action by G[s,S].\n3. The effective action and Langevin equation\nWe summarize the main steps in the derivation of the Langevin equation for the smean field from an\neffective action using the closed time path (CTP) formalism [ 40,41]. In the CTP formalism, one evolves the\nfields over the Schwinger-Keldysh contour, an oriented time path C=C+[C\u0000, in that the time variable t\nruns from an initial time \u0000tto a time talongC+and going back to \u0000talongC\u0000. One identifies fields on\nC+with an index +, whereas those on C\u0000with\u0000, i.e. sa(x)and Sab(x,y)with a=\u0006. A time instant on\nC\u0000is posterior to any time instant on C+. The fields onC+and those onC\u0000are not independent fields;\nthey couple through a CTP boundary condition in that they coincide at large tfor all values of the spatial\ncoordinate [ 40]. To set notation and make the paper self-contained, we mention that we set the speed of\nlight c, the reduced Planck constant ¯h=h/2p, and the Boltzmann constant kBto unity, and define Sab(x,y)\nas\nS++(x,y) =hq(x)q(y)iq(x0\u0000y0)\u0000hq(y)q(x)iq(y0\u0000x0), (3)\nS\u0000\u0000(x,y) =hq(x)q(y)iq(y0\u0000x0)\u0000hq(y)q(x)iq(x0\u0000y0), (4)\nS+\u0000(x,y) = S<(x,y) =\u0000hq(y)q(x)i, (5)\nS\u0000+(x,y) = S>(x,y) =hq(x)q(y)i, (6)\nwhereh\u0001\u0001\u0001i stands for averaging with respect to a density matrix specifying the initial state.\nThe propagator S++(x,y)is nothing else the causal Feynman propagator and S\u0000\u0000(x,y)the corresponding\nanti-causal propagator; from the above definitions, one has:\nS++(x,y) = S\u0000+(x,y)q(x0\u0000y0) +S+\u0000(x,y)q(y0\u0000x0), (7)\nS\u0000\u0000(x,y) = S+\u0000(x,y)q(x0\u0000y0) +S\u0000+(x,y)q(y0\u0000x0). (8)\nThe semiclassical action is given by\nG[s,S] =Gcl[s] +iTr ln S\u0000iTr(i/D\u0000m0)S+G2[s,S], (9)\nwhere Gclis the classical action, m0=gs0, andG2[s,S]contains the sum of 2PI diagrams. Here, Tr stands\nfor a spatial integration over the Schwinger-Keldysh contour and sums over Dirac, color and flavor indices.\nAlthough one deals with two fields, s+ands\u0000, as mentioned above they are not independent, there is a\nsingle mean field s(x), and a single equation of motion [40]:\ndG[s,S]\nds+(x)\f\f\f\f\f\ns\u0000=s+=s=\u0000dG[s,S]\nds\u0000(x)\f\f\f\f\f\ns+=s\u0000=s=0. (10)\nWe need also the equation of motion for Sab(x,y):Symmetry 2021 ,xx, 5 5 of 24\ndG[s+,s\u0000,S]\ndSab(x,y)=0, (11)\nor, equivalently:\n(i/D\u0000m0)Sab(x,y)\u0000Z\nCd4zdG2[s,S]\ndSac(x,z)Scb(z,y) =idabd(4)(x\u0000y), (12)\nhereCindicates that the integration runs over the Schwinger-Keldysh contour. Only one 2PI diagram\ncontributes to G2[s,S], a single one-loop diagram that involves the trace over the magnetic field dependent\nquark propagator, namely:\nG2[s,S] =gZ\nCd4xtrCDF\u0002\nS++(x,x)s+(x) +S\u0000\u0000(x,x)s\u0000(x)\u0003\n, (13)\nwhere tr Dc findicates trace over Dirac, color and flavor indices.\nReplacing Eq. (13) into Eqs. (9) and (12), the last two terms in Eq. (9) cancel; but to complete the\nderivation of G[s,S], one still needs to solve Eq. (12) for Sab. However, to solve Eq. (12) for Sabis not an\neasy task, even for the zero magnetic field case due to the spatiotemporal dependence of s(x). Fortunately,\nthe problem with magnetic field is still tractable within the spirit of the semiclassical approach we use here.\nSpecifically, by assuming that long wavelength modes dominate the s(x)dynamics [ 29], in that dynamical\nfluctuations dsbuild on a s0background mean field, with s0governed by a locally equilibrated quark\nheath bath described by a thermomagnetic quark propagator Sthm. The propagator Sthmdepends on a\nlocal temperature and magnetic field, quantities that also drive a spatiotemporal dependence for the s0\nmean field. In practice, this amounts to split sa(x)as follows:\nsa(x) =sa\n0(x) +dsa(x), (14)\nand write Sabas a functional power series in dsa(x), with Sab\nthmthe zeroth order term:\nSab(x,y) =Sab\nthm(x,y) +dSab(x,y) +d2Sab(x,y) +\u0001\u0001\u0001. (15)\nWhen one replaces these expansions into Eq. (12) and takes into account Eq. (13), one determines dSab(x,y),\nd2Sab(x,y),\u0001\u0001\u0001recursively. Specifically, the zeroth order propagator Sab\nthmobeys the equation\n[i/D\u0000m0\u0000gs0(x)]Sab\nthm(x,y) =\u0000idabd(4)(x\u0000y), (16)\nwhereas the fluctuating contributions, up to the second order in ds, read:\ndSab(x,y) =\u0000igZ\nSKd4z Sac\nthm(x,z)dsc(z)Scb\nthm(z,y), (17)\nd2Sab(x,y) =\u0000g2Z\nSKd4zd4z0Sac\nthm(x,z)dsc(z)Scd\nthm(z,z0)dsd(z0)Sdb\nthm(z0,y). (18)\nEquation (16) evinces the role played by the s0(x)background field, it gives quarks a local effective mass\nmq(x) =gs0(x)determined by local temperature and magnetic field. To obtain the equation of motion for\nthe mean field, one can now replace Eqs. (14)-(18) into Eq. (9) and trail the steps in Ref. [ 29]. Although\nthe magnetic field introduces new features into the Langevin dynamics, the generic form of the equation\nis the same as for zero magnetic field, in that Sab\nthmcontains all the effects of the magnetic field on the sSymmetry 2021 ,xx, 5 6 of 24\ndynamics. Therefore, for now, we do not need the explicit expression for Sab\nthmto write down the Langevin\nequation—we obtain the explicit form of Sab\nthmin the following section.\nBut before writing down the Langevin equation for the smean field, we comment on two points in\nthe derivation of the equation, namely: the lack of independence of the fields on C+from those onC\u0000,\nand the appearance of the noise source in the dsequation of motion. To account for the first point, one\nperforms a change of basis [ 40], a.k.a. Keldysh rotation [ 62]. We apply the Keldysh rotation to s=s0+ds,\nwhich implies for the fluctuating dsfield needed here:\nd¯s(x) =1\n2\u0000\nds+(x) +ds\u0000(x)\u0001\n, Ds(x) =ds+(x)\u0000ds\u0000(x). (19)\nThis transformation makes transparent the physics behind the doubling of fields: it reflects the need for\nboth response (Ds) and fluctuating (d¯s) fields to describe time-dependent fluctuating phenomena [ 63]. The\nsecond point refers to the fact that G[s,S]contains an imaginary part associated with dissipation, a feature\nthat obstructs the straightforward variation implied by Eq. (10). A way to obtain a real action uses the\nFeynman-Vernon trick [ 64], in that one replaces the imaginary part of the action by a noise source coupling\nlinearly to the field; this turns the equation of motion into a stochastic equation—we refer to the book\nof Ref. [ 40] for a thorough discussion on this and other aspects of the CTP formalism. In summary, after\nusing Eqs. (14)-(18) into Eq. (9) and rewriting the action in terms of the Keldysh-rotated fields, replacing\nthe resulting imaginary part in the action by a noise source, and varying w.r.t. Dsand setting ¯s(x) =s(x)\nas implied by Eq. (10), one obtains a stochastic differential equation for s(x), namely [29]:\n¶m¶ms(x) +dU[s]\nds(x)+grs(s0)\u0000Ds(x) =xs(x), (20)\nwhere rs(s0)is the scalar density:\nrs(s0) =trDc fS++\nthm(x,x), (21)\nand Ds(x)thedissipation kernel :\nDs(x) =ig2Z\nd4yq(x0\u0000y0)M(x,y)d¯s(y), (22)\nwith\nM(x,y) =trDc f\u0002\nS+\u0000\nthm(x,y)S\u0000+\nthm(y,x)\u0000S\u0000+\nthm(x,y)S+\u0000\nthm(y,x)\u0003\n, (23)\nandxs(x)is a colored-noise field with the properties:\nhxs(x)ix=0,hxs(x)xs(y)ix=N(x,y), (24)\nwith the noise kernel N (x,y)given by:\nN(x,y) =\u00001\n2g2trDc f\u0002\nS+\u0000\nthm(x,y)S\u0000+\nthm(y,x) +S\u0000+\nthm(x,y)S+\u0000\nthm(y,x)\u0003\n. (25)\nIn Eq. (24),h\u0001\u0001\u0001i xmeans functional average with the probability distribution\nP[x] =exp\u0014\n\u00001\n2Z\nd4xd4yx(x)N\u00001(x,y)x(y)\u0015\n. (26)Symmetry 2021 ,xx, 5 7 of 24\nOur summary on the the derivation of the Langevin equation ends here. To proceed with the study\nof magnetic field effects on the sdynamics, we need the explicit form of the thermomagnetic quark\npropagator Sab\nthm—we derive Sab\nthmin the next section.\n4. The thermomagnetic quark propagator\nOne can obtain the CTP thermomagnetic quark propagator at temperature Tfrom the corresponding\nT=0propagator through a Bogoliubov transformation, much in the same way as done in thermofield\ndynamics (TFD) [ 65–68]. Let Sm(x,y)be the causal, zero temperature quark propagator in a constant\nmagnetic field of strength Bpointing along the ˆzdirection. Sm(x,y)can be written as the product of a\ngauge-dependent Schwinger phase f(x,y)and a gauge-independent, translation invariant propagator\nSm(x\u0000y), namely [69]:\nSm(x,y) = q(x0\u0000y0)hq(x)¯q(y)i\u0000q(y0\u0000x0)i¯q(y)y(y)i=S++\nm(x,y) (27)\n=q(x0\u0000y0)S\u0000+\nm(x,y) +q(y0\u0000x0)S+\u0000\nm(y,x) (28)\n=f(x,y)Sm(x\u0000y) =f(x,y)Zd4p\n(2p)4e\u0000ip\u0001(x\u0000y)S++\nm(p). (29)\nThe phase factor is irrelevant for us since it cancels out in all terms appearing in the Langevin equation in\nEq. (20) due to the properties f(x,z)f(z,y) =f(x,y)andf(x,x) =1. We can also choose the symmetric\ngauge, Am= (0,\u0000y,x, 0)B/2, for which f(x,y) =1. In any case, one can focus on the translation invariant\npiece of the propagator which, from now on will be the object of interest.\nWe use the Landau level representation of Sm(p)and work with the lowest level contribution, the\ndominant contribution for strong fields. The lowest Landau level (LLL) contribution to Sm(p)can be\nwritten as [58]:\nS++\nm(p) =i e\u0000p2\n?/jqfBj2\u0010\n/pk+mq\u0011\np2\nk\u0000m2q+ieP+, (30)\nwhere p2\n?=p2\nx+p2\ny,p2\nk=p2\n0\u0000p2\nz,/pk=g0p0\u0000g3pz, and P+= [1+ig1g2sign(qB)]/2. The presence of\nthe operator P+in Eq. (30) reflects the spin-polarized nature of the lowest Landau level, as P+projects out\none of the two spin directions. From this result, one obtains the off-diagonal CTP components S+\u0000\nmand\nS\u0000+\nmby using Eqs. (7) and (28) and the identity:\niZ¥\n\u0000¥dp0/pk+mq\np2\nk\u0000m2q+iee\u0000ip0(x0\u0000y0)=q(x0\u0000y0)Z¥\n\u0000¥dp0(/pk+mq)2pd(p2\nk\u0000m2\nq)q(p0)e\u0000ip0(x0\u0000y0)\n+q(y0\u0000x0)Z¥\n\u0000¥dp0(/pk+mq)2pd(p2\nk\u0000m2\nq)q(\u0000p0)e\u0000ip0(x0\u0000y0). (31)\nTherefore, the CTP components Sab\nm(p)of the zero-temperature propagator can be written as:Symmetry 2021 ,xx, 5 8 of 24\nS++\nm(p) = e\u0000p2\n?/jqfBjA(p)i\np2\nk\u0000m2q+ie, (32)\nS+\u0000\nm(p) = e\u0000p2\n?/jqfBjA(p)2pd(p2\nk\u0000m2\nq)q(\u0000p0), (33)\nS\u0000+\nm(p) = e\u0000p2\n?/jqfBjA(p)2pd(p2\nk\u0000m2\nq)q(p0), (34)\nS\u0000\u0000\nm(p) = e\u0000p2\n?/jqfBjA(p)\u0000i\np2\nk\u0000m2q\u0000ie, (35)\nwhere, to lighten the notation, we defined\nA(p) =2(/pk+mq)P+= (/pk+mq)h\n1+ig1g2sign(qB)i\n. (36)\nOne obtains the thermal propagator Sab\nthm(p)from Sab\nm(p)through the Bogoliubov transformation,\n S++\nthm(p)S+\u0000\nthm(p)\nS\u0000+\nthm(p)S\u0000\u0000\nthm(p)!\n=VCTP(T,p) S++\nm(p)S+\u0000\nm(p)\nS\u0000+\nm(p)S\u0000\u0000\nm(p)!\nVCTP(T,p). (37)\nA possible CTP transformation matrix VCTP(T,p)is the following:\nVCTP(T,p) =1\n2p\nsinhjp0j/T ejp0j/2T\u0000e\u0000jp0j/2T\n\u0000e\u0000jp0j/2Tejp0j/2T!\n. (38)\nVCTPis the fermionic counterpart to the bosonic Bogoliubov transformation matrix in Ref. [ 70], denoted\nUCT(T,p)in that reference. The individual Sab\nthm(p)components are then given by:\nS++\nthm(p) = e\u0000p2\n?/jqfBjA(p)\"\ni\np2\nk\u0000m2q+ie\u00002pnF(p0)d(p2\nk\u0000m2\nq)#\n, (39)\nS+\u0000\nthm(p) = e\u0000p2\n?/jqfBjA(p)2pd(p2\nk\u0000m2\nq)[q(\u0000p0)\u0000nF(p0)], (40)\nS\u0000+\nthm(p) = e\u0000p2\n?/jqfBjA(p)2pd(p2\nk\u0000m2\nq)[q(p0)\u0000nF(p0)], (41)\nS\u0000\u0000\nthm(p) = e\u0000p2\n?/jqfBjA(p)\"\n\u0000i\np2\nk\u0000m2q\u0000ie\u00002pnF(p0)d(p2\nk\u0000m2\nq)#\n, (42)\nwhere nF(p0)is the Fermi-Dirac distribution:\nnF(p0) =1\nejp0j/T+1. (43)\nHere, qu=2e/3,qd=\u0000e/3, and e=1/p\n137(we use Gaussian units). We note that one obtains\nthe same result for Sab\nthmwith the more standard TFD Bogoliubov transformation, by multiplying the\noff-diagonal elements of TFD propagator, S12(p)and S21(p), by e\u0000p0/2Tand e+p0/2T, respectively. The\ndiagonal elements of CTP and TFD propagators are the same, of course.Symmetry 2021 ,xx, 5 9 of 24\nWe note that the LLL approximation is suitable for strong magnetic fields only. Therefore, one cannot\nextrapolate B6=0results to recover B=0results. Such an extrapolation is possible when performing the\nsum over all Landau levels or using an alternative representation of the propagator—see, for example,\nAppendix A of Ref. [58].\nThis completes the derivation of Sab(x,y). In the next section we compute the different pieces entering\nthe Langevin equation in Eq. (20), namely, the scalar density rsand the dissipation D(x)and noise N(x,y)\nkernels. As mentioned before, our interest in on the long-wavelength physics of the smean field dynamics,\nthereby we neglect vacuum contributions to these quantities.\n5. The scalar density, dissipation and noise kernels\nWe start with the scalar density rs(s0). Although we have flavor symmetry at the level of the quark\nmasses, mu=md=mq=gs0, we still need to make explicit the flavor content of the propagator because\nof the quark electric charges. After taking the trace over Dirac, color and flavor indices, one can write\nrs(s0)as the sum of two contributions [ 71,72],rs(s0) =rB\ns(s0) +rBT\ns(s0), where rB\ns(s0)depends only on\nB:\nrB\ns(s0) =\u0000Nc\n2p2mqå\nf=u,djqfBj\u0014\nlnG(xf)\u00001\n2ln 2p+xf\u00001\n2(2xf\u00001)lnxf\u0015\n, (44)\nandrBT\ns(s0)that depends on Band T:\nrBT\ns(s0) =\u0000Nc\np2mq(jquBj+jqdBj)Z¥\n0dpznF(Eq(pz))\nEq(pz), (45)\nwhere Nc=3is the number of colors, xf=m2\nq/2jqfBj,G(x)the Euler gamma function, and Eq(pz) =q\np2z+m2q.\nNext, we consider the dissipation D(x)and noise N(x,y)kernels , Eqs. (22) and (25). To compute\nD(x), we need the function M(x,y), given in Eq. (23). The Schwinger phase f(x,y)cancels out in Eq. (22);\nas a result, M(x,y)becomes a function of x\u0000y:\nM(x\u0000y) = trfZd4p\n(2p)4d4q\n(2p)4e\u0000i(p\u0000q)\u0001(x\u0000y)trDc\u0002\nS+\u0000\nthm(p)S\u0000+\nthm(q)\u0000S\u0000+\nthm(p)S+\u0000\nthm(q)\u0003\n=Zd4p\n(2p)4e\u0000ip\u0001(x\u0000y)M(p), (46)\nwhere\nM(p) =å\nf=u,dZd4q\n(2p)4trDc\u0002\nS+\u0000\nthm(p+q)S\u0000+\nthm(q)\u0000S\u0000+\nthm(p+q)S+\u0000\nthm(q)\u0003\nf\u0011å\nf=u,dMf(p). (47)\nThe Schwinger phase also cancels out in Eq. (25) and N(x,y) =N(x\u0000y)can be written as:\nN(x\u0000y) =Zd4p\n(2p)4e\u0000ip\u0001(x\u0000y)N(p), (48)\nwhereSymmetry 2021 ,xx, 5 10 of 24\nN(p) =\u00001\n2g2å\nf=u,dZd4q\n(2p)4trDc\u0002\nS+\u0000\nthm(p+q)S\u0000+\nthm(q) +S\u0000+\nthm(p+q)S+\u0000\nthm(q)\u0003\nf\u0011å\nf=u,dNf(p). (49)\nNext, we use M’s translation invariance to write the dissipation kernel D(x)as [29]:\nD(x) = D(t,x) =ig2Zd4p\n(2p)4M(p0,p)Z\nd3ydy0q(x0\u0000y0)e\u0000ip0(x0\u0000y0)+ip\u0001(x\u0000y)ds(y0,y)\n=ig2Zd3p\n(2p)3eip\u0001xZ¥\n\u0000¥dp0\n2pM(p0,p)Z¥\n0dte\u0000ip0tds(t\u0000t,p). (50)\nHere, we made the change of variable x0\u0000y0=tand defined the spatial Fourier transform of the smean\nfield:\nds(t\u0000t,p) =Z\nd3y e\u0000ip\u0001yds(t\u0000t,y). (51)\nEq. (50) exposes the presence of memory in the sdynamics, in that the value of sat time tdepends\nupon the values of sat earlier times t\u0000t. This feature imposes technical difficulties to the analysis of\nthe Langevin equation as it requires numerical techniques to proceed. To maintain the pace with an\nanalytically tractable analysis, we follow Refs. [ 22,24,29] and use a linear harmonic approximation , whereby\nthe dynamics memory is captured by soft-mode harmonic oscillations around the mean field s0(t,p). This\napproximation amounts to assume an harmonic tdependence for ¯s(t\u0000t,p), namely:\ns(t\u0000t,p) = a(t)cos(Es(p)t) +b(t)sin(Es(p)t)\n=s0(t,p)cos(Es(p)t)\u00001\nEs(p)sin(Es(p)t)¶s(t,p)\n¶t\n\u0011s0(t,p) +ds(t,p)\n=s0(t,p) +\u001a\ns0(t,p)[cos(Es(p)t)\u00001]\u00001\nEs(p)sin(Es(p)t)¶s(t,p)\n¶t\u001b\n, (52)\nwhere Es(p)\u0019q\np2+m2sis a characteristic soft-mode frequency, where msis the sfield mass. The\nfunctions a(t)and b(t)were determined using as initial conditions s(t\u0000t,p)jt=0=s0(t,p)and¶s(t\u0000\nt,p)/¶tjt=0=\u0000¶s(t,p)/¶t. The first term within the curly brackets in Eq. (52), being linear in s0is a\nleading-order correction to grsand, since cos(Es(p)t)\u00001oscillates around zero, it is neglected; as such,\none obtains for D(t,x):\nD(t,x) =\u0000Zd3p\n(2p)3eip\u0001xh(p)¶s(t,p)\n¶t. (53)\nwhere h(p)is the momentum-dependent damping coefficient :\nh(p) =g21\n2Es(p)M(p). (54)Symmetry 2021 ,xx, 5 11 of 24\nTo lighten the notation, we denoted h(Es(p),p)byh(p)and M(Es(p),p)byM(p)—from this point on,\nthis notation will be used throughout the paper.\nThe harmonic approximation rendered the dissipation kernel local in time and in a form appropriate\nto work with the Langevin equation in momentum-space:\n¶2s(t,p)\n¶t2+p2s(t,p) +h(p)¶s(t,p)\n¶t+Fs(t,p) =xs(t,p), (55)\nwhere h(p)was defined in Eq. (54), and\nFs(t,p) =Z\nd3x e\u0000ip\u0001x\u0014dU[s]\nds(t,x)+grs(s0)\u0015\n. (56)\nThe momentum space colored noise field has zero mean hxs(t,p)ix=0 and correlation:\nhxs(t,p)xs(t,p)ix= (2p)3d(p+p0)N(t\u0000t0,p), (57)\nwhere\nN(t\u0000t0,p) =Z¥\n\u0000¥dp0\n2pe\u0000ip0(t\u0000t0)N(p0,p). (58)\nAlthough Eq. (55) involves colored noise, it can be solved efficiently by iteration on a discrete\nmomentum lattice using fast Fourier transformation to switch back and forth between coordinate space and\nmomentum space to compute the nonlinear term Fs(t,p)[73]. As our aim is to get analytic understanding\nas much as possible, we leave for a future publication the study of numerical solutions of Eq. (55). But we\nneed to simplify further the analysis to proceed with an analytical treatment. A common simplification\nrestricts the dynamics to a constant soft-mode frequency Es(p)\u0019q\np2+m2s\u0019ms[24,30,36,38]. We adopt\nanother simplification, one motivated by the dimensional reduction brought out by the magnetic field:\nwe restrict the dynamics to the plane orthogonal to the magnetic field, namely s(t,p) =s(t,px,py,pz)!\ns(t,px,py,pz=0)\u0011s(t,p?). Therefore, we need to compute the kernel Mf(p)forp= (p?,pz=0). We\nuse Eqs. (40) and (41) into Eq. (47), take the traces over Dirac and color indices and integrate over the\ntransverse momentum q?, to obtain for Mf(p?,pz=0)\u0011Mf(p?)the result:\nMf(p?) =2Nc\np2 \npjqfjB\n2!\ne\u0000p2\n?/2jqfBjIM(Es(p?)), (59)\nwhere IM(Es(p?))is the integral\nIM(Es(p?)) =Z¥\n\u0000¥dqzZ¥\n\u0000¥dq0n\nd((p+q)2\nk\u0000m2\nq)d(q2\nk\u0000m2\nq)h\nq0(p0+q0)\u0000qz(pz+qz)+m2\nqi\n\u0002\u0010\n[q(q0)\u0000nF(q0)] [q(\u0000p0\u0000q0)\u0000nF(p0+q0)]\n\u0000[q(\u0000q0)\u0000nF(q0)] [q(p0+q0)\u0000nF(p0+q0)]\u0011o\np0=Es(p?). (60)Symmetry 2021 ,xx, 5 12 of 24\nHere and in the following we suppress the explicit reference to the fact that pz=0inp-dependent\nfunctions. We first use the delta function d(q2\nk\u0000m2\nq) =d(q2\n0\u0000q2\nz\u0000m2\nq)to integrate over q0, then use the\nother delta function to integrate over qzto obtain:\nIM(Es(p?)) = [1\u00002nF(Es(p?)/2)] \nE2\ns(p?)\u00004m2\nq\n4E2s(p?)!Z¥\n\u0000¥dqzd(Eq(qz)\u0000Es(p?)/2)\n=[1\u00002nF(Es(p?)/2)] \nE2\ns(p?)\u00004m2\nq\n4E2s(p?)!\n2Es(p?)q\nE2s(p?)\u00004m2q\n=[1\u00002nF(Es(p?)/2)]1\n2Es(p?)q\nE2s(p?)\u00004m2q. (61)\nTherefore:\nMf(p?) =Nc\np\u0010\njqfjB\u0011\n[1\u00002nF(Es(p?)/2)]1\n2Es(p?)q\nE2s(p?)\u00004m2qe\u0000p2\n?/2jqfBj. (62)\nFrom this, one obtains for the momentum-dependent noise coefficient h(p?):\nh(p?) =g2Nc\n4p[1\u00002nF(Es(p?)/2)]1\nE2s(p?)q\nE2s(p?)\u00004m2qå\nf=u,djqfBje\u0000p2\n?/2jqfBj. (63)\nNext, we compute the noise kernel N(x,y)with the same simplifications used for M(x). We use\nEqs. (40) and (41) into Eq. (25), take the traces over Dirac and color indices, and integrate over the transverse\nmomentum q?to obtain:\nNf(p?) =\u00001\n2g22Nc\np2 \npjqfjB\n2!\ne\u0000p2\n?/2jqfBjIN(Es(p?)), (64)\nwith\nIN(Es(p?)) =\u0000[1\u00002nF(Es(p?)/2)]coth(Es(p?)/2T)1\n2Es(p?)q\nE2s(p?)\u00004m2q. (65)\nTaking into account Eq. (62), one can write:\nNf(p?) =1\n2g2coth(Es(p?)/2T)Mf(p?). (66)\nTherefore, after summing over flavor and using the result in Eq. (63), one can write for the momentum\nspace noise kernel N(p?):\nN(p?) =h(p?)Es(p?)coth(Es(p?)/2T). (67)\nFinally, replacing this result into Eq. (58), the p0integration leads to the Dirac delta d(t\u0000t0)andxsbecomes\na white noise field.\nThis concludes the derivation of the main ingredients entering the Langevin equation: rs,D(x)and\nN(x,y). In the next section, we examine the effects of a nonzero magnetic field on these quantities. There\nwe also need the equilibrium mean field s0and mass ms, which we discuss in the following.Symmetry 2021 ,xx, 5 13 of 24\nWe close this section deriving the equilibrium (constant and uniform) mean field solution by putting\nto zero the time and space derivatives and the dissipation and noise kernels in the Langevin equation in\nEq. (20), so that s=s0+ds!s0and:\ndU[s0]\nds0+grs(s0) =0. (68)\nThis equation is nothing else than the equation one obtains from the minimization of the equilibrium\neffective potential Veff[s0]:\nVeff[s0] =U[s0] +WB[s0] +WBT[s0], (69)\nwith [43,71,74]\nWB[s0] =\u0000Nc\n2p2å\nf=u,d\u0010\njqfBj\u00112\u0014\nz0(\u00001,xf)\u00001\n2\u0010\nx2\nf\u0000xf\u0011\nlnxf+1\n4x2\nf\u0015\n, (70)\nWBT[s0] =\u0000Nc\np2Tå\nf=u,djqfBjZ¥\n0dpzln\u0010\n1+e\u0000Eq(pz)/T\u0011\n, (71)\nwhere z0(\u00001,x) =dz(s,x)/dsjs=\u00001andz(s,x)the Riemann-Hurwitz zeta function. That is:\ngrB\ns(s0) =dWB[s0]\nds0and grBT\ns(s0) =dWBT[s0]\nds0. (72)\nWe used the result dz0(\u00001,x)/dx=\u00001/2+x+lnG(x)\u00001/2ln2pto obtain the expression for grB\ns(s0).\nWe obtain the temperature and magnetic field dependent mean field mass msfrom:\nm2\ns=d2Veff[s]\nds2\f\f\f\f\f\nsmin. (73)\n−150−75 075150225\n−150 −100 −50  0 50 100 150eB/m2\nπ = 15Veff[σ0] [MeV/fm−3]\nσ0 [MeV]T = 125 MeV\n = 150 MeV\n = 175 MeV\n−225−150−75 075150225\n−150 −100 −50  0 50 100 150eB = 0VB=0eff  [σ0] [MeV/fm−3]\nσ0 [MeV]T = 125 MeV\n = 150 MeV\n = 175 MeV\nFigure 1. Effective potential for temperatures close to the B=0 crossover temperature.\nIn the next section we present explicit results. We explore the dynamics under a magnetic field in a\ntemperature range around the B=0crossover temperature of the model, Tpc'150MeV . We choose this\nregion of temperature because of its phenomenological interest in a heavy-ion collision setting. The LSMqSymmetry 2021 ,xx, 5 14 of 24\nB=0crossover, in the mean field approximation, occurs for the parameter values g=3.3andl=20. The\ncorresponding (tree-level) vacuum values of the sand quark masses are ms=604MeV and mq=290MeV .\nWith a nonzero B, the chiral transition becomes a first order transition, with a critical temperature close\ntoTB\nc=180MeV; the precise value of TB\ncdepends on the value of B. Since we stay away from such a\ncritical point, these issues do not impact our results. In connection to the transition temperature, we note\nthat at the mean field level, the model does not realize a feature first observed by the lattice simulations\nof Refs. [ 75,76], in that the condensate has a nonmonotonic behavior as a function of Baround T=Tpc.\nBut for temperatures below to Tpc, the LSMq in mean field approximation model does reproduce the\nqualitative features of the lattice results [59].\nTo orientate the discussion of results in the next section, we show in Fig. 1 the effective potential\nVeff[s0]forB=15m2\np, Eq. (69), and B=0, and temperatures around T=150MeV . The effective potential\nfor zero magnetic field, VB=0\neff[s0], is given by [29]:\nVB=0\neff[s0] =U[s0]\u000024TZd3p\n(2p)3lnh\n1+e\u0000E(p)/Ti\n, (74)\nwhere E(p) =q\np2+m2s. The figure reveals that jVeffj<jVB=0\neffjforjs0j\u0014100MeV , a feature due to a\npartial cancellation between WBT[s0]andWB[s0], with the latter being positive for those values of s0.\n6. Dissipation and noise, short-time dynamics\nWe start examining the magnetic field impact on the damping coefficient h, the key quantity\ncontrolling the fluctuations in the smean field dynamics. The zero magnetic field his given in Ref. [ 24,29]\nfor the zero mode only, p=0, for which Es\u0019q\np2+m2s=ms:\nh0=g22Nc\np[1\u00002nF(ms/2)]1\nm2s\u0010\nm2\ns\u00004m2\nq\u00113/2\n. (75)\nPutting Es=msandp?=0in Eq. (54), one obtains for the magnetic field dependent damping coefficient:\nhB=g2Nc\n4p[1\u00002nF(ms/2)] (eB)1\nm2sq\nm2s\u00004m2q. (76)\nWe obtain msfrom Eq. (73). To have a real h, we must have ms>2mqin Eqs. (76) and (75), a constraint\nthat reflects the kinematical limit for the sdecay (at rest) into a quark-antiquark pair, s!q¯q, the only\nsource of dissipation in the model under the present approximations. We note that h0=0forT<150MeV\nin this calculation due to the absence of pions; in the presence of pions, the decay s!2pleads to a\nnonzero h. We recall that our results are valid for strong magnetic fields only. Therefore, one cannot\nextrapolate our results to B=0; for weak magnetic fields, one needs to use a different representation\nfor the magnetized quark propagator, as the LLL approximation is not valid in this case [ 58]. But, since\nweak fields (of strengthsp\neB\u001cLQCD) have little impact on chiral properties, we do not need alternative\nrepresentations for the quark propagator.\nFigure 2 displays the temperature dependence of the zero mode damping coefficient for B=0\nand three B6=0values. The magnetic field changes the qualitative temperature dependence of hclose\ntoT=150MeV . In a temperature quench scenario, T\u001dTB\nc!T\u001cTB\nc, the nonzero value of hBfor\nT<TB\ncdelays the start off of the condensate evolution after the quench. We extend the discussion on this\nissue at the end of this section, where we study explicit short-time solutions of the Langevin equation.\nThe magnetic field enters the expression for h, Eq (76), in two ways: through the multiplicative\neBterm, and through the values of msand mq. The latter dependence is subtle, Baffects msand mqSymmetry 2021 ,xx, 5 15 of 24\n0.01.02.03.04.05.0\n130 140 150 160 170η [fm−1]\nT [MeV]eB/m2\nπ = 0\n = 15\n = 20\n = 25\nFigure 2. Temperature and magnetic field dependence of the zero mode damping coefficient. Temperature\nrange chosen to include the B=0 pseudocritical temperature, Tpc=150 MeV .\nand thereby affects the inequality ms>2mq. The magnetic field modifies not only the position of the\nminimum of Veff(which determines mq) but also its curvature around the minimum (which determines\nms)—compare the B6=0and B=0effective potentials in Fig. 1. To appreciate this B-dependence of\nmsand mq, we show in Fig. 3 the temperature dependence of these masses for the values of Bused in\nFig. 2. It is important to notice the different temperature dependence of msand mq: the former increases\nfaster as the temperature decreases. This faster increase of msexplains the hBincrease at low temperatures.\n200400600800\n130 140 150 160 170mσ [MeV]\nT [MeV]eB/m2\nπ = 0\n = 15\n = 20\n = 25\n100200300400\n130 140 150 160 170mq [MeV]\nT [MeV]eB/m2\nπ = 0\n = 15\n = 20\n = 25\nFigure 3. Temperature and magnetic field dependence of the sand quark masses.\nContinuing with the aim of gaining analytic understanding, we consider s’s dynamics in a\ntemperature quench scenario. Before continuing, we spell out the required simplifications here. We\nneglect expansion of the system. Expansion is perhaps the most relevant trait of a heavy-ion collision that\nneeds to be taken into account when simulating a real laboratory event. But such a simulation is out of the\nscope of this work. We also assume a constant magnetic field in the course of the condensate evolution.\nAs such, we do not consider the complex magnetohydrodynamics that governs the magnetic field in the\nmedium expansion course. The magnetic field weakens as the system expands, but it also induces electric\ncurrents that can sustain a magnetic field of sizeable strength while the system exists [ 77–79]. This feature,\nto some extent, justifies the assumption of a constant field. Finally, we do no consider reheating, i.e. energy\ntransfer between the condensate and the background. Reheating changes the local temperature of theSymmetry 2021 ,xx, 5 16 of 24\nbackground and, as for zero magnetic fields, can effect the dynamics [ 30]. We reserve for a separate study\nthe inclusion of the neglected effects.\nIn a quench scenario, a sudden drop in the temperature drives the system out of a high temperature\nphase, in which s\u00190, and forces the system to evolve to a lower temperature phase in which s6=0.\nOne gets insight on how a nonzero Bimpacts such a quench by examining the time scale controlling\nthe short-time dynamics. That time scale, which we denote by ts, determines how quickly the system\nleaves the initial state. It depends, of course, on h, and also on the nature of the lower temperature phase,\nwhich the magnetic field affects as well. This interplay between hand the nature of the low temperature\nphase in a quench scenario is well known [ 18,19]. We take as lower temperature phase one around the B=0\npseudocritical temperature; that is, at t=0the system is brought to one of the local maxima of the Veffin\nFig. 1.\nAt short times, when s\u00190, one can linearize the Langevin equation, neglect the second-order time\nderivative, and solve the equation analytically. It is convenient [ 24] to rescale the fields by the volume\nV=L3, namely s=s/L3andxs=xs/L3. The Langevin equation for scan be written as:\nh(p?)¶s(t,p?)\n¶t\u0000\u0010\nm2\u0000p2\n?\u0011\ns(t,p?) +grs(s0)\u0000fpm2\np=xs(t,p?), (77)\nwhere\nm2=l\u0012\nf2\np\u0000m2\np\nl\u0013\n, (78)\nandxshas zero mean,hxs(t,p?)ix=0, and correlation\nhxs(t,p?)xs(t0,p0\n?)ix= (2p)2d(p?+p0\n?)Ld(t\u0000t0)N(p?), (79)\nwhere N(p?) = N(p?)/L6. We compute the equal-time correlation function (variance) of the field,\nhs2(t,p2\n?)ix. Taking as initial condition s(0,p?) =0, one obtains:\nhs2(t,p2\n?)ix=\u0002\ngrs(s0)\u0000fpm2\np\u00032\n(m2\u0000p?)2\u0010\nel(p?)t/ts\u00001\u00112\n+E(p?)coth(E(p?))\nL3(m2\u0000p2\n?)\u0010\ne2l(p?)t/ts\u00001\u0011\n, (80)\nwhere\nts=hB\nm2and l(p?) =1\u0000p2\n?/m2\nh(p?)/hB, (81)\nwith hBgiven by Eq. (76).\nFrom the definition of l(p?)one sees that the exponentials in Eq. (80) increase with time for long\nwavelength modes, p2\n?<m2, and decrease for short wavelengths, p2\n?>m2. That is, long wavelength\nmodes explode at short times, akin to the familiar phenomenon of spinodal decomposition [ 18,19]. We recall\nthat the quench we are considering brings the system to one of the local maxima of the effective potential\nVeffin Fig. 1; there are no barriers to overcome. The explosion is controlled by the time scale ts, which\ndepends on h(fluctuations) and m2(state). The first term in Eq. (80) exposes the role played by the\nlow temperature phase; it comes from dVeff[s]/ds. Notice that for small s, that term is nothing elseSymmetry 2021 ,xx, 5 17 of 24\ndVeff[s]/ds=0:(grs\u0000fpm2\np)/m2\u0011s2\ns, where ssstands for small s. The second term comes from the\nnoise source.\n0.000.501.001.502.00\n0.00 0.25 0.50 0.75 1.00 1.25 1.50p⊥ = 0\np⊥ = 〈 p⊥〉th.T = 150 MeV〈σ−〉(t, p⊥)/σs\nt [fm]eB/m2\nπ = 25\n = 20\n = 15\n0.000.751.502.253.00\n0.00 0.25 0.50 0.75 1.00 1.25 1.50p⊥ = 0\np⊥ = 〈 p⊥〉th. T = 120 MeV〈σ−〉(t, p⊥)/σs\nt [fm]eB/m2\nπ = 25\n = 20\n = 15\nFigure 4. Square root of the equal-time correlation function, normalized to ss—see text for definitions.\nNotice the different vertical axes ranges in two panels. There is no dashed-red curve for p?=0in the left\npanel because his zero for eB/m2p=15 and T=150 MeV (see Fig. 2).\nWe present results for the (square root of the) equal-time correlation function for two values of p?;\nthe zero mode p?=0, and a thermal average value hp?ith.=q\nhp2\n?ith., wherehp2\n?ith.is the average:\nhp2\n?ith.=R\nd2p?p2\n?nB(p?)R\nd2p?nB(p?), where nB(p) =1\neEs(p)/T\u00001, (82)\nwith Es(p) =q\np2+m2s. We take a volume of dimension L3= (10fm)3. Figure 4 shows results for\nhsi(t,p?)/ss, where we defined hsi(t,p?) =q\nhs2(t,p?)ix. The zero mode’s fast exponential growth\nstands out in the two panels of the figure. The magnetic field impact on the short-time growth also\nstands out, notably the explosion delay alluded to previously. Since our calculation does not take into\naccount expansion of the system, it is difficult to assess the phenomenological impact of such a delay,\ne.g. on the QGP disassemble into hadrons. However, the delay does not seem irrelevant in this respect,\nas it can reach 1fm(right panel of Fig. 4), being of the order of 10% of the total time the QGP takes to\ndisassemble into hadrons. We recall that the latter is on the average of the order of 10fm, time over\nwhich the temperature varies between Tch\u0018150MeV and TK\u0018100MeV [80,81]. Here, Tchand TK\nare respectively the chemical and kinetic freeze out temperatures; the former signals the end of inelastic\ncollisions and fixes the observed hadron abundances and the latter signals the end of elastic hadron\ncollisions and leads to the disassemble of the system into hadrons. Given that a magnetic field also affects\nhadron masses, there seems to be room for optimism for a possible experimental signal in hadron emission\nspectra from noncentral collisions. Certainly these results warrant further studies.\nWill pions change qualitatively the overall picture? Probably not. For B=0, pions have a significant\neffect on the sdynamics only close to the first-order transition of the LSMq [ 82]. The results we have shown\nhere refer to temperatures away from the first order transition temperature, which we recall, T\u0015180MeV .\nMoreover, results from lattice QCD [ 83] and phenomenological models [ 58,84,85] predict that a background\nmagnetic field leaves unchanged the p0mass and increases the p\u0006masses, as expected on general grounds,\nfeatures that will not change the results of Ref. [ 82]. An instance where pions will change quantitatively\nour results is in the value of h: pions bring further dissipation with the s!2pchannel, which implies aSymmetry 2021 ,xx, 5 18 of 24\npositive contribution to ts, i.e. the delay increases. However, the question will be answered only with a\ndetailed calculation.\n7. Conclusions and perspectives\nWe studied the impact of a strong magnetic field on the chiral quark condensate dynamics. We built\non the semiclassical framework developed for zero magnetic field developed in Ref. [ 29]. That framework\nbases the dynamics on a mean field Langevin equation derived from a microscopic chiral quark model.\nWe extended that Langevin equation to include the effects of a magnetic field. The Langevin equation\nwe derived features damping and noise modified by the magnetic field. Damping and noise reflect the\ncondensate’s interactions with an effective magnetized quark background in local thermal equilibrium.\nThe background results from integrating out quarks from a mean-field effective action defined by the linear\nsigma model. To integrate out quarks, we used the closed time path formalism of nonequilibrium quantum\nfield theory. We obtained numerical results using values of magnetic field strengths and space-time scales\nrelated to high-energy heavy-ion collision experiments. We presented results for the short-time condensate\ndynamics under temperature quenches. The quenches we used were from a high temperature, for which\nthe condensate is zero, to lower temperatures close to the zero magnetic field crossover temperature,\nT\u0018150MeV . The results we showed revealed that the magnetic field changes the dissipation pattern\nas compared to the zero magnetic field case, retarding condensate’s short-time evolution substantially, a\nfeature that can impact hadron formation at the QCD transition.\nOur study was a first incursion into a complex many-body problem. Our primary aim in this study\nwas to get insight into how a strong magnetic field affects condensate dynamics. We simplified the analysis,\nand sought an analytical understanding whenever possible. We also omitted physical effects peculiar to a\nheavy-ion collision. As such, before one can draw conclusions on phenomenological consequences in a\nrealist heavy-ion setting, one needs to extend the theoretical framework to include the omitted features.\nThese include pions, expansion, reheating, magnetohydrodynamics modes, and coupling to other order\nparameters. As in the case of zero magnetic field [ 29], the formalism developed here is flexible enough\nto tackle the more complex problem. Another extension of our study is to incorporate a confinement\nmechanism. A possibility is to couple a color dielectric field to the chiral sandpfields of the LSMq, a\npossibility very much explored in the context of bag and soliton models [ 86]. Such models can be extended\nto include explicit gluon degrees of freedom to realize dynamical chiral symmetry breaking and describe\nasymptotic freedom [87,88].\nThe framework developed in this paper can be adapted to study magnetic field effects on the QCD\nphase transition in the early universe and in the interior of magnetized compact stars (magnetars). Several\nmechanisms of strong magnetic field generation in the early universe have been suggested [ 7,8]; a very\nrecent, connected with the QCD phase transition, involves the collapse of domain walls related to the\nconfinement order parameter [ 89]. A marked difference between the early universe and heavy-ion collision\nsettings concerns the rate of change of the temperature dlnT/dtduring expansion of the system. In the\nearly universe, this rate is given by Hubble constant H\u001810\u000018s\u00001, which is much slower than that in a\nheavy-ion collision. Therefore, the primordial chiral condensate evolves in a slowly changing effective\npotential as the system expands. Such an evolution characterizes an annealing scenario for the phase\nchange, rather than of a quench, but it can be studied equally well with the Langevin equation framework\nof the present paper [ 90]. Regarding magnetars, the inner-core magnetic field can reach strengths varying\nbetween eB'm2\npand eB'50m2\np[91]. In this setting, the temperatures are very low, lower than 50 MeV ,\nand the phase conversion is driven by high baryon density. An issue of interest relates to the time\nscales associated with the phase conversion during the early stages of the magnetar formation after aSymmetry 2021 ,xx, 5 19 of 24\ncore-collapsing supernova process. In this case, the formalism used in this paper needs to be extended to\nnonzero baryon density [92].\nFunding: G.K was supported in part by: Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq,\nGrants No. 309262/2019-4, 464898/2014-5 (INCT Física Nuclear e Aplicações), Fundação de Amparo à Pesquisa do\nEstado de São Paulo - FAPESP , Grant No. 2018/25225-9. C.M. was supported by a scholarship from Coordenação de\nAperfeiçoamento de Pessoal de Nível Superior - CAPES.\nAcknowledgments: G.K. thanks Prof. Ashok Das for e-mail dicussions on the fermionic CTP Bogoliubov\ntransformation.\nConflicts of Interest: The authors declare no conflicts of interest. 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Larger domains of disoriented chiral condensate through annealing. Phys. Lett. B 1994 ,\n329, 486–492, [hep-ph/9312349]. doi:10.1016/0370-2693(94)91094-4.\n91. Ferrer, E.J.; Hackebill, A. Equation of State of a Magnetized Dense Neutron System. Universe 2019 ,5, 104.\ndoi:10.3390/universe5050104.\n92. Kroff, D.; Fraga, E.S. Nucleating quark droplets in the core of magnetars. Phys. Rev. D 2015 ,91, 025017,\n[arXiv:hep-ph/1409.7026]. doi:10.1103/PhysRevD.91.025017.\n©2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article\ndistributed under the terms and conditions of the Creative Commons Attribution (CC BY)\nlicense (http://creativecommons.org/licenses/by/4.0/)."    },    {        "title": "1604.07971v2.Influence_of_nonlocal_damping_on_the_field_driven_domain_wall_motion.pdf",        "content": "In\ruence of nonlocal damping on the \feld-driven domain wall motion\nH. Y. Yuan,1Zhe Yuan,1,\u0003Ke Xia,1and X. R. Wang2, 3\n1The Center for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, Beijing 100875, China\n2Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n3HKUST Shenzhen Research Institute, Shenzhen 518057, China\n(Dated: March 16, 2021)\nWe derive the complete expression of nonlocal damping in noncollinear magnetization due to the\nnonuniform spin current pumped by precessional magnetization and incorporate it into a generalized\nThiele equation to study its e\u000bects on the dynamics of the transverse and vortex domain walls (DWs)\nin ferromagnetic nanowires. We demonstrate that the transverse component of nonlocal damping\nslows down the \feld-driven DW propagation and increases the Walker breakdown \feld whereas\nit is neglected in many previous works in literature. The experimentally measured DW mobility\nvariation with the damping tuned by doping with heavy rare-earth elements that had discrepancy\nfrom micromagnetic simulation are now well understood with the nonlocal damping. Our results\nsuggest that the nonlocal damping should be properly included as a prerequisite for quantitative\nstudies of current-induced torques in noncollinear magnetization.\nI. INTRODUCTION\nGilbert damping,1which is spatially local and was in-\ntroduced to describe the energy dissipation in magne-\ntization dynamics, is a phenomenological parameter of\nmagnetic materials although a microscopic theory based\non the spin-orbit interaction and disorder scattering is\navailable.2The energy dissipation plays an important\nrole in both current-driven3and \feld-driven4magnetiza-\ntion dynamics. For example, in the \feld-driven domain\nwall (DW) propagation in a magnetic wire, the prop-\nagation speed is proportional to the energy dissipation\nrate.4Thus a comprehensive understanding of dissipa-\ntion (damping) in magnetic materials is not only funda-\nmentally interesting, but also technologically important\nsince the performance of many spintronic devices such as\nrace-track memory5is directly related to the DW prop-\nagation speed. In the past several decades, the progress\nin our understanding of the damping has greatly ad-\nvanced both theoretically and experimentally. Theoret-\nically, the Gilbert damping of real materials can be cal-\nculated by using the torque-correlation model,6{8Kubo\nformalism9{11and the scattering approach12{14in com-\nbination with the \frst-principles electronic structures.\nIn experiments, the value of Gilbert damping can be\nmeasured by ferromagnetic resonance (FMR).15{19For\na nonuniform magnetic structure (noncollinear magneti-\nzation) such as a DW in a ferromagnetic wire, FMR is not\napplicable and the Gilbert damping is usually extracted\nvia measuring the \feld-driven DW velocity.4,20{22This\ntechnique is based on the following general features of\nthe \feld-driven DW propagation: below a critical \feld, a\nDW propagates like a rigid body and its velocity is pro-\nportional to the external \feld and inversely proportional\nto the Gilbert damping.23Surprisingly, the extracted\nGilbert damping coe\u000ecient of permalloy (Ni 80Fe20) from\n\feld-driven DW motion is three times larger than the\nvalue of the same material from FMR measurement.19\nThe enhanced damping in magnetic DW has been at-tributed to the surface roughness of a ferromagnetic\nnanowire in combination with texture-enhanced Gilbert\ndamping arising from spin pumping.19\nSpin pumping was \frst proposed to understand the\nGilbert damping enhancement in a thin ferromagnetic\n\flm in contact with a nonmagnetic metal.24A preces-\nsional magnetization m(t) pumps an electron spin cur-\nrent of polarization js\npump\u0018m\u0002@m=@tinto the nonmag-\nnetic metal that dissipates via spin-\rip scattering.25,26\nSpin pumping does not have any observable e\u000bect in\na homogeneous magnetic structure because the net in-\n\row/out\row spin current anywhere in the system is zero\ndue to a precise cancellation of the pumped spin cur-\nrents in opposite directions. In noncollinear magnetiza-\ntion, the partial cancellation of the spatially dependent\nspin pumping gives rise to a nonzero net in\row/out\row\n\u0000r[m(r)\u0002@m(r)=@t]. This results in an extra torque\nthat has nonlocal damping in nature, di\u000berent from the\nlocal Gilbert damping due to spin-orbit interaction. The\ntotal damping of a nonuniform magnetic structure such\nas a spin spiral and a DW is enhanced through spin\npumping27{31or spin wave emission.32The enhanced\ndamping in nonuniform magnetic structure of permal-\nloy has been quantitatively calculated from the \frst-\nprinciples, which depends not only on the magnetization\ngradient, but also on the particular dynamical modes.33\nThe torque due to the pumped spin current from pre-\ncessional noncollinear magnetization can be decomposed\ninto a longitudinal and a transverse components, which\nwere independently predicted by several groups around\nthe same time. Foros et al.29and Zhang et al.31showed\nthat the longitudinal spin current increases the e\u000bective\ndamping in spin spirals and DWs. Hankiewicz et al.27,28\ndiscovered that the transverse spin current in\ruences the\ndissipation of spin waves. Many previous works in lit-\nerature only include the longitudinal component.19,34{36\nFor the \feld-driven transverse DW motion, the \frst-\nprinciples calculation showed that the transverse and\nlongitudinal spin currents are responsible for the rigid-arXiv:1604.07971v2  [cond-mat.mes-hall]  10 May 20162\nbody motion below a critical \feld and the oscillatory\nmotion above the critical \feld, respectively.33For more\ncomplicate magnetic structures with noncollinear mag-\nnetization, e.g. vortex DWs, how this extra nonlocal\ndamping in\ruences magnetization dynamics is not en-\ntirely clear. For example, it was predicted that the lon-\ngitudinal component of the pumped spin current has no\ne\u000bect on steady-state DW motion.31On the other hand,\nmicromagnetic simulation showed that the longitudinal\ncomponent can slow down the \feld-driven DW propaga-\ntion as what was observed in experiments.19The incon-\nsistency in the theory and experiment requires a better\nunderstanding of the e\u000bects of the nonlocal damping on\nthe DW motion.\nIn this paper, the complete form of the nonlocal damp-\ning is derived from physically transparent spin pumping\nformalism and is incorporated into a generalized Thiele\nequation (GTE), which describes the steady-state mo-\ntion of a nonuniform magnetic structure. We focus on\nthe consequences on propagation of di\u000berent types of\nDWs resulting from the nonlocal damping. The solu-\ntions of the GTE explicitly show the role played by the\nnonlocal damping in DW motion, where the longitudi-\nnal and transverse components are equally important in\ngeneral. For the steady motion of a DW driven by an\nexternal \feld, the longitudinal component vanishes while\nthe transverse one slows down the DW velocity and in-\ncreases the Walker breakdown \feld. The experimentally\nobserved dependence of the DW mobility on the damp-\ning parameter, which were systematically lower than the\nvalues from micromagnetic simulations, can be under-\nstood by including the nonlocal damping. As a prerequi-\nsite, quantitative extraction of current-induced torques in\nnoncollinear magnetization dynamics relies on an accu-\nrate description of nonlocal damping in analytical models\nand micromagnetic simulations. The paper is organized\nas follows. In Sec. II, we derive the nonlocal damping and\nincorporate it into the GTE. The GTE for both trans-\nverse and vortex DWs in a magnetic nanowire is then\nsolved in Sec. III under a static external magnetic \feld\nalong the wire. The solution is also used to determine the\nstrength of the nonlocal damping from available experi-\nmental data. The conclusions are summarized in Sec. IV.\nIn Appendix A, we provide a detailed derivation of the\nspin pumping and the resulting damping torque of non-\ncollinear magnetization in a ferromagnetic nanowire.\nII. GENERAL FORMALISM\nA. Nonlocal damping torque\nWe consider a nonuniform magnetic structure in a\nnanowire as schematically shown in Fig. 1. The x\u0000,\ny\u0000, andz\u0000axes are along the length, width and thick-\nness directions, respectively. A spin current polarized\nalong\u0018m(r;t)\u0002@tm(r;t) is pumped out towards all\ndirections25as magnetization m(r;t) changes with time.\nFIG. 1. (a) Top view of a head-to-head transverse DW in a\nmagnetic nanowire. The thick arrows denote the directions of\nmagnetization m. Thex\u0000,y\u0000,z\u0000axes are along the length,\nwidth and thickness directions of the nanowire, respectively.\n(b) Top view of a vortex DW in a nanowire. The circle in the\ncenter denotes the vortex core.\nWhile the detailed derivation is given in Appendix A,\nthe resulting spin current in a noncollinear magnetiza-\ntion reads\njs\ni(r;t) =\u0000~\n4\u0019\u0000\"#@i[m(r;t)\u0002@tm(r;t)]\n=\u0000~\n4\u0019\u0000\"#@im(r;t)\u0002@tm(r;t)\n\u0000~\n4\u0019\u0000\"#m(r;t)\u0002@i@tm(r;t)\n\u0011jsk\ni(r;t) +js?\ni(r;t): (1)\nHerei=x;y;z denotes the propagation direction of the\nspin current. \u0000 \"#is the (intralayer) spin-mixing conduc-\ntivity that has the dimension of the inverse of length. Its\nrelation to the conventional spin-mixing conductance is\ndisucssed in Appendix A. The longitudinal spin current\n(LSC) jsk\ni=\u0000~\n4\u0019\u0000\"#@im\u0002@tmis always aligned with\nthe local magnetization msince both@imand@tmare\nperpendicular to m.js?\ni=\u0000~\n4\u0019\u0000\"#m\u0002@i@tmis the\ntransverse spin current (TSC), which is perpendicular to\nthe local magnetization m.\nIn early works29,31with the adiabatic approximation,\nin which the polarization of the spin current is as-\nsumed to align with local magnetization, the transverse\nspin current is arti\fcially neglected. This approxima-\ntion is also used in some later works19,34{36without a\nproper justi\fcation. Taking permalloy as an example,\nthe \frst-principles calculations showed that the spin dif-\nfusion length and the transverse spin coherent length are\n5.5 nm13and 13.1 nm,33respectively, at low tempera-\nture. These lengths are not much smaller than the width\nof DWs,37,38as required for the adiabatic approximation.\nThus there are no reasons that the TSC in real materi-\nals can be neglected. Indeed, we will see that the TSC\nsubstantially in\ruences the magnetization dynamics in\nthe nonuniform magnetic structures, especially for the\nsteady-state DW motion below the critical \feld.\nThe dissipative torque generated by the precessional\nmotion of noncollinear magnetization is given by the di-3\nvergence of the spin current in Eq. (1),\n\u001cdamping =\r\nMs@i\u0010\njsk\ni+js?\ni\u0011\n=\u001cLSC+\u001cTSC;(2)\nwhere\r=g\u0016B=~is the gyromagnetic ratio in terms of\nLand\u0013 egfactor and Bohr magneton \u0016BandMsis the sat-\nuration magnetization. Einstein summation is assumed\nin this paper unless it is stated otherwise. The total\ntorque can also be decomposed into the longitudinal and\ntransverse components, \u001cLSCand \u001cTSC. Speci\fcally, we\nhave\n\u001cTSC=\u0000g\u0016B\u0000\"#\n4\u0019Ms@i(m\u0002@i@tm)\n=g\u0016B\u0000\"#\n4\u0019Ms\u0002\n(m\u0001@i@tm)m\u0002@im\u0000m\u0002@2\ni@tm\u0003\n;\n(3)\nand\n\u001cLSC=\u0000g\u0016B\u0000\"#\n4\u0019Ms@i(@im\u0002@tm)\n=g\u0016B\u0000\"#\n4\u0019Msm\u0002(A\u0001@tm); (4)\nwhere Ais dyadic tensor\nA= (m\u0002@im)(m\u0002@im): (5)\nEquation (4) reproduces the tensor form obtained in\nRef. 31. Alternatively, the second term of \u001cTSC could\nalso be obtained through an expansion of the Gilbert\ndamping term to the second-order spatial derivative\nof magnetization28or through the phenomenological\nLandau-Lifshitz-Baryakhtar equation.39In a very weak\ntexture like a spin wave, where the precessing magne-\ntization deviates slightly from the equilibrium direction\nwith a cone angle \u0012, the energy dissipation due to the\n\frst term of \u001cTSCis proportional to sin4\u0012while the sec-\nond term is proportional to sin2\u0012.33This is the reason\nwhy the \frst term in Eq. (3) can be neglected in spin\nwave dynamics.27,28It should not be neglected in strong\nmagnetization textures like a transverse DW.33B. Generalized Thiele equation\nTo describe the dynamics of noncollinear magnetiza-\ntion, we add the two torques \u001cLSCand \u001cTSC into the\nLandau-Lifshitz-Gilbert (LLG) equation, i.e.\n@tm=\u0000\rm\u0002He\u000b+\u000bm\u0002@tm+\u001cLSC+\u001cTSC:(6)\nHereHe\u000bis the e\u000bective \feld that the external applied\n\feld, the exchange \feld, anisotropy and demagnetization\n\felds.\u000bis the usual local Gilbert damping coe\u000ecient.\nFor simplicity, we rewrite the LLG equation in a more\ncompact form as40\n@tm=\u0000m\u0002(\rHe\u000b\u0000\u000b@tm)\u0000\u0011r2(m\u0002@tm);(7)\nwhere we have de\fned \u0011\u0011g\u0016B\u0000\"#=(4\u0019Ms) representing\nthe strength of nonlocal damping. Following the Thiele41\nanalysis for the \feld-driven rigid DW motion and by m\u0002\n[Eq:(7)]\u0001@im, we have\n(m\u0002@tm)\u0001@im=\rHe\u000b\u0001@im\u0000\u000b(@tm)\u0001(@im)\n+\u0011(@im\u0002m)\u0001r2(m\u0002@tm):(8)\nFor the rigid DW motion, the magnetization is only\nthe function of r\u0000vt41,42, i.e.\n\u0012(r) =\u0012(r\u0000vt); '(r) ='(r\u0000vt); (9)\nwhere\u0012and'are, respectively, the polar and azimuthal\nangles of m= (sin\u0012cos';sin\u0012sin';cos\u0012) in spherical\ncoordinates and vis velocity of the magnetic structure.\nThen the spatial and time derivatives of magnetization\ncan be written as\n@im=@i\u0012^\u0012+ sin\u0012@i'^';\n@tm=@t\u0012^\u0012+ sin\u0012@t'^'\n= (\u0000v\u0001r\u0012)^\u0012+ sin\u0012(\u0000v\u0001r') ^': (10)\nSubstituting Eq. (10) into Eq. (8), we obtain\nsin\u0012[(v\u0001r')@i\u0012\u0000(v\u0001r\u0012)@i'] =\r@i(m\u0001Hext) +\u000bv\u0001\u0000\nr\u0012@i\u0012+ sin2\u0012r'@i'\u0001\n\u0000\u0011h\u0010\n\u0000@i\u0012^'+ sin\u0012@i'^\u0012\u0011\n\u0001r2\u0010\nsin\u0012r'^\u0012\u0000r\u0012^'\u0011i\n\u0001v: (11)\nHere Hextis the external magnetic \feld. Multiplying\nMsto the both sides of Eq. (11) and integrating over the\nwhole nanowire, the generalized Thiele equation becomes\nF+G\u0002v+\u000bD\u0001v+\u0011D0\u0001v= 0; (12)\nwhere the vectors FandGand the tensors DandD0are, respectively,\nF=MsZ\nr(\u0000m\u0001Hext)d3r;\nG=\u0000Ms\n\rZ\n(sin\u0012r\u0012\u0002r')d3r;\nD=\u0000Ms\n\rZ\u0000\nr\u0012r\u0012+ sin2\u0012r'r'\u0001\nd3r;\nD0=Ms\n\rZ\u0010\nsin\u0012r'^\u0012\u0000r\u0012^'\u0011\n\u0001r2\u0010\nsin\u0012r'^\u0012\u0000r\u0012^'\u0011\nd3r:\n(13)4\nThe nonlocal damping appears in the new dissipation\nterm\u0011D0\u0001vin Eq. (12). The original Thiele equation41\nis reproduced for \u0011= 0.\nIII. FIELD-DRIVEN DW MOTIONS\nTo explicitly see the e\u000bects of nonlocal damping on the\nDW motion, we apply the GTE (12) to the propagation\nof transverse and vortex DWs. The analytical results\nfor transverse DWs are compared with micromagnetic\nsimulations of the LLG Eq. (6). We will also use the\navailable experimental data in literature to extract the\nnonlocal damping coe\u000ecient \u0011.\nA. Transverse DWs\nA transverse DW is energetically preferred in relatively\nnarrow and thin nanowires.37,38We consider a head-to-\nhead N\u0013 eel DW in a ferromagnetic nanowire of thickness\nTand width W, as illustrated in Fig. 1(a). The mag-\nnetization m(r\u0000vt) is only a function of x\u0000vxt, and\nm(\u00001) =\u0000m(+1) =^x. An external \feld Hext=H^x\nis applied that drives the DW to propagate along + x\ndirection. Under these conditions, Eq. (13) gives,\nF=\u0000TWMsH^xZ\ndx@xmx(x) = 2TWMsH^x;\nG= 0;\nD=\u0000TWMs\n\r^x^xZ\ndxh\n(@x\u0012)2+ sin2\u0012(@x')2i\n=\u0000TWMs\n\r^x^xZ\ndx[@xm(x)]2;\nD0=TWMs\n\r^x^xZ\ndxd0\nxx[\u0012(x);'(x)]; (14)\nwhered0\nxxis de\fned as\nd0\nxx=\u0010\nsin\u0012@x'^\u0012\u0000@x\u0012^'\u0011\n\u0001@2\nx\u0010\nsin\u0012@x'^\u0012\u0000@x\u0012^'\u0011\n:\n(15)\nSubstituting Eq. (14) into the GTE (12), we \fnd the DW\nsteady-state velocity\nvx=2\rHR\ndx[\u000b(@xm)2\u0000\u0011d0xx]: (16)\nFor a N\u0013 eel DW centered at x0and with a width \u0015,\nthe polar and azimuthal angles of the magnetization are\ngiven by the Walker pro\fle,23\n\u0012(x) =\u0019\n2;\n'(x) =\u0019\u0000arccos\u0014\ntanh\u0012x\u0000x0\n\u0015\u0013\u0015\n: (17)\nThe \feld-driven DW velocity of Eq. (16) can be explicitly\ncalculated,\nvx=\rH\u0015\n\u000b+\u0011=(3\u00152): (18)The e\u000bective damping of the Walker DW with the nonlo-\ncal damping is \u000be\u000b=\u000b+\u0011=(3\u00152) in agreement with the\ncalculated in-plane damping of transverse DWs using the\n\frst-principles scattering theory.33The DW propagation\nis slowed down by the nonlocal damping for a given \feld.\nThe so-called Walker breakdown HW,23above which\nthe solution for a rigid DW motion does not exist, is\nHW=\u000bKz=(\u00160Ms)4,23in a one-dimensional model and\nin the absence of the nonlocal damping, where Kzis\nthe total magnetic anisotropy energy along the hard\naxis including both the magnetocrystalline and shape\nanisotropy. In the presence of the nonlocal damping,\nthe e\u000bective Gilbert damping of a Walker DW becomes\n\u000b+\u0011=(3\u00152) and the Walker breakdown \feld becomes\nlarger,\nHW=Kz(3\u000b+\u0011=\u00152)\n3\u00160Ms: (19)\nThe corresponding velocity at the breakdown \feld can\nbe calculated from Eq. (18) and Eq. (19),\nvW\nx=\rKz\u0015\n\u00160Ms; (20)\nwhich is not a\u000bected by the nonlocal damping.\nThe LLG equation (6) is numerically solved for a trans-\nverse DW in a nanowire of 4 nm thick and 16 nm wide\nunder an external \feld. The system is discretized into\nuniform meshes of size 4 nm. The permalloy parameters\nare used with the saturation magnetization Ms= 8\u0002\n105A=m, the exchange sti\u000bness A= 1:3\u000210\u000011J=m, the\nin-plane crystalline anisotropy (along x)Kc= 500 J=m3\nand Gilbert damping \u000b= 0:01. The shape anisotropy\nis implicitly taken into account via including the dipole-\ndipole interaction in the simulation. Using the above\nmaterial parameters, we obtain the static DW width\n\u0015\u001912:6 nm. When an external \feld is applied, the\nwidth decreases23and approaches 9 nm near the break-\ndown \feldHW.\nFigure 2(a) shows the \feld-dependence of DW prop-\nagation velocity vxalong the + xdirection. At a low\nexternal \feld, vxis proportional to H, following Eq. (18)\n(straight lines). For the external \feld larger than 20 Oe,\nthe numerical velocity is slightly lower than the analytical\nvalue. This is because the Walker solution becomes un-\nstable as the \feld is close to the breakdown \feld and spin\nwave emission in this regime would further slow down the\nDW propagation.32,43\nThe critical \feld HWthat is obtained from numerical\nsimulations is plotted in Fig. 2(b) as a function of \u0011.\nThe linear relation in Eq. (19) is reproduced and a linear\nleast squares \ftting yields the intercept HW(\u0011= 0) =\n25:9\u00060:3 Oe and the slope HW=\u0011= 10:8\u00060:3 Oe=nm2.\nUsing Eq. (19), the values of both the intercept and slope\nconsistently lead to the e\u000bective anisotropy energy Kz=\n(2:1\u00060:1)\u0002105J=m3. Then we are able to calculate the\nanalytical value of the DW velocity at the breakdown\n\feld using Eq. (20), vW\nx= 414 m/s, as shown by the red5\n0.0 0.51.0 1.52.0d (nm2)253035404550HW (Oe)300350400450500vxW (m/s)(b)0 204060 80H (Oe)0100200300400500vx (m/s)d=0d=1 nm2(a)\nFIG. 2. (a) The velocity of a transverse DW as a function\nof the external \feld from numerical simulations without the\nnon-local damping (black open circles for \u0011= 0) and with\n\u0011= 1:0 nm2(red solid circles). The black dashed and red\nsolid lines illustrate Eq. (18) with \u0011= 0 and\u0011= 1:0 nm2,\nrespectively. (b) The Walker breakdown \feld HW(left axis)\nand the velocity of DW motion vW\nxat the Walker breakdown\n\feld as a function of the nonlocal damping strength \u0011. The\nblack dashed line is a linear least squares \ftting to the black\nsolid squares, which gives the total anisotropy Kz= (2:1\u0006\n0:1)\u0002105J=m3. The red dashed-dotted line is the calculated\nvelocity at the breakdown \feld using Eq. (20) with the \ftted\nKz.\nhorizontal dashed-dotted line in Fig. 2(b). The values\nofvW\nxobtained from the numerical simulation plotted in\nFig. 2(b) are in very good agreement with the analytical\nvalue.\nB. Vortex DWs\nA vortex DW is energetically more stable in relatively\nwide and thick nanowires.37,38Here a vortex DW is mod-\neled by an inner vortex core with out-of-plane magneti-\nzation and an outer curling structure.44The spatial de-\npendence of the polar and azimuthal angles can be ana-lytically described as45,46\n\u0012(x;y) =8\n><\n>:2 arctan\u0012r\nrc\u0013\n; r\u0014rc;\n\u0019\n2; r>r c;\n'(x;y) = arg[(x\u0000x0) +i(y\u0000y0)] +\u0019\n2; (21)\nwhere (x0;y0) is the center of the vortex core and r=p\n(x\u0000x0)2+ (y\u0000y0)2.rcis the radius of vortex core\nthat is comparable to the exchange length ( \u00185 nm for\npermalloy).\nThe motion of a vortex DW under an external \feld\nHext=H^xcan be described by applying the GTE (12).\nBecausercis much smaller than the outer curling struc-\nture (several tens to hundreds of nanometers) of a vortex\nDW, the dominant contribution to DandD0comes from\nthe spins in the curling structure,\nD=\u0000Ms\n\rZ\n(sin2\u0012r'r')d3r\n=\u0000TMs\n\rZ\u0012@m\n@x\u00132\ndxdy (^ x^ x+^ y^ y)\n=\u0000\u0019TMs\n\rlnW\n2rc(^ x^ x+^ y^ y): (22)\nHere we have used W=2 as the outer radius of the vortex\nDW. The tensor D0is di\u000ecult to derive analytically due\nto the high order spatial derivatives of \u0012and'but it is\nwell converged numerically,\nD0=\u00003:3TMs\n\rr2c(^ x^ x+^ y^ y): (23)\nSimilarly, it is a reasonable approximation to neglect\nthe contribution of the vortex core to force F. So we have\nF=\u0000TMsHZ\nr(sin\u0012cos')d2r\n\u0019TMsHZ2\u0019\n0ZR\nrc\u0012sin2\u001e\nr;\u0000cos\u001esin\u001e\nr\u0013\nrdrd\u001e\n=\u0019WTMsH\n2^ x: (24)\nFollowing He et al. ,46we add a restoring force that is\nlinear in the transverse displacement \u000eyof the vortex\ncore from the nanowire center, Fre=\u0000^y\u0014\u000ey, where the\ncoe\u000ecient\u0014depends on the magnetic anisotropy and the\nequilibrium width of the vortex DW.46\nFor the gyrovector G, the spins outside the vortex core\ndo not contribute because of \u0012=\u0019=2. The integral over\nthe vortex core can be analytically evaluated as46\nG=\u00002\u0019TMs\n\r^ z: (25)\nSubstituting the expressions of F;G;DandD0into6\nGTE (12), the coupled equations become\n\rHW\n2+ 2vy\u0000\u0012\n\u000blnW\n2rc+3:3\u0011\n\u0019r2c\u0013\nvx= 0;\n\u0000\r\u0014\u000ey\n\u0019TMs\u00002vx\u0000\u0012\n\u000blnW\n2rc+3:3\u0011\n\u0019r2c\u0013\nvy= 0:(26)\nUnder an external \feld, the vortex core moves both along\nthe \feld direction (longitudinal) and transverse to the\n\feld (ydirection) due to the gyro force. Then the vor-\ntex structure is deformed with the core displacement \u000ey\n(from the nanowire centre). At steady-state motion, the\ngyro e\u000bect is balanced by the restoring force Fre, i.e.\nvy= 0. Then the vortex core only moves in the longitu-\ndinal direction and the velocity can be obtained,\nvx=\rHW\n2\n\u000blnW\n2rc+3:3\u0011\n\u0019r2c\u0011\rH\u0015 e\u000b\n\u000be\u000b; (27)\nwhere the e\u000bective width is de\fned as\n\u0015e\u000b\u0011W\n2 ln[W=(2rc)]; (28)\nand the e\u000bective damping is\n\u000be\u000b\u0011\u000b+3:3\u0011\n\u0019r2cln[W=(2rc)]: (29)\nAccording to Eq. (29), the nonlocal damping enhances\nthe e\u000bective damping of a vortex DW. The enhancement\ndepends on the strength of nonlocal damping ( \u0011), the size\nof vortex core ( rc) and the width of nanowire ( W).\nSimilar to the case of transverse DWs, the Walker\nbreakdown \feld increases in the presence of the nonlo-\ncal damping. For steady-state motion with vy= 0, the\ntransverse displacement of the vortex core is given by\n\u000ey=\u00002\u0019TMsH\u0015e\u000b\n\u000be\u000b\u0014: (30)\nEquation (30) indicates that the transverse displacement\nincreases with the external magnetic \feld. When the\nvortex core reaches the edge of the nanowire \u000ey=\u0000W=2,\none has the Walker breakdown \feld\nHW=\u000be\u000b\u0014W\n4\u0019TMs\u0015e\u000b: (31)\nWith the nonlocal damping, the breakdown \feld in-\ncreases and is proportional to the e\u000bective damping \u000be\u000b.\nThe longitudinal velocity at the breakdown \feld is inde-\npendent of the nonlocal damping as in the case of trans-\nverse DWs.\nDetermining the value of \u0011from \feld-driven DW mo-\ntion and Eq. (27) is not straightforward due to the inho-\nmogeneity of samples. The disorder and surface rough-\nness increase e\u000bectively the Gilbert damping by adding\na factor\u000bR. Both\u000bRand the nonlocal damping slow\ndown the \feld-driven DW propagation.47,48Weindler et\n0 0.02 0.04 0.06 0.08α+αR02468η (nm2)Beach et al.\nParkin et al.\nWeindler et al.\n0 40 80 120\n1/α05101520v/H [m/(s Oe)]Simulation η=0\nExperiment\nFittingFIG. 3. Calculated \u0011as a function of Gilbert damping pa-\nrameter including the contributions from spin-orbit interac-\ntion\u000band roughness \u000bR, for the experimental DW velocities\ndriven by an external \feld taken from Ref. 49 (black dashed\nline), Ref. 5 (red solid line) and Ref. 19 (blue dashed-dotted\nline). In Ref. 19, \u000b+\u000bR= 0:011 is determined by \ftting ex-\nperimental data to the simulations with various DW pinning\nstrengths, and the corresponding nonlocal damping strength\n\u0011= 1:3 nm2is obtained50(blue solid triangle). Inset: Gilbert\ndamping dependence of DW mobility of Ho-doped permalloy\nnanowires. The orange open squares are experimental data\nand the magenta open triangles are the simulation without\n\u0011.22The green solid line shows a least squares \ftting using\nEq. (27), which yields the upper bound of the nonlocal damp-\ning strength 5 :1\u00061:6 nm2.\nal.measured the Gilbert damping of collinear permalloy\n\u000b= 0:008 and determined \u000bR= 0:003 by comparing the\nexperimental and simulated depinning magnetic \felds.\nBy using the measured e\u000bective damping for the \feld-\ndriven DW motion \u000be\u000b= 0:023, the nonlocal damping\nparameter in the system of Ref.19is\u0011= 1:3 nm2that is\ndenoted by the blue solid triangle in Fig. 3. We can also\nestimate\u0011by using the experimental data from Refs. 5\nand 49. The result is plotted in Fig. 3 as a function of the\nGilbert damping \u000b+\u000bR. The inset of Fig. 3 is the exper-\nimental data (orange open squares) of the DW mobility\nfor di\u000berent \u000bobtained by Moore et al. .22They tuned\nthe value of \u000bby doping permalloy with a rare-earth\nelement Holmium. Micromagnetic simulation in the ab-\nsence of the nonlocal damping results in a signi\fcantly\nlarger mobility (magenta empty triagles) than the mea-\nsured values.22We use Eq. (27) to \ft the experimental\ndata and perfectly reproduce the measured mobility as a\nfunction of \u000b. Since we assume \u000bR= 0 in the \ftting, the\nonly \ftted parameter \u0011= 5:1\u00061:6 nm2corresponds to\nthe upper bound of the nonlocal damping strength.\nThe values of \u0011can also be extracted from the wave\nvector dependence of the spin wave damping measured\nvia FMR.51{53The coexisting mechanisms such as the\neddy current and the radiative damping result in com-\nplexities in determining \u0011. Nembach et al.52found that7\n\u0011= 1:4 nm2in permalloy nanodisks. Li and Bailey51\nmeasured permalloy, cobalt and CoFeB alloy and ob-\ntained the value \u0011= 0:11\u00060:02 nm2for permalloy.\nLater Schoen et al.53discussed the contribution of radia-\ntive damping and determined that \u0011of permalloy from\nthe experimental measurement is less than 0.045 nm2.\nAll these experiments were performed at room temper-\nature while \frst-principles calculation at low tempera-\nture found \u0011= 0:016 nm2in permalloy but it could be\nsigni\fcantly enhanced by two orders of magnitude, up\nto 5.9 nm2, because of the \fnite propagating length of\ntransverse spin currents.33\nIn the derivation of the GTE (12), we have included\nboth the transverse and longitudinal torques [Eq. (3) and\nEq. (4)] due to the spin pumping. For the steady-state\nDW motion described by Eq. (10), the longitudinal com-\nponent \u001cLSC=\u0000\u0011@i(@im\u0002@tm) vanishes because @imis\naligned with @tm. In our numerical simulation for trans-\nverse DWs, it is con\frmed that \u001cLSCalone did not change\nthe DW velocity below breakdown \feld, consistent with\nconclusion in Ref. 31 where the expression of \u001cLSCwas\nderived.\nIn contrast, Weindler et al.19considered only \u001cLSCin\ntheir micromagnetic simulations with \u0011= 0:07 nm2and\nfound the texture-enhanced damping in the steady-state\nDW motion. In micromagnetic simulations, there might\nbe higher order e\u000bects that eventually a\u000bect the DW mo-\nbility. On the other hand, \u001cTSCcan have remarkable in-\n\ruence on the \feld-driven DW velocity, as predicted by\nEq. (27). This is partly the reason why we extracted a\ndi\u000berent\u0011= 1:3 nm2by using the same experimental\ndata in Sec. III B. It suggests that \u001cTSCmust be properly\nincluded to extract a reliable value of \u0011.\nIV. CONCLUSIONS\nWe have derived the nonlocal damping torque orig-\ninated from the pumped spin current by the preces-\nsional noncollinear magnetization. This nonlocal damp-\ning torque consists of a longitudinal and a transverse\ncomponents that both depend on the magnetization gra-\ndient and inevitably a\u000bects the noncollinear magnetiza-\ntion dynamics. We derive a generalized Thiele equa-\ntion (12) for the \feld-driven steady-state DW motion\nunder the in\ruence of the nonlocal damping. For both\ntransverse and vortex DWs in ferromagnetic nanowires,\nthe transverse component of the nonlocal damping slows\ndown the DW propagation and increases the Walker\nbreakdown \feld. The analytical results are further con-\n\frmed by numerical simulations for transverse DWs. In\naddition, our result compares well the experimentally re-\nported damping dependence of vortex DW mobility un-\nder an external magnetic \feld, while the LLG equation\nwithout the nonlocal damping signi\fcantly overestimates\nthe DW velocity.\nThe nonadiabatic spin transfer torque \fis one of the\nkey parameters in current-driven DW motion becausethe velocity is proportional to \fand inversely propor-\ntional to the damping \u000b. Previous works in literature\nextracted\fvalue from experimentally measured DW ve-\nlocity by assuming a constant \u000bin a DW. Thus the in-\n\ruence of the nonlocal damping revealed in this paper\nmay explain the large spread in the extracted \fvalues\nfor permalloy.54{57The measurement of current-induced\ntorques due to Rashba and Dzyaloshinskii-Moriya in-\nteractions in noncollinear magnetization is usually per-\nformed by comparing the experimental observation and\nmicromagnetic simulation. As a prerequisite, the non-\nlocal damping has to be appropriately included in the\nsimulations while the damping form may be more com-\nplicated because of the complex interactions.58\nAppendix A: Spin pumping and the damping torque\nin a noncollinear ferromagnetic nanowire\nIn this appendix, we derive spin pumping Eq. (1) and\nthe resulting damping torque Eq. (2) exerted on the local\nmagnetization in a noncollinear ferromagnetic nanowire.\nWithout loss of generality, we consider a one-dimensional\ncase with the magnetization m(x;t), in which a small\nsegment of magnetization centered at position xcan be\napproximated by a local collinear magnetization m(x).\nAsm(x) varies in time, it can pump out a spin current\njs\npump(x) =~\n4\u0019g\"#m(x)\u0002@tm(x); (A1)\nwhereg\"#is the conventional spin-mixing conductance\ncharacterizing the magnitude of spin pumping.25This\npumped spin current \rows both forward and backward,\nas schematically shown by the blue arrows in Fig. 4. Since\nthe magnetization in the nanowire is not uniform, the\nspin current pumped forward by the segment at position\nxis di\u000berent from that pumped backward by the next\nsegment at position x+\u0001x. The incomplete cancellation\n   jpumps(x)   jpumps(x+Δx)   jpumps(x−Δx)\n   jouts\n   jins\n  x−Δx2  x+Δx2\nFIG. 4. Sketch of spin pumping in a ferromagnetic nanowire\nwith noncollinear magnetization. A segment of magnetiza-\ntion pumps a spin current js\npump(x) =~\n4\u0019g\"#m(x)\u0002@tm(x).\nThe net spin current \rowing into (out of) the segment at x\nis given by the backward (forward) derivative of the pumped\nspin current. The torque exerted on the segment of magne-\ntization at xis then de\fned as the net spin current that is\nabsorbed by the segment of magnetization, as formulated by\nEq. (A8).8\nof these two pumped spin currents gives rise to a net spin\ncurrents across the cross section at x+\u0001x=2, which reads\njs\u0012\nx+\u0001x\n2\u0013\n=js\npump(x)\u0000js\npump(x+ \u0001x)\n=~\n4\u0019g\"#[m(x)\u0002@tm(x)\n\u0000m(x+ \u0001x)\u0002@tm(x+ \u0001x)]\n=\u0000~\n4\u0019g\"#\u0001x@x[m(x)\u0002@tm(x)]x+\u0001x\n2:\n(A2)\nIf we choose + xas the positive direction, js(x+ \u0001x=2)\ncorresponds to the out\row of the spin current for the seg-\nment of magnetization at x, which will later be referred\nto as js\nout.js\noutcan be rewritten as a superposition of\na longitudinal component jsk\noutthat is aligned with m(x)\nand a transverse one js?\noutthat is perpendicular to m(x),\njsk\nout=\u0000~\n4\u0019g\"#\u0001x[@xm(x)\u0002@tm(x)]x+\u0001x\n2;(A3)\njs?\nout=\u0000~\n4\u0019g\"#\u0001x[m(x)\u0002@x@tm(x)]x+\u0001x\n2:(A4)\nIn the same manner, we are able to \fnd the the net spin\ncurrent across the cross section at x\u0000\u0001x=2 corresponding\nto the in\row,\njs\nin\u0011js\u0012\nx\u0000\u0001x\n2\u0013\n=js\npump(x\u0000\u0001x)\u0000js\npump(x)\n=\u0000~\n4\u0019g\"#\u0001x@x[m(x)\u0002@tm(x)]x\u0000\u0001x\n2: (A5)\nIts longitudinal and transverse components can be re-\nspectively written as\njsk\nin=\u0000~\n4\u0019g\"#\u0001x[@xm(x)\u0002@tm(x)]x\u0000\u0001x\n2;(A6)\njs?\nin=\u0000~\n4\u0019g\"#\u0001x[m(x)\u0002@x@tm(x)]x\u0000\u0001x\n2:(A7)\nThe di\u000berence between in\row and out\row spin cur-\nrents is absorbed by the local magnetization resulting a\ndamping torque\n\u001cdamping =\u0000\rA(js\nin\u0000js\nout)\nMsA\u0001x=\r\nMs@xjs(x):(A8)whereAis the cross sectional area of the nanowire and\nthe prefactor \r=(MsA\u0001x) is to convert the torque in the\nunit of s\u00001. The torque in Eq. (A8) due to absorption\nof pumped spin current is only determined by the local\nmagnetization gradient, implying that the length scale of\nmagnetization variation, e.g. a DW width or a spin wave\nlength, is much larger than the propagating length of the\npumped spin current in the ferromagnetic nanowire. This\ncondition is in general satis\fed in disordered ferromag-\nnetic materials, but it is found in permalloy at low tem-\nperature that the spin coherent length is up to 13.1 nm,\nwhere the nonlocal absorption of the pumped spin cur-\nrent even changes the scaling of the e\u000bective damping\nwith the DW width.33\nWhile Eq. (A2) and Eq. (A5) are two particular exam-\nples, the net spin current in a noncollinear magnetization\nresulting from spin pumping can be generally expressed\nas\njs(x;t) =\u0000~\n4\u0019g\"#\u0001x@x[m(x;t)\u0002@tm(x;t)]\n=\u0000~\n4\u0019\u0000\"#@x[m(x;t)\u0002@tm(x;t)]:(A9)\nHere we de\fne the spin-mixing conductivity \u0000 \"#\u0011g\"#\u0001x\nthat is a proper parameter characterizing the (intralayer)\nspin pumping in a noncollinear ferromagnetic material.\nThe conventional spin pumping at a ferromagnet-normal\nmetal interface25can be recovered from Eq. (A9) by tak-\ning the e\u000bective thickness of the interface \u0001 x. Notice\nthatg\"#has the dimension of the inverse of area and \u0000 \"#\nhas the dimension of the inverse of length.\nACKNOWLEDGMENTS\nH. Y. Y. would like to thank Lei Wang for helpful\ndiscussions. 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B 92, 014418 (2015)."    },    {        "title": "1911.09803v3.Role_of_Element_Specific_Damping_on_the_Ultrafast__Helicity_Independent_All_Optical_Switching_Dynamics_in_Amorphous__Gd_Tb_Co_Thin_Films.pdf",        "content": "Role of element-specific damping on the ultrafast, helicity-independent all-optical switching\ndynamics in amorphous (Gd,Tb)Co thin films.\nAlejandro Ceballos,1, 2,\u0003Akshay Pattabi,3,\u0003Amal El-Ghazaly,3Sergiu Ruta,4Christian P Simon,5Richard F L Evans,4\nThomas Ostler,6, 7Roy W Chantrell,4Ellis Kennedy,1, 8Mary Scott,1, 8Jeffrey Bokor,3, 2and Frances Hellman1, 2, 5\n1Department of Materials Science and Engineering, University of California Berkeley, Berkeley, California 94720, USA\n2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA\n3Department of Electrical Engineering and Computer Sciences,\nUniversity of California, Berkeley, Berkeley, CA 94720, USA\n4Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom\n5Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA\n6Materials and Engineering Research Institute, Sheffield Hallam University, Howard Street, Sheffield S1 1WB, UK\n7Faculty of Science, Technology and Arts, Sheffield Hallam University, Howard Street, Sheffield, S1 1WB, UK\n8National Center for Electron Microscopy, Molecular Foundry,\nLawrence Berkeley National Laboratory, Berkeley, CA 94720\nUltrafast control of the magnetization in ps timescales by fs laser pulses offers an attractive avenue for appli-\ncations such as fast magnetic devices for logic and memory. However, ultrafast helicity-independent all-optical\nswitching (HI-AOS) of the magnetization has thus far only been observed in Gd-based, ferrimagnetic amor-\nphous ( a-) rare earth-transition metal ( a-RE-TM) systems, and a comprehensive understanding of the reversal\nmechanism remains elusive. Here, we report HI-AOS in ferrimagnetic a-Gd 22\u0000xTbxCo78thin films, from x = 0\nto x = 18, and elucidate the role of Gd in HI-AOS in a-RE-TM alloys and multilayers. Increasing Tb content re-\nsults in increasing perpendicular magnetic anisotropy and coercivity, without modifying magnetization density,\nand slower remagnetization rates and higher critical fluences for switching but still shows picosecond HI-AOS.\nSimulations of the atomistic spin dynamics based on the two-temperature model reproduce these results qual-\nitatively and predict that the lower damping on the RE sublattice arising from the small spin-orbit coupling of\nGd (with L=0) is instrumental for the faster dynamics and lower critical fluences of the Gd-rich alloys. An-\nnealing a-Gd 10Tb12Co78leads to slower dynamics which we argue is due to an increase in damping. These\nsimulations strongly indicate that acounting for element-specific damping is crucial in understanding HI-AOS\nphenomena. The results suggest that engineering the element specific damping of materials can open up new\nclasses of materials that exhibit low-energy, ultrafast HI-AOS.\nI. INTRODUCTION\nThe ability to control magnetism at short ps and sub-ps\ntimescales has tantalized scientists since the discovery in 1996\nof the ultrafast demagnetization of Ni following irradiation by\nfs laser pulses,[1] opening up the field of ultrafast magneti-\nzation dynamics. A major breakthrough was the discovery\nof helicity-independent all-optical switching (HI-AOS) of the\nmagnetization, sometimes referred to as thermally-induced\nmagnetization switching (TIMS), in ferrimagnetic a-Gd-Fe-\nCo alloys[2–4] in ps timescales by a single fs laser pulse. This\nmagnetization switching phenomenon is unique in the field of\nall-optical switching as it exhibits one of the fastest switching\nspeeds recorded[5], it is reversible and cyclable for 1000s of\nrepetitions, and it is thermally-driven by a single laser pulse\nirrespective of polarization, i.e. no transfer of angular mo-\nmentum from the laser is involved. HI-AOS offers the possi-\nbility of promising technological applications in high-speed,\nenergy-efficient and non-volatile magnetic memory and logic,\nwith two to three orders of magnitude higher operating speeds\ncompared to conventional spintronic devices that operate on\nmechanisms such as external field control,[6] spin-transfer-\ntorque,[7, 8] or spin-orbit torque.[9]\n\u0003These two authors contributed equallyDespite significant advances in the field, a complete under-\nstanding of the mechanism of HI-AOS still remains lacking.\nThus far, deterministic ultrafast toggle switching of the mag-\nnetization by a single laser pulse – where the switched area\nis controlled just by local heating from the spatial distribution\nof the laser pulse fluence – is predominantly found in Gd-\nbased a-RE-TM ferrimagnetic alloys and multilayers. These\ninclude a-Gd-Fe-Co,[2–4] a-Gd-Co,[10] Pt/Co/Gd,[11] and\nexchange coupled Co/Pt/Co/ a-GdFeCo[12] systems. Similar\nTb-based ferrimagnetic systems such as a-Tb-Co alloys[13]\nhave so far only shown helicity-dependent AOS (HD-AOS),\nrequiring multiple circularly polarized laser pulses over longer\ntimescales ( ms to ms), [14] or transient reversal with a single\nlaser pulse, [15] wherein the magnetization reverts back to\nits original direction after a short reversal of a few ps. How-\never, single-shot HI-AOS was demonstrated in Tb/Co multi-\nlayers when the layer thickness ratio of Co and Tb was within\n1.3 - 1.5,[16] although the switching dynamics were not re-\nported. HI-AOS was also demonstrated in an a-Tb 22Fe69Co9\nalloy,[17] but required patterning of nanoscale antennas to en-\nhance the optical field, thereby confining the switched region\nto less than 100 nm in areas near and around the antennas. The\nswitching in that experiment was strongly influenced by inho-\nmogeneities and control of the uniformity in the switched area\nunder the laser pulse fluence profile could not be achieved.\nA complete microscopic theory has yet to explain the intrin-\nsic fundamental physics of HI-AOS. In this work the role ofarXiv:1911.09803v3  [cond-mat.mtrl-sci]  16 May 20202\nGd22Co78\nGd18Tb4Co78\nGd14Tb8Co78\nGd10Tb12Co78\nGd7Tb15Co78\nGd4Tb18Co78\nTb22Co78Initial Pulse 1 Pulse 2 Pulse 3 Pulse 4a)\nc) b)\n0 5 10 15 20 250.05.0x1051.0x1061.5x1062.0x106\n Pt/Gd22-xTbxCo78/Pt\n Ta/Gd22Co78/Ta\n Pt/Gd19Co81/Pt\n SiN/Gd22-xTbxCo78/Ta\n SiN/Gd22-xTbxCo78/Pt\n Pt/Gd-Co-O/PtKui (erg/cm3)\nTb at.% x\nFIG. 1. a) MOKE microscopy images of Ta(3nm)/Pt(3)/ a-Gd 22\u0000xTbxCo78(10)/Pt(3) thin films following a series of single laser pulses at an\nincident fluence of 6.9 mJ/cm2, illustrate samples’ ability to all-optically reverse their magnetization upon irradiation. Samples with a Tb\nconcentration of up to 18 at.% exhibited HI-AOS while a-Tb22Co78only exhibited demagnetization as evidenced by the nucleation of random\ndomains. b) Incident and absorbed critical fluence increase with increasing Tb content. c) Intrinsic anisotropy constant K uivs Tb atomic\npercent in a-Gd 22\u0000xTbxCo78thin films with various capping and underlayers as shown in the legend. Anisotropy increases with increasing Tb\ncontent for fixed under and overlayers. Pt under or overlayers increased K uifor fixed x. The green box shows the compositions that exhibited\nHI-AOS; the red box those that did not. The effect of growing on different substrates and capping layers was tested but it had no effect on\nHI-AOS. Four a-Gd-Co samples with varying Gd/Co ratio are shown at x = 0. At the top, indicated with an orange star, is a-Gd 19Co81with\ncompensation T below RT exhibiting PMA and HI-AOS. Beneath it the black square a-Gd 22Co78which is part of the main study; it has Pt\nunder and over layers and exhibits PMA and HI-AOS. The black diamond is Gd-Co-O also with Pt over and underlayers, but grown in a\nreactive oxygen atmosphere, with K uivalue of 2.05 x 104erg/cm3and M Sof 40 emu/cm3. It exhibited HI-AOS though its dynamics were not\nstudied. In red is a-Gd 22Co78with Ta under and over layers; it has in-plane magnetization which cannot be probed using polar MOKE, hence\nit’s unknown if it exhibits HI-AOS.\nGd in enabling HI-AOS was investigated experimentally and\ntheoretically by studying a-Gd 22\u0000xTbxCo78thin films. Vapor\ndeposited a-RE-TM films have long been known to possess\nsignificant perpendicular magnetic anisotropy (PMA) leading\nthem to have been considered and sometimes used as mag-\nnetic recording media for decades. PMA is intrinsic to the\nstructure, a result of subtle structural ordering due to growth\nprocesses[18, 19]. Tb has significant orbital angular momen-\ntum compared to Gd ( L= 3 compared to L= 0) and greater\nspin-orbit coupling than Gd, leading to the large PMA and\nlarger magnetic damping[20, 21]. By systematically replacing\nthe Gd atoms with Tb atoms (such that the RE composition is\nkept constant), and by post-growth annealing, the anisotropy,\nmagnetic damping and spin-orbit coupling (SOC) are modi-fied, while the magnetization and the Curie and compensation\ntemperatures are held constant.\nHI-AOS was found in a-Gd 22\u0000xTbxCo78films up to x =\n18, while a-Tb 22Co78only demagnetized upon laser irradi-\nation. The threshold or critical fluence of the pulse needed\nfor switching increased linearly with increasing Tb concen-\ntration and the switching dynamics became slower, indicat-\ning increasing difficulties in the switching process with in-\ncreasing Tb. Annealing a-Gd 10Tb12Co78(a high x hence high\nmagnetic anisotropy film) at 300\u000eC for one hour reduced the\nanisotropy by an order of magnitude (without changing the\nmagnetization), due to increased randomization of the local\nuniaxial anisotropy fields that lead to PMA. This then in-\ncreases the damping and is here shown to yield slower re-3\nmagnetization dynamics but no change in critical fluence.\nThese slower dynamics with no change in critical fluence are\nreproduced in simulations with increased damping. Simu-\nlations of the atomistic spin dynamics using the VAMPIRE\nsoftware package[22, 23] combined with a two-temperature\nmodel (2TM)[4] indicate that these results can be explained\nby an increased damping on the RE site when Gd is replaced\nby Tb. The results and simulations suggest that engineering\nthe relative element-specific damping of the RE and TM sub-\nlattices can be used to find new materials that exhibit HI-\nAOS. Increased anisotropy with Tb content, and decreased\nanisotropy by annealing both led to slower switching dynam-\nics, indicating that anisotropy is not a significant factor in HI-\nAOS. The high perpendicular magnetic anisotropy in Tb-rich\nfilms make them attractive candidates for magnetic devices\nwith higher memory storage densities and retention times that\nexploit the fast read-write speeds of HI-AOS.\nII. RESULTS\nA. Single-shot HI-AOS in a-GdTbCo alloys\nAmorphous, ferrimagnetic thin-films of Ta(3)/Pt(3)/ a-\nGd22\u0000xTbxCo78(10)/Pt(3) (thicknesses are in nm) het-\nerostructures were sputter deposited onto substrates of\nSi(525 mm)/SiO 2(50nm)/SiN x(300 nm). The a-Gd-Tb-Co\nfilms were co-deposited from separate Tb, Gd and Co tar-\ngets with Pt or Ta over and underlayers grown in situ (Meth-\nods section at the end). Energy dispersive spectroscopy im-\nages taken with a scanning transmission electron microscope\nfound no evidence of inhomogeneities at the 10 nm scale\nas had been reported in previous work on a-Gd-Fe-Co[24]\n(See suppl. matls.). The magnetization was measured with a\nQuantum Design MPMS SQUID magnetometer as a function\nof field and temperature. Room temperature saturation M s\n(\u0018100 emu/cm3) and compensation temperature T M(\u0018400\nK) were independent of x, due to the fixed 22 at.% RE con-\ntent. The intrinsic perpendicular magnetic anisotropy constant\nKuiincreased with increasing x, from 4 \u0002105ergs/cm3for\na-Gd 22Co78to 2\u0002106ergs/cm3fora-Tb 22Co78. At room\ntemperature, the magnetization of all the films studied is RE-\ndominant. The Curie temperature of all films was found to be\ngreater than 600 K.\nThe magnetization of the samples is initialized at room tem-\nperature with an external out of plane magnetic field of \u00180.7\nT, fully saturating all samples. The field is then turned off, and\nthe samples are irradiated with 100 fs full-width half maxi-\nmum (FWHM) optical pulses from a regeneratively amplified\nTi-Sapphire laser. Magneto Optical Kerr Effect (MOKE) mi-\ncroscope images depict single shot switching of the magne-\ntization of all films except a-Tb 22Co78as shown in Fig. 1a.\nFilms with as much as 18 at.% Tb (and hence as little as 4\nat.% Gd) have deterministic magnetization reversal upon irra-\ndiation with a single laser pulse. Amorphous Tb 22Co78only\nshows demagnetization, evidenced by the nucleation of ran-\ndom magnetic domains. Further growth and characterization\ndetails are available in supplemental materials.Fig. 1b shows that increasing the Tb content increases\nthe incident critical fluence, starting from 4.4 mJ/cm2for\na-Gd 22Co78and linearly increasing to 6.2 mJ/cm2fora-\nGd4Tb18Co78. The absorbed critical fluence calculated from\nellipsometry measurements of the optical properties of the\nfilms (see suppl. matls.) increases linearly from 1.8 mJ/cm2to\n2.5 mJ/cm2fora-Gd 22Co78anda-Gd 4Tb18Co78respectively.\nThe intrinsic anisotropy constant (K ui) was measured at\nroom temperature and is plotted as a function of Tb at.% in\nFig. 1c. The anisotropy constant increases systematically with\nincreased Tb due to its large single ion anisotropy. The effect\nof growing on Ta or SiN, and of capping with Ta or Pt are\nalso shown in Fig. 1c; these over and under layers, particu-\nlarly Pt, increase the magnetic anisotropy of these thin films,\nnearly certainly due to interfacial anisotropy effects of these\nhigh SOC elements. The HI-AOS was unaffected by these\nbuffer layers, clear evidence that K uialone is not the relevant\ndriving parameter for HI-AOS. We also note that several films\nof other RE/TM ratio were tested, with varying magnetization\nvalues M, including TM-dominant (low RE/TM ratio). These\nalso exhibited HI-AOS, demonstrating that at least over a lim-\nited range of M, M is not a determining factor for the ability\nto show HI-AOS.\nForx=0, i.e. a-Gd-Co, a number of different types of\nfilms were prepared and measured, with different Gd/Co ra-\ntios and different K ui, as shown in the legend and data of\nFig 1c. With Pt over and underlayer, a-Gd-Co has PMA and\nshows HI-AOS. With Ta over and underlayer, a-Gd-Co has in\nplane anisotropy and polar MOKE cannot measure HI-AOS.\nA Gd-Co-O sample (black diamond in Fig. 1c) was grown\nvia oxygen reactive sputtering and exhibited both PMA and\nHI-AOS, as well as a distinct blue hue compared to the metal-\nlic gray of all other samples. The origin of PMA in a-Gd-Co\nis still an open question[25], although this reactively-grown\nsample suggests that oxidation plays a significant role, and we\nhave found that a-Gd-Co grown under very clean conditions\nand without Pt over or underlayers are magnetized in-plane,\nwhile slightly worse background pressure or deliberate O in-\ntroduction yields PMA. Rutherford backscattering spectrome-\ntry (RBS) measurements of a-Gd 22Co78of the main study do\nnot reveal an oxygen peak, but its presence is likely hidden\nby the overlap of the Si signal from the substrate. Simula-\ntions with SIMNRA[26] indicate that if oxygen is present in\na-Gd 22Co78it is at most 5 at.% O. Therefore it’s likely that the\nPMA in a-Gd-Co studied here is a combination of slight get-\ntering of residual oxygen in the chamber, plus an interfacial\ncontribution from heavy metal layers.\nB. Time dynamics of HI-AOS of a-GdTbCo alloys\nThe temporal dynamics of the magnetization of the a-\nGd22\u0000xTbxCo78films as they undergo HI-AOS was measured\nby time-resolved magneto optical Kerr effect (TR-MOKE)\n(see suppl. matls.). An incident fluence of 6.9 mJ/cm2was\nchosen for all TR-MOKE experiments as it slightly exceeds\nthe incident critical fluence of a-Gd 4Tb18Co78, the sample\nwith the highest critical fluence that was able to be switched.4\nNote that the dynamics are not affected by the fluence as\nshown in supplemental materials. Fig. 2a shows the ultra-\nfast magnetization dynamics of all samples. The magnetiza-\ntion reversal process follows a two-step behavior. In the first\nstep a rapid initial drop in the magnetization ocurs in which all\nfilms share a similar demagnetization process within the first\npicosecond post irradiation from the pump pulse. The second\nstage consists of remagnetization in the opposite direction as\nthe system cools down, except for a-Tb 22Co78.a-Gd 22Co78\nexhibits the fastest remagnetization time; with increasing Tb\nconcentration, the remagnetization systematically slows. The\nremagnetization rate plateaus with 15% and 18% Tb sam-\nples exhibiting similar dynamics. Finally, a-Tb 22Co78only\ndemagnetizes upon irradiation and then recovers its magneti-\nzation along its initial direction upon cooling. It is possible\nthata-Tb 22Co78exhibits a transient switching in the first few\nps following irradiation, similar to the behavior reported by\nAlebrand et al.,[15] and modeled by Moreno et al.,[27], sug-\ngesting that switching could ocur at a higher fluence. How-\never, utilizing higher fluences led to irreversible damage of\nthe sample as the laser ablated or burned the sample surface.\nBy 200 ps all samples had remagnetized to about 80% of the\nsaturation value as shown in supplemental materials.\nFig. 2b is a close up of the experimental data of Fig. 2a,\nand shows, after the initial fast demagnetization step, behavior\nthat deviates from exponential decay functions that appears to\nbe more linear in character. The duration of this more linear\nbehavior increases with increasing Tb before resuming expo-\nnential decay characteristics.\nC. Simulation of HI-AOS in a-GdTbCo alloys\nAtomistic spin dynamics simulations using the VAMPIRE\nsoftware package[22, 23] combined with a two-temperature\nmodel (2TM)[4] (details discussed in suppl. matls.) were per-\nformed to simulate the experimental magnetization dynamics.\nThese simulations incorporate the Hamiltonian of the system,\nwhich includes the exchange and anisotropy energies of Gd,\nTb and Co, into the Landau-Lifshitz-Gilbert equation to com-\npute the dynamics. As shown in Fig. 2c the simulations are\nin excellent agreement with the experiments, reproducing the\ncharacteristic behavior of similar demagnetization dynamics\nfollowed by increasingly slow remagnetization times with in-\ncreasing Tb content. The bump in the magnetization following\nthe initial demagnetization step is clearly seen in simulation,\nand exhibits a more linear character with increasing Tb as seen\nexperimentally. The approximately factor of two discrepancy\nin the time scales between experiment and simulation is due\nto both the small size of the simulated system, which does not\nallow for domain dynamics to be taken in consideration, and\nalso due to heat dissipation effects.\nThe simulations were only able to reproduce the experi-\nmental data when the element specific damping of the Gd,\nTb and Co sites were assigned separately, rather than when\nusing a net damping of the system as is typically done when\nsimulating such RE-TM systems [27–29] (see suppl. matls.\nfor comparison). This is consistent with previous experimen-\na)\nb)\nc)Experimental\nExperimental\nSimulated\na-Gd22Co78\n4 at.% Tb\n8 at.% Tb\n12 at.% Tb\n16 at.% Tb\na-Tb22Co78a-Gd22Co78\n4 at.% Tb\n8 at.% Tb\n12 at.% Tb\na-Tb22Co7818 at.% Tb15 at.% Tba-Gd22Co78\n4 at.% Tb\n8 at.% Tb\n12 at.% Tb\na-Tb22Co7818 at.% Tb15 at.% TbFIG. 2. a) Time-resolved magnetization dynamics of a-\nGd22\u0000xTbxCo78thin films following irradiation with laser pulses of\n6.9 mJ/cm2fluence and 100 fs pulewidth. The initial rapid drop in\nmagnetization is similar across all samples, but upon entering the re-\nmagnetization regime different rates are observed. b) Close up of\nthe experimental data showing a bump in the magnetization (at \u00181\nps) following the initial demagnetization step. c) Simulation results\nof the magnetization dynamics of Gd 22\u0000xTbxCo78after laser irradi-\nation obtained with a two-temperature model neglecting spin-lattice\ncoupling as described in the text. In this simulation, the damping co-\nefficients for Tb and Co were 0.05[27] and the damping of Gd aGd\nwas fixed at 0.035.5\nFIG. 3. Simulated critical fluence of a-Gd 22\u0000xTbxCo78films as a\nfunction of Tb at.% x for different damping values of Gd, aGd. The\nTb and Co damping parameters are constant (0.05). The critical flu-\nence increases as aGdincreases, indicating that lowering aGdre-\nduces the threshold switching condition of a-Gd-Tb-Co alloys below\nthe laser ablation limit.\ntal and theoretical work on the role of damping in systems\ndoped with RE[21, 30]. The experimental work of Radu et al.\n[21] on permalloy doped with RE showed increased damping\nfor doping with Tb but no significant increase when doping\nwith Gd. Ellis et al. [30] showed that element-specific damp-\ning is required to reproduce the macroscopic damping in such\nsystems. In this work we varied the Gd/Tb ratio, as in the\nexperiments, and fixed the element-specific damping value at\n0.05 of the Tb and Co sites as in ref. [27]. The damping of Gd\nwas taken to be lower than Tb and was varied between 0.005\nand 0.05 in order to study its effect on the dynamics. In Fig.\n2c, it is set at 0.035.\nFig. 3 shows the simulation results of critical fluence as a\nfunction of Tb concentration and varying Gd damping. The\ncritical fluence for switching in the simulation is determined\nby the transition from non-deterministic to deterministic ther-\nmally induced switching. It shows that increasing the Tb con-\ntent increases the critical fluence as observed experimentally,\nfor all values of Gd damping. It also shows, that for a given\nconcentration, increasing the damping on the Gd site increases\nthe critical fluence.\nD. Effect of annealing on switching dynamics\nAnnealing was used to test the influence of anisotropy and\ndamping on the dynamics; the results are shown in Fig. 4.\nAnnealing a-Gd 10Tb12Co78at 300\u000eC for one hour results in a\nsignificant reduction in coercivity H cand anisotropy K ui(from\n4.6\u0002105erg/cm3to 2.5\u0002105erg/cm3) while maintaining\nthe composition and M Sconstant as seen in Fig 4b. Further\nannealing at 350\u000eC eliminated the PMA. The fact that M S\nwas unchanged by annealing strongly indicates that inhomo-\ngeneities such as phase segregation or crystallization have not\nocurred. The annealed sample shows a significantly slower re-magnetization time, as seen in Fig. 4b. Atomistic simulations\nof the magnetization dynamics of this sample as a function\nof Gd damping are shown in Fig. 4c, which shows that in-\ncreased damping increases the remagnetization time. It can\nbe seen that increasing the damping on the Gd sites explains\nthe slower remagnetization time, suggesting that the exper-\nimentally annealed sample has increased damping, in addi-\ntion to reduced anisotropy. Work from Malinowski[31] et al.\nshowed that introducing local variations of the anisotropy in\namorphous CoFeB leads to an increase in the damping pa-\nrameter. Since the origin of anisotropy in RE-TM alloys is\ndue to a combination of pair-ordering and Tb’s single ion\nanisotropy[18, 19], annealing of a-RE-TM alloys leads to\na structural relaxation of pair-ordering that introduces local\nanisotropy variations. This in turn leads to higher damping\nand the slower remagnetization time observed. Fig. 2 showed\nthat the samples with higher anisotropy (larger Tb at.%) show\nslower switching dynamics in both experiments and simula-\ntion. But the annealing study in Fig. 4 shows that the film\nwith lower anisotropy exhibits slower switching. These ob-\nservations lead us to conclude that it is the damping of the\nsystem, and not the anisotropy, that is the significant contrib-\nutor to the ability to exhibit HI-AOS, consistent with the ob-\nservation made in connection with Fig. 1 that HI-AOS was\nnot determined by the magnitude of K ui. Simulations shown\nin Fig. 4c with varying damping support this conclusion.\nIII. DISCUSSION\nThe experimental results show increasing critical fluence\nand remagnetization times with increasing Tb content, while\npost-growth annealing slowed the remagnetization rates. The\nsimulations strongly indicate that the relative element specific\ndamping of the rare earth site compared to the cobalt site\nis the key factor that influences the critical fluence required\nswitching and the speed of remagnetization. As verified in\nthe model for the annealed sample, at a fixed composition the\nkey parameter that leads to slower remagnetization is the in-\ncreased elemental damping. In simulation the damping con-\nstant is a phenomenological parameter that combines a host\nof diverse effects. Increasing the Tb composition leads to a\ngreater spin-orbit interaction in the system, which is consid-\nered the intrinsic source of damping[32] and is proportional\ntox2=W, where xis the spin orbital coupling energy and W\nis the d-band width[33]. Thus as the system becomes Tb-rich\nit experiences increased spin-orbit coupling which increases\ndamping and thus leads to the dynamics observed in Fig. 2.\nFor the dynamics observed in the annealed sample, the local\nspin-orbit coupling is very unlikely to have changed, but the\nanisotropy changed due to the structural relaxation and conse-\nquent randomization of local anisotropy axes induced by an-\nnealing, thereby increasing the macroscopic damping. The\nsimulation does not directly acount for spin-orbit coupling,\nbut it indirectly simulates the effects of stronger spin-orbit\ncoupling via increases in the damping parameter. As men-\ntioned before the deterministic switching is independent of the\nanisotropy and therefore the main contribution of spin-orbit6\na)\nMagnetization (emu/cm3)b)\nc)\nFIG. 4. a) Magnetization dynamics of a-Gd 10Tb12Co78in the as-\ngrown state (blue curve) and after annealing (red curve) at 300\u000eC for\n1 hour. Annealing leads to a slower remagnetization time. b) Magne-\ntization loops in the out-of-plane orientation show that annealing re-\nduces the coercivity and anisotropy while M Sis unchanged. c) Sim-\nulated time-resolved magnetization dynamics of a-Gd 10Tb12Co78as\na function of increasing Gd damping. Increasing the damping leads\nto a slower remagnetization time, indicating that annealing leads to a\nhigher damping value.coupling for all optical switching is the damping. Therefore\nalthough the simulations reveal the critical role of damping\nin modifying the ultrafast magnetization dynamics, it is likely\nthat the underlying physical mechanism is rooted in the spin-\norbit interaction.\nIV . CONCLUSION\nIn conclusion, we have shown that a-Gd 22\u0000xTbxCo78thin\nfilms exhibit HI-AOS from x = 0 to x = 18, displaying a two-\nstep reversal process with an identical fast demagnetization\nstep and second slower remagnetization. The remagnetiza-\ntion time and the critical fluence increase as the Tb content\nincreases, indicating that replacing Gd with Tb atoms hin-\nders the switching process speed, but Tb helps increase PMA\nwhich is required for competitive memory storage technolo-\ngies. Atomistic simulations explain the switching dynamics of\nour material system only after taking into acount the individ-\nual element specific damping of each RE and TM sites. The\nslower remagnetization dynamics and the increased critical\nfluence as the Tb atomic percentage increases are explained\non the basis of increased damping on the RE site upon ad-\ndition of Tb. Annealing of an a-Gd 10Tb12Co78film, which\nreduces K uiand Hc without changing M, led to a slowing of\nthe remagnetization rate without changing the critical fluence;\nthese results are captured in simulations which indicate that\nreduced damping is responsible for modifying the ultrafast\nmagnetization dynamics. The experimental inability to ob-\nserve HI-AOS in a-TbCo is due to the high critical fluence\nthat under the present conditions is inacessible without ablat-\ning the film. This suggests that better management of the laser\nthermal load may lead to HI-AOS beyond Gd-TM alloys. Our\nresults indicate that the engineering of element specific damp-\ning of RE-TM systems will be crucial in endeavors to uncover\nmaterial systems exhibiting HI-AOS with favorable properties\n(such as high anisotropy and large magnetization) for applica-\ntions in ultrafast spintronic devices.\nV . ACKNOWLEDGMENTS\nThis work was primarily supported by the Director, Office\nof Science, Office of Basic Energy Sciences, Materials Sci-\nences and Engineering Division, of the U.S. Department of\nEnergy under Contract No. DE-AC02-05-CH11231 within\nthe Nonequilibrium Magnetic Materials Program (KC2204).\nUltrafast laser measurements were supported by the NSF\nCenter for Energy Efficient Electronics Science. A.C. ac-\nknowledges support by the National Science Foundation un-\nder Grant No. DGE 1106400. This project has received fund-\ning from the European Unions Horizon 2020 research and\ninnovation programme under Grant Agreement No. 737093\n(FEMTOTERABYTE). 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